11. Spontaneous Emission¶
PDF pages 495–510
Chapter 11 Spontaneous Emission 11.1 Atom–Field Coupling Here we will consider the spontaneous decay of an atomic excited level due to coupling to the vacuum field, according to the treatment of Weisskopf and Wigner1. We will also consider corrections and extensions to this result as well as the implications of this treatment for the spontaneous-emission master equation. The uncoupled Hamiltonian for a two-level atom and the field (including a sum over all field modes) is
X k,\zeta ¯h\omegak a\dagger
, (11.1) where the ground state has zero energy, \omega0 is the atomic transition frequency, the wave vector k labels the field modes of different frequency and orientation, the index \zeta labels the two independent polarizations, and ak,\zeta is the annihilation operator for the (k, \zeta) mode. We will write the eigenstates that we need of the free Hamiltonian in the form |\alpha, nk,\zeta\rangle , which means that the atom is in state |\alpha\rangle , while the field mode (k, \zeta) has n photons (other modes not explicitly labeled are in the vacuum state). We use the dipole form of the atom–field interaction Hamiltonian
(11.2) where as before the dipole operator is
=: dge
. (11.3) Here, \sigma := |g\rangle \langle e| is the atomic lowering operator, and dge := \langle g|d|e\rangle is the dipole matrix element for the atomic transition. We can write the electric field modes as
r ¯h\omegak 2ϵ0
(11.4) where the fk,\zeta(r) are normalized (classical) mode functions. Thus, the interaction Hamiltonian becomes HAF = − X k,\zeta r ¯h\omegak 2ϵ0
dge \cdot
k,\zeta . (11.5) 1V. Weisskopf and E. Wigner, ‘‘Berechnung der natürlichen Linienbreite auf Grund der Diracschen Lichttheorie,’’ Zeitschrift für Physik 63, 54 (1930) (doi: 10.1007/BF01336768). See also Marlan O. Scully and M. Suhail Zubairy, Quantum Optics (Cambridge, 1997) (ISBN: 9780511813993) (doi: 10.1017/CBO9780511813993), p. 206; or Peter W. Milonni, The Quantum Vacuum (Academic Press, 1993) (ISBN: 0124980805) (doi: 10.1016/C2009-0-21295-5), p. 204.
Chapter 11. Spontaneous Emission In the rotating-wave approximation, we drop the energy nonconserving terms, so that HAF = − X k,\zeta r ¯h\omegak 2ϵ0 dge \cdot
k,\zetaf∗ k,\zeta(r) = X k,\zeta ¯h
k,\zeta , (11.6) where the coupling factor (one-photon Rabi frequency) is defined as
r \omegak 2ϵ0¯h dge \cdot fk,\zeta(r) (11.7) for each mode. 11.2 Evolution Again, we will write the eigenstates that we need of the free Hamiltonian in the form |\alpha, nk,\zeta\rangle , which means that the atom is in state |\alpha\rangle , while the field mode (k, \zeta) has n photons (other modes not explicitly labeled are in the vacuum state). We will assume that the atom is initially excited, and the field is in the vacuum state. According to the interaction (11.6), the only states coupled to the initial state are where the atom is in the ground state and one photon is present. Thus, we may write the state of the atom and field as
X k,\zeta
(11.8) The evolution is given by the Schrödinger equation,
(11.9) which gives the following coupled equations for the amplitudes:
X k,\zeta
k,\zetace. (11.10) Now we will define slowly varying amplitudes ˜ce := ceei\omega0t,
(11.11) in terms of which the equations of motion (11.10) become
X k,\zeta
(11.12) Integrating the second equation, we find
k,\zeta Z t dt′ ˜ce(t′) ei(\omegak−\omega0)t′. (11.13) We can use this in the first of Eqs. (11.12) to decouple the equations,
X k,\zeta |gk,\zeta|2 Z t dt′ ˜ce(t′) e−i(\omegak−\omega0)(t−t′), (11.14) so that now we must simply evaluate this expression to uncover the decay process.
11.3 Large-Box Limit 11.3 Large-Box Limit Now we can put in the explicit form of the coupling factor from Eq. (11.7):
2ϵ0¯h X k,\zeta
Z t dt′ ˜ce(t′) e−i(\omegak−\omega0)(t−t′), (11.15) where we used
(11.16) and where ˆ\epsilonk,\zeta is the local polarization unit vector for the field mode. Also, note that for the spherically symmetric atom,
3d2 ge, (11.17)
symmetric atom is not valid. Because we are summing the quantities |ˆ\epsilonk,\zeta \cdot dge|2 over all field modes, and the electromagnetic vacuum is isotropic, the final result can’t be changed by a termwise average over the relative orientation of atom and field polarization. Now we can put in the explicit (k, \zeta) modes. In free space, from Section 8.4.2, the mode functions are
\sqrt V
(11.18) and so
V , (11.19) where V is the quantization volume, and summing over both polarizations simply gives an extra factor of 2:
d2 ge 3ϵ0¯hV X k \omegak Z t dt′ ˜ce(t′) e−i(\omegak−\omega0)(t−t′). (11.20) The next step is to evaluate the wave-vector sum in the continuum limit. When the box becomes large (V −\rightarrow \infty), the spacing between the modes becomes small. (We covered this procedure when working out the free-space commutators in Section 8.6.1.2, but we’ll review it to keep this self-contained.) In this limit, an integral of a function is equivalent to a sum weighted by the mode spacings. Since the modes exist only for positive and negative k\alpha, we can write X k f(k) ∆kx ∆ky ∆kz −\rightarrow Z \infty −\infty dkx Z \infty −\infty dky Z \infty −\infty dkz f(k) (11.21) for an arbitrary function f(k). Since
L , (11.22) we can thus make the formal replacement X k −\rightarrow V (2\pi)3 Z \infty −\infty dkx Z \infty −\infty dky Z \infty −\infty dkz, (11.23) where V = L3. Thus, we can write the equation of motion as
d2 ge 3ϵ0¯h(2\pi)3 Z d3k \omegak Z t dt′ ˜ce(t′) e−i(\omegak−\omega0)(t−t′) = −d2 gec 6\pi2ϵ0¯h Z \infty dk k3 Z t dt′ ˜ce(t′) e−i(\omegak−\omega0)(t−t′) = − d2 ge 6\pi2ϵ0¯hc3 Z \infty d\omega \omega3 Z t dt′ ˜ce(t′) e−i(\omega−\omega0)(t−t′), (11.24)
Chapter 11. Spontaneous Emission where we have now carried out the angular integrals in spherical coordinates, and we are using \omega = \omegak = ck. Note here the characteristic \omega3 dependence, due partially to the frequency dependence of the vacuum density of states and also due to the explicit frequency dependence of HAF, which will become manifest as an \omega 3 dependence of the spontaneous decay rate (see Section 11.6.1 for an explicit calculation of the spontaneous decay rate in terms of the density of states). 11.4 Decay Rate To proceed, we can now note that ˜ce(t′) varies slowly on optical time scales. Also, \omega3 is slowly varying compared to the exponential factor in Eq. (11.24), which oscillates rapidly (at least for large times t) about zero except when t \approx t′ and \omega \approx \omega0. Thus, we will get a negligible contribution from the \omega integral away from \omega = \omega0. Thus, we will make the replacement \omega3 −\rightarrow \omega 3 0 :
0 d2 ge 6\pi2ϵ0¯hc3 Z \infty d\omega Z t dt′ ˜ce(t′) e−i(\omega−\omega0)(t−t′). (11.25) The same argument gives Z \infty
Z \infty −\infty
(11.26) We can see from this that our argument here about the exponential factor is equivalent to the Markovian approximation. Thus,
0 d2 ge 3\piϵ0¯hc3 Z t dt′ ˜ce(t′) \delta(t −t′)
0 d2 ge 3\piϵ0¯hc3 ˜ce(t) . (11.27) Here, we have split the \delta-function since the upper limit of the t′ integral was t, in view of the original form (11.25) for the t′ integral, where the integration limit is centered at the peak of the exponential factor. We can rewrite the final result as
2 ˜ce, (11.28) where the spontaneous decay rate is given by \Gamma := \omega 3 0 d2 ge 3\piϵ0¯hc3 . (11.29) (spontaneous decay rate) This decay rate is of course defined so that the probability decays exponentially at the rate \Gamma:
(11.30) Also, note that
−i\omega0 −\Gamma ce (11.31) after transforming out of the slow variables. Thus, we now have the an expression for the spontaneous decay rate in terms of the atomic parameters, which comes from a fully quantum treatment of the atom–field interaction. Recall that we derived this same expression earlier in Eq. (5.242) by comparing the optical Bloch equations to the rate equations; that result was correct because the rate equations are valid for the thermal quantum state of the field. The above argument, using Eq. (11.26) is actually a bit sloppy because it misses another effect. A more
replacement: \partial t˜ce \approx − d2 ge 6\pi2ϵ0¯hc3 ˜ce(t) Z \infty d\omega \omega3 Z t
(11.32)
11.4 Decay Rate and then considers the integral Z t¶
\omega −\omega0
\omega −\omega0 . (11.33) The integral of the sin term over all frequencies is \pi for t > 0, and for long times this term is arbitrarily narrowly peaked in frequency as t increases, as in our previous argument. The imaginary term is zero for
integral, we can make the replacement Z t dt′ e−i(\omega−\omega0)(t−t′) −\rightarrow −iP \omega −\omega0
(11.34) where P denotes that the \omega integral should handle the singularity at \omega = \omega0 in the sense of a Cauchy principle value (Section 14.1.4.2). Making this replacement in Eq. (11.32), we have \partial t˜ce \approx − d2 ge 6\pi2ϵ0¯hc3 ˜ce(t) \pi\omega 3 0 −i – Z \infty d\omega \omega3 \omega −\omega0 . (11.35) This can be written
(11.36) where we have the constants \Gamma = \omega 3 0 d2 ge 3\piϵ0¯hc3 ,
d2 ge 6\pi2ϵ0¯hc3 – Z \infty d\omega \omega3 \omega −\omega0 = − \Gamma 2\pi\omega 3 – Z \infty d\omega \omega3 \omega −\omega0 . (decay rate and Lamb shift) (11.37) The decay rate is the same rate (11.29) that we obtained before, but the frequency shift \delta\omega0 of the transition frequency is the effect we missed (this also pops up in the course of solving Problem 11.4). This is the Lamb shift, which is an ac Stark shift due to the coupling of the excited state to the vacuum electromagnetic field modes. This quantity involves a divergent integral that must be handled carefully, and we will defer this procedure to Section 13.12. The Lamb shift can justifiably be ignored (despite being divergent here!), because \omega0 is typically assumed to be the observed frequency, which already includes effects like the Lamb shift. Also, note that the ground state also experiences a Lamb shift, but here we have ignored it by declining to work out the equation of motion for the ground-state amplitude (we have also ignored couplings of the ground state to other important states besides |e\rangle ). So what is the decay rate, typically? If we assume an optical transition, \omega0/2\pi = 400 THz, and a dipole matrix element of order dge ∼ea0, where the Bohr radius a0 \approx 0.5 Å, then we get a decay rate of \Gamma ∼5\times106 s−1. This is a reasonably good estimate, although for the larger alkalis, the decay rate is slightly larger (around 30 \times 106 s−1), since the dipole matrix elements have larger magnitudes for these large atoms. However, this justifies our assertion that ˜ce is a slowly varying amplitude (slow compared to the optical frequency). Finally, note that we are only treating the decay of two-level atoms here, whereas real atoms are more complicated. We already treated the case of degeneracy due to angular momentum in Section (7.3.7.4). There we considered the decay of the Jg −\rightarrow Je fine-structure transition (with Je being the excited state as usual). Then the decay rate from sublevel |Je me\rangle −\rightarrow |Jg mg\rangle is just given by Eq. (11.29) with the appropriate matrix element: \GammaJg,mg;Je,me = \omega 3 3\piϵ0¯hc3 |\langle Jg mg|d|Je me\rangle |2. (11.38) With the normalization conventions assumed for the reduced matrix element, we found \GammaJgJe = \omega 3 3\piϵ0¯hc3 2Jg + 1
(spontaneous decay rate, fine-structure transition) (11.39) for the decay rate in terms of the reduced dipole matrix element. This same formula also applies to hyperfine transitions.
Chapter 11. Spontaneous Emission 11.5 Master Equation for Spontaneous Emission In order to arrive at the usual damping part of the master equation for the atom due to spontaneous emission (i.e., the damping part of the optical Bloch equations), we want to consider the reduced density operator for the evolution of the atomic state, tracing over the state of the field. Here we will compute the individual matrix elements
(11.40) for the atomic state. The easiest matrix element to treat is the excited-level population,
e. (11.41) Differentiating this equation and using (11.31) gives
(11.42) The matrix element for the ground-state population follows from summing over all the other states:
X \zeta Z dk ˜ck,\zeta˜c∗ k,\zeta. (11.43) Notice that the states |e\rangle and |g\rangle are effectively degenerate, but when we eliminate the field, we want |e\rangle to have ¯h\omega0 more energy than the ground state. The shortcut for doing this is to realize that the latter situation corresponds to the ‘‘interaction picture’’ with respect to the field, where we use the slowly varying ground-state amplitudes ˜ck,\zeta (which have been boosted down in energy by ¯h\omegak to where we expect the atomic ground state should be) but the standard excited-state amplitude ce. This explains why we use
(11.44) Finally, the coherences are
e,
ge, (11.45) where we have extended the pure state to include a component in the ground state without spontaneously emitted light:
X k,\zeta
(11.46)
conclusions go through with this change.) The corresponding equation of motion is
i\omega0 −\Gamma c∗ e = i\omega0 −\Gamma \rhoge. (11.47) The time derivatives of the cg is zero here, because nothing in the Hamiltonian causes it to evolve. Notice that what we have derived are exactly the same matrix elements generated by the master equation
(11.48) where once again the form of the Lindblad superoperator D[\sigma]\rho is given by
, (11.49) and the atomic Hamiltonian is
(11.50) That is, the damping term here represents the same damping as in the optical Bloch equations.
11.6 Fermi’s golden rule 11.6 Fermi’s golden rule Now we will rederive the spontaneous-emission rate from a more general approach that applies to any quantum decay problem or transition to a continuum of states. We will then recover the result (11.29) for the rate of spontaneous emission in free space. Later, in Section (14.3.10), we will also use this approach to see how the rate of spontaneous emission is modified in the presence of a macroscopic body, such as a mirror or a cavity. Let’s consider a transition from an initial state |i\rangle to a final state |f\rangle , where |i\rangle and |f\rangle are eigenstates of some background Hamiltonian H0. The transition is due to the constant perturbation Hamiltonian Hint. We assume the state of the system at t = 0 to be |i\rangle , and then consider the effect of Hint for t > 0. We will solve this problem by noting that we have already solved it in Section 5.2.2. In the setup of that problem, we showed that a two-level atom interacting with a monochromatic field is equivalent to a two-state system interacting with a dc perturbation, after making the rotating-wave approximation and transforming to the rotating frame of the field. Thus, we will make some formal identifications with the two-level atom problem. First, we can identify the free atomic Hamiltonian ˜HA for the two-level atom in the rotating frame with the background Hamiltonian in the present problem,
\leftarrow \rightarrow
(11.51) where ¯h\omegafi := Ef −Ei is the energy difference of the initial and final states, and we take Ei = 0. We can further identify the atom–field interaction Hamiltonian ˜HAF for the two-level atom with the perturbation Hamiltonian Hint, ˜HAF = ¯h h
\leftarrow \rightarrow
(11.52) where we are taking the form of ˜HAF generalized to a complex Rabi frequency Ω, as in Eq. (5.406), and \sigma is the usual atomic lowering operator. Note that we ignore diagonal matrix elements of Hint, as is appropriate for the dipole-interaction Hamiltonian (i.e., we can absorb any diagonal matrix elements of Hint into H0). We showed that the solution to the two-level atom problem is [Eq. (5.60)]
|Ω|2 |Ω|2 + ∆2 sin2 1 p |Ω|2 + ∆2 t , (11.53) which gives the excitation probability given that the atom is initially in the ground state. We can map this solution to the current problem by making the identifications |g\rangle −\rightarrow |i\rangle , |e\rangle −\rightarrow |f\rangle , ∆−\rightarrow \omegaif, and ¯hΩ−\rightarrow 2\langle f|Hint|i\rangle . We will further note that we are treating the interaction as a weak perturbation, so that |Ω| ≪|∆|. Thus, the solution to the present perturbation problem is the transition probability to |f\rangle :
¯h2\omega 2 if sin2 \omegaift ,
(11.54) Now consider the part of the above expression that depends on \omegaif. It is normalized such that Z \infty −\infty d\omegaif \omega 2 if sin2 \omegaift
2 , (11.55) and the integrand is a localized function in \omegaif with a width that scales as 1/t. Thus, for large t, when the integrand becomes a very narrow function of frequency, we may replace the integrand by a delta function: \omega 2 if sin2 \omegaift −\rightarrow \pit
(11.56) Thus, Eq. (11.54) becomes
(11.57)
11.6.1 Free-Space Decay Rate¶
Chapter 11. Spontaneous Emission Thus, in the long-time limit, the excitation probability increases linearly with time. This is clearly valid only for ‘‘weak’’ perturbations and short times such that the excitation probability is small; on the other hand, we had to make the long-time assumption such that the time was long enough to justify the approximation by a delta function. We will return to these constraints below. In any case, the transition rate from |i\rangle to |f\rangle is simply the time derivative of the transition probability, and we can thus write the transition rate as
¯h |\langle i|Hint|f\rangle |2\delta(Ei −Ef). (11.58) (Fermi’s golden rule) The transition rate in this regime of intermediate times is time-independent, and this expression for the transition rate is Fermi’s golden rule. This statement can also be regarded as a statement of energy conservation: transitions only occur when the energies of the initial and final states match. The delta function in this expression really only makes sense under an integral over energies, since it represents the transition probability summed over a range of energies. This is a crucial point: it is the existence of a continuum of energy levels that causes the time-independent transition rate; otherwise, the transition rate to a discrete state oscillates in time, due to coherent Rabi flopping. Thus, suppose we consider the transition from |i\rangle to a continuous set F of states. Then we must sum over the transition rate to all final states |f\rangle \in F. We will carry out this sum only over a narrow range (Ei −\epsilon/2, Ei + \epsilon/2) of final states, where \epsilon defines a range of energies over which Hint is constant. Letting n(E) denote the number of states with energy less than E, the sum over transition rates is
¯h
dn(E′) |\langle i|Hint|f\rangle |2\delta(Ei −E′)
¯h
(11.59)
the integral and taking Ef = Ei, we arrive at an alternate form of Fermi’s golden rule:
(11.60) (Fermi’s golden rule) In deriving this, we also had to assume that the density of states was approximately constant over the range of integration, which sets another upper bound on \epsilon. For the delta-function approximation to hold, we needed that the frequency width of the function in the expression (11.56) must be small compared to \epsilon/¯h, and thus that t ≫¯h/\epsilon, quantifying the long-time constraint we mentioned above. The short-time constraint is that \Gammai\rightarrow Ft ≪1, but typically this expression is valid to much longer times by accounting explicitly for depletion of |i\rangle : the decay rate holds so long as the decayed population in the states F do not influence the decay rate. 11.6.1 Free-Space Decay Rate Now we show that Fermi’s golden rule leads to the correct spontaneous-emission rate in free space. From the Weisskopf–Wigner treatment of spontaneous emission, the dipole interaction Hamiltonian takes the form [see Eq. (11.5)] HAF = − X k,\zeta r ¯h\omegak 2ϵ0
dge \cdot
k,\zeta . (11.61) The field modes here are labeled as usual by the wave vector k and the polarization index \zeta \in {1, 2}, and in free space, from Section 8.4.2, the mode functions are given by
\sqrt V
(11.62)
11.7.1 Short Times¶
11.7 Corrections to Exponential Decay when quantized in the fictitious quantization volume V . Then considering the transition from the initial state |e\rangle to the set of final states of the form |g, 1k,\zeta\rangle , we can write down the matrix element
r ¯h\omegak
(11.63) We can then compute the density of states as follows, considering only the final states with the atom in the ground state and one photon in some mode. If we assume a cubic quantization volume with V = L3, then the wave vectors are constrained to be kx,y,z = 2\pinx,y,z L
3\sqrt V , (11.64) where the nxj are any integers, as a result of the periodic boundary conditions on the quantization box. Thus, in k-space, the states form a cubic lattice with spacing 2\pi/ 3\sqrt V . We can thus associate a cubic volume of (2\pi)3/V with each state in k-space, where this mini-volume ‘‘surrounds’’ its particular state. Now the set of all states with energy less than E is given by the set of all states in k-space that fall within a radius of kE = E/¯hc of k = 0. The volume of the sphere of this radius is 4\pik 3 E/3, and thus the number of states is given by dividing this volume by (2\pi)3/V , and then multiplying by 2 to count independent polarizations:
E V
E3V 3\pi2¯h3c3 . (11.65) Then the density of states is
dE = E2V \pi2¯h3c3 . (11.66) The relevant initial and final energy in this problem is E = ¯h\omega0, being the energy of the initially excited atom, so that
0 V \pi2¯hc3 . (11.67) Putting these pieces together in the golden rule (11.60) we find the rate of spontaneous emission in free space
3\piϵ0¯hc3 , (11.68) (spontaneous decay rate in free space) upon taking \omegak \approx \omega0, and taking |ˆ\epsilonk,\zeta \cdot dge|2 = |\langle g|d|e\rangle |2/3 for a spherically symmetric atom. This result agrees with Eq. (11.29) from our previous Weisskopf–Wigner calculation. 11.7 Corrections to Exponential Decay The above results of exponential decay of the atomic excited state is a universal result of unstable quantum systems. However, it is also an approximation, and under physically reasonable assumptions the exponential decay law fails for very short and very long times. This was first discussed by Khalfin on very general grounds.2 Fonda, Ghirardi, and Rimini have given a comprehensive review, and we follow their treatment for short-time deviations from exponential decay.3 11.7.1 Short Times For short times, we will show that the decay rate in fact vanishes. This result applies broadly, even beyond atomic-level decay. Let us denote the survival probability by
(11.69) 2L. A. Khalfin, ‘‘Contribution to the Decay Theory of a Quasi-Stationary State,’’ Soviet Physics JETP 6, 1053 (1958). 3L. Fonda, G. C. Ghirardi, and A. Rimini, ‘‘Decay theory of unstable quantum systems,’’ Reports on Progress in Physics
Chapter 11. Spontaneous Emission where c(t) is the amplitude of the initial state (|e\rangle in the spontaneous-emission problem). Then
(11.70) Now let’s consider the eigenstates of the Hamiltonian H. For a system with several degrees of freedom, there will in general be multiple eigenstates for a given energy, and we label them by an extra index a. The index a can, for example, represent the set of simultaneous eigenvalues of a complete set of observables commuting with each other and H, if the system has enough symmetry to permit this, but the existence of such observables is not required here. Then we can write the eigenstates as
(11.71) and the completeness relation becomes Z dE Z
(11.72) Now we can write the coefficient c(t) in terms of the unitary time-evolution operator (assuming a time- independent system, as we have already done by assuming energy eigenstates) as
= Z \infty −\infty dE Z
= Z \infty −\infty dE \omega(E)e−iEt/¯h, (11.73) where
Z
(11.74) is the Fourier transform of c(t). Now we make the reasonable assumption that the energies E are bounded from below (which happens, for example, if there is a ground state):
Z \infty Emin dE \omega(E)e−iEt/¯h, (11.75) Note that the integral here is uniformly convergent for all t, since \omega(E) > 0, and the integral converges by assumption at t = 0, so that Z \infty Emin dE |\omega(E)| (11.76) is convergent. (The integral for c(t) is hence absolutely convergent.) Thus, we may extend the integral to negative times, and the integral is uniformly convergent for any t \in R. Differentiating (11.75), we find dc(t) dt = −i ¯h Z \infty Emin dE \omega(E)Ee−iEt/¯h. (11.77) If we assume finite average energy of the initial state, Z \infty Emin dE \omega(E)E < \infty, (11.78)
we have the Fourier-transform property
(11.79)
11.7 Corrections to Exponential Decay and then differentiating¶
(11.80) we find dP(t) dt
dt c(−t) + c(t)dc(−t) dt . (11.81)
lim t\rightarrow 0+ dP(t) dt = lim t\rightarrow 0+ dc(t) dt c(−t) + c(t)dc(−t) dt = lim t\rightarrow 0+ dc(t) dt −dc(t) dt = 0. (short-time nonexponential decay) (11.82)
at t = 0, and so the true decay is always slower than exponential for very short times. 11.7.1.1 Quantum Zeno Effect This slower-than exponential decay can lead to an interesting effect: if the decay is ‘‘interrupted’’ by a quantum measurement that distinguishes the initial state from the final states, the evolution is ‘‘reset’’ to t = 0. With continual, rapid measurements, the system never settles into the usual exponential decay, and the decay rate for the observed system is slower than without observation. This is the essence of the quantum Zeno effect.4 To see this more explicitly, assume that for short times, the survival probability may be written as
(11.83) for some constant a > 0. Then suppose we make a measurement at time ∆t. A single quantum system is projected back into the initial state with probability P(∆t), and an ensemble of identically prepared systems will have fraction P(∆t) in the initial state after the measurement. In either case, the system is in the same initial state, and then we can repeat the process with more measurements spaced at intervals of duration ∆t. After n such measurements, the survival probability is
1 −a(∆t)2 n
(11.84) Noting that the time is t = n∆t, we can write
(11.85) The decay rate in the presence of the measurements has been modified from a to a ∆t. As the frequency of the measurements increases, then ∆t −\rightarrow 0 and thus the decay rate is also reduced to zero. Thus, decay is almost completely inhibited for sufficiently frequent measurements. Over what time scale do we expect the exponential-decay law to be invalid? The relavent time scale is set by the emitted energy during the decay: \taunonexp ∼ h ∆E . (11.86) This sets the time scale over which the excited and ground states cannot be ‘‘resolved.’’ For an optical transition, this time scale is 2\pi/\omega0, or just the optical period. For visible-wavelength transitions, this time scale is only a couple of fs. This type of nonexponential decay is thus very difficult to observe (and has not been observed thus far), since optical detectors are typically far too slow. 4B. Misra and E. C. G. Sudarshan, ‘‘The Zeno’s paradox in quantum theory,’’ Journal of Mathematical Physics 18, 756 (1977) (doi: 10.1063/1.523304); C. B. Chiu, E. C. G. Sudarshan, and B. Misra, ‘‘Time evolution of unstable quantum states and a resolution of Zeno’s paradox,’’ Physical Review D 16, 520 (1977) (doi: 10.1103/PhysRevD.16.520).
11.7.2 Long Times¶
Chapter 11. Spontaneous Emission 11.7.2 Long Times For long times, the decay is also nonexponential. A very general argument5 appeals to the Paley–Wiener theorem, which states that if the frequency spectrum of c(t) cuts off below some minimum frequency \omegamin (corresponding to a lower energy bound), then c(t) must satisfy Z \infty −\infty dt log |c(t)|
1 + t2 < \infty. (11.87) In particular, for the integral to converge, we must have log |c(t)| ∼tq (11.88) for q < 1 at large times, in which case
(11.89) where \alpha > 0. Thus, the probability must decay more slowly than exponential at late times. In particular, for the two-level atom, an extension of the Weisskopf–Wigner calculation6 shows that at late times, the decay goes as P(t) ∼ \Gamma 2\pi\omega 3 2 1 t4 , (11.90) (long-time nonexponential decay) once the exponential part of the decay has damped away. (We will defer this calculation until Section 15.5.4.) Since \Gamma/\omega0 ≪1 for optical transitions, the correction is very small. We can estimate the crossover time by setting \Gamma 2\pi\omega 3 2 1 t4 ∼e−\Gammat, (11.91) which gives \Gammat ∼130 for a typical ratio \Gamma/\omega0 \approx 1.2 \times 10−8. The correction here is very small. 5L. A. Khalfin, op. cit.; L. Fonda, et al., op. cit. 6P. L. Knight and P. W. Milonni, ‘‘Long-Time Deviations from Exponential Decay in Atomic Spontaneous Emission Theory,’’ Physics Letters 56A 275 (1976) (doi: 10.1016/0375-9601(76)90306-6).
11.8 Exercises 11.8 Exercises Problem 11.1 The Lamb–Dicke effect7 occurs as a narrowing of the radiation spectrum of an atom if it is confined to very small volumes (as can happen for a trapped ion or a neutral atom in an optical lattice). This effect is closely related to the Mössbauer effect8 for scattering from atoms bound in solids. In this problem you will work out a fairly simple model for this effect. (a) Assume that the center-of-mass component of the particle is described by the Hamiltonian HCM = p2 2m + V (r), (11.92) where V (r) is the trapping potential. To model the emission process, assume the two-level atom is initially excited, and the field is initially in the vacuum state. Then the emission process for a photon into the (k, \zeta) mode is the transition
(11.93) where the |\psin\rangle are the energy eigenstates of HCM with eigenvalues En, and we are assuming that the vibrational state of the atom changes from n −\rightarrow l during the transition. (We will always assume the vibrational energies are small compared to the optical-transition energy.) Assume that the field is quantized in free space, and show that the corresponding transition amplitude is
r ¯h\omegak
(11.94) where |i\rangle and |f\rangle are the initial and final states, respectively. (b) What are the different possible frequencies in the radiation spectrum? (c) From the above arguments, the strength of the n −\rightarrow l transition is proportional to the squared transition amplitude Snl :=
2 . (11.95) Show that the sum of the line strengths for all possible transitions from the initial state is independent of position. In doing so, you have shown that the total decay rate is independent of V (r), and thus equal to the free-space value. (d) Show that if the atomic wave packet is confined to a region much smaller than \lambda0 in each direction, where \lambda0 is the resonance wavelength, that the dominant emission line is for the n −\rightarrow n vibrational transition. Problem 11.2 By modifying the Weisskopf–Wigner derivation of the spontaneous emission rate, derive an expression for the spontaneous decay rate for a spherically symmetric atom a distance z from a perfect, planar mirror. To do this, you will need the half-space mode functions from Section 8.4.3. Also, be careful when taking the continuum limit to obtain the correct prefactor for the integral over k-space. Problem 11.3 An optical analogue to the quantum Zeno effect occurs in propagation through ideal polarizers. A photon can be regarded as a two-state quantum system, with state |\psi\rangle = cV|V\rangle + cH|H\rangle , where |V\rangle 7R. H. Dicke, ‘‘The Effect of Collisions upon the Doppler width of Spectral Lines,’’ Physical Review 89, 472 (1953) (doi: 10.1103/PhysRev.89.472). 8Rudolf L. Mössbauer, ‘‘Kernresonanzfluoreszenz von Gammastrahlung in Ir191, Zeitschrift für Physik 151, 124 (1958) (doi: 10.1007/BF01344210).
Chapter 11. Spontaneous Emission indicates vertical (linear) polarization, and |H\rangle indicates horizontal polarization. With idealized po- larizers, we can model the action of a vertical polarizer via the projector |V\rangle \langle V|, and the action of a horizontal polarizer by |H\rangle \langle H|. Note that the probability for a photon to transmit through two crossed polarizers is zero. However, there is a nonzero transmission probability if, for example, another polarizer is inserted between the crossed polarizers, but oriented at 45◦with respect to them. Derive an expression for the transmission probability if N such polarizers are inserted between the crossed polarizers, with the nth polarizer making an angle of n\pi/2(N + 1) radians with respect to the frontmost polarizer. (That is, the intermediate polarizers uniformly and gradually sweep the polarization through 90◦.) Your expression should assume that the photon transmitted through the frontmost polarizer (i.e., the first of the crossed polarizers). Show that your expression converges to unity as N −\rightarrow \infty. This can be interpreted as ‘‘dragging’’ the polarization through 90◦by making many, slightly different measurements. Problem 11.4 In this problem, you will work out the theory of spontaneous emission in the presence of a thermal electromagnetic field at temperature T. (a) Use the formalism for the general Born–Markov master equation in Section 4.5, and apply it to the atom–field interaction in the rotating-wave approximation (as in the Weisskopf–Wigner derivation) to derive the master equation
(11.96) for an atom interacting with the electromagnetic field at temperature T, where
(11.97) represents the energy shift of the atomic transition (after moving the shifted ground state to zero energy), with the divergent, temperature-independent Lamb shift ∆0 = \Gamma 2\pi\omega 3 – Z \infty d\omega \omega3
(11.98) and the temperature-dependent shift
\Gamma \pi\omega 3 – Z \infty
(11.99) Here, ¯n(\omega) is the mean photon number for a mode with frequency \omega. (b) Give an interpretation for all the terms in the master equation (write out equations for the density- matrix elements if you need to). (c) Write down an expression for ¯n(\omega) (nothing fancy, just basic statistical mechanics and Boltzmann statistics). (d) Argue that ∆(T) scales as T 4 at low temperatures. (e) How do the decay terms, ∆0, and ∆(T) change if you do not make the rotating-wave approximation? Give a qualitative explanation for your results, and indicate whether the low-temperature scaling of ∆(T) has changed. (f) Estimate the temperature-dependent shift for the 780 nm transition in 87Rb (making the crude approximation of treating it as a two-level atom), at room temperature.
11.8 Exercises Problem 11.5 In the Weisskopf–Wigner derivation of the spontaneous emission rate that we covered in this chapter, we considered the interaction of an atom with the vacuum field, but there was no mention of the Lamb shift, which should also arise from this interaction (as in Problem 11.4). Pinpoint the exact step in the derivation where the Lamb shift disappeared. Explain. Also explain how to modify the derivation to obtain an expression for the Lamb shift. Problem 11.6 (a) A spherically symmetric atom is located halfway between two perfectly reflecting, infinite planar mirrors a distance L apart as shown. L Derive an expression for the decay rate in the limit of small L, such that the cavity acts as a two- dimensional optical waveguide. (Write your answer in terms of the free-space decay rate \Gamma.) Obviously, any physically correct treatment of this problem must involve three dimensions, so part of the problem is to figure out what ‘‘two-dimensional’’ means in this context. (b) Derive an expression for the decay rate if the atom is at the center of a long, perfectly reflecting, rectangular cavity as shown, in the regime where the cavity acts as a one-dimensional optical waveguide. Lx Ly Again, part of the problem is to figure out what ‘‘one-dimensional’’ means in this context. Problem 11.7 Derive an expression for the spontaneous-emission rate (Einstein A coefficient) for an atom located a distance a from one of a pair of parallel, infinite, perfectly conducting plates separated by a distance L. Use whatever (quantum-mechanical) formalism you like. Write your result in terms of the free-space rate \Gamma0, and plot the resulting rate as a function of L for the case a = L/2. Problem 11.8 (a) Compute the decay rate due to the magnetic-dipole transition for the 6.8 GHz ground-state hyper-
You should proceed by mapping the magnetic-dipole Hamiltonian HAF = −µ\cdotB onto the electric-dipole Hamiltonian, then adapt the spontaneous-emission results from this chapter as appropriate to obtain the magnetic-dipole decay rate
0 µ2 ge 3\piϵ0¯hc5 . (11.100) For the relevant matrix element of the magnetic-dipole moment, use µB as an estimate, or better yet,
(b) How is the spontaneous-emission rate modified given that the surrounding environment is at room temperature (298 K)? Use the results of Problem 11.4 to help in this calculation. Problem 11.9 Consider an atom coupled to a single mode of an optical cavity with coupling coefficient g. Assume the intensity-decay of the (empty) cavity is exponential with rate \kappa. Ignore coupling to other (non-cavity) modes, and do not assume that the atomic and cavity resonances necessarily coincide. Use Fermi’s
Chapter 11. Spontaneous Emission Golden Rule to derive an expression for the decay rate of the atom in the ‘‘bad-cavity’’ limit \kappa ≫g. Hint: in this limit you can treat the atomic decay directly to continuum, where the cavity has modified the density of vacuum states.