14. QED with Dielectric Media¶
PDF pages 579–652
14.1.1 Effective Sources¶
Chapter 14 QED with Dielectric Media 14.1 Classical Electrodynamics in Dielectric Media Our starting point will be Maxwell’s equations for the electromagnetic fields in a medium:
(14.1) (Maxwell’s equations) Here, D is the electric flux density or electric displacement, B is the magnetic flux density, E and H are the usual electric and magnetic fields, respectively, \rho is the source charge density, and j is the source current density. We will ignore magnetic effects, so that B = µ0H, (14.2) and the electric fields are related by
(14.3) where P is the polarization density of the medium (i.e., dipole moment per unit volume). 14.1.1 Effective Sources There are multiple ways to treat the polarization field here. One is to treat it in terms of equivalent, effective sources. To see this, we can put Eq. (14.3) into the first Maxwell equation, with the result
ϵ0
(14.4) Defining the effective ‘‘bound charge density’’ for the polarization field by
(14.5) (bound charge density) we see that the first Maxwell equation becomes
ϵ0
(14.6) This is the form for the free-space Maxwell equation, with an extra source charge density.
14.1.2 Linear, Dispersive Media¶
Chapter 14. QED with Dielectric Media Similarly, from the continuity equation
(14.7) the existence of the bound charge density implies a ‘‘bound current density:’’
(14.8) Thus, we can define
(14.9) (bound polarization current density) as the effective polarization current density. We then see that the last Maxwell equation,
(14.10) can be written
(14.11) This is the corresponding free-space Maxwell equation, with an extra source current density jP. If we were not ignoring magnetic media, there would also be a second bound current associated with the magnetization. These effective sources are useful ways to think about the medium response. In particular, one route to quantizing the field in absorptive media is to realize that in quantum mechanics, absorption (dissipation) is always accompanied by noise (fluctuations) to preserve commutation relations at all times. The noise can be explicitly put in via noise source fields. However, for now, we will specialize to a linear, dispersive medium, and instead treat the medium response by frequency-dependent response functions. 14.1.2 Linear, Dispersive Media 14.1.2.1 Frequency Domain To continue, we will consider the Fourier transforms of the Maxwell equations (14.1)
(Maxwell’s equations, frequency domain) (14.12) We are now explicitly marking the dependence of each field on space and frequency, with frequency \omega of course representing an implicit time dependence of the form e−i\omegat. Note that in our notation here, we are writing fields such as E(t) and E(\omega) as different functions comprising a Fourier-transform pair:
Z \infty −\infty dt E(r, t) ei\omegat
2\pi Z \infty −\infty
(14.13) Again ignoring magnetic effects, we can write
(14.14) while the electric fields are still related by
(14.15)
14.1.3 Classical Green Tensor¶
14.1 Classical Electrodynamics in Dielectric Media A linear, dispersive medium is defined such that the medium polarization is given by the relation
(14.16) where \chi(r, \omega) is the dimensionless, linear, frequency dependent susceptibility of the medium. In this context, ‘‘linear’’ means that the susceptibility is independent of the electric-field amplitude, and ‘‘dispersive’’ means that the polarization response is a simple proportionality at each frequency. Thus, the electric fields are related by
(14.17) where
(14.18) is the (linear) dielectric permittivity (the dimensionless ratio ϵ/ϵ0 is the dielectric constant). 14.1.2.2 Time Domain The description of dispersive media in the time domain is more complicated. According to the convolution theorem (Section 17.1.2), the time-domain version of Eq. (14.16) is the convolution
Z \infty dt′ g\chi(r, t′) E(r, t −t′), (14.19) where the susceptibility \chi(r, \omega) is the Fourier transform of the correlation function g\chi(r, t):
Z \infty dt g\chi(r, t) ei\omegat. (14.20) Note that in writing down Eqs. (14.19) and (14.20), we are only integrating over positive times, where normally the integration should extend over the entire real axis. We do this to avoid an unphysical feature: otherwise, the medium response P(r, t) would depend on the input field E(r, t′), even for future times t′ > t. It is thus physically reasonable to impose the causality requirement that the polarization field only depends on the electric field in the present or in the past. We can also just as well write the causality requirement as
(14.21) where \Theta(t) is the Heaviside step function. Based on these relations, we see that the electric fields are related by
Z \infty dt′ g\chi(r, t′) E(r, t −t′). (14.22) Of course, we could also write this relation down in terms of a permittivity kernel
(14.23) so that
Z \infty 0−dt′ ϵ(r, t′) E(r, t −t′). (14.24) Here, the integration limit 0−denotes a limit of \delta, where the limit \delta −\rightarrow 0 is taken from below. 14.1.3 Classical Green Tensor Now we can ask, what are the decoupled wave equations corresponding to the above Maxwell equations in linear, dispersive media? It is most convenient to stick to the frequency domain, as we have seen above.
(14.25)
Chapter 14. QED with Dielectric Media Then using Eq. (14.17) and the last Maxwell equation, we arrive at the wave equation for the electric field:
(wave equation in dielectric media) (14.26)
replace the iterated curl in terms of the Laplacian. However, this doesn’t help much here because \nabla \cdot E̸ = 0 in general for a dielectric, even without a source charge. Note that this wave equation in this form hides the source charge density \rho. In general, this is not a concern because any time-dependent charge (such as a dipole) will generate a field through the current density j, which is tied to the charge density by the continuity constraint. Unfortunately the magnetic field H does not decouple as cleanly into its own wave equation, so we will stick to analyzing the electric field E as the primary object. Now as a general solution of the wave equation (14.26), we will introduce the classical Green tensor G(r, r′, \omega). Specifically, we will define the Green tensor component G(0) \alpha\beta(r, r′, \omega) to be the solution E\alpha(r), given a localized source current
(14.27) That is, it is the \alpha component of the electric-field solution to Eq. (14.26), assuming we replace the right-hand side by a delta function, with an orientation along the \beta direction (the factor of µ0\omega2 is arbitrary, but we will see the reason for this choice later). The Green tensor is thus the solution of the impulse-driven wave equation
(Green tensor wave equation) (14.28) where the delta function on the right-hand side, being set equal to a tensor object, is understood to be proportional to the identity tensor. We can write this same equation in components as
(14.29) where the box operator is defined by
(14.30) or in components,
(14.31) where \epsilon\alpha\beta\gamma is the Levi-Civita permutation symbol [which is zero if any two indices have the same value, or 1 or −1 if (\alpha\beta\gamma) is respectively an even or odd permutation of (xyz)]. The first form follows from the component representation of the vector curl as
(14.32)
field ϵ(r, \omega), we are implicitly assuming an isotropic medium, without ‘‘preferred’’ directions. In general, to handle something such as a dielectric medium, we would have to make the generalization to a permittivity tensor
(14.33) to give the most general linear response (in terms of direction) of the medium to the field. As with any Green function, the Green tensor here is convenient to know, as it in principle represents a very general solution to the wave equation. Because the wave equation is linear, the solution for an arbitrary source iµ0\omegaj(r, \omega) can then be written as the integral over the Green tensor and the source current,
\omega Z
(14.34)
14.1 Classical Electrodynamics in Dielectric Media or to disambiguate the tensor product,
\omega Z d3r′ G(0)
(14.35) This solution follows from the fact that the source can be viewed as a weighted sum over delta functions:
\omega Z d3r′
(14.36) The Green tensor gives the solution in the case of each of the component delta functions µ0\omega2\delta\alpha\beta\delta(r −r′), and the full solution is just a sum over the Green tensors, since the wave equation is linear. 14.1.3.1 Example: Green Tensor in Free Space As it turns out, the electromagnetic Green tensor is quite simple in the case of free space: essentially, it is just the electric field due to a single dipole. To see this, consider what is the meaning of an oscillating current density localized to a single point. A simple model of an oscillating dipole is two opposite but equal charges, one fixed (as in an atomic nucleus), and one oscillating in space. The moving charge, however, implies an oscillating current density. Furthermore, the dipole idealization involves the contraction of the distance r between the two charges to zero, while increasing the charge q such that the dipole moment d = qr remains fixed. In this limit, the current density likewise becomes localized to a single point.
from the identity
Z
(14.37) along with the expansion [Eq. (9.110)]
(14.38) so that
Z
\delta3(r) − Z d3r′ r′
1 Z d3r′ r′ \alphar′
(14.39) Repeated indices imply summations as usual here. The quantities in square brackets, from left to right, are the monopole moment (total charge), the dipole moment vector, and the quadrupole moment tensor. When the charge density is multiplied by another function under an integral, the moments act as coefficients of a Taylor expansion, weighting each of the derivatives of the other function. Now that we have defined the multipoles, we can compute the multipole expansion corresponding to the localized current density
(14.40)
origin is a multipole expansion about r′. The Fourier transform of the continuity constraint (14.7) is
(14.41) so that the localized current implies the charge density
(14.42) Clearly, the monopole moment of this charge density vanishes, since the integral of the derivative of the delta function vanishes. This is perhaps more evident by examining the identity Z \infty −\infty
(14.43)
Chapter 14. QED with Dielectric Media which is easy to prove via integration by parts. Now to compute the dipole moment:
Z
= − Z
Z d3r \delta3(r)
(14.44) To get rid of the gradient here, we simply integrated by parts. Thus, the localized current density implied by the Green tensor is simply a dipole, oriented in the same direction as the source delta-function vector, of unit magnitude (now we see the reason for the factor of µ0\omega2 in front of the delta function in the Green- tensor equation (14.28); without this factor, the dipole here would have amplitude 1/µ0\omega2). A calculation analogous to this one shows that all higher-order multipoles vanish, because they involve higher powers of r. After integration by parts, they will leave a final integral of the form R
Thus, the Green-tensor elements G(0) \alpha\beta(r, r′, \omega) in free space are constructed simply as follows: it is the \alpha component of the electric field vector at position r, due to an oscillating dipole located at r′, which is oriented along ˆr\beta and oscillates with unit ‘‘amplitude.’’ Note that the effective dipole amplitude, instead of having the expected SI units of (C m), is in fact dimensionless, so that the resulting electric field, normally of units (N/C), has units of N/C2m. Thus, the Green tensor has units of N/C2m, which agrees with what we find by examining the defining relation (14.28), or by noting that the product with a dipole moment G\alpha\beta d\beta should have the dimension of an electric field. The units we have chosen here are rather odd, but they will yield a particularly simple form for the Green tensor (14.129) that will parallel other, familiar Green functions that we have already used but not specifically identified as such. We have written down the electric field for a single dipole, located at the origin and oscillating along an arbitrary direction ˆ\epsilon before in Eqs. (1.42) and (1.115):
4\piϵ0
1 r3 −i k r2
r
(14.45) Recall that ˆr is a unit vector in the r direction, and ˆ\epsilon is a unit vector marking the dipole orientation. This completely determines the form of the present Green tensor, and evidently from our discussion above, we may write the free-space Green tensor as1 G(0)
4\piϵ0
1 r3 −i k r2
r eikr, (free-space Green tensor) (14.46) where k = \omega/c, ˆr\alpha ≡r\alpha/r is the component of the unit vector ˆr along the \alpha direction, and the naught superscript indicates that this is the Green tensor for free space. To obtain this expression, we have set d(+)(\omega) −\rightarrow 1, associated the index \alpha with the field orientation, and associated the index \beta with the dipole orientation ˆ\epsilon. To obtain the general case of the Green function for a dipole located at r′ instead of the origin, we may clearly write G(0)
(14.47) since the Green function here only depends on the difference between the two coordinates (i.e., free space is invariant under translations). In the presence of a material of some shape, the Green tensor is modified by reflections of fields at interfaces, changes of wavelength due to dispersion, and attenuation via absorption. In general, it will be
there should be an additional ‘‘contact’’ term, corresponding for example to −(1/3ϵ0)\delta\alpha\beta \delta3(r) in the Green tensor (14.46). However, in the following we will not write this term, because it is typically not useful for Casimir–Polder calculations.
14.1 Classical Electrodynamics in Dielectric Media quite a bit more complicated than in the simple free-space case. However, under the restrictions of our assumptions about the media, the Green tensor completely characterizes the response of the medium to external perturbations, even within quantum electrodynamics. 14.1.3.2 Green Tensor in Free Space: Alternate Forms A couple of other forms of the free-space Green tensor (14.46) will be useful later on, and they provide a bit more insight now, so we’ll go ahead and derive them. The first alternate form is G(0)
4\piϵ0
eikr r . (14.48) (free-space Green tensor) The equivalence of this expression to Eq. (14.46) can be verified directly by evaluating the derivatives in this expression (see Problem 8.8). The factor here of (\delta\alpha\beta + \partial \alpha\partial \beta/k2) is characteristic of a transverse field, as we see by comparison with the momentum-representation expression for the transverse delta function \delta⊥
Eq. (8.178). In fact, this is essentially the projection operator for a transverse wave at frequency ck, so from this we see that the dipole field is just the transverse part of the scalar spherical wave. This form for the Green tensor will be useful in deriving the Green tensor for a planar material interface (Section 14.3.5). To arrive at the other alternate form, we can use \nabla r = ˆr and \nabla \cdot ˆr = 2/r to write \nabla 2 eikr r
\nabla 1 r \cdot \nabla eikr
ˆreikr + 2 −ˆr r2 \cdot ikˆreikr
−k2 eikr r + 2ik eikr r2 + −2ik eikr r2
r , (14.49)
function term, so that we arrive at the representation G(0)
4\piϵ0
r −1 ϵ0
(14.50) (free-space Green tensor) We will use this form in deriving the Green tensor in the presence of a second atom (Section 14.3.7). 14.1.3.3 Derivation of the Formula for the Dipole Radiation Field We derived the above forms for the free-space Green tensor based on the knowledge of the dipole radia- tion field (14.45), which we simply wrote down. Now we are in a position to justify the above forms for G(0)(r, r′, \omega) and thus for the dipole radiation field. We start with the wave equation Eq. (14.28) with
ϵ0 \delta3(r −r′). (14.51)
ϵ0 \delta3(r −r′). (14.52) In components, this relation reads
G(0)
ϵ0
(14.53)
14.1.4 Permittivity Properties¶
Chapter 14. QED with Dielectric Media Guided by our comments above, we make the ansatz that the dipole radiation field is the transverse part of a scalar field, and thus we assume G(0)
\psi(r −r′), (14.54) where \psi(r) is a scalar function to be determined. Putting this ansatz into the wave equation for the Green tensor, we find
\psi(r −r′) −k2 ϵ0
\psi(r −r′) −k2 ϵ0
(14.55) Cancelling terms, we find
ϵ0
(14.56) For nonzero k, we then find
ϵ0 \delta3(r). (14.57) Evidently, \psi(r) is essentially the Green function for the scalar wave equation (Helmholtz equation). Note that we already showed in Eq. (14.49) that
r
(14.58) and thus it follows that
4\piϵ0 eikr r . (14.59) That is, \psi(r) is a spherical wave. This result, together with the ansatz (14.54), establishes the validity of the form (14.48) for the Green tensor, and thus of the other associated forms (14.46) and (14.50) and the dipole radiation field (14.45). Note that the scalar Green function (14.59), and thus the Green tensor (14.48), can correspond to outgoing or incoming waves, if k is respectively positive or negative. 14.1.4 Permittivity Properties The medium itself is described solely by the function ϵ(r, \omega) (or equivalently, the susceptibility), within the assumptions of a medium that is: linear, dispersive, isotropic, spatially nondispersive, nonmagnetic, and describable in terms of macroscopic variables (i.e., where one can coarse-grain away any microscopic structure). It is thus worth spending some time looking at this function more carefully.2 First, since g\chi(t), as in Eq. (14.19), relates the real fields P(r, t) and E(r, t), g\chi(t) itself is a real function. This implies that
(14.60)
components at \omega and −\omega∗, being complex conjugates, must be present with the same amplitude. From the definition (14.18) of the permittivity, it has the same frequency constraint as the susceptibility:
(14.61) In particular, this implies that Re[ϵ(\omega)] is an even function of \omega \in R, while Im[ϵ(\omega)] is an odd function. Furthermore, from the Fourier-transform relation (14.20), we can see that under the physically reason- able assumption that g\chi(t) is bounded, \chi(r, \omega) is analytic in the upper half of the complex-frequency plane, 2For further reading, see John David Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), section 7.10; or L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960), §58-62.
14.1 Classical Electrodynamics in Dielectric Media Im[\omega] > 0. This is because the Fourier integral (14.20) is always cut off exponentially, and thus guaranteed to converge. Thus, ϵ(\omega) is also analytic in the upper half-plane. But what about right on the real axis? For dielectrics, ϵ(\omega) is analytic on the real axis, if we make the reasonable assumption that g\chi(t) −\rightarrow 0 as t −\rightarrow \infty, so that the medium response ‘‘forgets’’ the input field in the distant past. In this case the oscillatory integral (14.20) converges because of this cutoff. However, for conductors, which are also modeled by a (complex) permittivity ϵ(\omega), the situation is slightly more complicated. Starting with the assumption of linear, dispersive conduction,
(14.62) where \sigma(\omega) is the conductivity, the wave equation (14.26) becomes
(14.63) The source term can then be combined with the permittivity term as
\omega
(14.64) The bracketed quantity in the second term can then be interpreted as a complex permittivity,
\omega , (14.65) (permittivity for a conductor) and we can henceforth regard ϵ(\omega) as modeling conductors as well as dielectrics. Since the conductivity tends to a finite constant at zero frequency, we see that for a conductor, the permittivity ϵ(\omega) (including conduction effects) has a simple pole at the origin. Thus, except for a possible simple pole at the origin,
14.1.4.1 Energy Loss and Poynting’s Theorem Before, we argued that the total energy of the field in free space was [Eq. (8.34)] U = 1 Z d3r ϵ0E2 + µ0H2 , (14.66) so that the field energy density is u = 1 ϵ0E2 + µ0H2 . (14.67) As we recall, in considering loss of transported energy, we need only consider transverse fields here, though we will drop the usual superscript denoting this. To generalize this expression to the dispersive-dielectric case, we must decompose the fields into frequency components, and then make the replacement ϵ0 −\rightarrow ϵ(\omega) at each frequency \omega. We can thus write the dielectric energy spectral density as
Z d3r h
i + c.c., (14.68) such that the total energy density is u = Z \infty
(14.69) This comes from writing the real fields at frequency \omega as, e.g., E(+)(r, \omega)e−i\omegat + E(−)(r, \omega)ei\omegat, and then discarding the fast-rotating terms to effect a time average over short times corresponding to optical oscilla- tions. Note that the electric-field part of the energy, represented by E \cdot D = ϵ0E2 + P \cdot E explicitly contains both the free-field energy as well as the coupling of the medium polarization to the field.
zeroes in the upper half-plane. See L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed. (Pergamon, 1980), §123.
Chapter 14. QED with Dielectric Media In fact, for a dispersive dielectric, this energy is not constant. However, differentiating this expression is difficult, as in the time domain we recall from Eq. (14.24) that such a differentiation will not be simple. Thus, we will start from a different direction. The force on a charge q due to the electric field is F = qE, and
since the force is normal to the velocity v). For a charge density \rho, the rate per unit volume at which the electromagnetic field does work on the charge is \rhov\cdotE = j\cdotE. This is the rate at which field energy is converted to heat by the electromagnetic force, and thus represents an energy dissipation. Using the Maxwell equations
we may write
(14.70) Rearranging terms and using the definition S := E \times H of the Poynting vector, we arrive at Poynting’s theorem:
(14.71) (Poynting’s theorem) To see what this means, we can integrate over a volume and use the divergence theorem to write Poynting’s theorem as − I
Z
Z
(14.72) (Poynting’s theorem) Recalling that the Poynting vector represents the energy flux density of the field, the surface integral on the left represents the rate of energy entering the volume. The stuff on the right-hand side must therefore tell us where this incoming energy is going. We have already decided that the first term on the right-hand side represents loss of energy due to motion of free charges, or equivalently conversion of electromagnetic energy to heat. Now the important interpretation here is that the second term also represents field energy loss, but due to bound charges, in the form of the polarization of the dielectric medium (and the magnetization in the magnetodielectric case). Thus we may define
(dielectric dissipation rate per unit volume) (14.73) as the rate of energy loss per unit volume due to dielectric absorption.
\cdot
E(+)
E(+) . (14.74) Note that in the second step, we discarded terms rotating at optical frequencies, replacing them by their zero average values. The same calculation for the magnetic-field term gives
H(+) = 0 (14.75) for a nonmagnetic medium. Thus, the dissipation rate for a field at frequency \omega is
E(+) . (dielectric dissipation rate per unit volume) (14.76)
14.1 Classical Electrodynamics in Dielectric Media Thus, we see that Im[ϵ(\omega)] controls the dielectric dissipation of energy. Assuming a passive medium (i.e.,
(\omega > 0, passive medium). (14.77)
Eq. (14.61) that Im[ϵ(\omega)] be an odd function of \omega. 14.1.4.2 Kramers–Kronig Relations At this point we will need a couple of results from complex analysis. Consider a closed, simply connected contour in the complex plane, and let f(z) be a function analytic everywhere inside the contour. Then an integral around the contour vanishes, I
(14.78) and Cauchy’s integral formula states that on the same contour,
2\pii I f(z) z −a dz (14.79) if a is a point interior to the contour and the integration proceeds counterclockwise around the contour (a clockwise integration implies an extra minus sign). The common terminology is that the function f(z)/(z−a)
both results are independent of the contour, so long as the above conditions hold. We can rewrite Cauchy’s integral formula as
2\pii I ϵ(\omega′) −ϵ0 \omega′ −z d\omega′, (14.80) where z is a point in the upper half-plane, where the contour runs along the real axis and along a semicircle in the upper half-plane, with the radius of the semicircle expanding to infinity. Re[w] Im[w] z Our goal will be to keep only the part of the contour on the real axis, and get rid of the semicircular part. If we integrate the Fourier-transform relation for \chi(\omega) by parts, we find ϵ(\omega) −ϵ0 ϵ0
Z \infty dt g\chi(t) ei\omegat
\omega −g′
\omega2 −ig′′
\omega3
(14.81)
continuous,
(14.82) so that the first term in the asymptotic expansion vanishes. The continuity requirement is physically rea- sonable because g\chi(t) represents a temporal response of the medium polarization to the input field. This
Chapter 14. QED with Dielectric Media signal involves the displacement of electrons, which cannot happen instantaneously in response to a sudden change in the field. Thus, we have the asymptotic behavior
(14.83) It is evident as the semicircle is taken out to infinity, its contribution to the contour integral vanishes. Thus, we may write the contour integral (14.80) as
2\pii Z \infty −\infty ϵ(\omega′) −ϵ0 \omega′ −z d\omega′. (14.84) Now if we take the point z and move it to the real axis, z −\rightarrow \omega + i0+, so that
2\pii Z \infty −\infty ϵ(\omega′) −ϵ0
(14.85) Notice that by our contour construction, the 0+ says that the integration contour must be deformed below the real line to avoid the pole at the real frequency \omega. If we instead deform the contour above the real line, the integral vanishes, because the semicircular contour no longer contains a pole: 0 = 2\pii Z \infty −\infty ϵ(\omega′) −ϵ0
(14.86) Adding together Eqs. (14.85) and (14.86), we can write
\pi – Z \infty −\infty ϵ(\omega′) −ϵ0
d\omega′, (14.87) where the cut integral sign represents the Cauchy principal value integral, – Z \infty −\infty f(z′) z′ −z dz′ := 1 Z \infty −\infty f(z′)
Z \infty −\infty f(z′) z′ −z −i0−dz′ , (14.88) which averages the results of avoiding the pole by deforming the contour on either side of it. Note that the Cauchy principal value can be alternately defined as a symmetric excision of the pole from the real-axis integral, – Z \infty −\infty f(z′) z′ −z dz′ = lim \delta−\rightarrow 0 "Z z−\delta −\infty f(z′) z′ −z dz′ + Z \infty
f(z′) z′ −z dz′ # , (14.89) as can be seen from the contour definition by considering semicircular deformations of the contour around the pole. In either case, we may write out the real and imaginary parts of the complex relation (14.87) to obtain the Kramers–Kronig relations
\pi – Z \infty −\infty Im[ϵ(\omega′) −ϵ0]
d\omega′
\pi – Z \infty −\infty Re[ϵ(\omega′) −ϵ0]
d\omega′, (14.90) (Kramers–Kronig relations) relating the real and imaginary parts of ϵ(\omega) −ϵ0. These relations can be written more compactly in terms of the Hilbert transform, defined as
\pi – Z \infty −\infty f(x′) x′ −x dx′, (14.91)
14.1 Classical Electrodynamics in Dielectric Media so that¶
(14.92) or separated into real and imaginary parts,
H {Im [ϵ(\omega) −ϵ0]}
(14.93) Note that the we derived the Kramers–Kronig relations just on the assumption that ϵ(\omega) −ϵ0 represents a causal response of the medium (polarization field) to an input stimulus (electric field). Any causal response function, such as the gain spectrum of an amplifier circuit, must satisfy the same relations. From the Kramers–Kronig relations, some features of the permittivity are immediately apparent. First, notice that the Hilbert transform acts ‘‘something like’’ a derivative. To see this, note that the Hilbert transform of a constant vanishes: H [c] = c \pi – Z \infty −\infty dx′ x′ −x = 0. (14.94) This is because the convolution kernel 1/(x′ −x) is antisymmetric about x, and even though the integrand is singular the contributions to the integral on either side of x cancel. This is precisely how the Cauchy principal value avoids problems with the singular integrand. However, as soon as the integrand has a slope, the antisymmetry of the kernel ‘‘picks it out,’’ because it upsets the cancellation in the above integral. In terms of the Kramers–Kronig relations, this implies that any variation in the real part of ϵ implies a nonzero imaginary part, and vice versa. We can rephrase this statement as: the existence of dispersion implies the existence of absorption (though possibly at some other frequency). This is quite a general result, being only
analytic in the upper half-plane, which allowed us to the whole contour-integral calculation. Of course, these conclusions apply to any causal, dispersive response function. The other obvious implication of the Kramers–Kronig relations is that for the integrals in Eqs. (14.93) to converge, the integrands must vanish at large frequencies, so that ϵ(\omega) −\rightarrow ϵ0 as \omega −\rightarrow \infty. Physically, this means that any medium must behave like the vacuum (i.e., become perfectly transparent) for very large frequencies. This is reasonable in the context of Lorentz-type models for media, since the atom will refuse to respond to fields with frequencies well over the atomic resonance frequencies. But we see here that this property is also implied by causality. Finally, we note that in the above derivation, we ignored any possible pole at \omega = 0. In fact, the Kramers–Kronig relations written above are valid only for dielectric media, and must be adapted to the case of conductors (Problem 14.1). 14.1.4.3 Imaginary Frequencies It turns out to be useful to know the behavior of the permittivity ϵ(\omega) on the positive imaginary axis, ϵ(is) for s > 0. First, from the reality constraint (14.61), we can let \omega −\rightarrow −is to find
(14.95) and thus ϵ(is) \in R for s > 0. We can go even further along this line. Using the first of the Kramers–Kronig relations (14.93) in the form (Problem 14.2),
\pi – Z \infty
d\omega′, (14.96) we can let \omega −\rightarrow is so that
\pi Z \infty
d\omega′. (permittivity for imaginary frequencies) (14.97)
Chapter 14. QED with Dielectric Media
where the equality only holds in vacuum. Furthermore, the nonnegative integrand decreases monotonically with s, and thus ϵ(is) is a monotonically decreasing function of s, so long as s > 0. Finally, from this expression it is apparent that for any imaginary frequency is, ϵ(is) contains information about ϵ(\omega) over the whole real axis. Specifically, the imaginary part of ϵ(\omega) is involved here, although of course information about the real part is implicitly included as well in view of the Kramers–Kronig relations. As an example, consider the general permittivity for an atomic vapor corresponding to the Lorentz model (1.32):
m X j f0j \omega 2
(14.98) For the imaginary frequency, this expression becomes
m X j f0j \omega 2
(14.99) which is obviously positive, real, and monotonically decreasing with s. These same results obviously hold for closely related functions such as the polarizability \alpha(is) and the susceptibility \chi(is). Another way of thinking about the imaginary-axis permittivity comes from Eq. (14.20): ϵ(is) −ϵ0 ϵ0
Z \infty dt g\chi(t) e−st. (14.100) Hence ϵ(is) [and of course \chi(is)] is related to the Laplace transform of g\chi(t). This is of course because \chi(\omega) is the Fourier transform of g\chi(t), but the Laplace integral is properly cut off at zero here because g\chi(t) is a causal response function. 14.2 Generalized Susceptibility and Linear-Response Theory Recall that the usual electromagnetic susceptibility \chi(r, \omega) gives the dispersive response of the dielectric medium polarization P(r, \omega) to an applied field E(r, \omega):
(14.101) Of course, this is only the linear response, valid for small applied fields. Now we want to generalize this notion of the susceptibility to the quantum case, and to more general input-response situations.4 Consider the interaction Hamiltonian
(14.102) where x is an operator coupling to a time-dependent ‘‘force’’ function F(t). We can write the linear response of the expectation value \langle x\alpha(t)\rangle in general as
Z \infty −\infty
(14.103) assuming that the perturbation has been in effect since the distant past (corresponding to the +\inftyintegration limit). Again, the integration limits ensure a causal response to the perturbation. The Fourier transform of this relation is, by the convolution theorem,
(14.104) 4Ryogo Kubo and Kazuhisa Tomita, ‘‘A General Theory of Magnetic Resonance Absorption,’’ Journal of the Physical Society of Japan 8, 888 (1954); R. B. Stinchcombe, ‘‘Kubo and Zubarev Formulations of Response Theory,’’ in Correlation Functions and Quasiparticle Interactions in Condensed Matter, J. Woods Halley, Ed. (Plenum, 1978), p. 3; A. D. McLachlan, ‘‘Retarded dispersion forces between molecules,’’ Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 271, 387 (1963).
14.2.1 Proof¶
14.2 Generalized Susceptibility and Linear-Response Theory where
Z \infty
Z \infty −\infty
Z \infty −\infty dt F\alpha(t) ei\omegat. (14.105) What we want to show is that the temporal response can be written as the correlation function
¯h
\Theta(\tau), (14.106) where x\alpha(\tau) is in the interaction picture. Note that the Heaviside function enforces a causal response in Eq. (14.103). In what follows, we will want to show that the generalized susceptibility can be written
¯h Z \infty d\tau
(14.107) (generalized susceptibility) That is, in the linear regime, the generalized susceptibility can be written as a Fourier transform of a quantum correlation function. 14.2.1 Proof To prove these results, we will start by representing the quantum state by the density operator
(14.108) where \rho0 represents the quantum state in the presence of only the background, time-independent Hamiltonian H0, and \delta\rho is the linear-order correction due to Hint. Then the equation of motion for the quantum state is
(14.109) Note that the use of Hamiltonian evolution here is generally valid even with dissipation, since we can always extend the system to a larger Hilbert space where the evolution is Hamiltonian. Then canceling the zeroth- order parts and discarding the second-order term [Hint, \delta\rho], the equation of motion becomes
¯h [Hint, \rho0] = −i
(14.110) Transforming to the interaction picture by defining
(14.111) this equation of motion becomes
(14.112) Integrating this equation gives
¯h Z t −\infty
(14.113)
14.2.2 Atom and Field Susceptibilities¶
Chapter 14. QED with Dielectric Media and then transforming out of the interaction picture, the formal solution for the perturbed state is
¯h Z t −\infty dt′ eiH0(t′−t)/¯h [x\alpha, \rho0] F\alpha(t′) e−iH0(t′−t)/¯h. (14.114)
the distant past. Now the perturbation to the mean response it
= i ¯h Z t −\infty dt′ Tr n
F\beta(t′) = i ¯h Z t −\infty
= i ¯h Z t −\infty
= i ¯h Z 0 −\infty
= i ¯h Z \infty
(14.115) Here we have used t′ = \tau + t, we have defined the interaction-picture response operator
(14.116) and, to linear order, we see that the expectation value is taken with respect to the background state \rho0. Comparison to Eq. (14.103) thus allows us to identify
(14.117) which is our desired result. 14.2.2 Atom and Field Susceptibilities For the atom–field interaction, corresponding to the interaction Hamiltonian Hint = −d\cdotE, we can work out two important generalized-susceptibility formulae. First thinking about the dipole moment as an operator coupled to a classical electric field, we can consider the dipole response due to the field. In fact, we already have treated this in detail, as the mean dipole induced by a classical field is the atomic polarizability \alpha(\omega). In the general case, we can write the polarizability as a tensor,
(14.118) so that we do not necessarily assume a spherically symmetric atom (molecule). Thus, Eq. (14.107) implies that
¯h Z \infty
(polarizability in terms of dipole correlation) (14.119) It is clear from this expression that if dµ and d\nu represent completely independent degrees of freedom, then the commutator vanishes and thus the particular component \alphaµ\nu likewise vanishes. This happens, for example, when µ and \nu refer to different principal axes of the atom. In fact, if the coordinate system
14.2 Generalized Susceptibility and Linear-Response Theory is aligned with the principal atomic axes, then it follows that \alphaµ\nu is a diagonal tensor. For a spherically symmetric atom, all orthogonal coordinate systems form principal axes, and hence
(14.120) (scalar polarizatibility) where \alpha(\omega) is the scalar polarizability. It is also useful to invert the Fourier transform here, so that
¯h 2\pii Z \infty −\infty
(14.121) To eliminate the step function here, we can write down the complex conjugate of this equation, using [A, B]\dagger = −[A, B] if A and B are Hermitian operators, with the result
¯h 2\pii Z \infty −\infty
(14.122)
¯h 2\pii Z \infty −\infty
(14.123) Now reversing the operators in the commutator and advancing the time of both operators in the commutator by \tau (recall the correlation function only depends on time differences, since we aren’t considering transients),
2\pii Z \infty −\infty
(14.124) Now adding this equation to Eq. (14.121), we find
\pi Z \infty −\infty
(dipole correlation in terms of polarizability) (14.125) Now, what about the symmetry of the polarization tensor? Recall that according to the Hamiltonian Hint = −d \cdot E, the energy is the work required to adiabatically turn on the field in the presence of the induced dipole: U = − Z E
(14.126) This is valid at any for amplitudes of any component of frequency \omega, though we will suppress the frequency dependence. It is also valid for linear polarization (i.e., it omits circular polarization), so this argument is not completely general. The change in energy due to a small field change is
(14.127) Of course, we can equivalently write this as
(14.128)
and thus the polarizability tensor is symmetric. Similarly, thinking of the field at r as responding to a classical dipole at position r′, we may write the generalized susceptibility for the field. Recall that the classical Green tensor was precisely this response function for a dipole of unit amplitude, and thus we may write the Green tensor in terms of the correlation function as
¯h Z \infty
(Green tensor in terms of field correlation) (14.129)
14.3.1 Kramers-Heisenberg Formula¶
Chapter 14. QED with Dielectric Media The inverse relation is correspondingly
\pi Z \infty −\infty
(field correlation in terms of Green tensor) (14.130) As it turns out, the expressions here are very useful in treating interactions between an atom and the quantum electromagnetic field. One more note is in order here. The permittivity ϵ(\omega) −ϵ0 is another generalized susceptibility. Thus, the properties derived in Section 14.1.4 that were not specific to the electromagnetic field apply also to \alphaµ\nu(\omega) and G\alpha\beta(r, r′, \omega). To summarize, G\alpha\beta(r, r′, \omega) is analytic in the upper half-plane,
\pi – Z \infty −\infty
d\omega′
\pi – Z \infty −\infty
d\omega′, (14.131) and G\alpha\beta(r, r′, is) is real and monotonically decreasing with s > 0. Of course, all these properties apply to
14.3 Atom–Surface Potentials Near Dielectric Media Now we can use the above formalism to compute the energy shift for a ground-state atom due to the electromagnetic vacuum, modified by a dielectric medium. The potential shift is given by second-order perturbation theory as
X j X k,\zeta
Eg,0 −Eej,1k,\zeta , (14.132) just as in our treatment of the Casimir–Polder potential in the conducting-plane case of Chapter 13, where |g\rangle is the ground state, the excited states are |ej\rangle , and the field modes are labeled by the wave vector k and the polarization index \xi. We are again considering the usual dipole interaction Hamiltonian HAF = −d \cdot E. Both the zeroth- and first-order terms vanish, leaving VCP = − X j X k,\zeta
, (14.133) (second-order level shift)
to derive expressions for the atom and field susceptibilities up to a consistent order in perturbation theory. 14.3.1 Kramers–Heisenberg Formula Recall that the atomic dipole susceptibility (polarizability) gives the mean induced dipole from the interaction
= − X j h
j i \cdot h ˆϵE(+)
ei\omegati (14.134) with the classical electric field E(t), which is of the same form as the linear-response interaction (14.102). Here, \sigmaj := |g\rangle \langle ej| is the usual atomic lowering operator for the |g\rangle −\rightarrow |ej\rangle transition. Now we wish to
14.3 Atom–Surface Potentials Near Dielectric Media compute the generalized susceptibility (atomic polarizability) for the mean dipole response to the applied classical field. We can treat this interaction in time-dependent perturbation theory by deriving the perturbed ground state |g\rangle + \delta|g\rangle , where the state response
X j a(+) j
j ei\omegat e−iE0t/¯h|ej\rangle , (14.135) to the perturbation represents the mixing in of the excited states by the interaction (E0 is the ground- state energy). The coefficients a(\pm) j remain to be determined. The idea is to then put this ansatz into the Schrödinger equation \partial t h
i = −i ¯h(H0 + HAF) h
i , (14.136) so that to first order
¯hHAF|g\rangle , (14.137)
X j h
j
j ei\omegati e−iE0t/¯h|ej\rangle = −i ¯h X j Ej a(+) j
j ei\omegat e−iE0t/¯h|ej\rangle + i ¯h X j
h ˆϵE(+)
e−i\omegati e−iE0t/¯h|ej\rangle . (14.138) Matching coefficients of e\pmi\omegat|ej\rangle ,
j = −i ¯hEja(+) j + i
j = −i ¯hEja(−) j + i
, (14.139) and thus the perturbation coefficients are a(+) j
a(−) j
, (14.140)
ground state due to the perturbation is
(14.141) to first order in the perturbation, assuming that \langle d\rangle = 0 in the unperturbed ground state. Thus,
X j
ˆϵE(+)
ˆϵ∗E(−)
! + X j
ˆϵ∗E(−)
ˆϵE(+)
!
(14.142) Given that there is no circular polarization, the polarization vector ˆϵ is real, and thus
X j
¯h(\omega 2 j0 −\omega2) \cdot ˆϵ h E(+)
ei\omegati . (14.143)
Chapter 14. QED with Dielectric Media Writing out only the positive-frequency amplitude, D
E = X j
¯h(\omega 2 j0 −\omega2)
(14.144) and then using tensor-component notation, D d(+) µ (\omega) E = X j
)\nu ¯h(\omega 2 j0 −\omega2) , (14.145) we can compare this expression to the defining relation D d(+) µ (\omega) E
)\nu (14.146) for the polarizability tensor to arrive at the following expression for the polarizability in terms of the dipole matrix elements:
X j
¯h(\omega 2 j0 −\omega2) . (14.147) (Kramers–Heisenberg formula) For a spherically symmetric atom, all dipole-operator components dµ are identical, and to be consistent with
X j
¯h(\omega 2 j0 −\omega2) . (14.148) (scalar Kramers–Heisenberg formula) Either expression is referred to as a Kramers–Heisenberg formula for the atomic polarizability. We derived this expression for the atomic ground state, but of course it is valid for any atomic state |q\rangle , by letting
which is negative for states of lower energy than |q\rangle . Actually, we obtained the polarizability by calculating the physical dipole moment, which only involves the real part of \alphaµ\nu(\omega). Thus, we have so far only computed the real part:
X j
¯h(\omega 2 j0 −\omega2) . (14.149) According to the Kramers–Kronig relations, variation in Re[\alphaµ\nu(\omega)] leads to nonzero values of Im[\alphaµ\nu(\omega)].
\pi – Z \infty −\infty
d\omega′. (14.150) However, it turns out to be easier to guess Im[\alphaµ\nu(\omega)] from the inverse relation,
\pi – Z \infty −\infty
d\omega′, (14.151) if we compare it to Eq. (14.149) in the form
X j
¯h
. (14.152) The two expressions are consistent if
¯h X j
h
i . (14.153)
14.3.2 Green Tensor¶
14.3 Atom–Surface Potentials Near Dielectric Media These expressions are equivalent to the classical polarizability expression (from Chapter 1, if we write the oscillator strength in terms of the dipole matrix elements)
\Gammaj\rightarrow 0 X j
¯h (\omega 2
j \omega2 (\omega 2
, (14.154) if we take the limit of zero damping, being careful to note the emergence of the delta function in the imaginary part. What happens if we don’t assume that the field is linearly polarized? Going back to Eq. (14.147), this is still the correct formula for the tensor and scalar [Eq. (14.148)] parts. For the vector part, going back to Eq. (14.142), we can use the fact that the dipole combination is antisymmetric and the polarization conjugate ˆϵ∗= −ˆϵ to write
X j
ˆϵE(+)
ˆϵE(−)
! − X j
− ˆϵE(−)
ˆϵE(+)
! , (14.155) so that (note the \omega instead of the \omegaj0 in the numerator)
X j
¯h(\omega 2 j0 −\omega2) \cdot ˆϵ h E(+)
ei\omegati . (14.156) The same line of reasoning gives \alpha(1)
X j
¯h(\omega 2 j0 −\omega2) (Kramers–Heisenberg formula, vector part) (14.157) for the vector part of the polarizability tensor (in the case of circular polarization). 14.3.2 Green Tensor The interaction is symmetric with respect to the dipole and field operators. Thus, thinking of the mean, linear quantum response at r of all the electromagnetic field modes (labeled by k, \zeta) to a dipole at r′, we may write down the analogous Kramers–Heisenberg formula for the Green tensor as
X n X k,\zeta
¯h (n2\omega 2 k −\omega2) . (14.158) Again, the numbers here denote the number of photons in the labeled mode, and the energy of |nk,\zeta\rangle is n¯h\omegak. Since the field is a harmonic oscillator, the electric field operators are effectively a sum over creation and annihilation operators. Thus, when considering the response of the vacuum, only the one-photon states contribute to this sum:
X k,\zeta
¯h (\omega 2 k −\omega2) . (Kramers–Heisenberg formula for Green tensor) (14.159) Correspondingly, the imaginary part of the Green tensor is
¯h X k,\zeta
h
i , (14.160) for consistency with causality.
14.3.3 Interaction Energy¶
Chapter 14. QED with Dielectric Media 14.3.2.1 Mode Expansion of the Green Tensor A particular application of the Kramers–Heisenberg formula is to write down a Green tensor in terms of electromagnetic field modes, assuming such modes exist. This happens, for example, when quantizing in an enclosed domain, as in a cavity with perfectly conducting walls. Recall that in such cases, the electric-field operator has the form [Eq. (8.61)]
X k,\zeta − r ¯h\omegak 2ϵ0
(14.161) where the fk,\zeta(r) are the normalized classical mode functions for wave vector k and polarization \zeta. Putting this field operator into Eq. (14.159), we find the general expression
ϵ0 X k,\zeta \omega 2 k \omega 2
(mode expansion of Green tensor) (14.162) The Green tensor is in principle a more general object, however, and simple mode functions may not exist in particular when the Green tensor represents dispersive (and thus absorbing) media. However, this general approach below to the interaction energy doesn’t depend on the existence of mode functions in this simple sense; in principle, when absorption is present, the system can be extended to include extra, effective ‘‘absorbing’’ fields that account for absorption, while making the total, extended system Hamiltonian. Mode functions that span multiple fields exist in this case, and though possibly difficult to write down, their existence allows our arguments below to go through. 14.3.3 Interaction Energy We will now claim that the second-order interaction energy (14.133) may be written in the form5
2\pi Z \infty ds Tr h \alpha(is) \cdot G(r, r, is) i = −¯h 2\pi Z \infty
(14.163) (Casimir–Polder potential) where r denotes the location of the atom. This is a rather remarkable result, as although it assumes that both the atom and the field are quantized, it does not require us to state exactly how the field was quantized. This simplifies things dramatically, as quantization with dissipation is not simple. This result is also remarkable in that it only involves the classical susceptibilities, although to derive this we restricted ourselves to a linear (and thus lowest-order) interaction Hamiltonian between the atom and the quantum electromagnetic field. To confirm this result, first note that Im[\alphaµ\nu(is)] and Im[G\alpha\beta(r, r′, is)] will vanish for s \in R, because they involve delta functions of the form \delta(is −\omegaj), which will always vanish for real \omegaj. Thus, on the imaginary axis, both susceptibilities are given only by their real parts, which again for any generalized susceptibility (Section 14.1.4.3) turn out to be real, monotonically decreasing functions of s. Now, we can 5A. D. McLachlan, ‘‘Retarded dispersion forces between molecules,’’ Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 271, 387 (1963); A. D. McLachlan, ‘‘Van der Waals forces between an atom and a surface,’’ Molecular Physics 7, 381 (1963) (doi: 10.1080/00268976300101141); J. M. Wylie and J. E. Sipe, ‘‘Quantum electrodynamics near an interface,’’ Physical Review A 30, 1185 (1984) (doi: 10.1103/PhysRevA.30.1185).
14.3.5 Planar Interface¶
14.3 Atom–Surface Potentials Near Dielectric Media substitute the above expressions for the susceptibilities into (14.163) to obtain VCP = −¯h 2\pi Z \infty ds X j
¯h(\omega 2 j0 + s2) X k,\zeta
¯h (\omega 2 k + s2) = −2 \pi¯h X j X k,\zeta
Z \infty ds
(\omega 2
k + s2) = − X j X k,\zeta
. (14.164) Here, we have used the identity Z \infty dx ab
\pi 2(a + b), (a, b > 0) (14.165) which follows by noting the integrand is an even function of x, changing to a contour integral over the great semicircle in the upper half-plane, and using the Cauchy integral formula, since the contour encloses the
perturbation expression (14.133) that we started off with, so we see that Eq. (14.163) gives the correct atom–material interaction energy. The only result that we really needed to derive Eq. (14.163), beyond the contour integral, was the Kramers–Heisenberg formula (applied to both the atom and field susceptibilities). 14.3.4 Renormalization We have already considered the form for the free-space Green tensor in Section 14.1.3.1. Thus, there is an interaction energy associated with an atom in free space,
2\pi Z \infty ds Tr h \alpha(is) \cdot G(0)(r, r, is) i , (14.166) where G(0)(r, r, \omega) is the free-space Green tensor, Eq. (14.46). We have already seen (Section 13.12) that this energy diverges and after renormalization gives the Lamb shift. To investigate the vacuum interaction energy of an atom with a dielectric body, we will need to subtract this contribution to obtain the energy difference in the presence vs. the absence of the body. The correct (observed) renormalized interaction potential is thus V (r)
VCP = −¯h 2\pi Z \infty ds Tr h \alpha(is) \cdot G(s)(r, r, is) i , (renormalized Casimir–Polder potential) (14.167) after dropping the superscript, where
(14.168) (scattering Green tensor) is the scattering part of the Green tensor, i.e., the part of the Green tensor due specifically to the presence of the dielectric body. 14.3.5 Planar Interface To compute the atom–surface interaction with a planar interface, we will first start by expanding the free- space Green function into plane waves.6 The idea is that in a very general way, we can compute the effect 6A. D. McLachlan, ‘‘Retarded dispersion forces between molecules,’’ Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 271, 387 (1963); J. E. Sipe, ‘‘The Dipole Antenna Problem in Surface Physics: A New Approach’’ Surface Science 105, 489 (1981) (doi: 10.1016/0039-6028(81)90014-5); T. Setälä, M. Kaivola, and A. T. Friberg, ‘‘Decomposition of the point-dipole field into homogeneous and evanescent parts,’’ Physical Review E 59, 1200 (1999) (doi: 10.1103/PhysRevE.59.1200).
Chapter 14. QED with Dielectric Media of the planar interface on a plane wave, via the usual optical reflection coefficients. Then, summing over all plane-wave contributions, we can obtain the effect of the surface on a dipole field, and thus obtain the scattering Green tensor for the surface. We start by recalling the free-space Green tensor in the form (14.48): G(0)
4\piϵ0
eikr r , (14.169) The spherical-wave factor eikr/r has the plane-wave expansion7 eikr r = i 2\pi Z \infty −\infty dkx Z \infty −\infty dky kz ei(kxx+kyy+kz|z|), (14.170) where the magnitude of the wave vector k must be simply k = \omega/c, so kz = q k2 −k 2 x −k 2 y if k 2 x + k 2 y \le k2 i q k 2 x + k 2 y −k2 if k2 x + k2 y > k2. (14.171) Note that kx and ky correspond to transverse wave numbers in a wave-vector expansion, while kz corresponds to the longitudinal wave number, thinking about propagation along the z direction. Small values of kx and ky (the upper expression for kz) thus correspond to propagating waves (with a longitudinal phase factor of eikzz), while, large values (the lower expression for kz) correspond to evanescent waves8 (with longitudinal exponential decay of e−kzz). Thus, we may write the expansion for the free-space Green function as G(0)
i 8\pi2ϵ0 Z \infty −\infty dkx Z \infty −\infty dky kz
ei(kxx+kyy+kz|z|). (free-space Green tensor, plane-wave decomposition) (14.172) We will generally imagine the interface to be below the dipole (i.e., smaller values of z), so that we will need the Green tensor in the region z < 0 G(0)
i 8\pi2ϵ0 Z \infty −\infty dkx Z \infty −\infty dky kz
\alpha k− \beta ei(kxx+kyy−kzz), (14.173) where k−:= kxˆx + kyˆy −kzˆz, to determine the scattering from the surface. Further, we will ultimately need a Green tensor of the form G\alpha\beta(0, 0, \omega), so we can discard the transverse spatial dependence by setting x = y = 0, since the planar surface is invariant under transverse displacement. Further restoring the explicit coordinate z′ for the dipole, G(0)
i 8\pi2ϵ0 Z \infty −\infty dkx Z \infty −\infty dky kz
\alpha k− \beta e−ikz(z−z′), (14.174) where again this expression is valid for z < z′. 14.3.5.1 Reflection Coefficients Now we will introduce the reflection coefficients for the plane-wave field amplitudes r⊥(\theta, \omega) and r∥(\theta, \omega). These are defined such that in incoming and reflected field amplitudes are related by Eout = rEin. The subscripts denote whether the incoming field polarization is perpendicular or parallel to the plane of incidence (the plane containing both the surface-normal vector and the incoming wave vector), and \theta is the angle
coefficients are in general complex, reflecting phase shifts from the surface reflection. Note that these can be readily calculated by matching electromagnetic boundary conditions for bodies with planar-type symmetry, 7This is the Weyl representation for the spherical wave; see Leonard Mandel and Emil Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995), Eq. (3.2-61). 8The evanescent modes are plasmonic modes in the case of a metal and polaritonic modes in the case of a dielectric.
14.3 Atom–Surface Potentials Near Dielectric Media including simple interfaces or stacks of thin films, including conducting films.9 For example, for plane waves incident from vacuum onto a planar interface to a medium of dispersive permittivity ϵ(\omega), the reflection coefficients are
cos \theta − q
q
q
q
, (14.175) (Fresnel reflection coefficients)
at normal incidence,
(14.176) whereas a common alternate convention is to flip the sign of r∥compared to what we have here. These coefficients are then sufficient to determine the scattering part of the Green tensor in the presence of the interface, because we can now calculate how each plane-wave component of the free-space Green tensor is reflected by it. 14.3.5.2 Scattering Green Tensor Due to the Planar Interface Now let us define our coordinates such that ˆz is normal to the planar interface at z = 0, and we may assume the atom to be located at x = y = 0. We want the scattering Green tensor at the same location as the source dipole, G(s) \alpha\beta(z, z, \omega). In this particular case, the off-diagonal components of the scattering Green tensor vanish: the reflection of an oscillating dipole in the r\alpha-direction will be polarized in the r\alpha-direction, from the point of view of the original dipole. This is just due to the symmetry of the surface, since the effect of the reflection on a wave vector k is the transformation k −\rightarrow k−. We can start with the dipole oscillating in the z-direction. The scattering part of the Green tensor is then given by multiplying the dipole field by −r∥, since the z-axis is always in the plane of incidence, but the extra minus sign is present because the z-component of the polarization vector is always reversed upon reflection from the surface. The phase factor is given by that of a wave traveling the distance z down to the surface and back, and the result is G(s)
i 8\pi2ϵ0 Z \infty −\infty dkx Z \infty −\infty dky kz k2 −k 2 z
(14.177) Switching the integration to cylindrical coordinates, we can carry out the angular integral and write G(s)
i 4\piϵ0 Z \infty dkT k 3 T kz
(14.178) where the transverse part of the wave vector is kT = kxˆx + kyˆy, (14.179) so that kT = q k 2 x + k 2 y = p k2 −k 2 z , (14.180) and now the incidence angle \theta can be defined in the integration coordinates by tan \theta = kT/kz. To evaluate the transverse components of the Green tensor, first pretend that kT = kTˆx. That is, choose the transverse part of the wave vector to point in the x-direction. 9Daniel A. Steck, Classical and Modern Optics (2006), Chapters 9-10, available online at http://steck.us/teaching/. Note that r⊥≡rS and r∥≡rP in the notation there.
Chapter 14. QED with Dielectric Media x xoáo|| yoáo\pi k kT plane of incidence y z Then a dipole oscillating about x has a field polarized parallel to the incidence plane, and a dipole in the y- direction produces a perpendicular polarization. Thus, the appropriate scattering Green tensor components are G(s)
i 8\pi2ϵ0 Z \infty −\infty dkx Z \infty −\infty dky kz k2 −k 2 T
G(s)
i 8\pi2ϵ0 Z \infty −\infty dkx Z \infty −\infty dky kz k2
(14.181) Switching to cylindrical components and carrying out the angular integral amounts to simply replacing the xx and yy components by their average: G(s)
i 8\piϵ0 Z \infty dkT kT kz k 2
e2ikzz. (14.182) These Green-tensor components are all we need to characterize the surface in terms of the atom–surface potential, which we now evaluate. 14.3.5.3 Explicit Expressions for the Atom–Surface Potential Putting these Green-tensor components into the expression (14.167) for the atomic potential due to the planar surface, we find VCP = −¯h 2\pi Z \infty
\nuµ(r, r, is) = − i¯h 8\pi2ϵ0 Z \infty ds Z \infty dkT kT kz
k 2
−\alphazz(is)k 2 T r∥(\theta, is) e2ikzz, (14.183) where k = is/c, and kz = i p
T . Defining the real quantity
p
T , (14.184) we may rewrite this expression as VCP = ¯h 8\pi2ϵ0 Z \infty ds Z \infty dkT kT \kappa
c2 r⊥(\theta, is)
T r∥(\theta, is) e−2\kappaz, (Casimir–Polder potential in terms of reflection coefficients) (14.185) so that now the potential is written entirely in terms of the reflection coefficients, the polarizability, real (non- negative) integration variables, and real functions of the integration variables. Notice that for a medium with n > 1, the reflection coefficients are in fact negative at normal incidence. So long as they stay predominantly negative, the overall potential is negative, and thus attractive (as the contributions decay exponentially with distance).
14.3 Atom–Surface Potentials Near Dielectric Media Suppose now that the atom is spherically symmetric, as is appropriate for typical ground-state atoms. Then the polarizability tensor becomes a scalar, and so10 VCP = ¯h 8\pi2ϵ0c2 Z \infty ds s2 \alpha(is) Z \infty dkT kT \kappa
1 + 2k 2 T c2 s2 r∥(\theta, is) e−2\kappaz. (Casimir–Polder potential, spherically symmetric atom) (14.186) Note that the integrand here is rather smooth, and not oscillatory. As a function of the imaginary part s of the frequency, \alpha(is) decreases monotonically owing to its function as a susceptibility. The same can be said of r⊥,∥(\theta, is), which can also be regarded as a susceptibility (linear response) of the reflected-field amplitude to the input field. Similarly \kappa increases smoothly with s, and any explicit dependence on s is smooth. 14.3.5.4 Perfect-Conductor Limit To gain some more insight into the basic result, we can examine the potential (14.186) in various limits. The first limit we can examine is that of a perfect conductor, corresponding to perfect reflections: r⊥,∥(\theta, is) −\rightarrow −1. (14.187) In this case, the potential (14.186) becomes VCP = − ¯h 4\pi2ϵ0c2 Z \infty ds s2 \alpha(is) Z \infty dkT kT \kappa 1 + k 2 T c2 s2 e−2\kappaz. (14.188) Using Eq. (14.148) for the polarizability in the form
X j 2kj0|\langle g|dz|ej\rangle |2
j0), (14.189) and putting in the explicit form for \kappa, we find VCP = − 2\pi2ϵ0c X j d 2 j,zkj0 Z \infty ds Z \infty dkT kT p
T
j0 e−2z q
T = 4\pi2ϵ0c X j d 2 j,zkj0 \partial z Z \infty ds
j0 Z \infty dkT kT e−2z q
T = 8\pi2ϵ0c X j d 2 j,zkj0 \partial z Z \infty ds
j0 Z \infty d(k 2 T ) e−2z q
T , (14.190) where again d 2 j,z is shorthand for |\langle g|dz|ej\rangle |2. We can evaluate the second integral using Z \infty dx e−2z p
2z2 e−2zs/c, (14.191) so that VCP = 16\pi2ϵ0c X j d 2 j,zkj0 \partial z z2 Z \infty
j0 e−2zs/c = − 16\pi2ϵ0c X j d 2 j,zkj0 \partial 2 z z Z \infty ds
j0 e−2zs/c. (14.192) 10cf. Werner Vogel and Dirk-Gunnar Welsch, Quantum Optics, 3rd ed. (Wiley, 2006), Eq. (10.84), noting the alternate sign convention for r∥. Also see A. D. McLachlan, ‘‘Van der Waals forces between an atom and a surface,’’ Molecular Physics 7, 381 (1963) (doi: 10.1080/00268976300101141), Eq (3.9) for the case of a simple dielectric interface.
Chapter 14. QED with Dielectric Media Now, we can evaluate the last integral, using the integral formula (Problem 13.1)11 Z \infty dx e−µx
\beta (Re[\beta] > 0, Re[µ] > 0), (14.193) where f(z) is an auxiliary function to the sine and cosine integrals [see Eq. (13.26)]. Thus, the Casimir–Polder potential in this case becomes VCP = − 16\pi2ϵ0 X j d 2 j,z \partial 2 z z f(2kj0z), (Casimir–Polder potential, perfect conductor) (14.194) which of course agrees exactly with the result we computed before, Eq. (13.51). 14.3.5.5 Near-Field Limit
atom. First, note that due to the exponential factor in Eq. (14.186), only relatively small values of \kappa,
c2 + k 2 T ≲ 4z2 , (14.195) will contribute in the integral to the potential. However, due to the form of the factor s2\alpha(is)/\kappa, the most important values of s in the integral are of the order of \omegaj0. But then in the near-field regime, we can conclude that s c ≪1 2z . (14.196) This means that typically in the integral, \kappa ∼kT ∼1 2z , (14.197) and these are all large compared to s/c. Physically, since |s/c| is the optical wave vector, this means that the evanescent-wave modes—modes that propagate along the surface, decaying exponentially away from the surface—dominate the atom–surface potential in this regime. This is sensible, as surface modes should be most important in the near field. We will proceed by considering an atom in the near field of a simple dielectric interface, so that the reflection coefficients are given by the Fresnel expressions (14.175). In the near field, we can use the relations above to simplify the Fresnel coefficients, since we may represent the incidence angle by
k = −ickT s
s . (14.198) 11See I. S. Gradstein and I. M. Ryzhik, Table of Integrals, Series, and Products, English translation 6th ed., A. Jeffrey and D. Zwillinger, Eds. (Academic Press, 2000), Formula 3.354.1.
14.3 Atom–Surface Potentials Near Dielectric Media In the case of transverse polarization, then, using s/c ≪kT, \kappa,
cos \theta − q n2(is) −sin2 \theta
q n2(is) −sin2 \theta
p
T
p
T \approx \kappa −s2n2/2c2kT −kT
\approx (s2/2c2kT)(1 −n2)
\approx s2 4c2k2 T (1 −n2), (14.199) where we have used
r s2 c2 + k 2 T \approx s2 2c2kT + kT. (14.200) Thus, in the near field, r⊥≪1, and we will in fact neglect it compared to the longitudinal reflection coefficient, which is
q n2(is) −sin2 \theta −n2(is) cos \theta q n2(is) −sin2 \theta + n2(is) cos \theta = p n2 + c2k 2
p n2 + c2k 2
1 + n2(is)
ϵ0 + ϵ(is). (14.201) Putting this coefficient into Eq. (14.186) and ignoring the transverse reflection, VCP = − ¯h 8\pi2ϵ0c2 Z \infty ds s2 \alpha(is) ϵ(is) −ϵ0 ϵ(is) + ϵ0 Z \infty dkT kT \kappa 1 + 2k 2 T c2 s2 e−2\kappaz = − ¯h 8\pi2ϵ0 Z \infty ds \alpha(is) ϵ(is) −ϵ0 ϵ(is) + ϵ0 Z \infty s/c d\kappa s2 c2 + 2k 2 T e−2\kappaz \approx − ¯h 4\pi2ϵ0 Z \infty ds \alpha(is) ϵ(is) −ϵ0 ϵ(is) + ϵ0 Z \infty
(14.202) The second integral is easy to carry out, with the result12 VCP = − ¯h 16\pi2ϵ0z3 Z \infty ds \alpha(is) ϵ(is) −ϵ0 ϵ(is) + ϵ0 (14.203) (van der Waals potential) for the near-field (van der Waals) force near a dielectric surface of permittivity ϵ(\omega). This is the dipole-dipole interaction of an atom with its image, located at distance z inside the dielectric. In the static case, the image charge distribution has a charge of (ϵ −ϵ0)/(ϵ + ϵ0) times the original charge.13 The expression here is the extension of that image concept to all frequencies, and the contribution to the atomic energy is weighted by the response \alpha(\omega), and then the energy is summed over all frequencies. 12A. D. McLachlan, op. cit., Eq. (3.7). 13See John David Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), p. 157.
Chapter 14. QED with Dielectric Media 14.3.5.6 Far-Field Limit In the far-field limit, where z becomes very large, the values of \kappa, and thus s and kT, that contribute to the integral are very small. In particular, we can replace \alpha(is) by the static polarizability
X j 2d 2 j,z ¯h\omegaj0 , (14.204) and we can replace the reflection coefficients by their dc values as well. Then the potential becomes VCP = ¯h\alpha0 8\pi2ϵ0c2 Z \infty ds s2 Z \infty dkT kT \kappa
1 + 2k 2 T c2 s2 r∥(\theta, 0) e−2\kappaz = ¯h\alpha0 8\pi2ϵ0c2 Z \infty ds s2 Z \infty s/c d\kappa
2c2\kappa2 s2 −1 r∥(\theta, 0) e−2\kappaz = ¯h\alpha0 8\pi2ϵ0c3 Z \infty d\xi
2\xi2 −1 r∥(\theta, 0) Z \infty ds s3 e−2s\xiz/c, (14.205) where we have introduced \xi = c\kappa/s. Carrying out the s integral, VCP = 3¯hc\alpha0 64\pi2ϵ0z4 Z \infty d\xi \xi4
2\xi2 −1 r∥(\theta, 0) , (14.206) where again the angle \theta is determined by the relation cos \theta = \xi. In the limit of a perfect conductor, the reflection coefficients are replaced by −1, and thus
32\pi2ϵ0z4 Z \infty d\xi \xi2 , (14.207) and since the integral evaluates to unity,
32\pi2ϵ0z4 . (14.208) (far field, perfect conductor) This is precisely the far-field (retarded) Casimir–Polder potential that we obtained before in Eq. (13.60). Note, however, that the z−4 scaling in this regime is universal, as we see from Eq. (14.206): the distance scaling is independent of the material properties. To be a bit more general, we can write out Eq. (14.207) in the case of a simple dielectric interface, in which case Eqs. (14.175) gives the reflection coefficients as
p
p
p
p
, (14.209)
64\pi2ϵ0z4 Z \infty d\xi \xi4 "p
p
- 1 −2\xi2 p
p
, (14.210)¶
the Casimir–Polder potential for a dielectric surface, and the integral can be evaluated explicitly, with the
14.3 Atom–Surface Potentials Near Dielectric Media result14¶
(14.211) (far field, dielectric) where we have defined the ‘‘efficiency’’
2\chi
2\chi3/2
sinh−1p
, (14.212) which ranges from 0 to 1, and measures the strength of the far-field Casimir–Polder potential compared to the perfect-conductor case, Eq. (14.208). This function is plotted below, as the strength of the Casimir–Polder potential for a surface of dc susceptibility \chi, compared to a perfectly reflecting surface, \chi −\rightarrow \infty. c 10-6 10-4 10-2 ho(c) 10-7 10-1 10-2 10-3 10-4 10-5 10-6 The function \eta(\chi) varies smoothly with \chi, and of course the potential here for finite \chi is always weaker
asymptotic forms are also plotted in the figure. For example, for synthetic fused silica (pure optical glass), the refractive index n tends to a value near 2.0 at low frequencies, so we may take \chi \approx 1.0 and conclude that the amplitude of the long-range Casimir–Polder potential is about 25% of that of the perfect conductor. Schott LaSFN9 glass has a refractive index tending to 4.2 near dc, so that VCP in the far field is about 47% of the perfect-conductor amplitude. Note that in view of the zero-frequency pole in the permittivity for a conductor, Eq. (14.65), the limit \chi −\rightarrow i\infty, where \eta(\chi) −\rightarrow 1, is appropriate even for an imperfect conductor. The efficiency (14.212) can also be separated into the relative contributions by the TE (⊥) and TM (∥) polarizations as
(14.213) where
6 + 1 \chi −
2\chi
2\chi3/2
2\chi
2\chi3/2
sinh−1p
, (14.214) 14E. M. Lifshitz, ‘‘The Theory of Molecular Attractive Forces between Solids,’’ Soviet Physics JETP 2, 73 (1956); for the explicit form, see I. E. Dzyaloshinskii, E. M. Lifshitz, L. P. Pitaevskii, ‘‘The general theory of van der Waals forces,’’ Advances in Physics 10, 165 (1961), Eqs. (4.37)-(4.38).
Chapter 14. QED with Dielectric Media by integrating separately the two terms in Eq. (14.210). The relative contributions are plotted below, compared to the total. TE TM TE + TM c 10-6 10-4 10-2 ho(c) 10-7 10-1 10-2 10-3 10-4 10-5 10-6 Interestingly, the contribution from the TM polarization is much larger than from the TE polarization— note that this is true even though the reflection coefficient for TM polarization is always smaller for the same angle than for TE polarization (the reflection coefficient for TM polarization vanishes at Brewster’s angle, for example). This is evidently due to the importance of the weighting factor (1 + 2k 2 T c2/s2) for the TM contribution in Eq. (14.186). In the perfect-conductor limit \chi −\rightarrow \infty, the relative contributions
14.3.5.7 Asymmetric Atoms and Molecules Going back to Eq. (14.185) VCP = ¯h 8\pi2ϵ0 Z \infty ds Z \infty dkT kT \kappa
c2 r⊥(\theta, is)
T r∥(\theta, is) e−2\kappaz, (14.215) let us consider this for asymmetric atoms and molecules, instead of spherically symmetric atoms as in
y)/2 + (a −b)(x −y)/2, gives VCP = ¯h 8\pi2ϵ0 Z \infty ds Z \infty dkT kT \kappa 1
c2 r⊥+ k 2 T r∥ e−2\kappaz + ¯h 8\pi2ϵ0 Z \infty ds Z \infty dkT kT \kappa 1
−\alphazz(is)
c2 r⊥−k 2 T r∥ e−2\kappaz. (14.216) Cleaning up this expression a bit gives VCP = ¯h 16\pi2ϵ0c2 Z \infty ds s2
Z \infty dkT kT \kappa r⊥+ 1 + 2k 2 T c2 s2 r∥ e−2\kappaz + ¯h 16\pi2ϵ0c2 Z \infty ds s2
−\alphazz(is) Z \infty dkT kT \kappa r⊥+ r∥ e−2\kappaz. (Casimir–Polder potential, asymmetric atom) (14.217) The first term is the symmetric potential that we derived earlier, and the second term is due to the asymmetric contribution of the atom.
14.3 Atom–Surface Potentials Near Dielectric Media¶
VCP = − ¯h 8\pi2ϵ0c2 Z \infty ds s2
−\alphazz(is) Z \infty dkT kT
(14.218) The polarizability is (with no sum over \gamma)
X j
j0), (14.219) and so the potential becomes15 VCP = − 4\pi2ϵ0c3 X j
d 2 ∥,j −d 2 ⊥,j ! Z \infty ds s2 kj0
j0 Z \infty s/c
= − 8\pi2ϵ0c3z X j
d 2 ∥,j −d 2 ⊥,j ! Z \infty ds s2 kj0
j0 e−2sz/c = − 8\pi2ϵ0z X j
d 2 ∥,j −d 2 ⊥,j ! Z \infty ds′ s′2 kj0 s′2 + k 2 j0 e−2s′z = − 32\pi2ϵ0z X j
d 2 ∥,j −d 2 ⊥,j ! \partial 2 z Z \infty ds′ kj0 s′2 + k 2 j0 e−2s′z = − 32\pi2ϵ0z X j
d 2 ∥,j −d 2 ⊥,j ! \partial 2 z f(2kj0z). (14.220) This is equivalent to what we had before in Eq. (13.33), but this sum includes the spurious 1/z2 contribution that vanishes under the Thomas–Reiche–Kuhn sum rule. Thus, consider V 0 CP = ¯h 16\pi2ϵ0c2 Z \infty ds s2 Z \infty dkT kT \kappa \alpha>
yy(is) −\alpha> zz(is) r⊥(\infty) + r∥(\infty) e−2\kappaz (14.221) where \alpha> is the same as \alpha, but in the asymptotic limit of large s, and the reflection coefficients are evaluated with the indices of refraction at infinite frequency. This becomes V 0 CP = 8\pi2ϵ0c3 X j
d 2 ∥,j −d 2 ⊥,j ! r⊥(\infty) + r∥(\infty) Z \infty ds s2 kj0 s2/c2 Z \infty s/c
= 8\pi2ϵ0c X j kj0
d 2 ∥,j −d 2 ⊥,j ! r⊥(\infty) + r∥(\infty) Z \infty ds Z \infty s/c
= 16\pi2ϵ0cz X j kj0
d 2 ∥,j −d 2 ⊥,j ! r⊥(\infty) + r∥(\infty) Z \infty ds e−2sz/c = 32\pi2ϵ0z2 X j kj0
d 2 ∥,j −d 2 ⊥,j ! r⊥(\infty) + r∥(\infty) (14.222) This expression vanishes due to the TRK sum rule. Thus the first expression (14.221) can be subtracted from the initial expression (14.217). We see here that the last expression is an artifact of a double-limit at infinity, because the reflection coefficients should go to zero at \omega −\rightarrow \infty. 15After using the integral formula from I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, English translation 7th ed., A. Jeffrey and D. Zwillinger, Eds. (Academic Press, 2007), Formula 3.354.1, or Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions (Dover, 1965), Eq. (5.2.12).
Chapter 14. QED with Dielectric Media 14.3.5.8 Dielectric Thin Films Suppose we consider the ground-state interaction of an atom with a thin dielectric film, surrounded on either side by vacuum.16 The film is described by the reflection coefficient17 r(∥,⊥)
1 −r 2
(14.223) for the two polarizations in terms of the appropriate Fresnel coefficients, where the round-trip phase \phi in the film is given by
q n2(is) −sin2 \theta. (14.224) The other parameters here are the refractive index n(\theta, is) of the film and the film thickness d. For a very thin film, we may expand the above expression to first order in the film thickness, with the result r(∥,⊥)
1 −r 2 ∥,⊥ , (14.225)
cients given by Eq. (14.225) for thin films, or by by Eq. (14.223) in the more general case. Here, we will only consider the limit of very thin films. Obviously, this situation is much more complicated than for a simple dielectric interface, and so we will again concentrate on evaluating the near- and far-field limits. In the near-field regime,
q
c r n2(is) + c2k 2 T s2 \approx i2kTd \approx i2\kappad. (14.226) Recalling that \kappa ≲1/2z, the thin-film approximation will be good if d ≪z. Thus, this treatment will be valid for atom–surface distances much smaller than any resonance wavelength, but still far enough away that the film appears to be thin. Physically, because the evanescent modes are most important in the near field, evanescent modes whose skin depths are of the order of the film thickness will be modified appropriately. The film reflection coefficients scale to lowest order as r∥,⊥, so just as in the case of the simple dielectric interface, we will ignore the small contribution of r⊥compared to r∥. Then we arrive at the same result (14.202), but with the replacement
ϵ0 + ϵ(is) −\rightarrow r∥ 1 −r 2 ∥ 2\kappad. (14.227) Explicitly, then, we have VCP \approx ¯hd 2\pi2ϵ0 Z \infty ds \alpha(is) r∥(is) 1 −r 2 ∥(is) Z \infty
(14.228) After evaluating the second integral, we have the result VCP \approx 3¯hd 16\pi2ϵ0z4 Z \infty ds \alpha(is) r∥(is) 1 −r 2 ∥(is). (14.229) Writing out the Fresnel reflection coefficient explicitly, e.g., as in Eq. (14.227), we finally have VCP \approx − 3¯hd 64\pi2ϵ0z4 Z \infty ds \alpha(is) ϵ(is) ϵ0 − ϵ0 ϵ(is) . (dielectric thin film, near field) (14.230) 16The results here were derived by Yu. S. Barash, ‘‘Van der Waals interaction of thin conducting layers,’’ Soviet Physics— Solid State 30, 1578 (1988), Eqs. (25)-(26); and Fei Zhou and Larry Spruch, ‘‘van der Waals and retardation (Casimir) interactions of an electron or an atom with multilayered walls,’’ Physical Review A 52, 297 (1995), Eqs. (4.57)-(4.58) (doi: 10.1103/PhysRevA.52.297). 17Daniel A. Steck, Classical and Modern Optics, available online at http://steck.us/teaching.
14.3 Atom–Surface Potentials Near Dielectric Media Thus, we see that the thin film has a different scaling of z−4 in the near field, as compared to the z−3 scaling for the bulk material. The dependence on the dielectric material has been modified as well. In the far field of a thin dielectric film, we again consider the limit of \omega = is −\rightarrow 0 in the response functions. However, in this limit, the thin-film phase \phi defined in Eq. (14.224) also vanishes, leading to vanishing thin-film reflection coefficients, as in Eq. (14.223). Thus, for the thin-film reflection coefficients, we should keep them to lowest nonvanishing order in s:
q
c p
c p
(14.231) Again, we are using the notation \xi = c\kappa/s = cos \theta, and \chi is the dc susceptibility of the film medium. Then our analysis from Section 14.3.5.6 carries through with the replacements r∥,⊥−\rightarrow −i r∥,⊥ 1 −r 2 ∥,⊥
r∥,⊥ 1 −r 2 ∥,⊥ 2sd c p
(14.232) Making this replacement in Eq. (14.205), we find VCP = ¯h\alpha0d 4\pi2ϵ0c4 Z \infty d\xi p
" r⊥(\theta, 0) 1 −r 2
2\xi2 −1 r∥(\theta, 0) 1 −r 2 ∥(\theta, 0) # Z \infty ds s4 e−2s\xiz/c. (14.233) The final integral is easy to evaluate, with the result
16\pi2ϵ0z5 Z \infty d\xi p
\xi5 " r⊥(\theta, 0) 1 −r 2
2\xi2 −1 r∥(\theta, 0) 1 −r 2 ∥(\theta, 0) # . (dielectric thin film, far field) (14.234) Of course, we could write the integrand out explicitly in terms of \xi, eliminating \theta, using the far-field expres- sions (14.209). However, we won’t be quite so masochistic right now. The important thing to notice from this relation is again, the different scaling of the far-field potential with distance due to a thin, dielectric film is z−5, compared to the z−4 scaling that we derived for a bulk material, whether a perfect conductor or a dielectric. 14.3.5.9 Metallic Thin Films For a thin metallic film,18 we have to modify the above calculation, because now the permittivity may become quite large in relevant frequency ranges due to the existence of the dc pole. Furthermore, we can take as an explicit model of the metal’s response the Drude–Lorentz model (Problem 1.4) ϵ(is) ϵ0
\omega 2 p
(14.235) where \omegap is the plasma frequency, and \gamma is a material damping rate, given by \gamma = ϵ0\omega 2 p /\sigma0, where \sigma0 is the dc conductivity. In the near field, we may take the limit \gamma −\rightarrow 0, as the relevant frequencies that contribute to the result are of the order of the atomic resonances, which are generally much larger than \gamma. Thus, we will in fact use the plasma model for the metal, ϵ(is) ϵ0
p s2 . (14.236) As we argued in the dielectric case, the near field corresponds to kT ≫s/c. Furthermore, we will note
p )/c2. But we will consider the case where the plasma frequency is of comparable 18These results were derived by Yu. S. Barash, op. cit., Eqs. (28) and (30); and M. Boström and Bo E. Sernelius, ‘‘van der Waals energy of an atom in the proximity of thin metal films,’’ Physical Review A 61, 052703 (2000), Eqs. (21) and (23) (doi: 10.1103/PhysRevA.61.052703).
Chapter 14. QED with Dielectric Media magnitude to the atomic resonance frequencies, which is reasonable for real atoms and metals, so that
T ≫n2s2/c2, just as in the dielectric case. In particular, this implies that as before, we can write the Fresnel reflection coefficients as r⊥\approx 0 and
\omega 2 p
p . (14.237) Then we can use Eq. (14.223) for the thin-film reflection coefficient to write r(⊥) film = 0 and r(∥)
1 −r 2
p ) 1 −e−2kTd
p )2 −\omega 4 p e−2kTd , (14.238) where we used the film round-trip phase \phi = i2d p
T \approx i2kTd, and again d is the film thickness. Note that we are not yet expanding this expression in d: unlike the dielectric case, the Fresnel reflection coefficient can be close to unity, leading to an unphysical divergence in the thin-film reflection coefficient unless we keep factors of e−2kTd around for now. We can put these bits into Eq. (14.186), replacing the polarizability by the dc value \alpha(is) −\rightarrow \alpha0, since for a very thin film, the reflection coefficient will be small except near zero frequency due to the dc pole in ϵ(is). We thus obtain
p \alpha0 4\pi2ϵ0 Z \infty
ds
p
p )2 −\omega 4 p e−2\kappad , (14.239)
are related by \kappa2 = s2/c2 + k 2 T , but they are independent as far as integration variables are concerned, since s represents the (imaginary) optical ‘‘frequency,’’ whereas \kappa (or kT) parameterizes the (independent) wave vector, whose magnitude is unconstrained when we consider the entire family of propagating and evanescent modes. Now we can evaluate the second integral as Z \infty ds
p
p )2 −\omega 4
\pi \sqrt 8 \omegap 1 + \sqrt 1 −e−2\kappad \sqrt 1 −e−2\kappad \sqrt
\sqrt
\approx \pi \sqrt
, (14.240) where we have kept only the lowest-order term in d in the last expression. Thus, taking the d −\rightarrow 0 limit in the rest of Eq. (14.239), we find
\sqrt d \sqrt 2 \piϵ0 Z \infty
(14.241) After evaluating the final integral, we find the result
\sqrt d
(14.242) (metallic thin film, near field) Thus, for a metallic thin film, the main difference with the case of a bulk metal or dielectric is the fractional- power scaling of z−7/2, which is faster than z−3 for the bulk case and slower than z−4 for the dielectric-thin- film case. The other obvious difference from the case of the thin dielectric film is the \sqrt d scaling with the metallic-film thickness, as opposed to the linear scaling in the dielectric case. The far-field case of a metallic film is fairly easy, by comparison. Again, the conductivity leads to a dc pole in ϵ(\omega), which dominates the far-field response. Thus, in the far-field limit of a metallic film, the Casimir–Polder potential is given by the usual perfect-conductor expression (14.208):
32\pi2ϵ0z4 . (14.243) (metallic thin film, far field) Physically, this is because as the relevant frequencies decrease to zero, so does the depth of penetration of the field modes into the film, and so the thinness of the film becomes irrelevant.
14.3.6 Perfectly Conducting, Spherical Cavity¶
14.3 Atom–Surface Potentials Near Dielectric Media 14.3.6 Perfectly Conducting, Spherical Cavity A geometrically more challenging example of the Casimir–Polder potential is the potential felt by an atom within a metallic, spherical cavity. For simplicity we consider only the limit of perfect conductivity. Inside the spherical cavity, we can use the mode-expansion formula (14.162) for the Green tensor and the appropriate mode functions from Section (8.4.4) to obtain the Green tensor
ϵ0 X nlm \omega 2 nl \omega 2 nl −\omega2 f (TE) nlm,\alpha(r)f (TE)∗
ϵ0 X nlm \omega 2 nl \omega 2 nl −\omega2 f (TM) nlm,\alpha(r)f (TM)∗ nlm,\beta(r′), (spherical-cavity Green tensor) (14.244) where the TE and TM modes are given from Eqs. (8.111) and (8.118) in terms of spherical Bessel functions and vector spherical harmonics by f (TE)
(TE) nl jl(knlr) Xm
f (TM)
(TM) nl knl \nabla \times [jl(knlr) Xm
where \partial r r jl(knlr)
(spherical-cavity modes) (14.245) with normalization factors N (TE) nl = R3 2 j 2 l+1(knlR) −1/2 N (TM) nl = R3 1 −l(l + 1) k 2 nlR2 j 2 l (knlR) −1/2 . (14.246) (normalization factors) However, this calculation in the spherical-cavity case turns out to miss a singular term −\delta\alpha\beta\delta3(r −r′)/ϵ0.19 However, this doesn’t matter for the Casimir–Polder-type calculations below, since it corresponds to a term independent of the size of the cavity, which thus disappears after renormalization, as discussed below. In fact, for Casimir–Polder-type calculations, what we will need is the Green tensor evalutated at r′ = r and \omega = is. Also, for spherically symmetric atoms we only need the trace of the Green tensor. For example, for the spherical cavity we thus need
ϵ0 X nlm \omega 2 nl \omega 2 nl + s2 f (TE) nlm(r) 2 + 1 ϵ0 X nlm \omega 2 nl \omega 2 nl + s2 f (TM) nlm (r) 2 . (14.247) We then write out the TE and TM squared modes using Eqs. (8.124), (8.124), and (8.108) as f (TE) nlm(r) 2 = [N (TE) nl ]2j 2 l (knlr) |Xm
f (TM) nlm (r) 2 = [N (TM) nl ]2 " l(l + 1) jl(knlr) kr 2 |Y m
\partial r[rjl(knlr)] kr 2 |Xm
. (14.248) Now employing the sum rules (8.88) and (8.110) for the scalar and vector spherical harmonics, we can compute the sum over m as l X m=−l f (TE) nlm(r)¶
4\pi [N (TE) nl ]2j 2 l (knlr) l X m=−l f (TM) nlm (r)
4\pi [N (TM) nl ]2 " l(l + 1) jl(knlr) kr 2 + \partial r[rjl(knlr)] kr 2# , (14.249) 19Chen-To Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, 1994), Section 10-4, pp. 218-220.
14.3.7 Ground-State Atom-Atom Potentials¶
Chapter 14. QED with Dielectric Media and so the Green tensor finally becomes
4\piϵ0 ( X nl \omega 2 nl \omega 2
(TE) nl ]2j 2 l (knlr) + X nl \omega 2 nl \omega 2
(TM) nl ]2 " l(l + 1) jl(knlr) kr 2 + \partial r[rjl(knlr)] kr 2# ) . (14.250) Recall the two summations are different the first referring to TE-mode boundary conditions, while the second refers to TM modes. The remaining sums are more cumbersome, and can be carried out numerically or by using simpler but approximate asymptotic forms. To compute the potential, we will assume a spherically symmetric atom and use the Kramers–
X j 2\omegaj0d 2 j,z ¯h(\omega 2 j0 −\omega2), (14.251)
write VCP = −¯h 2\pi Z \infty ds \alpha(is) Gµµ(r, r, is) = −
X j d 2 j,z Z \infty ds \omegaj0 \omega 2 j0 + s2 ( X nl \omega 2 nl \omega 2
(TE) nl ]2j 2 l (knlr) + X nl \omega 2 nl \omega 2
(TM) nl ]2 " l(l + 1) jl(knlr) kr 2 + \partial r[rjl(knlr)] kr 2# ) . (14.252) Again using the formula (14.165) to carry out the imaginary-frequency integral, VCP = − 2(4\piϵ0) X j d 2 j,z ( X nl \omeganl
(2l + 1)[N (TE) nl ]2j 2 l (knlr) + X nl \omeganl
(2l + 1)[N (TM) nl ]2 " l(l + 1) jl(knlr) kr 2 + \partial r[rjl(knlr)] kr 2# ) . (Casimir–Polder potential, spherical cavity) (14.253) In general, this sum must be performed numerically to obtain an answer. Furthermore, this expression is divergent; it must still be renormalized by subtracting off the same expression, but in the limit R −\rightarrow \infty. However, the potential turns out again to be negative and divergent as the atom nears the cavity surface, while the potential becomes weakest at the center of the sphere.20 14.3.7 Ground-State Atom–Atom Potentials This formalism can handle not only the interaction of atoms with macroscopic bodies, but also with other atoms. To see this, we will consider the vacuum atom–atom interaction potential in otherwise free space. We suppose the atoms to have polarizability tensors \alpha(1)
µ\nu , and without loss of generality we may assume them to be separated along the z axis at a distance r. From Eq. (14.167), we may write the interaction potential from the point of view of the first atom as
2\pi Z \infty ds \alpha(1) µ\nu (is) G(s) \nuµ(0, 0, is), (14.254) 20W. Jhe and K. Jang, ‘‘Cavity quantum electrodynamics inside a hollow spherical cavity,’’ Physical Review A 53, 1126 (1996) (doi: 10.1103/PhysRevA.53.1126).
14.3 Atom–Surface Potentials Near Dielectric Media in terms of the scattering Green tensor describing the influence of the second atom, assuming atom 1 to be located at the origin. To compute the Green tensor, we begin with the free-space Green tensor in the form of Eq. (14.50): G(0)
4\piϵ0
r −1 ϵ0
(14.255) For our purposes, this will describe the electric field at atom 2 due to a unit dipole at the origin (i.e., atom 1). Since we will assume a nonzero separation between the two atoms, we will henceforth drop the last (contact) term in this Green tensor. Atom 2 then responds to this field according to its polarizability, producing a dipole moment given by \alpha(2)
\gamma\nu (r, 0, \omega). Another multiplication with the free-space Green tensor gives the field back at atom 1, and thus we may write the scattering Green tensor as G(s)
(14.256) Thus, the interaction potential is
2\pi Z \infty ds \alpha(1) µ\nu (is) G(0)
\betaµ(r, 0, is). (14.257) This result is correct, but there is a subtlety with the overall coefficient. Because the second dipole moment is induced by the first, there is an overall factor of 1/2 because this is an induced-dipole energy. However, this cancels the factor of 2 from summing over the (identical) energy shifts of both atoms. This induced-dipole factor is justified more formally later [see the second term in Eq. (14.416)]. Now putting in the form (14.255) for the free-space Green tensor, we find the resulting expression
¯h
Z \infty ds \alpha(1)
r
r (atom–atom potential) (14.258) for the atom–atom potential. 14.3.7.1 Near-Field van der Waals–London Potential If we consider this potential in the near field, we can set e−sr/c \approx 1, so that
r \approx
r\beta
r5
Dropping the contact term in this result, the near-field potential becomes
¯h
r5
r3 −3r\betarµ r5 Z \infty ds \alpha(1)
(14.260) and then using the Kramers–Heisenberg formula (14.147),
X j
¯h(\omega 2 j0 −\omega2) , (14.261) we may write the near-field potential as
X j,j′
d(1) j \cdot d(2)∗ j′ r3 − d(1) j \cdot r d(2)∗ j′ \cdot r r5
2 Z \infty ds
(\omega 2
j′0 + s2), (14.262)
Chapter 14. QED with Dielectric Media
(4\piϵ0)2r6 X j,j′ d(1) j \cdot d(2)∗ j′ −3 d(1) j \cdot ˆr d(2)∗ j′ \cdot ˆr
(near-field van der Waals–London potential) (14.263) that the near-field potential scales at r−6. Comparing this to the classical interaction energy between two dipoles,
(4\piϵ0)r3 , (14.264) we see a similar dependence on the dipole, but the present interaction behaves more like V 2 dip. This is because the ground-state interaction is an interaction of atom 1 with the dipole of atom 2 induced by atom 1’s ground-state fluctuations. Hence, the much weaker interaction at long ranges. In the simpler case of identical, isotropic atoms, where the dipole matrix elements are independent of direction, the result (14.263) becomes
(4\piϵ0)2r6 X j,j′ |dz,j|2|dz,j′|2
(4\piϵ0)2r6 X j,j′ |dz,j|2|dz,j′|2
(14.265) where as usual dz,j ≡ˆz \cdot dj. If we assume that only one transition makes the dominant contribution to the force, then we can make a two-level-atom approximation and write the potential as21
3|dz|4
(4\piϵ0)24r6 , (near-field van der Waals–London potential, identical two-level atoms) (14.266) where the two-level static polarizability is \alpha0 = 2d 2
Note that at very close separations, the dipole approximation breaks down, and the atoms should repel due to the overlap of their electron clouds. This is commonly modeled by the Lennard–Jones potential,22 which is literally a kludge of adding a repulsive r−12 term (or some other high-order power, but r−12 is most common) to the r−6 van der Waals–London potential to model the repulsion. 14.3.7.2 Far-Field Potential The atom–atom potential is also simple in the far-field regime where retardation effects are important. As for the atom–mirror interaction, the exponential factors in (14.258) indicate that only modes with small frequencies contribute to the potential at large separations. Thus, we may replace the atomic polarizabilities with their dc values and pull them out of the integral:
¯h \alpha(1)
\gamma\beta (0)
Z \infty ds
r
r (14.267) Then we can evaluate the integral, with the result
¯hc \alpha(1)
\gamma\beta (0)
\partial ′
rr′(r + r′)
r′=r , (14.268) 21This result originally derived by F. London, ‘‘Über einige Eigenshaften und Anwendungen der Molekularkräfte,’’ Zeitschrift für Physikalische Chemie 11, 222 (1930), Eq. (6). For the more general short-range expression (14.263), see H. B. G. Casimir and D. Polder, ‘‘The Influence of Retardation on the London-van der Waals Forces,’’ Physical Review 73, 360 (1948) (doi: 10.1103/PhysRev.73.360), Eq. (51); and A. D. McLachlan, ‘‘Retarded dispersion forces between molecules,’’ Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 271, 387 (1963), Eq. (6.6). 22J. E. Lennard-Jones, ‘‘Cohesion,’’ Proceedings of the Physical Society 43, 461 (1931) (doi: 10.1088/0959-5309/43/5/301).
14.3 Atom–Surface Potentials Near Dielectric Media where the normal derivatives act on r only, the primed derivatives act on r′ only, and r′ is set to r after the derivatives are evaluated. To continue, we specialize to the case of scalar polarizabilities, \alphaµ\nu = \alpha\deltaµ\nu, so that
0 \alpha(2)
\partial ′
rr′(r + r′)
r′=r
0 \alpha(2)
rr′(r + r′)
r′=r . (14.269) The derivatives are cumbersome but easy to carry out with the help of a computer, with the result23
0 \alpha(2)
(retarded atom–atom potential, spherically symmetric atoms) (14.270) Thus, in the far field, where retardation is important, the atom–atom potential scales as r−7, compared to the near-field r−6 dependence. This is similar to the case of an atom near a planar mirror, where retardation caused the near-field power-law dependence to gain an extra power (there, from r−3 to r−4). 14.3.7.3 General Form for Scalar Polarizabilities To evaluate the atom–atom interaction potential more generally, we can write out Eq. (14.258) in the case of scalar polarizabilities,
¯h
rr′ Z \infty
r′=r = −
X jj′ |d(1) j d(2)
"
rr′ Z \infty ds
j′0) # r′=r , (14.271)
\sqrt 3\langle g|dz|ej\rangle . Thus, we need to evaluate an integral of the form
Z \infty ds
=
Z \infty ds
, (14.272)
Z \infty dx e−µx
\beta (Re[\beta] > 0, Re[µ] > 0), (14.273) and f(z) is an auxiliary function to the sine and cosine integrals [see Eq. (13.26)]. Note that I(\omega, \omega′) has a removable singularity at \omega = \omega′, so that we may write
2\omega3 . (14.274) Thus, the atom–atom potential finally becomes
X jj′ |d(1) j d(2)
r′=r . (scalar, ground-state atom–atom potential) (14.275) 23H. B. G. Casimir and D. Polder, op. cit., Eq. (56); A. D. McLachlan, op. cit., Eq. (6.10). 24See I. S. Gradstein and I. M. Ryzhik, Table of Integrals, Series, and Products, English translation 6th ed., A. Jeffrey and D. Zwillinger, Eds. (Academic Press, 2000), Formula 3.354.1.
Chapter 14. QED with Dielectric Media In principle we have obtained the full potential, in terms of analytic functions and derivatives. The derivatives are unfortunately cumbersome. However, we can see that we recover our former results. For example, in the far field, we can use the large-z forms f(z) ∼1/z and g(z) ∼1/z2 so that
c
(14.276) Using this result and restricting to a single, dominant resonance while using \alpha0 = 2d2/3¯h\omega0, we recover the far-field result in the form (14.269) from the general form (14.275). For small separations, we can set
\pi
(14.277) Putting this into Eq. (14.275), we recover the near-field result (14.265) by evaluating the derivatives in the same way as before in the near-field case. 14.3.7.4 Three-Atom Potential Another interesting example is the Casimir–Polder potential due to three atoms. By the same reasoning leading to the two-atom potential (14.257), we obtain
¯h 2\pi Z \infty ds \alpha(1) µ\nu (is) G(0)
\sigmaµ(r3, 0, is), (14.278) for scattering from the first atom at r = 0 to atom 3 at r3, to atom 2 at r2, and back to the first atom. The factor of 1/3 is because this is an induced-dipole energy. This induced-dipole factor is justified more formally later [see the thirdterm in Eq. (14.416)]. The factor of 3! counts the permutations of the three atoms (all of which are equivalent as far as the energy is concerned). Using Eq. (14.255) again for the free-space Green tensor, we find the resulting expression
0 \alpha(2) 0 \alpha(3)
\times Z \infty ds
r2
r32
r3 (14.279) for the three-atom potential, where we are simplifying the calculation by assuming scalar polarizibilities for the atoms, and we are considering the far-field limit, where only the dc limit of the atomic polarizibilities matter. Note also that r32 := r3 −r2. Carrying out the s integral, we obtain
0 \alpha(2) 0 \alpha(3)
\partial ′
\partial ′′
r=r2 r′=r32 r′′=r3 . (14.280) Multiplying out the derivatives leads to
0 \alpha(2) 0 \alpha(3)
\partial ′′
µ\partial ′′ µ\partial ′
\times
r=r2 r′=r32 r′′=r3 , (14.281)
14.3.8 Temperature Dependence¶
14.3 Atom–Surface Potentials Near Dielectric Media or in index-free notation,
0 \alpha(2) 0 \alpha(3)
\times
r=r2 r′=r32 r′′=r3 . (14.282) At this point it is best to consider specific configurations. For example, for three atoms in a straight line with distance r between adjacent atoms, we have r2 = rˆz, r3 = 2rˆz, and r32 = rˆz, in which case evaluating the derivatives above leads to
0 \alpha(2) 0 \alpha(3)
0 \alpha(2) 0 \alpha(3)
(14.283) equilateral triangle: r2 = rˆz, r3 = ˆzr/2 + ˆy \sqrt
\sqrt 3r/2
0 \alpha(2) 0 \alpha(3)
0 \alpha(2) 0 \alpha(3)
. (14.284) Note that at the same order of accuracy there are other lower-body interactions, for example propor- tional to (\alpha(1) 0 )3 or (\alpha(1) 0 )2\alpha(2) that we are ignoring; we are only focusing on the true three-body contribution to the potential. 14.3.8 Temperature Dependence In our treatment above, we have computed energy expectation values always respect to the ground/vacuum state of the combined atom/field system. However, we can also extend this formalism to cover other states, in particular thermal states
X n
(14.285) where the occupation probability of each energy eigenstate |n\rangle is
Z e−En/kBT , (14.286) where Z := X n e−En/kBT (14.287) is the partition function. Of course, we can compute the energy shifts for other states, but the thermal state is a reasonable equilibrium state that accounts to some extent for excited states, and reduces to the vacuum-state results that we have derived above in the limit T −\rightarrow 0. 14.3.8.1 Fluctuation–Dissipation Relation To look at the fluctuations of a physical quantity, we can recall as motivation the optical Wiener–Khinchin theorem (Sections 2.2 and 5.7), and then write down the power spectral density as a Fourier transform of a correlation function. Specifically, we will write the two-sided power spectral density for the dipole fluctuations for the system in state |n\rangle in terms of a symmetrized correlation tensor as ˜S(n)
4\pi Z \infty −\infty
(14.288) and then write the one-sided tensor spectral density as S(n)
(14.289)
Chapter 14. QED with Dielectric Media We are as usual assuming steady state, so we suppress any explicit dependence on the absolute time t. This is somewhat different from the spectrum we have written down before, but it clearly represents some fluctuation at frequency \omega, and the sum over all frequencies properly represents the total dipole fluctuation in state |n\rangle , Z \infty d\omega S(n)
Z \infty −\infty d\omega ˜S(n)
= 1 4\pi Z \infty −\infty d\tau Z \infty −\infty
= 1 Z \infty −\infty
= 1
(14.290) where in the second step we used the fact that the correlation function is an even function of \tau, if we assume the power spectral density S(n) µ\nu (\omega) to be symmetric. In particular, the trace of this relation is mostly what we would associate with the total fluctuations, Z \infty d\omega S(n)
(14.291) (with the usual implied summation). Technically speaking, the diagonal elements of the spectral tensor represent the fluctuations, while the off-diagonal elements represent correlations (covariances) between fluc- tuations of different dipole-operator components. Now our goal will be to relate these fluctuations to the dissipation (absorption) in the system. We will start by rewriting the spectral (fluctuation) tensor as 2 ˜S(n)
2\pi Z \infty −\infty
h
i |n\rangle = 1 2\pi X j Z \infty −\infty
i = 1 2\pi X j Z \infty −\infty
= X j h
i = X j
h
i . (14.292) Here, the frequency interval \omeganj is as usual
¯h (14.293) in terms of the eigenstate energies. Now we can compute the spectral tensor in the case of a thermal state by summing over all states |n\rangle ,
14.3 Atom–Surface Potentials Near Dielectric Media with each term weighted by the occupation probability (14.286).
X nj
h
i = X nj P(n) + P(j)
= X nj P(n) h
= X nj P(n) h
(14.294) In the second step here we interchanged (relabeled) the summation indices, and in the last step we used the projection property of the delta function. Now that we have the spectral tensor in this form, we will turn our attention to the dissipation. Recall from Section 14.1.4.1 that the imaginary part of a response function is responsible for the loss or dissipation from energy. This result certainly applies to the polarizability, and we showed explicitly this to be the case in its classical treatment, as in Eq. (1.85). From Eq. (14.153), the imaginary part of the atomic polarizability for an atom in state |n\rangle is Im[\alpha(n)
¯h X j
h
i . (14.295) Similarly averaging this expression over the thermal state (14.285), we find
¯h X nj
h
i
¯h X nj P(n) −P(j)
¯h X nj P(n)
(14.296) Comparing this result to Eq. (14.294), we can see the similarity and identify
¯h\omega 2kBT . (fluctuation–dissipation relation) (14.297) This result is known as the fluctuation–dissipation relation,25 relating the spectral density Sµ\nu(\omega) of fluctuations at frequency \omega to the dissipation part of the response function Im[\alphaµ\nu(\omega)]. Again, summing Eq. (14.288) over all levels,
4\pi Z \infty −\infty
(14.298) where the expectation value here is again taken with respect to the thermal state at temperature T. Note that coth ¯h\omega 2kBT
1 2 +
, (14.299) where the last quantity in brackets represents the mean thermal energy of a quantum harmonic oscillator at frequency \omega, in units of ¯h\omega—the first term is the zero-point energy, while the second term represents the thermal contribution. 25Herbert B. Callen and Theodore A. Welton, ‘‘Irreversibility and Generalized Noise,’’ Physical Review 83, 34 (1951) (doi: 10.1103/PhysRev.83.34); L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed. (Pergamon, 1980), §124.
Chapter 14. QED with Dielectric Media We can invert the Fourier transform in the fluctuation–dissipation relation (14.297) and use Eq. (14.298) to write the fluctuations directly in terms of the dipole autocorrelation function:
2\pi Z \infty −\infty
¯h\omega 2kBT . (fluctuation–dissipation relation) (14.300) Then the covariance matrix for the dipole fluctuations is given by taking \tau = 0 in this expression, with the result
\pi Z \infty
¯h\omega 2kBT . (fluctuation–dissipation relation) (14.301) As written here, the covariance matrix is obviously symmetric, so we have dropped the anticommutator. Thus, any absorptive character of the atomic dipole necessarily leads to dipole fluctuations. Of course, any dispersion implies some absorption by the Kramers–Kronig relations, so dispersion also implies fluctuations. Note that coth x −\rightarrow 1 as x −\rightarrow \infty, so fluctuations persist even as T −\rightarrow 0. These zero-temperature fluctuations are obviously quantum-mechanical in nature. However, for high temperatures, we can use coth x −\rightarrow 1/x for small x to write
\pi Z \infty d\omega
(14.302) (large T) which no longer involves ¯h. (We used the Kramers–Kronig relations (14.90) to evaluate the integral here.) In this case, the quantum fluctuations are negligible compared to the (classical) thermal fluctuations. Of course, all of these results apply as well to any observable and its linear response function, assuming a linear interaction Hamiltonian. In particular, for the electric field,
2\pi Z \infty −\infty
¯h\omega 2kBT . (fluctuation–dissipation relation) (14.303) The imaginary part of the Green tensor represents dissipation (material absorption) of the electromagnetic field, which again implies field fluctuations. 14.3.8.2 Fluctuation–Dissipation Example: Johnson Noise As a simple example and application of the fluctuation–dissipation relation, we can consider Johnson noise,26 the intrinsic noise in a any resistor, independent of the details of its material composition, geometry, and so on. The resistance obviously represents the dissipation, and we will show that it implies fluctuations in the form of voltage noise. We start with an interaction Hamiltonian in the form of (14.102), which will represent the energy of electrons in a resistor: Hint = V X j qjxj L . (14.304) Here, V is the ‘‘voltage operator,’’ L is the length of the conduction path of the resistor, and qj and xj are respectively the charge and position of particle j. In linear-response theory, the voltage operator will respond to the classical ‘‘force’’
X j qjxj L . (14.305) 26Johnson noise is named after the first person to measure it: J. B. Johnson, ‘‘Thermal Agitation of Electricity in Conductors,’’ Physical Review Letters 32, 97 (1928) (doi: 10.1103/PhysRev.32.97). It is also called Johnson–Nyquist noise, named additionally after the first to describe it theoretically: H. Nyquist, ‘‘Thermal Agitation of Electric Charge in Conductors,’’ Physical Review Letters 32, 110 (1928) (doi: 10.1103/PhysRev.32.110).
14.3 Atom–Surface Potentials Near Dielectric Media The current represents the flow of charge as a rate of charge per unit time, and thus
X j qj ˙xj L . (14.306)
voltage. Expressed as a Fourier transform,
(14.307) The Fourier transform of Ohm’s law reads
(14.308) where Z(\omega) is the frequency-dependent impedance of the resistor. In terms of the ‘‘force’’ function,
(14.309) and thus i\omegaZ(\omega) is the generalized susceptibility for the resistor. Now using the fluctuation–dissipation relation in the high-temperature limit (14.302), we can write the variance of the voltage fluctuations as
V 2 = 2kBT \pi Z \infty
\omega = 2kBT \pi Z \infty
(14.310) The real part of the impedance is the resistance, and thus27
V 2 = 2kBT \pi Z \infty
(14.311) (Johnson noise) These voltage fluctuations are what are referred to as Johnson noise. The voltage noise is typically measured only over some bandwidth ∆\nu. Changing to a ‘‘regular’’ frequency from the angular frequency,
V 2 = 4kBT Z ∆\nu
(14.312) and if the resistance is roughly constant over the measurement bandwidth, we arrive at the well-known expression
V 2
(14.313) (Johnson noise, ∆\nu bandwidth limit) The noise is proportional to temperature (in the classical limit of high temperature), and has the character
a 10 kΩresistor measured over a 10 kHz bandwidth has an intrinsic, thermal rms voltage noise of 1.2 µV. At the same temperature, a 1 MΩresistor measured over a 1 MHz bandwidth has an rms voltage noise of 0.12 mV, which is starting to become significant on the scale of laboratory voltages. If we use the fluctuation–dissipation relation in the more general form (14.301), we similarly find the general result
V 2 = ¯h \pi Z \infty
¯h\omega 2kBT . (14.314) (Johnson noise, arbitrary T) If we take the limit of small temperature, we can replace the coth by 1:
V 2 = ¯h \pi Z \infty
(14.315) (Johnson noise, small T) 27Herbert B. Callen and Theodore A. Welton, op. cit., Eq. (4.11).
Chapter 14. QED with Dielectric Media The same bandwidth limit leads in this case to zero-temperature noise of
V 2
(Johnson noise, small T, ∆\nu bandwidth limit) (14.316) assuming a constant resistance over the bandwidth. Clearly, quantum fluctuations persist even at zero temperature, producing ‘‘quantum Johnson noise.’’28 14.3.8.3 Temperature-Dependent Shifts Now on to the level shift.29 In the zero-temperature case, we used the second-order perturbation expression
X j
En −Ej (14.317) for the shift of level |n\rangle due to the interaction energy Hint, where the indices label combined states of the atom and field. Now, for a thermal state, we must perform the average
X nj P(n) |\langle n|Hint|j\rangle |2 En −Ej (14.318) over the thermal occupation probabilities (14.286) to treat the shift at nonzero temperature. We will now claim that this second-order shift may be written in terms of the correlation function
2¯h Z 0 −\infty
(14.319) (perturbative energy shift) This result is, of course, valid for any stationary state, not just the thermal ones, and gives a nice, representation-independent expression for the shift. Strictly speaking, this result assumes that the per- turbation is turned on adiabatically in the distant past, so that we may insert a convergence factor to guarantee a sensible result:
i 2¯h Z 0 −\infty
(14.320) To see this, we perform algebraic steps that are similar to what we used for the fluctuation–dissipation relation:
2¯h X n P(n) Z 0 −\infty
= i 2¯h X nj P(n) Z 0 −\infty
i |n\rangle = i 2¯h X nj P(n) Z 0 −\infty
= i 2¯h X nj P(n) −P(j) Z 0 −\infty
(14.321) 28For a measurement of the coth dependence of Johnson noise at low temperature, see R. Movshovich, B. Yurke, P. G. Kaminsky, A. D. Smith, A. H. Silver, R. W. Simon, and M. V. Schneider, ‘‘Observation of zero-point noise squeezing via a Josephson-parametric amplifier,’’ Physical Review Letters 65, 1419 (1990) (doi: 10.1103/PhysRevLett.65.1419). 29The derivation here closely follows A. D. McLachlan, ‘‘Retarded Dispersion Forces in Dielectrics at Finite Temperatures,’’ Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 274, 80 (1963).
14.3 Atom–Surface Potentials Near Dielectric Media As we mentioned, we should really have a convergence factor here, so
2¯h X nj P(n) −P(j) lim
Z 0 −\infty
= 1 2¯h X nj P(n) −P(j)
\omeganj = X nj P(n)|\langle n|Hint|j\rangle |2 ¯h\omeganj . (14.322) This last expression is equivalent to the second-order expression Eq. (14.318).
−dµEµ. We can then write
= 1
+
i = 1
, (14.323) where we used the shorthands dµ ≡dµ(\tau), d\nu ≡d\nu(0), Eµ ≡Eµ(\tau), and E\nu ≡E\nu(0), and we have used the fact that under unperturbed evolution, d(\tau) and E(\tau ′) commute even at different times. We can then take the expectation value of the commutator [Hint(\tau), Hint(0)] and then use the commutator-correlation-function expressions in the forms of Eqs. (14.125) and (14.130), as well as the fluctuation–dissipation relations in the forms (14.300) and (14.303), to write
= ¯h2 2\pi2 Z \infty −\infty d\omega Z \infty −\infty
( coth ¯h\omega 2kBT + coth ¯h\omega′ 2kBT )
(14.324) The energy shift (14.319) represents the finite-temperature version of the Casimir–Polder potential, and thus we can now write VCP = i¯h 4\pi2 Z 0 −\infty d\tau Z \infty −\infty d\omega Z \infty −\infty
( coth ¯h\omega 2kBT + coth ¯h\omega′ 2kBT )
= −¯h 4\pi2 Z \infty −\infty d\omega Z \infty −\infty
( coth ¯h\omega 2kBT + coth ¯h\omega′ 2kBT ) , (14.325) where we implicitly used the usual convergence factor in the time integral. We can then use the Kramers– Kronig relations (14.90) adapted to the dipole and field response functions,
\pi – Z \infty −\infty
d\omega′
\pi – Z \infty −\infty
d\omega′, (14.326)
Chapter 14. QED with Dielectric Media to carry out the \omega′ integral (changing variables beforehand in the second term), with the result VCP = −¯h 4\pi Z \infty −\infty d\omega h
i coth ¯h\omega 2kBT = −¯h 4\pi Z \infty −\infty
¯h\omega 2kBT , (14.327) where we have used the fact that the real parts of the response functions are even functions of the real frequency \omega. We may rewrite this last expression as VCP = −¯h 4\pii – Z \infty −\infty
¯h\omega 2kBT , (14.328)
so that it leads to a vanishing contribution in the principal-value integral. Now we will reduce this integral to a summation as follows. Since coth ix = −i cot x, coth x has simple
frequencies are given by
¯h . (14.329) (Matsubara frequencies) These discrete frequencies are called the Matsubara frequencies.30 Furthermore, the residues of the
Eq. (14.328) to a contour integral over the great semicircle in the upper half-plane, as we did to derive the Kramers–Kronig relations in Eq. (14.1.4.2). The semicircular part of the contour vanishes because \alphaµ\nu(\omega) decays at least as fast as 1/|\omega|2 for large |\omega|, as we saw in our derivation of the Kramers–Kronig relations, Eq. (14.83). Thus, by Cauchy’s integral formula, the integral in Eq. (14.328) changes to 2\pii times the sum over residues at frequencies isn, with the result VCP = −kBT 2 \alphaµ\nu(is0) G\nuµ(r, r, is0) −kBT \infty X n=1
(14.330) Notice that the pole at \omega = 0 only contributes half its residue because of the principal value that we take in Eq. (14.328) knocks out half the contribution of any real-axis pole. Using the original notation of Lifshitz,31 we may write this sum as VCP = −kBT \infty X ′ n=0
(temperature-dependent Casimir–Polder shift) (14.331) where the primed summation symbol P′ denotes that the n = 0 term is accompanied by an extra factor of 1/2. Again, to avoid a divergence and to focus only on the interaction of the atom with a macrocopic body, we should remove the free-field contribution and use only the scattering part of the Green tensor: VCP = −kBT \infty X ′ n=0
\nuµ(r, r, isn). (14.332) (renormalized form) Thus, compared to the zero-temperature expression (14.163), which involved an integral over imaginary frequency of the product of the dipole and field susceptibilities, the finite-temperature case involves a discrete sum over the Matsubara frequencies. 30After Takeo Matsubara, ‘‘A New Approach to Quantum Statistical Mechanics,’’ Progress in Theoretical Physics 14, 351 (1955) (doi: 10.1143/PTP.14.351). 31E. M. Lifshitz, ‘‘The Theory of Molecular Attractive Forces between Solids,’’ Soviet Physics JETP 2, 73 (1956).
14.3 Atom–Surface Potentials Near Dielectric Media 14.3.8.4 Imaginary Time and the Low-Temperature Limit Recalling from Eq. (14.129) that the Green tensor is given as a correlation function as
¯h Z \infty
(14.333) we can see that for imaginary frequencies \omega = is, the Green tensor amounts to a Laplace transform:
¯h Z \infty
(14.334) Shifting to an imaginary time \tau −\rightarrow −i\tau, we return to the Fourier-type integral expression
¯h Z \infty
(14.335) However, the point of the discussion above is that the Green tensor effectively vanishes everywhere except at the Matsubara frequencies isn, VCP = −¯h 2\pi Z \infty
(14.336) where
¯h \infty X ′ n=0
(14.337) Thus, the finite-temperature potential takes on the same form as the zero-temperature potential (14.163), under the replacement G\alpha\beta(r, r, is) −\rightarrow ˜G\alpha\beta(r, r, is). Since the spectrum G\alpha\beta(r, r, is) is then effectively discrete and periodic, we may refer to our discussion of sampling and the sampling theorem (Section 25.1) to note that we may think of its Fourier transform as a periodic function in the imaginary time, with the ‘‘samples’’ given by
¯h Z ¯h/kBT
(14.338) That is, the values (kBT/¯h)G\alpha\beta(r, r′, isn) are the Fourier components of the imaginary-time Green tensor
(14.339) which we may regard as time-periodic with period ¯h/kBT. In the low-temperature limit as T −\rightarrow 0, the period diverges, and under an integral sign we see from the definition (14.337) that ˜G\alpha\beta(r, r, is) −\rightarrow G\alpha\beta(r, r, is), so that we obtain the previous expression for the zero-temperature Casimir–Polder shift. 14.3.8.5 High-Temperature Limit The high-temperature limit of the potential (14.332) comes by noting that the spacing between the Matsubara frequencies becomes very large for large T. Since the summand of Eq. (14.332) decreases monotonically with frequency, for sufficiently high temperature only the dc term will make a substantial contribution. Thus, to leading order, VCP = −1
\nuµ(r, r, 0). (14.340) (large T) This expression is evidently the classical Stark shift of the atom due to the presence of thermal photons, where the quantum zero-point contribution is negligible, since ¯h is absent in this expression. (In the general time-dependent expression, ¯h appears in the definition of the frequencies sn.) In the case of a nondegenerate
Chapter 14. QED with Dielectric Media ground state, the imaginary parts of the susceptibilities vanish at \omega = 0, since there is no mechanism for dissipation. Then32 VCP = −1 2kBT Re[\alphaµ\nu(0)] Re[G(s)
2kBT = −
H 2 AF
2kBT , (14.341) where we used the high-temperature fluctuation–dissipation relation (14.302) and its analog for the field. Again the expectation values here are relative to their values in free space, since we have discarded the divergent free-space contribution. 14.3.8.6 Planar Boundaries at Nonzero Temperature As an example of temperature-dependent effects, we will consider an atom near a planar surface. For simplicity, we will consider the high-temperature limit in the far field, and also a spherically symmetric atom. Comparing the general expression (14.167) at zero temperature to the high-temperature expression (14.340) we see that we can obtain the high-temperature result from the zero-temperature result by omitting the integral over s, setting s = 0 in the remaining integrand, and multiplying by \pikBT/¯h. Doing this in Eq. (14.205), we obtain
4\piϵ0 Z \infty
(14.342) where with s = 0, we can use kT = \kappa. Furthermore, for s = 0, the reflection coefficient is replaced as in the far-field limit by [ϵ0 −ϵ(0)]/[ϵ0 + ϵ(0)]. Then evaluating the remaining integral, we find
16\piϵ0z3 ϵ(0) −ϵ0 ϵ(0) + ϵ0 . (planar dielectic, high-temperature/large-distance limit) (14.343) For a conductor, ϵ(0) −\rightarrow \infty, and thus
16\piϵ0z3 . (conductor, high-temperature/large-distance limit) (14.344) When do these high-temperature expressions hold? Essentially, the second term in the Matsubara sum
is damped as e−2\kappaz \le e−2sz/c, this will occur for z ≫ c 2s1 = ¯hc 4\pikBT . (14.345) (high-temperature condition) At room temperature, this condition amounts to z ≫0.6 µm. The low-temperature limit corresponds to the opposite regime z ≪¯hc/4\pikBT, where the terms in the Matsubara sum are closely spaced in frequency and can be well approximated by the zero-temperature integral. Thus, even for normal temperatures corre-
state, it is still possible to be in a regime of ‘‘high temperature’’ if the distance is sufficiently large.33 Thus, for a ground-state atom near the planar surface of a bulk dielectric, the interaction potential scales as z−3
Then, in the very long-distance regime of z ≫¯hc/4\pikBT, the potential scales again as z−3. Comparing the general expression (14.167) at zero temperature to the temperature-dependent ex- pression (14.332) we can write the temperature-dependent result by adapting the zero-temperature result 32A. D. McLachlan, op. cit. 33For the experimental observation of the large-distance temperature-dependent corrections, but in a nonequilibrium ther- mal state, see J. M. Obrecht, R. J. Wild, M. Antezza, L. P. Pitaevskii, S. Stringari, and E. A. Cornell, ‘‘Measurement of the Temperature Dependence of the Casimir-Polder Force,’’ Physical Review Letters 98, 063201 (2007) (doi: 10.1103/Phys- RevLett.98.063201).
14.3 Atom–Surface Potentials Near Dielectric Media as follows: replace the integral over s by an appropriate sum over Matsubara frequencies, and multiply- ing by 2\pikBT/¯h. Doing this in Eq. (14.186), we find a general expression for the temperature-dependent Casimir–Polder potential near a planar surface in terms of the reflection coefficients: VCP = kBT 4\piϵ0c2 \infty X ′ n=0 s 2 n \alpha(isn) Z \infty dkT kT \kappan
1 + 2k 2 T c2 s 2n r∥(\theta, isn) e−2\kappanz. (Lifshitz formula: temperature-dependent potential, planar surface) (14.346)
p
T , and the angle \theta still depends on sn and kT. This is the Lifshitz expression for the atom–surface potential, after the original treatment of Lifshitz for the temperature-dependent potential between two surfaces,34 from which this expression for the atom–surface force may be deduced.35 Again, in the limit of small temperature, the sum goes over to an integral, and we recover the zero- temperature expression (14.186). Then at small temperatures, by how much does the sum differ from the integral? To obtain a perturbative correction due to nonzero temperature, we can use the Euler–Maclaurin summation formula in the form36 \infty X ′ j=0
∆t Z \infty dt f(t) − \infty X j=1 (∆t)2j−1 (2j)! B2jf (2j−1)(0) = 1 ∆t Z \infty dt f(t) −∆t 12 f ′(0) + (∆t)3
(14.347) to look at precisely this difference, where Bn are the Bernoulli numbers. Then in the general case, we can approximately evaluate the Matsubara sum in Eq. (14.332) for small T by keeping only the correction terms shown explicitly in Eq. (14.347): VCP \approx −¯h 2\pi Z \infty
\nuµ(r, r, is)
12¯h h \partial s
\nuµ(r, r, is) i
720¯h3 h \partial 3 s
\nuµ(r, r, is) i
(small T expansion) (14.348) The first term is the usual zero-temperature expression, while the rest are effectively a power series in the temperature. Let’s evaluate these perturbative corrections for a perfect conductor, in which case Eq. (14.346) becomes VCP = kBT 2\piϵ0c2 \infty X ′ n=0 s 2 n \alpha(isn) Z \infty dkT kT \kappan 1 + k 2 T c2 s 2n e−2\kappanz. (14.349) To expand this expression as in Eq. (14.348), we first consider the zero-temperature formula (14.192), which we may write in the form VCP = − ¯h 16\pi2ϵ0z3 Z \infty ds \alpha(is) 1 + 2sz c + 2s2z2 c2 e−2sz/c. (14.350) To write the temperature expansion in Eq. (14.348), the zeroth-order term is simply given by this expression. The order T 2 correction is given by this same expression, if we remove the integral sign, hit the integrand with \partial s, set s −\rightarrow 0, and multiply by −(2\pikBT/¯h)2/12. The order T 4 correction is given by this same expression, if we remove the integral sign, hit the integrand with \partial 3 s , set s −\rightarrow 0, and multiply by (2\pikBT/¯h)4/720.
1 + 2sz c + 2s2z2 c2
3c3 + O(s4), (14.351) 34E. M. Lifshitz, op. cit. 35See J. F. Babb, G. L. Klimchitskaya, and V. M. Mostepanenko, ‘‘Casimir–Polder interaction between an atom and a cavity wall under the influence of real conditions,’’ Physical Review A 70, 042901 (2004) (doi: 10.1103/PhysRevA.70.042901). 36E. M. Lifshitz, op. cit.; Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions (Dover, 1965), p. 806, Eq. (23.1.30).
14.3.9 Excited-Level Shifts¶
Chapter 14. QED with Dielectric Media to see that the order T 2 correction vanishes. Then with the order T 4 correction the potential becomes
90ϵ0¯h3c3 + O(T 6), (14.352) where VCP(z, 0) is given by Eq. (14.194). The lowest-order nonvanishing temperature correction thus just amounts to a z-independent offset. If we compare the correction to the far-field expression (14.208), the temperature-dependent corrections should be negligible so long as z ≪ r ¯hc \pikBT . (14.353) (zero-temperature validity condition) At room temperature, this condition amounts to z ≪4 µm. 14.3.9 Excited-Level Shifts Thus far, we have mainly treated level shifts only of the ground state, but what happens to excited levels due to the presence of some extra body? The shift of level |n\rangle is given in second-order perturbation theory by the same expression (14.133) as for the ground state, Vn = − X j X k,\zeta
, (14.354) but now the difference is that \omegajn may be negative for states |j\rangle lower in energy than |n\rangle , whereas for the ground state this frequency was always positive. To see how the level shift changes due to this sign change, we can try out the ground-state expression (14.163) V (1) n = −¯h 2\pi Z \infty ds \alpha(n) µ\nu (is) G\nuµ(r, r, is), (‘‘ground-state part’’ of level shift) (14.355) where now \alpha(n) µ\nu is the polarizability tensor for |n\rangle , given by rewriting the Kramers–Heisenberg formula (14.147) as \alpha(n)
X j
¯h(\omega 2 jn −\omega2) . (14.356) Substituting this expression and also Eq. (14.159) for the Green tensor into this relation as before, we find V (1) n = −¯h 2\pi Z \infty ds X j
¯h(\omega 2 jn + s2) X k,\zeta
¯h (\omega 2 k + s2) = −2 \pi¯h X j X k,\zeta
Z \infty ds
(\omega 2
k + s2) = − X j X k,\zeta
, (14.357) where we have used the generalization of the integral formula (14.165) Z \infty dx ab
\pi 2(a2 −b2)(a sgn b −b sgn a)
(14.358) (See Problem 14.5.) Thus, we don’t quite recover the perturbation expression (14.355), due to the presence of the extra sgn \omegajn, whenever \omegajn < 0. However, note that we may write the total shift as
n + V (2) n , (14.359) (total level shift)
14.3 Atom–Surface Potentials Near Dielectric Media where V (1) n is the expression (14.357) that describes the energy shift of |n\rangle , and V (2) n is the difference between the full shift (14.354) and V (1) n in the form (14.357): V (2) n = − X j X k,\zeta
¯h
−
= − X j \Theta(\omeganj) X k,\zeta
¯h
−
= 2 X j \Theta(\omeganj) X k,\zeta
¯h \omega 2 jn −\omega 2 k . (14.360) Notice that the Heaviside function \Theta(\omeganj) ‘‘activates’’ whenever En > Ej, that is, whenever |n\rangle acts as an excited state with respect to |j\rangle . Then we may use the Kramers–Heisenberg formula (14.159) for the Green tensor to eliminate the electric-field matrix elements and the sum over modes to obtain V (2) n = − X j
(extra level shift for excited states) (14.361) The extra shift here is due to the interaction of the atomic dipole with fields at the atomic resonance frequencies for every transition where |n\rangle is the excited state. In fact, we may regard this shift as the Stark shift of the atom due to coupling to its own field. We have already treated this in the case of a perfectly conducting plane in terms of a mode sum in Section 13.7 along with the comparison to the same predictions of the Lorentz model in Section 1.5.1. From our previous discussion, we may conclude that this shift is a classical shift, which is consistent with its form: it is simply the coupling of the dipole covariance matrix to the Green tensor, which represents the light backscattered to the atom by the external body. We can see this directly from our construction of the Green tensor in Section (14.1.3). Given a classical dipole d\beta at location r and oscillating at frequency \omega, d\beta G\alpha\beta(r, r, \omega) gives the electric field at r due to the dipole and any other bodies that may reflect or otherwise influence the dipole’s radiated field. The interaction energy will then be the product of the original dipole with the field, or d\alpha d\beta G\alpha\beta(r, r, \omega). The real part is then taken in Eq. (14.361) since the Green tensor represents a complex field amplitude, and the Heaviside function limits this mechanism to radiative (excited) states. The Green tensor is then evaluated only at the transition frequencies, since those are the frequencies of the dipole radiation; the radiation rates of each transition are given by the magnitudes of the corresponding dipole matrix elements. Again, we must renormalize this shift to remove the divergent free-field contribution, with the result37
n + V (2) n V (1) n = −¯h 2\pi Z \infty ds \alpha(n) µ\nu (is) G(s) \nuµ(r, r, is) V (2) n = − X j
(14.362) (renormalized level shift) for the shift of an arbitrary atomic level |n\rangle . 14.3.9.1 Example: Spherically Symmetric Atom, Perfectly Conducting Plane As an example of applying this formalism, let’s consider a spherically symmetric atom near a perfectly conducting plate. The spherically symmetric atom has a scalar polarizability, and hence the shift reduces 37cf. J. M. Wylie and J. E. Sipe, ‘‘Quantum electrodynamics near an interface. II,’’ Physical Review A 32, 2030 (1985) (doi: 10.1103/PhysRevA.32.2030), Eqs. (4.3)-(4.4); also Werner Vogel and Dirk-Gunnar Welsch, Quantum Optics, 3rd ed. (Wiley-VCH, 2006), Eqs. (10.75)-(10.77), noting the different normalization convention there for the Green tensor.
Chapter 14. QED with Dielectric Media to
n + V (2) n V (1) n = −¯h 2\pi Z \infty ds \alpha(n)(is) G(s) µµ(r, r, is) V (2) n = − X j \Theta(\omeganj)|\langle n|dz|j\rangle |2 Re[G(s)
(spherically symmetric atom) (14.363) The V (1) n part of the shift is already given in terms of the reflection coefficients by (14.186), V (1) n = ¯h 8\pi2ϵ0c2 Z \infty ds s2 \alpha(n)(is) Z \infty dkT kT \kappa
1 + 2k 2 T c2 s2 r∥(\theta, is) e−2\kappaz, (14.364) while we now must compute V (2) n . The diagonal Green-tensor components are given by Eqs. (14.178) and (14.182) and the trace of the Green tensor is thus given by G(s)
i 4\piϵ0 \omega2 c2 Z \infty dkT kT kz
1 −2k 2 T c2 \omega2
ei2kzz, (14.365) where kz = p
T . Then V (2) n can be written directly in terms of this form. For a perfect conductor, we replace the reflection coefficients by unity, and so G(s)
i 2\piϵ0 \omega2 c2 Z \infty dkT kT kz 1 −k 2 T c2 \omega2 ei2kzz = 8\piϵ0 \partial z Z \infty d(k 2 T ) ei2z q
T = 16\piϵ0 \partial z i2\omegaz c −1
z2 = 16\piϵ0 \partial 2 z
z . (14.366) Putting this into (14.363), we find V (2) n
16\piϵ0 X j
z z cos(2kjnz), (14.367)
V (1) n
16\pi2ϵ0 X j
z z f(2|kjn|z), (14.368) though we have had to modify it here by introducing the factor sgn \omegajn and introducing the absolute value of kjn in the argument of f(z). This is because due to the form (14.356) of the polarizability, the integral for V (1) n in Eqs. (14.363) is of the form Z \infty ds \omegajn h(s) \omega 2
Z \infty −\infty ds \omegajn h(s)
= 1
2i|\omegajn|
2 h(i|\omegajn|) sgn \omegajn (14.369)
14.3 Atom–Surface Potentials Near Dielectric Media for the appropriate (even) function h(s), where we have used Cauchy’s integral formula applied to the contour around the upper half-plane. Of course, for the ground state \omegajn > 0, so the absolute value and sgn function were unnecessary, but they are needed now. Putting these parts together, the total level shift is
n (z) + V (2) n
16\pi2ϵ0 X j
z z h f(|2kj0|z) −\Theta(\omeganj)\pi cos(2|kjn|z) i , (level shift near perfectly conducting plane, spherically symmetric atom) (14.370) This agrees with the result of our previous mode-summation calculation, Eq. (13.68), if we restrict that result to a spherically symmetric atom. Of course, the formalism here covers the case of an anisotropic atom, with only a bit more work. 14.3.10 Lifetime Shifts Now we will consider the complementary problem to the body-induced level shifts: the shifts in the lifetimes or decay rates of atomic excited levels. In our classical treatment (Section 1.5) of this problem in the special cases of an atom near a planar mirror or another atom, we saw that the shifts of the decay rate and the transition frequency were different aspects of the same effect. Here we treat the decay-rate shifts separately from the level shifts, due to the additional complexity of the formalism here. 14.3.10.1 Decay Rate Near a Macroscopic Body To treat the general decay problem in the presence of a macroscopic body, we start with Fermi’s golden rule in the form (11.58)
¯h |\langle i|Hint|f\rangle |2\delta(Ei −Ef). (14.371) We are interested in the decay from atomic state |i\rangle to state |f\rangle due to the interaction with the field; we should thus also include initial and final field states:
¯h X I F P(I) |\langle i I|Hint|f F\rangle |2\delta(Ei + EI −Ef −EF ). (14.372)
occupation probability for the field state |I\rangle . If we use the integral representation of the delta function, this becomes \Gammai\rightarrow f = 1 ¯h2 X I F P(I) Z \infty −\infty d\tau |\langle i I|Hint|f F\rangle |2ei(Ei+EI−Ef−EF )\tau/¯h = 1 ¯h2 X I F P(I) Z \infty −\infty
(14.373)
\Gammai\rightarrow f = 1 ¯h2 X I F P(I) Z \infty −\infty d\tau
= 1
X I F P(I) Z \infty −\infty
= 1
X I F P(I) Z \infty −\infty
= 1
X I P(I) Z \infty −\infty
= 1
Z \infty −\infty
(14.374)
Chapter 14. QED with Dielectric Media where the final expectation value is an ensemble average over initial states, and we have transformed the field operators to the Heisenberg picture (technically, the interaction picture, since they evolve as if they were unperturbed). To evaluate the field correlation function, we will need Eq. (14.130) for the commutator correlation function,
\pi Z \infty −\infty
(14.375) along with the fluctuation–dissipation theorem (14.303),
\pi Z \infty −\infty
¯h\omega 2kBT . (14.376) It follows, for example, from the Kramers–Heisenberg formulae (14.159) and (14.160) that the Green tensor
rewrite the fluctuation–dissipation relation as
\pi Z \infty −\infty
¯h\omega 2kBT . (14.377) We can alternately see this since the left-hand side is real, the time dependence on the right-hand side is of the form cos(\omega\tau); thus the correlation function is an even function of the time difference \tau. Now adding Eqs. (14.375) and (14.377), we obtain
2\pi Z \infty −\infty
1 + coth ¯h\omega 2kBT = ¯h \pi Z \infty −\infty
1 −exp −¯h\omega kBT . (14.378) Inverting the Fourier transform leads to Z \infty −\infty
1 −exp −¯h\omega kBT , (14.379) and putting this into Eq. (14.374), we find38 \Gammai\rightarrow f = 2
1 −exp −¯h\omegaif kBT . (atomic spontaneous decay rate, finite T) (14.380) Notice that this rate diverges linearly with T when kBT ≫¯h\omegaif. Taking the T −\rightarrow 0 limit, we find \Gammai\rightarrow f = 2
(14.381) for the rate of spontaneous decay for the atomic |i\rangle −\rightarrow |f\rangle transition. Comparing this result to the excited level shift V (2) from Eqs. (14.362) we see that the two expressions have the nearly same form. The obvious 38cf. J. M. Wylie and J. E. Sipe, ‘‘Quantum electrodynamics near an interface,’’ Physical Review A 30, 1185 (1984) (doi: 10.1103/PhysRevA.30.1185), Eq. (2.5). The derivation here is essentially identical to theirs. For the zero-temperature limit, see also J. M. Wylie and J. E. Sipe, ‘‘Quantum electrodynamics near an interface. II,’’ Physical Review A 32, 2030 (1985) (doi: 10.1103/PhysRevA.32.2030), Eq. (4.5); and Werner Vogel and Dirk-Gunnar Welsch, Quantum Optics, 3rd ed. (Wiley, 2006), Eq. (10.27), again noting the different normalization convention there for the Green tensor.
14.3 Atom–Surface Potentials Near Dielectric Media differences here are: the extra factor of 2/¯h here because we are considering a decay rate of population rather than a level shift; the absence here of a level sum and a Heaviside function, although we are implicitly considering only transitions of unstable excited states to lower-energy states, and the total decay rate from any level follows from summing over all posssible decay paths; and finally, the important difference is the presence of the imaginary part of the Green tensor, as opposed to the real part from the level shift. We saw precisely this dependence on the field quadratures before in the classical treatment of these problems [cf. Eq. (1.125)]. This is also what we expected from what we know about generalized susceptibilities from Section 14.1.4.1; the imaginary part alone leads to dissipation. 14.3.10.2 Free Space: Green-Tensor Example Now we can work out the general decay rate (14.381) for the case of free space. Recall that the free-space Green tensor is, from Eq. (14.46), G(0)
4\piϵ0
1 r3 −i k r2
r eikr. (14.382) In computing the decay rate, we need a product of the form d\alphaG(0) \alpha\betad\beta. The first (near-field) term of the free-
taking the imaginary part and then letting r −\rightarrow 0 gives a factor lim r\rightarrow 0 k2Im[eikr] r = k3. (14.383) Putting all the pieces together, we find the free-space decay rate for a two-level atom: \Gamma0 = 2
3\piϵ0¯hc3 . (free-space atomic spontaneous decay rate) (14.384) This is the same result that we found directly from Fermi’s golden rule, Eq. (11.68), and of course our result, Eq. (11.29), from our previous Weisskopf–Wigner calculation. In general, then, we can write the decay rate in the presence of a macroscopic body as
(atomic spontaneous decay rate) (14.385) so that the deviation from the free-space decay rate is given in terms of the scattering part of the Green tensor. 14.3.10.3 Planar Reflector As another example, we take the case of an atom near a planar interface. We recall from Eqs. (14.182) and (14.178) the nonvanishing Green-tensor components G(s)
i 8\piϵ0 Z \infty dkT kT kz k 2
e2ikzz G(s)
i 4\piϵ0 Z \infty dkT k 3 T kz
(14.386)
Chapter 14. QED with Dielectric Media Taking the matrix elements to be the same in the x and y directions, the shift in the decay rate due to the interface is
= 2\piϵ0¯hRe Z \infty dkT kT kz h (d 2 ge,∥/2) k 2
−d 2 ge,z k 2 eg −k 2 z
i e2ikzz (decay-rate shift near planar interface) (14.387) where k 2 T + k 2 z = k 2
regarding the angle dependence of the reflection coefficients. Near a perfectly conducting interface, we may set both reflection coefficients to unity. Then we change integration variables to \kappa = −ikz, so that k 2
eg, with the result
k 2 eg 2\piϵ0¯h d 2 ge,∥/2 −d 2 ge,z − 4k 2 eg d 2
ge,z \partial 2 z Re ( i Z \infty −ikeg
) = k 2 eg 4\piϵ0¯h d 2 ge,∥/2 −d 2 ge,z − 4k 2 eg d 2
ge,z \partial 2 z 1 z Re ie2ikegz = −k 2 eg 4\piϵ0¯h d 2 ge,∥/2 −d 2 ge,z − 4k 2 eg d 2
ge,z \partial 2 z sin(2kegz) z = −k 3 eg 2\piϵ0¯h h d 2 ge,∥/2 −d 2 ge,z − d 2
ge,z \partial 2 z′ i sin z′ z′ , (14.388)
we may write the shift as
2\Gamma0 h ˆ\epsilon 2
z − ˆ\epsilon 2
z \partial 2 z′ i sin z′ z′ , (decay-rate shift near perfect mirror) (14.389) where ˆ\epsilon 2 ∥:= d 2 ge,∥/d 2 ge and ˆ\epsilon 2 z := d 2 ge,z/d 2 ge. This result is identical to what we derived using the classical model, Eq. (1.132). Recall that the spatial dependence of the quantum modes is purely classical, and that’s what determines the spatial dependence of the shift. Evidently, all the ‘‘quantumness’’ in this relation is buried in the details of the free-space decay rate \Gamma0. 14.4 Casimir Energy Thus far, we have concentrated on the evaluation of Casimir–Polder energies—energies of an atom (or molecule or other point particle) interacting with a dielectric body or bodies. Here we will (relatively briefly) consider the analogous problem of Casimir energies, or interactions involving only macroscopic bodies. Of course, the Casimir–Polder effect is just a special case of this more general Casimir effect, and we will take advantage of this fact. In the case where we neglect dispersion, it is relatively straightforward to come up with an expression for the Casimir energy within linear-response theory. For example, for a dispersionless medium with dielectric and magnetic response, we can write the field Hamiltonian as [see Eq. (8.309) for the dielectric case, and we are also appropriately generalizing the magnetic response here] H = Z
Z d3r ϵ(r)E2 + µ(r)B2 . (electromagnetic Hamiltonian) (14.390) The Casimir energy VC is then just the expectation value in the vacuum state: VC = Z d3r
ϵ(r)E2 + µ(r)B2
. (14.391)
14.4.1 Casimir Energy of Dispersive Dielectric Bodies¶
14.4 Casimir Energy Formally this is a divergent quantity, as in the Casimir–Polder case, where we subtracted the vacuum contribution to the energy in Eq. (14.167). Here, we will simply note that it is important to compute the difference in energy between two configurations of the dielectric bodies in order to obtain a sensible result.
and at equal times, so that
\pi Z \infty −\infty
(14.392)
\pi Z \infty −\infty
(14.393) we obtain
\pi Z \infty
(14.394) For the magnetic contribution, we can similarly write
\pi Z \infty
(14.395) or
\pi Z \infty
\pi Z \infty d\omega
\leftarrow − \nabla ′. (14.396) In these expressions, \nabla operates only on r, \nabla ′ operates only on r′, and the arrow on the last \nabla ′ indicates that it operates to the left on the Green tensor. Putting these expressions into Eq. (14.391), we have39 VC = ¯h \pi Z d3r Z \infty d\omega
\leftarrow − \nabla ′ r=r′ , (Casimir energy, no dispersion) (14.397) where the repeated µ’s are summed. Although it is reasonably simple to obtain this expression in the absence of dispersion, the case where we include dispersion is more subtle, and must be handled with some care. For example, simply replacing ϵ(r) by ϵ(r, \omega) does not work in either Eq. (14.391) or Eq. (14.390). 14.4.1 Casimir Energy of Dispersive Dielectric Bodies To incorporate dispersion into the Casimir energy, we will take advantage of the fact that we have already incorporated dispersion into the Casimir–Polder energy, which in terms of an imaginary-frequency integral is VCP = −¯h 2\pi Z \infty ds Tr h \alpha(is) \cdot G(s)(r, r, is) i , (14.398) from Eq. (14.167), or as a real-frequency integral from Eq. (14.327), VCP = −¯h 2\pi Im Z \infty
(14.399) after taking the zero-temperature limit, exploiting the even nature of the integrand, and renormalizing by subtracting the free-space Green tensor. 39A similar expression can be found in K. A. Milton, The Casimir Effect: Physical Manifestations of Zero-Point Energy (World Scientific, 2001), p. 84, Eq. (5.27a) (ISBN: 9810243979).
Chapter 14. QED with Dielectric Media Although these corrections correctly model a dispersive interaction of an atom with a macroscopic body (or collection of bodies), it implicitly assumes that the effect of introducing the atom has a negligible effect on the vacuum field (e.g., via the Green tensor). In introducing a macroscopic body in the context of the Casimir effect, this is no longer true, and we have to be more careful with the notion of the interaction energy. This procedure will allow us to generalize these expressions into the correct expressions to model Casimir energies for dispersive objects.40 To simplify this treatment somewhat, we will only consider dielectric bodies, ignoring any magnetic effects. 14.4.1.1 Induced Interaction Energies First, we will have a short digression to treat the general notion of an ‘‘induced’’ interaction energy.41 That is, consider the Hamiltonian
(14.400) which is parameterized by \lambda, which satisfies 0 \le \lambda \le 1. Then \lambda smoothly interpolates between the ‘‘unper- turbed’’ Hamiltonian H0 and the ‘‘perturbed’’ Hamiltonian H0 + V . Unlike perturbation theory, however, we will not make any assumptions about having a weak perturbation: the interaction V may be as strong as we like. We will assume we know the energy E\lambda corresponding to an eigenstate |E\lambda\rangle , for every possible value of \lambda:
(14.401) Often this technique will be useful for analyzing ground-state shifts (such as Casimir energies), but we need not assume this explicitly. Now differentiating the energy E\lambda with respect to \lambda, we obtain
= \partial
\partial
\partial
\partial
\partial
= 1
(14.402) Note that in the next-to-last equality, we used \langle E\lambda|E\lambda\rangle = 1. The result above is often called the Hellmann– Feynman theorem;42 we will write this in integral form here as E1 −E0 = Z 1 d\lambda
(14.403) (Hellmann–Feynman theorem) This gives the energy difference E1 −E0 between the system in the presence and absence of the interaction V . 40From here through Section 14.4.1.7, we are closely following the derivation of F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, ‘‘Electromagnetic energy, absorption, and Casimir forces. II. Inhomogeneous dielectric media,’’ Physical Review A 84, 053813 (2011) (doi: 10.1103/PhysRevA.84.053813). 41Alexander L. Fetter and John Dirk Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, 1971), pp. 69-70 (ISBN: 0070206538). 42For a history of this theorem, see David Wallace, An Introduction To Hellmann–Feynman Theory, Master’s thesis (University
14.4 Casimir Energy As a simple example, consider the energy due to an induced dipole in a static electric field. Setting
X j
¯h\omegaj0 . (14.404) The interaction Hamiltonian for the dipole–field interaction is V = −d\cdotE. The perturbation-theory analysis
that V ∼d but \alpha0 ∼d2. Thus the energy shift (14.403) becomes ∆E = Z 1 d\lambda
(14.405) and so putting in the explicit factor of \lambda2, we find ∆E = Z 1
2\alpha0E2. (14.406) This is exactly what we expect for the energy of an induced dipole, specifically the factor of 1/2. For example, we can obtain the same result by bringing the electric field up from zero to the final value, integrating the work in doing this along the way, we obtain ∆E = − Z E
Z E
2\alpha0E2, (14.407) which gives the same factor of 1/2. However, Eq. (14.405) is a more general expression than (1/2)\alpha0E2 of the induced dipole energy, because different functional forms for \alpha0(\lambda), due to more complicated interactions, can lead to different final expressions for the energy. This is precisely the path we will follow in adapting the Casimir–Polder potential to the Casimir energy. 14.4.1.2 Assembling a Dielectric Medium, Atom by Atom Now we will model the dielectric media as a collection of N atoms. Since a dipole responds to an applied electric field via the tensor polarizability \alpha as
(14.408) we can write the dipole response dn(\omega) at frequency \omega for each atom at rn as
(14.409) We assume the electric field to comprise an externally applied field E0 as well as the induced field from each dipole. The latter contribution can be written in terms of the free-space Green tensor G0, so that the total dipole response for the nth atom is
N X j=1
(14.410) Grouping the dipole moments together, we may formally write this system of equations in matrix form as h
i
(14.411) where the matrices encode both polarization and ‘‘which-atom’’ degrees of freedom. That is, the ‘‘1’’ in the left-hand matrix should be interpreted as the 3N \times 3N identity matrix I3N, and \alpha(\omega) and G0(\omega) are
Chapter 14. QED with Dielectric Media similarly 3N \times 3N, while d(\omega) and E0(\omega) are column vectors of dimension 3N. In this form, we may just as well introduce the freedom to assign a different polarizability tensor to each atom, which corresponds to replacing \alpha(\omega) by \alpha(n)(\omega) in Eq. (14.409). Then solving for the collection of dipole moments, we obtain
h
i−1
(14.412) The collective polarizability of the N atomic dipoles is just the entire matrix that acts on the field vector E0(\omega) on the right-hand side. Thus, to adapt the unrenormalized form of the single-atom interaction energy (14.399), we can replace the polarizability in that expression with the collective polarizability and the Green tensor with the free-space tensor, with the result VC = −¯h \pi Z 1 d\lambda \lambda Im Z \infty d\omega Tr h
i−1
, (14.413) which follows from introducing the correct integral over the interaction parameter \lambda as in Eq. (14.403). As in
so that VC = −¯h \pi Im Z \infty d\omega Tr Z 1
h
i−1
. (14.414) Note again that the trace here extends over not only the polarization degrees of freedom, but also the coordinates of all N atoms. Hence, this expression corresponds to a Casimir energy of the entire collection of atoms. Carrying out the \lambda integral then gives the expression VC = ¯h 2\pi Im Z \infty d\omega Tr log h
i (14.415) for the Casimir energy. Note that since the matrices do not commute, technically we should verify this
of (1 −\lambda2A)−1 is \lambda2nAn, so the nth term in the expansion of (1 −\lambda2A)−1\lambdaA is \lambda2n+1An+1. Integrating over \lambda gives An+1/2(n + 1). This matches the series expansion of (−1/2) log A. The Taylor expansion of Eq. (14.415) is itself interesting, VC = −¯h 2\pi Im Z \infty d\omega Tr
- 1
- 1
, (14.416) since the terms correspond to one-body, two-body, three-body, etc. interactions. As a simple example, consider a collection of N = 2 atoms, in which case the two-body term gives the two-atom interaction V12 = −¯h 4\pi Im Z \infty d\omega Tr h
i = −¯h 4\pi Im Z \infty d\omega X n,k=1 Tr h
i = −¯h 2\pi Im Z \infty d\omega Tr h
i (14.417) In the second step, we wrote out the spatial part of the trace, so that the remaining trace refers only
interactions. The result gives the Casimir–Polder potential between two atoms; this result is equivalent via Wick rotation to the result (14.257) that we derived directly.
14.4 Casimir Energy 14.4.1.3 Conversion to Permittivity In going over from a collection of atoms to the continuum limit, we can, for example, take the two-body interaction term from Eq. (14.416) and modify it via Tr
−\rightarrow Tr Z d3r Z
= Tr Z d3r Z d3r′ ϵ 2
= Tr
2, (14.418) where note that in the middle two expressions, the overall spatial integration is written out explicitly, while it is buried in the the trace in the other two expressions. Also, the dielectric susceptibility tensor \chi is related to the polarizability tensor [in a tensor generalization of Eq. (1.17)] via
ϵ0
(14.419) where N(r) is the atomic number density, and recall that the susceptibility is related to the dielectric
the series expansion of the logarithm, Eq. (14.415) becomes VC = ¯h 2\pi Im Z \infty d\omega Tr log h
i , (dispersive Casimir energy, with free-space Green tensor) (14.420) where again the trace in this expression encompasses position integral. 14.4.1.4 Introduction of the Dielectric Green Tensor Thus far, the expression (14.420) represents the field in terms of the vacuum Green tensor, but more it would be more sensible to have the Green tensor in the presence of the dielectric to represent the field. Recall the defining relation (14.28) for the Green tensor:
(14.421) Taking the difference between the dielectric and vacuum cases gives
G −G(0) −\omega2µ0
= 0. (14.422) Then rearranging and subtracting \omega2µ0ϵ0G from both sides,
G −G(0) −\omega2µ0ϵ0 G −G(0)
ϵ(\omega) −ϵ0
(14.423) This has the same form as the defining wave equation (14.421), but in vacuum, and with ‘‘source’’ \omega2µ0ϵ0\chi(\omega)G. The solution is thus G(0) ‘‘convolved’’ with the ‘‘source’’ ϵ0\chi(\omega) \cdot G:
(14.424) With explicit position dependence, this solution is
Z
(14.425)
Chapter 14. QED with Dielectric Media Now a transpose gives
(14.426) while operating with the left-inverse of G gives
(14.427) Using this result in Eq. (14.420) gives the energy expression VC = −¯h 2\pi Im Z \infty d\omega Tr log hG(0)−1 \cdot G i (dispersive Casimir energy, trace-log form) (14.428) after trading an overall minus sign for the inverse of the matrix inside the log. We can also write this expression more simply as VC = −¯h 2\pi Im Z \infty d\omega Tr log G(\omega), (dispersive Casimir energy, trace-log form) (14.429) with the understanding that we should renormalize the energy (which in this expression is divergent) by subtracting the energy of the vacuum or some other comparable dielectric configuration. Although we have been working in the space of real frequencies \omega, any of these expressions may be Wick-rotated to imaginary frequency. For example, Eq. (14.429) becomes VC = −¯h 2\pi Z \infty ds Tr log G(is), (dispersive Casimir energy, Wick-rotated trace-log form) (14.430) upon letting \omega −\rightarrow is. 14.4.1.5 Explicit Permittivity Dependence We can arrive at an alternate expression for the Casimir energy that highlights the direct dependence on the dispersive permittivity and allows a direct comparison to the nondispersive expression (14.397). Continuing with the unrenormalized expression (14.429), and keeping in mind the necessity of renormalizing the final result, we can integrate by parts to obtain VC = ¯h 2\pi Im Tr Z \infty
2\pi Im Tr Z \infty
(14.431) To evaluate \partial \omegaG(\omega), we can multiply through by \omega−2 and then differentiate the wave equation (14.421) to obtain
(14.432) In analogy with Eq. (14.424), we can write the solution as \partial \omega \omega−2G
\cdot G, (14.433) or simplifying this,
\omega2 \cdot G. (14.434) Putting this into Eq. (14.432), but discarding the first term (which produces a constant term that vanishes under renormalization), we arrive at VC = ¯h 2\pi Im Tr Z \infty d\omega \omega h
i \cdot G, (14.435) (dispersive Casimir energy)
14.4 Casimir Energy or writing out the trace in the spatial degrees of freedom, VC = ¯h 2\pi Im Tr Z d3r Z \infty d\omega \omega h \partial \omega
i
(dispersive Casimir energy) (14.436) In the limit of a nondispersive, isotropic dielectric, [\omega2ϵ]′ = 2\omegaϵ, and this relation reduces to the first (electric-field) term of the nondispersive expression (14.397), if we also ignore any imaginary part of the permittivity. However, the nondispersive expression (14.397) does not generalize in a completely obvious way to a dispersive medium, because of the presence of a term of the form \omegaϵ′(\omega). 14.4.1.6 Free-Space Energy A simple check on the Casimir-energy expression is to examine the energy in free space, by letting ϵ −\rightarrow ϵ0 in Eq. (14.436): VC = ¯hϵ0 \pi Z d3r Z \infty d\omega Tr Im G(0)(r, r, \omega). (14.437) Using the free-space Green tensor (14.46), G(0)
4\piϵ0
1 r3 −i k r2
r eikr, (14.438) we can discard the first portion in the trace and keep only the imaginary part to find
2\piϵ0 k2 r sin kr. (14.439) Then at the same source and effect points, we have
k3 2\piϵ0 = \omega3 2\piϵ0c3 . (14.440) In Eq. (14.437), this leads to the free-space energy VC = Z d3r Z \infty
Z d3r Z \infty d\omega 1 2¯h\omega \omega2 2\pi2c3 , (14.441) where we can interpret the last expression as counting zero-point energies ¯h\omega/2 for each mode, with 2 polarizations per frequency and a vacuum mode density \omega2/2\pi2c3. We can verify this energy by direct mode summation in a large quantization volume V , in analogy with the steps in the calculation of the spontaneous-emission rate in Section 11.3: V0 := X k,\zeta ¯h\omegak = ¯h X k \omegak −\rightarrow ¯hcV 8\pi3 Z d3k k = ¯hcV 2\pi2 Z dk k3 = ¯hV 2\pi2c3 Z
(14.442) The vacuum mode density here also agrees with the density of states from the Fermi-Golden-rule calculation
Chapter 14. QED with Dielectric Media 14.4.1.7 Reduction to the Casimir–Polder Potential Now to make contact again with the Casimir–Polder potential, consider a small variation \deltaϵ in the permit- tivity ϵ. Then the variation in the Casimir energy (14.435) is
2\pi Im Tr Z \infty d\omega \omega h
i
h
i
. (14.443) Note that it is critical here to consider the variation \deltaG in the Green tensor, which depends implicitly on the permittivity, as well as the direct variation \deltaϵ. The same variation in the wave equation (14.421) gives
(14.444) which has solution
(14.445) We can use this result to eliminate \deltaG in the integrand of Eq. (14.435): h
i
h
i
h
i
h
i
(14.446) Then using Eq. (14.434) in the form
(14.447) we can rewrite the last term of Eq. (14.446), with the result h
i
h
i
h
i
= h
i
(14.448) Then Eq. (14.435) becomes
2\pi Im Tr Z \infty d\omega \omega
(14.449) after cyclic permutation of matrices under the trace. Simplifying further gives
2\pi Im Tr Z \infty
, (14.450) and then integration by parts gives
2\pi Im Tr Z \infty
2\pi Im Tr Z d3r Z \infty
(Casimir-energy variation) (14.451) where we have written out the spatial part of the trace in the second expression. Now using Eq. (14.419) to identify the permittivity variation with a single localized atom at rA,
ϵ0
ϵ0
(14.452) the delta function removes the spatial integral, and we are left with
2\pi Im Tr Z \infty
(14.453) in agreement with Eq. (14.399), if we renormalize by subtracting the same expression with the vacuum Green tensor. Note that the variation expression (14.451) is also useful in other situations with a small change in the dielectric, such as computing the force between two planar dielectric slabs (where the idea is to compute the difference between two slightly different distances, so \deltaϵ corresponds to a thin film on one of the surfaces).
14.4 Casimir Energy 14.4.1.8 Temperature Dependence In generalizing these expressions for the Casimir energy to the case of nonzero temperature, we can appeal to the following expression, VCP = −¯h 2\pi Im Z \infty d\omega Tr
coth ¯h\omega 2kBT , (14.454) which follows from the temperature-dependent Casimir–Polder expression(14.327), exploiting the odd nature of the integrand (specifically, the imaginary parts of the response functions). Note that this is precisely the same as the starting expression (14.399) for the Casimir–Polder energy at zero temperature, except for the presence of the coth factor here. Thus, all the results here carry through by judiciously inserting this factor. For example, the trace-log expression (14.428) becomes43 VC = −¯h 2\pi Im Z \infty d\omega Tr log hG(0)−1 \cdot G i coth ¯h\omega 2kBT . (dispersive Casimir energy, trace-log form) (14.455) The same contour integration that we used in the Casimir–Polder case applies here, leading to the Matsubara frequencies appearing due to the poles in the coth function. 43Compare to, for exmaple, Francesco Intravaia and Ryan Behunin, ‘‘Casimir effect as a sum over modes in dissipative systems,’’ Physical Review A 86, 062517 (2012), Eq. (42) (doi: 10.1103/PhysRevA.86.062517).
Chapter 14. QED with Dielectric Media 14.5 Exercises Problem 14.1 Suppose we denote the permittivity including conduction by
\omega , (14.456)
relations (14.90)
\pi – Z \infty −\infty Im[ϵ(\omega′) −ϵ0]
d\omega′
\pi – Z \infty −\infty Re[ϵ(\omega′) −ϵ0]
d\omega′. (14.457) Show that ˜ϵ(\omega) satisfies the modified Kramers–Kronig relations
\pi – Z \infty −\infty Im[˜ϵ(\omega′) −ϵ0]
d\omega′
\pi – Z \infty −\infty Re[˜ϵ(\omega′) −ϵ0]
\omega , (14.458)
\omega −\rightarrow \infty. Hint: the algebra is simple if you keep the relations as much as possible in terms of the Hilbert-transform operator H . Problem 14.2 Show that the Kramers–Kronig relations for the permittivity
\pi – Z \infty −\infty Im[ϵ(\omega′)]
d\omega′
\pi – Z \infty −\infty Re[ϵ(\omega′)] −ϵ0
d\omega′ (14.459) can be written in the equivalent form
\pi – Z \infty
d\omega′
\pi – Z \infty Re[ϵ(\omega′)] −ϵ0
d\omega′. (14.460)
Problem 14.3 Consider the (classical) forced, damped harmonic oscillator
(14.461) subject to the forcing function f(t).
14.5 Exercises (a) Find the Green function g(t, t′) for the initially undisturbed system, defined as the solution x(t)
is, g(t, t′) satisfies
(14.462)
derive the solution to Eq. (14.461) (look it up from a reputable source). (b) Write down the general solution to Eq. (14.461) [by integrating Eq. (14.462) over t′] for an arbitrary
that the integral is in fact a convolution of f(t) with g(t). (c) Derive the frequency-space Green function ˜g(\omega), defined as the amplitude ˜x(\omega), where
(14.463) and x(t) is the solution to Eq. (14.461) due to a unit-amplitude, monochromatic forcing at frequency \omega,
(14.464) (d) For an arbitrary forcing function f(t) with Fourier transform ˜f(\omega), use the convolution theorem to show that the solution x(t) may be written
2\pi Z \infty −\infty
(14.465) Thus, ˜g(\omega) is the transfer function for the damped harmonic oscillator, because it gives the ‘‘trans- fer efficiency’’ for the forcing amplitude at frequency \omega through the system. Of course ˜g(\omega) is also the generalized susceptibility for the damped harmonic oscillator, and thus, for example, obeys the Kramers–Kronig relations. Problem 14.4 In Section 14.1.4, we argued that g\chi(t −\rightarrow \infty) −\rightarrow 0, where g\chi(t) is the inverse Fourier transform of the susceptibility \chi(\omega),
2\pi Z \infty −\infty
(14.466) was a reasonable requirement, since a dielectric medium ‘‘forgets’’ any electromagnetic perturbations in the distant past. However, this conclusion is not valid for a conductor. (a) For a conductor with permittivity
\omega , (14.467)
Note: this is not a technically difficult problem, but it is rather tricky for two reasons. First, when you set up the semicircular contour integral, the contour you choose will depend on the sign of t, in
consistent with a causal response. (b) Since g\chi(t −\rightarrow \infty) −\rightarrow 0 fails to hold for a conductor, this means that a conductor doesn’t forget perturbations in the distant past. Give a physical interpretation of this statement by considering a (near-dc) pulsed electric field applied to a medium (assume that the sign of the electric field is always positive). What is the net effect of the pulse long after it finishes for a dielectric vs. a conductor?
Chapter 14. QED with Dielectric Media Problem 14.5 Prove the integral formula (14.358) Z \infty dx ab
\pi 2(a2 −b2)(a sgn b −b sgn a)
(14.468) by considering the contour around the great upper half-plane. Problem 14.6 Justify the formula lim ϵ\rightarrow 0+ x \pm iϵ = P 1
(14.469) Here, the ‘‘P’’ denotes the Cauchy principal value, i.e., Z \infty −\infty dx P f(x) ≡P Z \infty −\infty dx f(x) ≡– Z \infty −\infty dx f(x). (14.470) Note: the delta function is defined in terms of its action on a test function f(x) under an integral, i.e. R
on the test functions f(x) for which this formula is valid? Problem 14.7 A linear circuit has the following frequency response for the output voltage amplitude |Vout(\omega)| as a function of the constant input-voltage amplitude Vin(\omega). Sketch the corresponding phase lag \delta(\omega) of
w |Vouto(w)o| Problem 14.8 Understand and use the expression (14.210)
64\pi2ϵ0z4 Z \infty d\xi \xi4 "p
p
- 1 −2\xi2 p
p
(14.471) for the Casimir–Polder energy, and adapt it to the magnetic-field case as follows. First, replace the dc electric polarizability with the analogous dc magnetizability \alpham0, defined such that the induced magnetic-dipole moment is µ = \alpham0B, making sure to keep the dimensions correct. Then use the considerations of the dielectric-mode-sum analysis of Section 13.13.4 to modify the integrals for the magnetic field. To keep things simple, consider only the limits of very small and very large \chi.¶
14.5 Exercises Problem 14.9 Beginning with the expression (14.217) VCP = ¯h 16\pi2ϵ0c2 Z \infty ds s2
Z \infty dkT kT \kappa r⊥+ 1 + 2k 2 T c2 s2 r∥ e−2\kappaz + ¯h 16\pi2ϵ0c2 Z \infty ds s2
−\alphazz(is) Z \infty dkT kT \kappa r⊥+ r∥ e−2\kappaz. (14.472) for the Casimir–Polder force between an atom and a planar dielectric interface, take the large-distance (long-wavelength) limit. Show that this may be written VCP = − 3¯hc 64\pi2ϵ0z4
−\alpha0,zz
, (14.473) where the \alpha0,\beta\beta are diagonal components of the static polarizability tensor, and the functions \eta(\chi)
Problem 14.10 Beginning with the same expression as in Problem 14.9 [Eq. (14.472)], VCP = ¯h 16\pi2ϵ0c2 Z \infty ds s2
Z \infty dkT kT \kappa r⊥+ 1 + 2k 2 T c2 s2 r∥ e−2\kappaz + ¯h 16\pi2ϵ0c2 Z \infty ds s2
−\alphazz(is) Z \infty dkT kT \kappa r⊥+ r∥ e−2\kappaz, (14.474) compute the Casimir–Polder potential in the near-field approximation.