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Lectures on LightNonlinear and Quantum Optics using the Density Matrix$

Stephen Rand

Print publication date: 2010

Print ISBN-13: 9780199574872

Published to Oxford Scholarship Online: September 2010

DOI: 10.1093/acprof:oso/9780199574872.001.0001

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Appendix F Frequency Shifts and Decay due to Reservoir Coupling

Appendix F Frequency Shifts and Decay due to Reservoir Coupling

Source:
Lectures on Light
Publisher:
Oxford University Press

Appendix F

Frequency Shifts and Decay due to Reservoir Coupling

A single parameter such as the decay constant γ introduced in Chapter 3 is typically thought to account for a single process, such as the rate of population decay in two‐ level atoms. However, here we show on the basis of perturbation theory that as many as four fundamentally different physical processes are linked to the decay constants in our description of atomic dynamics.

In Section 6.3 an exponential form was derived for the population decay of a two‐level system via spontaneous emission. This justified the assumed form of phenomenological decay terms in Chapter 3, where perturbation theory was used to solve for the probability amplitudes of a two‐level system undergoing radiative decay after being prepared in the excited state.

C 2 ( 1 ) ( t ) = exp ( γ 2 t / 2 ) ,
(F.1)

for times short compared to lifetime γ 1 1 but long compared to field periods. The process of emission from ǀ2> to ǀ1> also excites one of many radiation modes of the electromagnetic field. These modes are labeled by index λ. Then, setting (Ω21)λ = μ21 Eλ/ħ in the rate equations of Chapter 3, one finds

( C ˙ 1 ) λ = λ i 2 ( Ω 21 ) λ exp ( i [ ω 0 ω λ ] t ) ( C 2 ) λ ,
(F.2)
( C ˙ 2 ) λ = λ i 2 ( Ω 21 ) λ exp ( i [ ω 0 ω λ ] t ) ( C 1 ) λ
(F.3)

by ignoring the effect of decay at short times. Substitution of Eq. (F.1) into Eq. (F.2) then yields

( C ˙ 1 ) λ = λ i 2 ( Ω 21 ) λ exp ( i [ ω 0 ω λ ] t γ 2 t / 2 ) .
(F.4)

(p.279) This is easily solved to give

( C 1 ( 1 ) ) λ ( t ) = λ i 2 ( Ω 21 ) λ [ exp ( i [ ω 0 ω λ ] t γ 2 t / 2 ) 1 i [ ω 0 ω λ ] 1 2 ( γ 2 γ 1 ) ] .
(F.5)

Now we use the first order perturbation results to extract a general form of γ2. Upon substitution of Eqs. (F.1) and (F.5) into Eq. (F.3), we find

γ 2 2 exp ( γ 2 t / 2 ) = λ ( i 2 ) 2 | ( Ω 12 ) λ | 2 [ exp ( γ 2 t / 2 ) exp ( i [ ω 0 ω λ ] t ) i [ ω 0 ω λ ] 1 2 ( γ 2 γ 1 ) ]

or

γ 2 = 1 2 λ | ( Ω 12 ) λ | 2 [ 1 exp ( i [ ω 0 ω λ ] t + γ 2 t / 2 ) i [ ω 0 ω λ ] 1 2 ( γ 2 γ 1 ) ] .
(F.6)

Replacing the summation over modes with an integral over the density of modes D(ω) = Vω22 c 3 available per unit frequency interval according to

λ V ρ ( ω ) d ω ,
(F.7)

Eq. (F.6) becomes

γ 2 = 1 2 | Ω 12 ( ω ) | 2 [ 1 exp ( i [ ω 0 ω ] t + γ 2 t / 2 ) i [ ω 0 ω ] 1 2 ( γ 2 γ 1 ) ] D ( ω ) d ω .
(F.8)

For convenience we define the factor

f ( ω ) 1 2 | Ω 12 ( ω ) | 2 D ( ω ) ,
(F.9)

in terms of which our general expression for γ2 becomes

γ 2 = i ω f ( ω ) [ 1 exp ( i [ ω 0 ω ] t + γ 2 t / 2 ) [ ω 0 ω ] i 2 ( γ 2 γ 1 ) ] d ω .
(F.10)

If (γ2 − γ1) << ω the denominator in Eq. (F.10) simplifies to ~ [ω0 − ω]−1, and at short times the portion of the integrand in square brackets simplifies to

L i m t [ 1 exp ( i [ ω 0 ω ] t + γ 2 t / 2 ) [ ω 0 ω ] i 2 ( γ 2 γ 1 ) ] [ 1 cos ( ω 0 ω ) t ( ω 0 ω ) ] + i [ sin ( ω 0 ω ) t ( ω 0 ω ) ] = P { 1 ( ω 0 ω ) } + i π δ ( ω 0 ω ) ,
(F.11)

where P denotes the principal part. With this simplification, Eq. (F.10) becomes

γ 2 ω { i P [ 1 ω 0 ω ] π δ ( ω 0 ω ) } f ( ω ) d ω .
(F.12)

(p.280) This development shows (with only first‐order perturbation theory) that finite lifetime, and the coupling to radiation modes that causes it, leads to a separation of γ b into real and imaginary parts. Thus,

γ 2 = γ r + i γ i m ,
(F.13)

where the associated expressions are given by

γ i m 1 2 P { | Ω 12 ( ω ) | 2 ω 0 ω D ( ω ) d ω } ,
(F.14)
γ r π 2 D ( ω 0 ) | Ω 12 ( ω 0 ) | 2 .
(F.15)

The decay processes in Eqs. (F.14) and (F.15) are both induced by real optical fields, as is apparent by their mutual dependencies on ǀΩ12ǀ2. Now we are in a position to gain some perspective on the purely quantum mechanical effects that result from quantization of the radiation field by taking into account that the total electric field includes the vacuum state which does not depend on mode occupation number n λ, and a part which does depend on n λ. By recognizing that according to Sections 6.2 and 6.3 fluctuations of the vacuum field can induce spontaneous decay in much the same manner that the photon field induces stimulated relaxation, we see that the imaginary and real rate constants above split into two additional parts corresponding to spontaneous and stimulated processes. That is,

γ 2 = γ r ( spont ) + γ r ( induced ) + i γ i m ( spont ) + i γ s m ( induced ) .
(F.16)

The physical effects of relaxation are reflected in the frequency and time dependence of the emitted radiation. That is, the field leaving the atom has the form

E exp ( i ω 0 t ) · exp ( γ 2 t ) = exp [ i ( ω 0 + γ i m ( spont ) + γ i m ( induced ) ) ] · exp [ ( γ r ( spont ) + γ r ( induced ) ) t ] .
(F.17)

As a result of atom-field coupling four important processes take place, namely spontaneous decay, stimulated decay, the Lamb shift, and the AC Stark shift. This is illustrated in Fig. F.1.

Appendix F Frequency Shifts and Decay due to Reservoir Coupling

Figure F.1 Four fundamental, distinguishable physical processes that are intimately linked to field‐driven decay of atoms