9. Atomic Interaction with the Quantized Field¶
PDF pages 457–484
9.1 Lorentz Force¶
Chapter 9 Atomic Interaction with the Quantized Field Up till now, we have been using the dipole interaction Hamiltonian HAF = −d\cdotE to describe the coupling of the atom and field. However, this interaction is approximate, being valid only in the dipole approximation for nearly stationary atoms. Here, we will address the fundamental question, what is the fundamental Hamiltonian for the atom–field interaction? The answer turns out to have some subtleties, and this subject has historically been the source of substantial confusion. 9.1 Lorentz Force The classical force on an electron of charge q = −e in an electromagnetic field is given by1
(9.1) (Lorentz force law) Instead of the fields, we can write this in terms of the vector and scalar potentials A and \phi, respectively:
(9.2) But the vector identity
(9.3) gives
(9.4) The particular combination dA(r, t) dt
dx\alpha
(9.5) is known as the convective derivative, and allows us to write F = e
. (9.6) (Lorentz force law) Of course, we also make the identification F = m¨r. 1See David J. Griffiths, Introduction to Electrodynamics, 4th ed. (Prentice-Hall, 2013), Section 5.1.2, p. 212.
9.1.2 Hamiltonian¶
Chapter 9. Atomic Interaction with the Quantized Field 9.1.1 Lagrangian We note that we can derive this force law from the Lagrangian L = 1
(9.7) (Lorentz-force Lagrangian) To see this, we simply evaluate the Euler–Lagrange equation \partial L \partial r −d dt \partial L
(9.8) which gives
(9.9) It is easy to see (Problem 9.2) that this reproduces Eq. (9.6), but we can also write this in the form d
(9.10) which suggests that m˙r −eA plays the role of the momentum, while −e[\phi −(v \cdot A)] plays the role of the (velocity-dependent) potential. 9.1.2 Hamiltonian We can see that this is indeed the case by deriving the Hamiltonian. Since the potential is velocity-dependent, the canonical momentum involves the vector potential:
\partial ˙r = m˙r −eA = pkinetic −eA. (Lorentz-force canonical momentum) (9.11)
is the Legendre transform of the Lagrangian,
(9.12) so that the Hamiltonian for the Lorentz force is
2m −e\phi. (9.13) (Lorentz-force Hamiltonian) Now the magnetic-field (\nabla \timesA) and transverse-electric-field (\partial tA) parts of the interaction is included in the kinetic part, while the longitudinal-electric part of the interaction (due to \phi) is in the potential term. 9.2 Quantization and Minimal Coupling The total Hamiltonian for a system of particles of charge q\alpha and mass m\alpha interacting with the electromagnetic field is then2 H = X \alpha
2m\alpha + ϵ0 Z d3r E2 + c2B2 , (9.14) where we now explicitly include the field Hamiltonian, and for the moment we do not explicitly consider any contribution due to a scalar potential \phi, since we have in a sense already included it. This is because of the 2For further reading, see Peter W. Milonni, The Quantum Vacuum (Academic Press, 1993), Section 4.2, p. 115.
9.2 Quantization and Minimal Coupling Helmholtz theorem, which says that the electric field can be decomposed into transverse and longitudinal components as
(9.15) where \nabla \cdot E⊥= 0 and \nabla \times E∥= 0. This decomposition is obvious in the Coulomb gauge, since E⊥= −\partial tA and E∥= −\nabla \phi. Then using Z
(9.16) we can write the electric-field contribution to the Hamiltonian as Z d3rE2 = Z d3r E⊥2 + E∥2 = Z d3r E⊥2 + Z
= Z d3r E⊥2 − Z
= Z d3r E⊥2 + 1 ϵ0 Z
(9.17) where \rho is the source charge density, and we have dropped surface terms. Suppose now that the source charges come in the form of localized point particles,
N X \alpha=1
(9.18) Then the scalar potential is
Z d3r′ \rho(r′, t) 4\piϵ0|r −r′|. (9.19) So, we can now evaluate the integral in the last term of Eq. (9.17), Z
Z d3r Z
X
(9.20) so that we can write the total Hamiltonian (9.14) as H = X \alpha
2m\alpha + 4\piϵ0 X
Z d3r E⊥2 + c2B2 . (9.21) Thus, we see that we can associate the longitudinal field E∥with the fields due to the charged particles. Now suppose that we take all but one of the particles to be fixed, with the moveable atom an electron
interaction with the field). Then the Hamiltonian becomes
2me
Z d3r E⊥2 + c2B2 , (9.22) where pe and re are the canonical coordinates of the electron. In an atom, we interpret the potential \phi due to the other charged particles to give the binding potential V (re). We can also see then that when quantizing the field, it is only necessary to quantize the transverse field. To describe the atom–field interaction, we can associate the longitudinal field with the atom itself. This is true of the magnetic field, since the magnetic field is already transverse (since there are no magnetic monopoles).
9.3.1 Power-Zienau Transformation¶
Chapter 9. Atomic Interaction with the Quantized Field Thus, the quantization of the total Hamiltonian, including the atomic coupling to the electromagnetic field, proceeds as follows. Take the quantized Hamiltonian for the atom and the field, in the uncoupled limit, which we already know: H = p 2 e 2me
Z d3r E⊥2 + c2B2 . (9.23) Now, to include the atom–field interaction, we make the minimal-coupling replacement pe −\rightarrow pe + eA in the above Hamiltonian, to obtain the minimal-coupling Hamiltonian
2me
Z d3r E⊥2 + c2B2 (minimal-coupling Hamiltonian) (9.24) describing the coupled atom-field system within quantum electrodynamics. 9.3 Dipole Interaction Now we will move towards recovering the usual dipole-interaction Hamiltonian.3 Consider the first (kinetic) term in the minimal-coupling Hamiltonian (9.24): [pe + eA(re)]2 2me = p 2 e 2me + e me
2me A2. (9.25)
commute here, since we are in the Coulomb gauge where \nabla \cdot A = 0. Within the electric-dipole approxi- mation, we take the vector potential A to be independent of position, evaluating A at the nuclear position
size. This approximation is also called the long-wavelength approximation. Then the minimal-coupling Hamiltonian becomes H = p 2 e 2me
me
2me A2(0) + ϵ0 Z d3r E⊥2 + c2B2 (minimal-coupling Hamiltonian, long-wavelength approximation) (9.26) in the long-wavelength approximation. Comparison to the uncoupled Hamiltonian (9.23) gives HAF = e me
2me A2(0) (minimal-coupling interaction, long-wavelength approximation) (9.27) as the interaction Hamiltonian in terms of the vector potential. Here, the pe \cdot A term plays a role similar to the familiar d \cdot E Hamiltonian, as we will discuss in more detail below, while the A2 term is atomic-level- independent and for many purposes may be ignored. 9.3.1 Power–Zienau Transformation The atom–field interaction here is still in terms of the vector potential, and so we would like to see the connection to the usual interaction with the electric field. We thus use the unitary Power–Zienau trans- formation4 (again, in the long-wavelength approximation)
(Power–Zienau transformation, long-wavelength approximation) (9.28) 3For further reading, see Peter W. Milonni, The Quantum Vacuum (Academic Press, 1993), Sections 4.3-4.4, pp. 119-125; and J. R. Ackerhalt and P. W. Milonni, ‘‘Interaction Hamiltonian of quantum optics,’’ Journal of the Optical Society of America B 1, 116 (1984). 4E. A. Power and S. Zienau, ‘‘Coulomb Gauge in Non-Relativistic Quantum Electro-Dynamics and the Shape of Spectral Lines,’’ Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 251, 427 (1959); R. G. Woolley, ‘‘Molecular Quantum Electrodynamics,’’ Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 321, 557 (1971).
9.3 Dipole Interaction to transform the Hamiltonian. The unitary transformation here amounts to a gauge transformation, and we will refer to the situations before and after the unitary transformation as being different gauges. The new Hamiltonian is
(9.29) Using the identity
(9.30) we can see that the new momentum is
(9.31) so that we can take care of the transformation of the kinetic energy by writing it in terms of the untrans- formed momentum (which becomes both the canonical and the kinematic momentum in the Power–Zienau transformation):
e . (9.32) Also, the electric-field components transform as ˜E⊥
= E⊥
h iere \cdot A(0)/¯h, E⊥ \beta (r) i = E⊥
ϵ0
(9.33) where we used the commutation relation between the vector potential and electric field in free space from Eq. (8.207). Thus, the transformation of the electric-field energy gives U(re) Z d3r E⊥(r) 2
Z d3r U(re)E⊥(r)U \dagger(re) 2 = Z d3r E⊥(r) 2 + 2e ϵ0
ϵ 2 Z d3r P⊥(r) 2 , (9.34) where
(9.35) (atomic polarization density) is the polarization density for the atom, and the transverse polarization is the same but with the delta function replaced by the transverse delta function: P ⊥
(9.36) Thus, the transformed Hamiltonian is ˜H = p 2 e 2me
Z d3r E⊥2 + c2B2 + 1 2ϵ0 Z d3r P⊥(r) 2 , (transformed Hamiltonian) (9.37) which again is written only in terms of the untransformed coordinates pe and E(r). Comparison to the uncoupled Hamiltonian (9.23) gives an interaction Hamiltonian of
2ϵ0 Z d3r P⊥(r)
2ϵ0 Z d3r P⊥(r) 2 , (dipole interaction Hamiltonian) (9.38) where the atomic dipole moment is d = −ere, and the atomic center is located at r = 0. The second term, representing a (divergent) dipole self-energy, is commonly dropped, although sometimes it makes an
Chapter 9. Atomic Interaction with the Quantized Field important explicit contribution (e.g., in the calculation of the Lamb shift). We thus recover the familiar form for the electric-dipole Hamiltonian ˜HAF \approx −d \cdot E⊥ (9.39) (dipole interaction Hamiltonian) in the long-wavelength approximation. 9.3.1.1 Electric Displacement Let’s once again examine the transformed electric field.5 From Eq. (9.33), we have ˜E⊥
ϵ0
= E⊥ \alpha (r) −1 ϵ0 P ⊥ \alpha (r), (9.40) so that
(9.41) Thus, we see that the electric field in the original gauge (‘‘A gauge’’), or more precisely ϵ0E⊥, which is what couples to the atom, corresponds to the dielectric displacement in the new gauge (‘‘E gauge’’). Since the polarization density is localized, this is in fact the same as ϵ0 ˜E⊥away from the origin. 9.3.1.2 Active and Passive Viewpoints The viewpoint of the Power–Zienau transformation that we presented above—that the electric-dipole Hamil- tonian arises from a unitary transformation of the Hamiltonian—is the ‘‘active’’ view of the transformation. We can alternately use a ‘‘passive’’ view, where we can get the same form of the interaction Hamiltonian (9.37) without transforming it, if we use the coordinate transformation6
(9.42) which gives new coordinates p′
A′ = A r′ e = re B′ = B
ϵ0 P⊥. (9.43) Then the untransformed Hamiltonian is
2me
Z d3r E⊥2 + c2B2 = p′ e 2me + V (r′ e) + er′
Z d3r E′⊥2 + c2B′2 + 1 2ϵ0 Z d3r P⊥(r) 2 , (9.44) which has the same form as (9.37), but in transformed coordinates. (Before, we wrote the transformed Hamil- tonian in untransformed coordinates). Note here that ϵ0E′ is the dielectric displacement in the old variables. In both the active and passive viewpoints, the field that couples to the atom is in fact a displacement, not an electric field, although it is conventional to write it as an electric field (since they are the same outside the atom). 5For further discussion, see J. R. Ackerhalt and P. W. Milonni, ‘‘Interaction Hamiltonian of quantum optics,’’ Journal of the Optical Society of America B 1, 116 (1984). 6For further discussion, see J. R. Ackerhalt and P. W. Milonni, op. cit.
9.3.2 p · A vs. r · E¶
9.3 Dipole Interaction 9.3.1.3 Göppert-Mayer Transformation The Power–Zienau transformation was actually preceded by an equivalent canonical transformation derived by Göppert-Mayer.7 Recall that the classical equations of motion are unchanged if we add a total time derivative, say d dtS(q, t), (9.45) to the Lagrangian L(q, ˙q; t) in terms of the generalized coordinate q. Doing so induces a canonical trans- formation, and we can choose the generating function of the canonical transformation S(q, t) to be
L = 1 2me˙r 2 e −e˙re \cdot A(re) + e\phi(re), (9.46) which thus transforms to
dtere \cdot A
= 1 2me˙r 2
(9.47) We can see that the generating function exactly cancels the pe \cdot A term and adds in the re \cdot E term. Thus, this canonical transformation classically effects the transformation from H to ˜H. 9.3.2
Now it appears that, depending on the choice of gauge, we have two possible interaction Hamiltonians in the long-wavelength approximation. In the E gauge, we have from Eq. (9.38) H(E)
2ϵ0 Z d3r P⊥(r) 2 \approx ere \cdot E, (9.48) where again the polarization term is typically negligible. The A gauge, on the other hand, gives from Eq. (9.27) H(A) AF = e me
2me A2 \approx e me pe \cdot A, (9.49) where we have assumed that the A2 term is negligible, which is typically the case. Comparing these two Hamiltonians amounts to comparing pe \cdot A to mere \cdot E. This seems reasonable, as pe = me\partial tre, and E⊥= −\partial tA, so the two Hamiltonians seem to differ by moving a time derivative from one factor to the other, as in some sort of integration by parts. However, the matrix elements of these two Hamiltonians differ, as we will now show. First, we must derive a relation between matrix elements of r and p. Consider the commutator [re, HA] = i¯h pe me , (9.50) where the atomic Hamiltonian is as usual HA = p 2 e /2me + V (re). Then the momentum operator becomes pe = −ime ¯h [re, HA], (9.51) or in matrix elements in the basis of eigenstates |j\rangle of HA,
¯h \langle j|[re, HA]|j′\rangle
(9.52) 7M. Göppert-Mayer, ‘‘Über Elementarakte mit zwei Quantensprüngen,’’ Annalen der Physik 9, 273 (1931).
Chapter 9. Atomic Interaction with the Quantized Field where
¯h (9.53) is the transition frequency (and could be positive or negative depending on the ordering of the states). Thus, for the matrix elements of the two Hamiltonians, we may write
\langle j|H(E) AF |j′\rangle \langle j|H(A) AF |j′\rangle =
= \omega |\omegajj′|. (9.54) The matrix elements for the interaction Hamiltonians are different! This would seem to give different physical predictions, depending on which Hamiltonian we use. What gives? This problem has generated much discussion, and its resolution is somewhat tricky. One ‘‘resolution’’ states that a unitary transformation generated the second Hamiltonian from the first. Thus, to get the same answers in both gauges, one must also apply the same transformation to the states, and then trivially the matrix elements must be the same (matrix elements and expectation values are always invariant under unitary transformations). But that still doesn’t help much: given a particular pair of states, say atomic energy eigenstates, which is the appropriate interaction Hamiltonian to use? Using the passive viewpoint, which avoids the unitary transformation of the Hamiltonian—the Hamiltonian is the same in either gauge, just expressed in different coordinates—doesn’t actually help, because we would still need to find the action of the new variables on the old state, which is equivalent to making the unitary transformation. The point is, that physically measureable quantities are gauge-invariant, and thus should be calculable with either Hamiltonian.8 One ‘‘resolution’’ of this ‘‘paradox’’ asserts that because the E-gauge atomic energy operator is equivalent to the unperturbed Hamiltonian (and in particular, the kinematic and canonical momenta are equivalent), the usual energy eigenstates are associated with the E gauge, and the computation of matrix elements is most straightforward here.9 This interpretation is a bit tricky, since even in the A gauge, the kinematic and canonical momenta are equivalent in the absence of a perturbing field, and we have already counted the longitudinal binding field as part of the background, not the perturbation. The interpretation that we will prefer here is that the energy eigenstates can appropriately be used for either gauge, but only when we ask physical questions.10 But then what about the different matrix elements? Broadly speaking, there are two situations that have slightly different resolutions. 1. A matrix element between two stable states is physical only for an energy-conserving process (at least in the case of a time-independent Hamiltonian). In this case, the laser and transition frequencies are
the Hamiltonian treatment of spontaneous decay. 2. In cases where \omega̸ = \omega0, as can happen for a homogeneously broadened line or an inelastic process, the matrix element represents an intermediate transition in a larger, multiphoton process that conserves energy. We then regard the overall process as the physical one, and the combination of the matrix elements, summed over all intermediate states, is gauge-invariant. We will see examples of this in the Casimir–Polder effect and Lamb shift. The main idea here is that when the matrix elements differ between the gauges, then a physically relevant sum over the states is gauge-invariant. It may be the case that the sum has faster convergence in one gauge compared to another, so that for a specific calculation there may be a more convenient gauge. However, the final answer should always be the same. It’s possible that summing over all intermediate states and restricting yourself to physical results still doesn’t produce the same answer in both gauges, in which case the next step is to keep the extra self-energy 8For a good discussion of this point, see Marlan O. Scully and M. Suhail Zubairy, Quantum Optics (Cambridge, 1997), Appendix 5.A, p. 178. 9Marlan O. Scully and M. Suhail Zubairy, op. cit. 10Edwin A. Power, ‘‘A Review of Canonical Transformations as they Affect Multiphoton Processes,’’ in Multiphoton Processes: Proceedings of an International Conference at the University of Rochester, Rochester, N.Y., June 6-9, 1977, Joseph H. Eberly and Peter Lambropoulos, Eds. (Wiley, 1978); Claude Cohen–Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg, Photons & Atoms (Wiley, 1989), Complement BIV, p. 316; Zoltan Fried, ‘‘Vector Potential Versus Field Intensity,’’ Physical Review A 8, 2835 (1973) (doi: 10.1103/PhysRevA.8.2835).
9.4 Why the Vector Potential?¶
9.4 Why the Vector Potential? terms we ignored in the two interaction Hamiltonians (9.48) and (9.49). These extra terms are important in getting gauge-independent results, for example, in the Lamb shift, as we will see later in Section 13.12. 9.4 Why the Vector Potential? We have seen in the canonical quantization of the field, that the vector potential plays a central role. We have now also just seen that it plays a central role in the coupling of an atom to the electromagnetic field. But why the potential, and not the fields themselves? After all, the electromagnetic fields are far more intuitive, and the vector potential is somewhat ambiguous due to gauge freedom. In physics, it helps our intuition to have local interactions. For example, in classical electrodynamics, we can do away with electric and magnetic fields and regard electrodynamics as a theory of interacting charged particles. However, some strange things happen: the force between two moving, charged particles is not a central force. That is, it appears to violate Newton’s third law, that the electromagnetic force on one particle is not necessarily equal and oppose to the force on the other. Momentum conservation is saved by attributing some of the momentum to the electromagnetic fields.11 Another example comes again from considering the force between two initially stationary particles. You observe that when you start to wiggle one of them, the other doesn’t respond to the wiggle until a time r/c later, and thus the retarded time is important in electrodynamics. However, consider the direction of the force on a stationary particle due to one moving at constant velocity: you might be tempted to conclude that it points to the retarded location of the moving particle, when in fact it points to the instantaneous location.12 (If the motion is not of constant velocity, then the direction of the force is different still.) In classical physics, then, one function of introducing the electromagnetic fields is to avoid such counterintuitive, nonlocal interactions: one particle generates a field, which propagates to the other particle and thus influences it. In quantum mechanics, to preserve the same sort of locality of interactions, we are forced to include the vector potential at a fundamental level.13 The most striking example where this is the case is the Aharonov–Bohm effect,14 which deals with a charged particle moving in the exterior of a solenoid. In particular, suppose we set up an interference experiment, where the charged particle, after being split, travels on either side of the solenoid before being recombined. charged particle solenoid If the ideal solenoid is of radius R, is oriented along the z-axis, and has a linear density of turns N with current I, the magnetic field is15 B = µ0NIˆz, r < R 0, r > R, (9.55) 11See David J. Griffiths, Introduction to Electrodynamics, 4th ed. (Prentice-Hall, 2013), Section 8.2, p. 360. 12Richard Feynman, Robert B. Leighton, and Matthew L. Sands, The Feynman Lectures on Physics, Vol. II, Chapter 21 (Addison-Wesley, 1989). 13I learned this analogy to locality in classical physics from Tanmoy Bhattacharya. For more notes and a good discussion of the Aharonov–Bohm effect, see J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994), pp. 136-9. 14Y. Aharonov and D. Bohm, ‘‘Significance of Electromagnetic Potentials in the Quantum Theory,’’ Physical Review 115, 485 (1959) (doi: 10.1103/PhysRev.115.485). This effect was discussed earlier by W. Ehrenberg and R. E. Siday, ‘‘The Refractive Index in Electron Optics and the Principles of Dynamics,’’ Proceedings of the Physical Society. Section B 62, 8 (1949) (doi:
15David J. Griffiths, op. cit., p. 237.
Chapter 9. Atomic Interaction with the Quantized Field and thus vanishes outside the solenoid. The vector potential, on the other hand, is16 A = µ0NI r ˆ\phi, r < R µ0NI R2 r ˆ\phi, r > R, (9.56) so that the vector potential does not vanish outside the solenoid. Now consider the Schrödinger equation, including the minimal-coupling replacement, describing the particle motion in the presence of the magnetic field:
2m ¯h i \nabla −qA 2 \psi. (9.57) We will now effect the gauge transformation that we noted above as follows. Under the replacement
(9.58) where \chi(r) is some function (that defines the gauge transformation), we see that
¯h \psi
(9.59) and thus the Schrödinger equation is invariant if we also let
(9.60)
particular,
Z r r0 A \cdot ds. (9.61) With this choice of \chi, the vector potential goes away, and the Schrödinger equation becomes that of the free particle. That is, assuming a wave function
iq ¯h Z r r0 A \cdot ds , (9.62) then \psi0 is a solution to the free-particle wave equation. Thus, the phase shift accumulated by a moving particle due to the presence of the field is
¯h Z r r0 A \cdot ds, (9.63) where the integral is along the particle’s path, and in the interferometer above, we can write the phase difference of the two arms as a closed-path integral
¯h I A \cdot ds, (9.64) where the contour is around the total path of the interferometer. Note that we can now write
¯h Z
¯h Z
¯h\PhiB, (9.65) where \PhiB is the enclosed magnetic flux. For an electron with q = −e, the phase shift becomes
\Phi0 , (9.66) (Aharonov–Bohm phase) 16David J. Griffiths, op. cit., p. 247.
9.5.1 Atomic Polarization Field¶
9.5 Multipole Interactions where \Phi0 = h e \approx 4.14 \times 10−7 Wb (9.67) is a fundamental unit of magnetic flux, and 1 Wb = 1 T m2 = 1 V s. The whole point is this: we can observe interference fringes due to the magnetic field, even though the particle stays in regions of zero magnetic field. Even though this remarkable result can be explained in terms of flux of the magnetic field, it motivates the fundamental nature of the vector potential if we are to maintain a local interaction between particles and fields: evidently the quantum interaction of a particle and the magnetic field is nonlocal. 9.5 Multipole Interactions To generalize the above results for the dipole Hamiltonian (9.37) in the long-wavelength approximation, we will now consider the more general Power–Zienau transformation without making the long-wavelength approximation. In this way, we will derive general expressions for the atomic interaction with the electric and magnetic fields, and then we will expand these to generate the higher-order multipole couplings.17 9.5.1 Atomic Polarization Field We will start by making the approximation of a heavy nucleus, mnuc ≫me, so we identify the reduced electron mass with the normal electron mass and we will assume that the nuclear position rnuc = 0 defines the center of mass for the system. Then we can write the polarization field for the atom—here, a singly charged nucleus at rnuc = 0 and an electron at re—as the line integral
Z 1 ds \delta3(r −sre). (9.68) (atomic polarization density) To see that this is correct, recall that the polarization field corresponding to a charge density \rho satisfies18
(9.69) or since the transverse polarization does not contribute, this is really only a constraint on the longitudinal polarization.
(9.70) Computing the divergence of the atomic polarization (9.68),
Z 1 ds \delta3(r −sre) = e Z 1 ds \partial
(9.71) where the last equality holds if we identify the charge distribution
(9.72) 17E. A. Power and S. Zienau, op. cit.; R. G. Woolley, op. cit. See also Claude Cohen–Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg, Photons & Atoms (Wiley, 1989), Section IV.C, p. 280; and Werner Vogel and Dirk-Gunnar Welsch, Quantum Optics, 3rd ed. (Wiley, 2006). 18See, for example, David J. Griffiths, Introduction to Electrodynamics, 4th ed. (Prentice-Hall, 2013), p. 174.
9.5.2 Atomic Magnetization Field¶
Chapter 9. Atomic Interaction with the Quantized Field The first term is obviously the nuclear charge density, while the second term represents the electron charge density. Note that despite the expression here, we are not necessarily assuming a localized charge density for the electron, as re is an operator and thus is still subject to uncertainty and quantum fluctuations. You can visualize the above result (9.68) for the polarization as follows. The atom, in our simplified model, consists of two opposite and separated charges. The polarization is the dipole moment per unit volume, and here we represent it by a continuum of delta-function-localized dipoles, forming a line between the two charges. Each dipole moment is an idealized charge pair, and the charges for the successive dipole moments exactly cancel each other, except at the endpoints of the line. Of course we don’t want the endpoints of the line to cancel, since those are the atomic charges. The line of dipoles isn’t unique, since any path connecting the nucleus to the electron will do. That freedom is implicit since we did not constrain the curl of P, merely the divergence. However, we have chosen the simplest path, and it is consistent with the requirement (9.69). 9.5.2 Atomic Magnetization Field We can also define a magnetization field for the electron: while the polarization related to the atomic charge distribution, the magnetization (magnetic dipole moment per unit volume) summarizes the magnetic properties of the atom due to motion of the charge distribution. To motivate this field, we can differentiate Eq. (9.69) to obtain
(9.73) Comparing this to the continuity equation,
(9.74) we can see that we can identify j −\partial tP as an irrotational vector field:
(9.75) We can thus write this field as the curl of some other vector field. Recalling also that the curl of the magnetization M behaves as an effective current density,19
(9.76) we can conveniently interpret j −\partial tP as being the curl of the atomic magnetization:
(9.77) The longitudinal magnetization does not contribute here, so this is really only a constraint on the transverse magnetization:
(9.78) If we identify the atomic current density
(9.79) and differentiate the atomic polarization (9.68),
Z 1 ds \delta3(r −sre) −ere˙re \cdot Z 1 ds \nabla e\delta3(r −sre), (9.80) we can verify directly that the constraint (9.78) is satisfied by the expression
Z 1 ds s \delta3(r −sre), (9.81) (atomic magnetization density) 19David J. Griffiths, op. cit., p. 275.
9.5.3 Power-Zienau Transformation¶
9.5 Multipole Interactions¶
Z 1 ds s \nabla \delta3(r −sre)
Z 1 ds s \nabla \delta3(r −sre) −ere˙re \cdot Z 1 ds s \nabla \delta3(r −sre) = −e˙re Z 1 ds s \partial
Z 1 ds \nabla e\delta3(r −sre)
Z 1 ds \partial
Z 1 ds \nabla e\delta3(r −sre)
(9.82) Here, we used expressions (9.79) and (9.80) for the atomic current density and derivative of the polarization, respectively. Again, the choice of magnetization here is not unique, but is a simple choice that satisfies the constraint (9.78). 9.5.3 Power–Zienau Transformation The more general Power–Zienau transformation is then given by the unitary operator U = exp −i ¯h Z d3r P(r) \cdot A(r) . (9.83) (Power–Zienau transformation) We will see later [in Eq. (9.111)] that the dipole polarization is the first term in a multipole expansion of P(r), and so in the dipole approximation the Power–Zienau operator reduces to U = exp −i ¯h Z d3r P(r) \cdot A(r) \approx exp ie ¯h re \cdot A(0) , (9.84) which is precisely the operator (9.28) we used in the long-wavelength approximation. Clearly, the electron position and vector potential (hence, magnetic field) are still invariant under this transformation, so it remains to transform the electric field and canonical electron momentum, and of course, the minimal-coupling Hamiltonian (9.24)
2me
Z d3r E⊥2 + c2B2 , (9.85) in order to obtain the multipole interactions in terms of the electric and magnetic fields. 9.5.3.1 Electric Field Using the transformation (9.30) and the Jordan–Pauli commutation relation between the vector potential and electric field in free space from Eq. (8.207),
ϵ0 \delta⊥
(9.86) the electric field transforms as ˜E⊥
= E⊥
−i ¯h Z d3r′ P(r′) \cdot A(r′), E⊥ \beta (r) = E⊥ \beta (r) −i ¯h Z d3r′ P\alpha(r′) A\alpha(r′), E⊥ \beta (r) = E⊥ \beta (r) −1 ϵ0 Z
(9.87)
Chapter 9. Atomic Interaction with the Quantized Field so that ˜E⊥
\beta (r) −1 ϵ0 P ⊥ \beta (r). (9.88) (transformed field operator) We see here again that the transformed electric field still corresponds to the untransformed dielectric dis- placement. Thus, the electric-field part of the Hamiltonian becomes U(re) ϵ0 Z d3r E⊥(r) 2
Z d3r U(re)E⊥(r)U \dagger(re) 2 = ϵ0 Z d3r E⊥(r) −1 ϵ0 P⊥(r) 2 = ϵ0 Z d3r E⊥(r) 2 − Z
2ϵ0 Z d3r P⊥(r) 2 . (9.89) The first term is simply the energy of the transformed field, the second is the atomic interaction (via its polarization density) with the electric field, and the last term is again a dipole self-energy due to the dipole field. 9.5.3.2 Canonical Electron Momentum Now to carry out the transformation of the momentum. Using
(9.90) for an arbitrary function f, the momentum transforms as
−i ¯h Z d3r P(r) \cdot A(r), pe
Z d3r \nabla e [P(r) \cdot A(r)] = pe −e Z d3r \nabla e re \cdot A(r) Z 1 ds \delta3(r −sre) = pe −e Z d3r A(r) Z 1 ds \delta3(r −sre) −e Z d3r re \cdot A(r) Z 1 ds \nabla e\delta3(r −sre). (9.91)
−e Z d3r A(r) Z 1
Z 1 ds A(sre), (9.92)
9.5 Multipole Interactions and the third term is −e Z d3r re · A(r) Z 1¶
Z d3r re \cdot A(r) Z 1 ds s \nabla \delta3(r −sre) = −e Z
Z 1 ds s \delta3(r −sre) = −e Z
Z 1 ds s \delta3(r −sre) = −e Z
Z 1 ds s \delta3(r −sre) −e Z 1 ds s (re \cdot \nabla ) A(sre) = −e Z
Z 1 ds s \delta3(r −sre) −e Z 1 ds s \partial \partial sA(sre) = −e Z
Z 1 ds s \delta3(r −sre) −eA(re) + e Z 1 ds A(sre). (9.93) Collecting terms, we find
Z
Z 1 ds s \delta3(r −sre)
Z 1 ds s [\nabla \times A(sre)] , (9.94) and finally the new momentum is
Z 1 ds s B(sre). (9.95) (transformed momentum) Thus, the atomic part of the minimal-coupling Hamiltonian transforms as U(re)[pe + eA(re)]2 2me
2me pe −ere \times Z 1 ds s B(sre) 2 . (9.96) We can see that this part of the transformed Hamiltonian describes the coupling of the atom to the magnetic field. 9.5.3.3 Hamiltonian Collecting pieces from the last two sections, the minimal-coupling Hamiltonian (9.85) becomes
= 2me pe −ere \times Z 1 ds s B(sre) 2 + V (re) +ϵ0 Z d3r E⊥2 + c2B2 − Z
2ϵ0 Z d3r P⊥(r) 2 . (multipole Hamiltonian) (9.97)
Chapter 9. Atomic Interaction with the Quantized Field Note again that the canonical variables are old canonical variables; it is equally possible to obtain this result by writing the old Hamiltonian in terms of new variables. From its form, we can deduce (see Problem 9.3) that the classical canonical momentum for this Hamiltonian is
Z 1 ds s B(sre). (9.98) (multipole canonical momentum) That is, it differs from the kinematic momentum only by a term proportional to the local magnetic field. Although this argument is classical, we can also interpret this expression for the canonical momentum as the Heisenberg equation ˙re = −i ¯h[ ˜H, re] = pe −e me re \times Z 1 ds s B(sre), (9.99) giving the velocity operator in terms of the canonical momentum. It is convenient to multiply out the kinetic term in the above Hamiltonian and separate the terms in the Hamiltonian according to their ‘‘function:’’
(9.100) (multipole Hamiltonian) The isolated atomic Hamiltonian is as usual HA = p 2 e 2me + V (re), (9.101) (free-atom Hamiltonian) and the field Hamiltonian also has its usual form: HF = ϵ0 Z d3r E⊥2 + c2B2 . (9.102) (free field Hamiltonian) Recall here that the longitudinal electric field is already included in the Coulomb binding potential V (re). The coupling of the atom to the electric field is given by the interaction Hamiltonian HAE = − Z
2ϵ0 Z d3r P⊥(r) 2 . (9.103) (atom–E-field coupling) The first term gives the atom–field coupling via the atomic polarization density, while the second term represents an atomic self-energy from the coupling of the polarization to its own field. Note that the Coulomb binding potential may also be written a similar form in terms of P∥(r) and thus combined with the transverse self energy (just as it can be regarded as the energy of the longitudinal electric field), but we will separate the contributions here. Finally, the coupling to the magnetic field has the most complicated form: HAM = −e 2me pe \cdot re \times Z 1 ds s B(sre) + re \times Z 1 ds s B(sre) \cdot pe + e2 2me re \times Z 1 ds s B(sre) 2 . (9.104)
HAM = e 2me Z 1 ds s h (re \times pe) \cdot B(sre) + B(sre) \cdot (re \times pe) i + e2 2me re \times Z 1 ds s B(sre) 2 . (9.105)
9.5.4 Electric Multipole Expansion¶
9.5 Multipole Interactions To further understand the first term, note that we can rewrite it as HAM = −1 Z d3r h
i + e2 2me re \times Z 1 ds s B(sre) 2 , (atom–B-field coupling) (9.106) where the two quantum magnetizations are
re \times pe me Z 1 ds s \delta3(r −sre)
Z 1 ds s \delta3(r −sre) re \times pe me . (quantum atomic magnetizations) (9.107) This brings the first part of the magnetic interaction into the same form as the electric interaction (9.103). Note that if we identify pe −\rightarrow me˙re, both of these reduce to the classical magnetization,
Z 1 ds s \delta3(r −sre), (9.108) from Eq. (9.81) above. Of course, as we mentioned above, pe is not the kinematic momentum, but rather is given by Eq. (9.98). Thus, the magnetic field couples to an atomic quantity that is close to, but not exactly, the classical momentum. Further, because the magnetic field couples to the atomic momentum (or rather, the angular momentum), the symmetric ordering is important in the above interaction. The second term in the magnetic interaction Hamiltonian (9.105) is quadratic in the magnetic field, and we can interpret it as a diamagnetic energy of the atom in the magnetic field. 9.5.4 Electric Multipole Expansion Now we will effect the expansion of the electric-field interaction Hamiltonian (9.103) into multipole moments. The self-interaction term can also be expanded, but often it is dropped and we will do this here, expanding only the remaining atom–field interaction: HAE = − Z d3r P⊥(r) \cdot E⊥(r). (9.109) We begin by expanding the delta function in s, since we assume that the variation of the fields are slow over the length scale re:
h
i
(9.110) This odd expression is sensible since we are in a sense not expanding the delta function, but rather the test function on which the delta function acts. Multiplying by an arbitrary test function f(r) and integrating
(9.111) Thus, the interaction Hamiltonian becomes
(9.112) where note that the gradients operate on the electric fields. Then we can write the Hamiltonian in terms of the multipole moments as
(electric multipole expansion) (9.113)
9.5.5 Magnetic Multipole Expansion¶
Chapter 9. Atomic Interaction with the Quantized Field where note that there are implied summations over repeated indices. Here, the electric dipole moment is as expected
(9.114) (electric dipole moment) while the electric quadrupole moment is
2e
e
. (9.115) (electric quadrupole moment) The first term in the quadrupole operator follows directly from (9.112), while the Kronecker-delta term is added to remove the traces of the moments, Q\alpha\alpha = 0. The trace vanishes since \nabla \cdot E⊥= 0, which in index notation is \partial \alphaE⊥
\beta = 0. Note that the quadrupole moment as written here is the irreducible, rank-2 part of the symmetric, Cartesian tensor (−ere,\alphare,\beta/2) [see Eq. (7.206)]. Finally, the electric octupole moment is
6e
e
. (electric octupole moment) (9.116) Again, the first term is the physically important part, while the Kronecker deltas ensure that the tensor is traceless, giving the irreducible, rank-3 part20 of the symmetric, Cartesian tensor (−ere,\alphare,\betare,\gamma/6). The first two traces vanish, O\alpha\beta\alpha = O\beta\alpha\alpha = 0, in the same way as for the quadrupole moment. Note that for an octupole interaction with a longitudinal field of the form O\alpha\beta\gamma\partial \alpha\partial \beta\partial \gamma\phi in terms of the scalar potential, the remaining trace O\alpha\alpha\beta also vanishes by permutation symmetry of the indices. However, this is not the case for the interaction with the transverse field, where in general \partial \alpha\partial \alphaE⊥ \gamma (0) is nonvanishing. However, for a monochromatic interaction (as appropriate near resonance, when driving a narrow octupole transition), the electric field obeys the Helmholtz equation, and thus \partial \alpha\partial \alphaE⊥
\gamma (0). Thus, this trace of the octupole moment leads to an interaction of the form −O\alpha\alpha\beta\partial \alpha\partial \alphaE⊥
\beta (0). This is of the same form as the dipole interaction, so we should also remove this trace from the octupole moment, and
which now has a small correction at the level of only a part in 106. Due to the presence of additional factors of re with derivative operators, the quadrupole interaction is weaker than the dipole interaction by a factor kre, where k = \omega/c is the optical wave number. Generally this
and re ∼2a0 (a0 is the Bohr radius), kre ∼0.0051. The octupole interaction is a factor of kre weaker yet than the quadrupole term. However, octupole transitions have been driven in experiments.21 9.5.5 Magnetic Multipole Expansion Now expanding the magnetic field to lowest order in re, we find that the magnetic-field interaction Hamil- tonian (9.106) becomes HAM = e 2me
8me [re \times B(0)]2 . (9.117) Defining the magnetic dipole moment m := −e 2me (re \times pe), (9.118) (magnetic dipole moment) 20J. Jerphagnon, ‘‘Invariants of the Third-Rank Cartesian Tensor: Optical Nonlinear Susceptibilities,’’ Physical Review B 2, 1091 (1970) (doi: 10.1103/PhysRevB.2.1091). 21M. Roberts, P. Taylor, G. P. Barwood, P. Gill, H. A. Klein, and W. R. C. Rowley, ‘‘Observation of an Electric Octupole Transition in a Single Ion,’’ Physical Review Letters 78, 1876 (1997) (doi: 10.1103/PhysRevLett.78.1876).
9.6.1 Polarization¶
9.6 Center-of-Mass Röntgen Interaction we can then write the magnetic-interaction Hamiltonian in the dipole approximation as
8me [d \times B(0)]2 . (atom–B-field interaction, magnetic dipole approximation) (9.119) The first term is then the usual interaction of the magnetic dipole with the magnetic field, while the dia- magnetic term appears as a coupling of the electric dipole moment with the magnetic field. To compare the magnitude of the interaction with the electric multipoles, note that we can identify the magnitudes B ∼E/c and pe ∼meckre, so that the magnetic-dipole interaction is of the order ere(kre)E, which is the same as the order of the electric quadrupole interaction. The diamagnetic term, however, depends on r 2 e , but not on the wave number k, and thus its comparison to other terms depends on the strength of the field. Making the same order-of-magnitude replacements, the diamagnetic term is of order e2r 2 e E2/mec2. This is of the same order as the magnetic-dipole term for a field strength satisfying e\lambdaE ∼mec2, which would result in a magnetic-dipole interaction energy ∼(kre)2mec2, which is a very high (relativistic) energy. Hence for moderate (perturbative) field strengths, the diamagnetic term is negligible compared to the magnetic dipole term. 9.6 Center-of-Mass Röntgen Interaction Thus far, we have ignored the center-of-mass motion of the atom, assuming the nucleus to be fixed at r = 0. Motion of the center of mass generates additional multipole terms,22 and we will now consider them here. Consistently accounting for the center-of-mass velocity is important, for example, in obtaining physical results for the angular distribution of photons radiated by a moving atom.23 9.6.1 Polarization To include the center-of-mass motion in the Power–Zienau transformation, we generalize the atomic polar- ization field to the polarization due to an arbitrary system of point charges with respect to an arbitrary ‘‘atomic location’’ rA (which we will take below to be the center of mass):
X \alpha
Z 1 ds \delta3[r −rA −s(r\alpha −rA)]. (9.120) (atomic polarization) Here, q\alpha is the charge of the \alphath particle located at r\alpha. We will then use the unitary operator U = exp −i ¯h Z d3r P(r) \cdot A(r) (9.121) with this polarization to transform the suitably generalized minimal-coupling Hamiltonian H = X \alpha
2m\alpha
Z d3r E⊥2 + c2B2 , (minimal-coupling Hamiltonian, many particles) (9.122) where m\alpha is the mass of particle \alpha. 22E. A. Power and T. Thirunamachandran, ‘‘The Multipolar Hamiltonian in Radiation Theory,’’ Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 372, 265 (1980). 23Martin Wilkens, ‘‘Spurious velocity dependence of free-space spontaneous emission,’’ Physical Review A 47, 671 (1993) (doi: 10.1103/PhysRevA.47.671); Martin Wilkens, ‘‘Significance of Röntgen current in quantum optics: Spontaneous emission of moving atoms,’’ Physical Review A 49, 570 (1994) (doi: 10.1103/PhysRevA.49.570).
9.6.2 Center-of-Mass Coordinates¶
Chapter 9. Atomic Interaction with the Quantized Field 9.6.2 Center-of-Mass Coordinates We now introduce the usual center-of-mass coordinates as follows. The total atomic mass is mA := X \alpha m\alpha, (9.123) and the center-of-mass coordinates are rA := mA X \alpha
pA := X \alpha p\alpha. (9.124) Then the relative coordinates are
mA pA, (9.125) so that X \alpha
X \alpha
(9.126) Note that since the Coulomb binding potential is entirely composed of internal forces,
(9.127) From the commutation relation for the standard coordinates,
(9.128) we can see that the relative positions commute with the center-of-mass momentum,
r\alphaj − mA X \gamma
X \beta p\betak
X \alpha m\alpha mA
(9.129) and similarly the center-of-mass position commutes with the relative momenta,
(9.130) Thus, the center-of-mass coordinates act as an independent degree of freedom from the relative coordinates (which are themselves obey the above constraint equations which reduce the dimension of the relative- coordinate space). Furthermore, the commutation relation for the center-of-mass coordinates is [rAj, pAk] = 1 mA X \alpha
X \beta p\betak = mA X
mA X
(9.131) as we would expect. Furthermore, it is interesting to note that the commutator of the relative coordinates reads
mA [rAj, pAk] = i¯h\deltajk
mA , (9.132) and thus the relative coordinates themselves are not canonical. In these center-of-mass coordinates, the polarization (9.133) becomes
X \alpha
Z 1 ds \delta3(r −rA −s¯r\alpha). (9.133)
9.6.3 Transformation: Electric Dipole Approximation¶
9.6 Center-of-Mass Röntgen Interaction We can see that both the relative and center-of-mass positions are present here, so that moth the relative and center-of-mass momenta will become modified under the Power–Zienau transformation. Further, the minimal-coupling Hamiltonian in center-of-mass coordinates becomes simply H = X \alpha 2m\alpha m\alpha mA
2
Z d3r E⊥2 + c2B2 . (9.134) Partially multiplying out the momentum term gives H = pA2 2mA + X \alpha 2m\alpha
X \alpha q\alpha 2mA
Z d3r E⊥2 + c2B2 , (9.135) so that the center-of-mass and relative components are separated, with an interaction term between the center-of-mass momentum and the vector potential evaluated at the particle locations. 9.6.3 Transformation: Electric Dipole Approximation We must now transform the canonical momenta for the atom and field to obtain the center-of-mass multipole Hamiltonian. The electric field still transforms under the unitary Power–Zienau transformation (9.121) as in Eqs. (9.88), so that UE⊥
\beta (r) −1 ϵ0 P ⊥ \beta (r). (9.136) Thus, the electric-field part of the multipole interaction Hamiltonian is exactly the same as before. The relative momentum also transforms essentially as we worked out before in Eqs (9.95):
Z 1 ds s B(rA + s¯r\alpha). (9.137) We must also transform the center-of-mass momentum, with the result (see Problem 9.5)
X \alpha
X \alpha
Z 1 ds B(rA + s¯r\alpha). (9.138) Here, qA := X \alpha q\alpha (9.139) is the total atomic charge (which vanished for a neutral atom). The transformation of the atomic part of the minimal-coupling Hamiltonian is thus rather involved. To simplify our discussion here, we will make the dipole approximation and neglect the variation of the fields over the scale of the atom. In particular, the momenta now transform as
(9.140) and
X \alpha
(9.141) where the dipole operator is defined by d := X \alpha
(9.142) (electric dipole moment) as is consistent with our previous definitions.
Chapter 9. Atomic Interaction with the Quantized Field In the dipole approximation, we also make the replacement A(r\alpha) −\rightarrow A(rA) in the Hamiltonian, so that the minimal coupling Hamiltonian (9.135) becomes H = pA2 2mA + X \alpha 2m\alpha
qA 2mA
Z d3r E⊥2 + c2B2 . (9.143)
H = pA2 2mA + X \alpha 2m\alpha
Z d3r E⊥2 + c2B2 . (9.144) Thus, we may consider the transformations of the center-of-mass and relative momenta separately. Using Eq. (9.141), the transformation of the center-of-mass kinetic-energy part of the Hamiltonian is thus U pA 2 2mA
2mA
= pA2 2mA + 2mA h
i + 2mA [d \times B(rA)]2 . (9.145) Using Eq. (9.140), the relative-momentum part of the Hamiltonian transforms as U "X \alpha 2m\alpha
¶
X \alpha 2m\alpha h
i 2 = X \alpha ¯p 2 \alpha 2m\alpha
X \alpha q 2 \alpha 8m\alpha
(9.146) where we have defined the magnetic dipole moment as m := X \alpha q\alpha 2m\alpha
(9.147) (magnetic dipole moment) We can thus recognize the last two terms of Eq. (9.145) as the magnetic dipole and diamagnetic terms that we already discussed. We concluded that these were small compared to the electric-dipole interaction, so we will make the electric dipole approximation and drop these. We will similarly drop the last term of Eq. (9.145), which we can identify as a center-of-mass diamagnetic term, since it is quadratic in the atomic dipole moment and thus of higher order than the electric-dipole interaction. From what remains, we can write the full, transformed Hamltonian as
(9.148) (multipole Hamiltonian) where the isolated atomic Hamiltonian is HA = pA2 2ma + X p 2 \alpha 2m\alpha
(9.149) (free-atom Hamiltonian) the field Hamiltonian has its usual form, HF = ϵ0 Z d3r E⊥2 + c2B2 , (9.150) (free-field Hamiltonian) and we have the usual form for the electric-dipole interaction,
2ϵ0 Z d3r P⊥(r) 2 , (9.151) (electric-dipole interaction)
9.6.4 Full Transformation¶
9.6 Center-of-Mass Röntgen Interaction where the polarization (9.133) becomes
X \alpha
(9.152) in the dipole approximation. The new interaction, corresponding to the remaining term in (9.145) that we did not drop, is the Röntgen interaction,24 HR = 2mA h
i , (Röntgen interaction, electric dipole approximation) (9.153) which gives the coupling energy of the electric dipole and the magnetic field: recall that under a Lorentz boost, electric and magnetic fields interchange to some extent, so this energy can be interpreted as the electric dipole interaction with the transformed magnetic field. Naturally, the internal motion of the charges should likewise induce a Röntgen-type coupling to the magnetic field; this is what we have already identified as the magnetic-dipole interaction, and in fact we can obtain the magnetic-dipole Hamiltonian from the Röntgen Hamiltonian by making the replacements mA −\rightarrow m\alpha, pA −\rightarrow p\alpha, both inside the summation implicit in the definition of d. Thus, even in the electric dipole approximation, an extra interaction Hamiltonian must be considered if the atom is in motion for consistency with the minimal-coupling Hamiltonian. This comes up, for example, in atom optics. Consider the usual problem of radiation pressure, where a plane, traveling wave impinges on an atom, and causes a net force in the propagation direction of the field to to absorption. If the atom is moving, the Röntgen interaction can add another component to the radiation-pressure force that is along the polarization vector for the field, instead of along its propagation direction.25 Note, however, that this component is rather weak compared to the usual radiation pressure, and only occurs if the atomic dipole is not parallel to the field polarization (which is not possible for an S ground state of the resonant transition). The Röntgen interaction is generally weak, even though necessary for the consistency in the radiation of a moving atom as we mentioned above. 9.6.4 Full Transformation So then what does the completely general multipole Hamiltonian look like, if we account for center-of-mass motion but don’t make the electric-dipole approximation? You might regret that you asked that, but it is certainly possible to write down the answer.26 9.6.4.1 Effecting the General Transformation Our goal is to perform the Power–Zienau transformation on the minimal-coupling Hamiltonian in the form (9.135). The part we should concentrate on here is the atomic-momentum part, H = pA2 2mA + X \alpha 2m\alpha
X \alpha q\alpha 2mA
(9.154) since our transformation for the field as above is still valid. The transformation (9.137) for the relative momentum leads to the following transformation for the relative-momentum term, which we can derive 24Named for Wilhelm Conrad Röntgen, who figured out that charges moving in a magnetic field see an effective electric field EM = ˙r\timesB. See W. C. Röntgen, ‘‘Ueber die durch Bewegung eines im homogenen electrischen Felde befindlichen Dielectricums hervorgerufene electrodynamische Kraft,’’ Annalen der Physik und Chemie 35, 264 (1888). 25V. E. Lembessis, M. Babiker, C. Baxter, and R. Loudon, ‘‘Theory of radiation forces and momenta for mobile atoms in light fields,’’ Physical Review A 48 1594 (1993) (doi: 10.1103/PhysRevA.48.1594). 26C. Baxter, M. Babiker and R. Loudon, ‘‘Canonical Approach to Photon Pressure,’’ Physical Review A 47, 1278 (1993) (doi: 10.1103/PhysRevA.47.1278).
Chapter 9. Atomic Interaction with the Quantized Field simply by adapting our treatment from Section 9.5.3.3: U "X \alpha
¶
X \alpha ¯p 2 \alpha 2m\alpha −1 Z d3r h
i + X \alpha q 2 \alpha 2m\alpha
Z 1 ds s B(rA + s¯r\alpha) 2 . (9.155) The two quantum magnetizations are now suitably modified to include the center-of-mass coordinate to read
X \alpha q\alpha
m\alpha Z 1 ds s \delta3(r −rA −s¯r\alpha)
X \alpha q\alpha Z 1 ds s \delta3(r −rA −s¯r\alpha)
m\alpha . (9.156) We can recognize the terms in Eq. (9.155) as ones we have seen before: atomic kinetic energy, magnetic-field interaction with the atomic magnetization, and the diamagnetic energy. On the other hand, the transformation (9.138) for the center-of-mass momentum leads to a slightly more complicated transformation for the Hamiltonian. First, we can consider the transformation of the center-of-mass kinetic energy: U pA2 2mA
2mA " pA −qAA(rA) + X \alpha
X \alpha
Z 1 ds B(rA + s¯r\alpha) #2 . (9.157) Before multiplying this out, it simplifies things to consider the transformation of the remaining term in the Hamiltonian (9.154): U " − X \alpha q\alpha 2mA
¶
2mA X \alpha
- X \alpha qAq\alpha mA
mA "X \alpha
2 − mA X \alpha¶
X \beta
Z 1 ds B(rA + s¯r\beta). (9.158) These four terms exactly cancel the four terms in the expansion of (9.157) that involve the factor P
9.6 Center-of-Mass Röntgen Interaction Combining the center-of-mass parts of the transformed Hamiltonian thus gives U " pA2 2mA − X \alpha q\alpha 2mA
U \dagger = pA2 2mA −qA mA pA \cdot A(rA) + qA2 2mA A2(rA) + 2mA " pA \cdot X \alpha¶
Z 1 ds B(rA + s¯r\alpha) ! + X \alpha
Z 1 ds B(rA + s¯r\alpha) ! \cdot pA # −qA mA X \alpha
Z 1 ds B(rA + s¯r\alpha) + 2mA "X \alpha
Z 1 ds B(rA + s¯r\alpha) #2 = pA2 2mA −qA mA pA \cdot A(rA) + qA2 2mA A2(rA) + 2mA Z d3r h pA \cdot P(r) \times B(r) + P(r) \times B(r) \cdot pA i −qA mA A(rA) \cdot Z
2mA Z d3r P(r) \times B(r) 2 , (9.159) where we have used the polarization field in the form (9.133). We can identify the first three terms here as residual interaction of the center of mass with the vector potential, and the rest of the terms represent the generalized Röntgen interaction. 9.6.4.2 Final Result Collecting all terms after the transformation, we have the new Hamiltonian
(general multipole Hamiltonian, with center-of-mass motion) (9.160) The isolated atomic Hamiltonian HA = pA2 2mA + X \alpha ¯p 2 \alpha 2m\alpha
(9.161) (free-atom Hamiltonian) is the sum of the external and internal kinetic energies as well as the internal binding potential. The field Hamiltonian has its usual form, HF = ϵ0 Z d3r E⊥2 + c2B2 , (9.162) (free-field Hamiltonian) while we have the same interaction Hamiltonian for the coupling of the atom to the electric field via the atomic polarization, HAE = − Z
2ϵ0 Z d3r P⊥(r) 2 , (atom–E-field interaction Hamiltonian) (9.163) where the polarization P(r) is again given by Eq. (9.133). This Hamiltonian is again the generalization of the usual dipole interaction Hamiltonian, along with the electric self energy of the atom. The interaction Hamiltonian for the coupling of the internal degrees of freedom of the atom to the magnetic field is HAM = −1 Z d3r h
i + X \alpha q 2 \alpha 2m\alpha
Z 1 ds s B(rA + s¯r\alpha) 2 , (atom–B-field interaction Hamiltonian) (9.164)
Chapter 9. Atomic Interaction with the Quantized Field where the quantum magnetizations are given by Eqs. (9.156). The Hamiltonian here is the generalization of the magnetic-dipole interaction plus the diamagnetic energy of the atom in the magnetic field. Finally, the generalized Röntgen interaction coupling the center-of-mass motion of the atomic system to the field is HR = −qA mA pA \cdot A(rA) + qA2 2mA A2(rA) + 2mA Z d3r h pA \cdot P(r) \times B(r) + P(r) \times B(r) \cdot pA i −qA mA A(rA) \cdot Z
2mA Z d3r P(r) \times B(r) 2 , (center-of-mass (Röntgen) interaction Hamiltonian) (9.165) which is quite a complicated interaction involving the atomic center-of-mass momentum, the vector potential, and the magnetic field. In our treatment above of the Röntgen interaction in the electric dipole approxi- mation, we dropped all but the third term in the above expression for HR. Recalling that the total charge
case, and the first two terms of HR clearly represent a minimal-coupling-type interaction of the net atomic charge with the vector potential.
9.7 Exercises¶
9.7 Exercises 9.7 Exercises Problem 9.1 Prove the relation Z¶
(9.166) which we used in analyzing the minimal-coupling Hamiltonian. Problem 9.2 Consider the Lagrangian L = 1
(9.167) for a particle of charge q in electric and magnetic fields
\partial t ,
(9.168) written here in terms of the scalar potential \phi and the vector potential A. Show that the Euler–Lagrange equation, together with this Lagrangian, is equivalent to Newton’s Second Law with the Lorentz force. Problem 9.3 Show that a (classical) Hamiltonian of the form
2m + V (q) (9.169) has canonical momentum given by
(9.170) through correspondence with the Lagrangian L = 1 2m ˙q2 −˙q \cdot a(q) −V (q). (9.171) Problem 9.4 Consider a single-electron atom in the heavy-nucleus approximation. Go through the derivation of the coupling of the atom to the electric field, HAE = − Z
2ϵ0 Z d3r P⊥(r) 2 , (9.172) and the coupling of the atom to the magnetic field, HAM = −1 Z d3r h
i + e2 2me re \times Z 1 ds s B(sre) 2 , (9.173) without making the long-wavelength approximation. You should also go through the derivation of the atomic polarization field P(r) and (classical) magnetization density M(r), paying special attention to their physical interpretation. These results generalize the results we derived in class for the dipole interaction Hamiltonian to include all multipole orders (but they still neglect center-of-mass motion of the atom and the presence of more than one electron.)
Chapter 9. Atomic Interaction with the Quantized Field Problem 9.5 Carry out the Power–Zienau transformation of the center-of-mass momentum pA, with transformation operator U = exp −i ¯h Z d3r P(r) \cdot A(r) , (9.174) and polarization given by
X \alpha
Z 1 ds \delta3(r −rA −s¯r\alpha), (9.175) to obtain the result
X \alpha
X \alpha
Z 1 ds B(rA + s¯r\alpha) (9.176) where qA is the total atomic charge. Problem 9.6
(a) Assuming the transition may be driven by a monochromatic plane wave via the quadrupole in- teraction, show that the quadrupole interaction has the same form as the dipole interaction, with an
transition will be excited with an effective, quadrupole Rabi frequency
¯h , (9.177) (up to an arbitrary, overall phase that can be absorbed into the field), in analogy with the dipole Rabi
(b) Assuming that the plane wave propagates along the x-direction and is polarized along the z- direction, give an explicit expression for the relevant component of the effective dipole moment, in terms of a sum over products of dipole matrix elements (i.e., matrix elements like \langle a|dx|c\rangle , involving some auxiliary state |c\rangle ). Problem 9.7 Go through the steps in the derivation [Eq. (9.71)] to show that for an atom of charge density
(9.178) the polarization density
Z 1 ds \delta3(r −sre) (9.179) is consistent with the constraint \nabla \cdot P∥= −\rho.