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d„ ei(ωL ’ω)„ σ + („ + tr ) σ ’ (tr ) .
S (ω, tr ) = R (14.260)

Once again, we can use the ¬‚uctuation operators de¬ned by eqn (14.253) to split the
spectral density into a coherent contribution, due to oscillations driven by the external
Resonance ¬‚uorescence— ¿

laser ¬eld, and an incoherent contribution, due to quantum noise. Thus S (ω, tr ) =
Scoh (ω, tr ) + Sinc (ω, tr ), where

d„ ei(ωL ’ω)„ σ + („ + tr ) σ ’ (tr )
Scoh (ω, tr ) = R (14.261)

and
d„ ei(ωL ’ω)„ δσ + („ + tr ) δσ ’ (tr ) .
Sinc (ω, tr ) = R (14.262)

The assumption that tr is much larger than the atomic decay times means that
σ + („ + tr ) and σ ’ (tr ) are respectively given by the asymptotic steady-state values

σ + ss and σ + ss from eqn (14.243); consequently, the coherent contribution is

d„ ei(ωL ’ω)„ = 2πR | σ +
2 2
Scoh (ω, tr ) = R | σ + ss | ss | δ (ω ’ ωL ) . (14.263)

The ¬rst step in the calculation of the incoherent contribution is to write eqn
(14.262) as

Sinc (ω, tr ) = R d„ ei∆„ δσ + („ + tr ) δσ ’ (tr )
0
0
d„ ei∆„ δσ + („ + tr ) δσ ’ (tr ) ,
+R (14.264)
’∞

where ∆ = ωL ’ ω. In the second integral, one can change „ ’ ’„ and use time-
translation invariance to get

δσ + (’„ + tr ) δσ ’ (tr ) = δσ + (tr ) δσ ’ („ + tr )

= δσ + („ + tr ) δσ ’ (tr ) , (14.265)

so that

Sinc (ω, tr ) = 2R Re d„ ei∆„ δσ + („ + tr ) δσ ’ (tr ) . (14.266)
0

The correlation function in the integrand is one component of the matrix

(», µ = +, z, ’) ,
F»µ („, tr ) = δσ » („ + tr ) δσ µ (tr ) (14.267)

so

Sinc (ω, tr ) = 2R Re d„ ei∆„ F+’ („, tr )
0

d„ e’( ’i∆)„
= 2R lim Re F+’ („, tr )
’0+ 0
= 2R lim F+’ ( ’ i∆, tr ) , (14.268)
’0+


where F+’ (ζ, tr ) is the Laplace transform of F+’ („, tr ) with respect to „ .
Quantum noise and dissipation

The evaluation of the Laplace transform is accomplished with the techniques used
to prove the quantum regression theorem. We begin by subtracting eqns (14.240)
and (14.241) from eqns (14.238) and (14.239), to get the equations of motion for the
¬‚uctuation operators. By including the equation for δσ ’ (t)”the conjugate of eqn
(14.238)”the equations can be written in matrix form as

d
δσ » (t) = V»µ δσ µ (t) + ξ» (t) , (14.269)
dt µ


⎡ ¤
where
’ (“ ’ iδ) ’i„¦— 0
L
⎣ ’2i„¦L 2i„¦— ¦ .
’w
V= (14.270)
L
’ (“ + iδ)
0 i„¦L
After di¬erentiating eqn (14.269) with respect to „ , with tr ¬xed, and using eqn
(14.269) one ¬nds

F»µ („, tr ) = V»ν Fνµ („, tr ) , (14.271)
‚„ ν

where we have used the nonanticipating property ξ» („ + tr ) δσ µ (tr ) = 0 for „ > 0.
The Laplace transform technique for initial value problems”explained in Appendix
A.5”turns these di¬erential equations into the algebraic equations

ζ F»µ (ζ, tr ) ’ V»ν Fνµ (ζ, tr ) = F»µ (0, tr ) , (14.272)
ν


which determine the matrix F»µ (ζ, tr ). Since tr is large, the initial values F»µ (0, tr )
de¬ned by eqn (14.267) are given by the steady-state average

F»µ (0, tr ) = δσ » δσ µ ss
’ σ»
= σ»σµ σµ . (14.273)
ss
ss ss

The product of two Pauli matrices can always be reduced to an expression linear in
the Pauli matrices, so the initial values are determined by eqns (14.242) and (14.243).
The evaluation of the incoherent part of the spectral density by eqn (14.268) only
requires F+’ (’i∆, tr ), which is readily obtained by applying Cramers rule to eqn
(14.272) to ¬nd
N+’ (∆)
≡ (∆) .
F+’ (’i∆, tr ) = (14.274)
D (∆)
The numerator is a linear function of the initial values:

N+’ (∆) = ’2i„¦—2 F’’ (0, tr ) + i„¦— (∆ + i“) Fz’ (0, tr )
L L
2
+ i ∆2 ’ 2 |„¦L | ’ “w + i (“ + w) ∆ F+’ (0, tr ) , (14.275)

and the denominator is the product of three factors: D (∆) = D0 (∆) D+ (∆) D’ (∆),
where D0 (∆) = ∆ + i“,
Resonance ¬‚uorescence—

“+w
D± (∆) = ∆ ± 2„¦L + i , (14.276)
2
and
2
“’w
2
|„¦L | ’
„¦L = . (14.277)
4
The factorization of the denominator suggests using the method of partial fractions to
express (∆) as
C (∆) C (∆) C (∆)
(∆) = + + , (14.278)
D0 (∆) D+ (∆) D’ (∆)
with
N+’ (∆)
C (∆) = . (14.279)
D0 (∆) [D+ (∆) + D’ (∆)] + D+ (∆) D’ (∆)
The functions D0 (∆) and D± (∆) have zeroes at ∆0 = ’i“ and ∆± = “ 2„¦L ’
i (“ + w) /2 respectively, so (∆) has three poles in the lower-half ∆-plane. If the
laser ¬eld is weak, in the sense that
2
“’w
2
|„¦L | < , (14.280)
4
then eqn (14.277) shows that „¦L is pure imaginary. All three poles are then located
on the negative imaginary axis, so that Re (∆) will have a single peak at ∆ = 0,
on the real ∆-axis. For a strong laser, „¦L is real, and the poles at ∆± are displaced
away from the imaginary axis. In this case, Re (∆) will exhibit three peaks on the
real ∆-axis, at ∆+ = ’2„¦L , ∆0 = 0, and ∆’ = 2„¦L .
An explicit evaluation of eqn (14.278) can be carried out in the general case, but the
resulting expressions are too cumbersome to be of much use. One then has the choice
of studying the behavior of the spectral density numerically, or making simpli¬cations
to produce a manageable analytic result. We will leave the numerical study to the
exercises and impose three simplifying assumptions. The ¬rst is that the laser is exactly
on resonance with the atomic transition (δ = 0), and the second is that the laser ¬eld
is strong (|„¦L | “, w). The third simpli¬cation is to evaluate the numerator C (∆) at
the location of the pole in each of the three terms. This procedure will give an accurate
picture of the behavior of the function Sinc (ω, tr ) in the vicinity of the peaks, but will
be slightly in error in the regions between them. With these assumptions in place, one
¬nds
(+) (0) (’)
Sinc (ω, tr ) = Sinc (ω, tr ) + Sinc (ω, tr ) + Sinc (ω, tr ) , (14.281)
R “
(0)
Sinc (ω, tr ) = , (14.282)
2 (ω ’ ωL )2 + “2
R “+w
(±)
Sinc (ω, tr ) = . (14.283)
8 (ω ’ ωL “ 2 |„¦L |)2 + (“ + w)2 /4
This clearly displays the three peaks of the Mollow triplet. The presence of the side
peaks is evidence of persistent Rabi oscillations that modulate the primary resonance
at ω = ωL . The heights and widths of the peaks are related by
Quantum noise and dissipation

central peak height w
= 1 + (= 3 for the pure radiative case) , (14.284)
side peak height “

side peak width 1 w 3
= 1+ = for the pure radiative case , (14.285)
central peak width 2 “ 2
where the pure radiative case occurs when spontaneous emission is the only decay
mechanism. In this situation eqn (14.151) yields w = 2“. These features have been
experimentally demonstrated.

14.10 Exercises
14.1 Sample“environment coupling
Consider a single reservoir, so that the index J in eqn (14.14) can be suppressed. The
general ansatz for an interaction, HSE , that is linear in both reservoir and sample
operators is
v („¦ν ) O† bν ’ v — („¦ν ) b† O ,
HSE = i ν
ν

where v („¦ν ) is a complex coupling constant. Show that there is a simple unitary
transformation, bν ’ bν , that allows the complex v („¦ν ) to be replaced by |v („¦ν )|.

Multiplicative noise for the radiation ¬eld—
14.2
(1) Derive the evolution equation

dN (t)
= ’κN (t) + χ (t)
dt
for the number operator, where χ (t) = ξ † (t) a (t) + a† (t) ξ (t) is a multiplicative
noise operator.
(2) Combine the nonanticipating property (14.77), the delta correlation property
(14.74), and the end-point rule (A.98) for delta functions to ¬nd χ (t) = n0 κ.
Is this result consistent with interpreting the evolution equation as a Langevin
equation?
(3) Rewrite the equation for N (t) in terms of the new noise operator ξN (t) = χ (t) ’
χ (t) , and then derive the result

N (t) = N (t0 ) e’κ(t’t0 ) + n0 1 ’ e’κ(t’t0 )

describing the relaxation of the average photon number to the equilibrium value
n0 .
(4) Use the explicit solution (14.65) for a (t) to show that

ξN (t) ξN (t ) = CN N (t) δ (t ’ t ) ,

where CN N (t) approaches a constant value for κt 1.
Exercises

14.3 Approach to thermal equilibrium
The constant κ in eqn (14.61) represents the rate at which ¬eld energy is lost to the
walls, so it should be possible to recover the blackbody distribution for radiation in a
cavity with walls at temperature T . For this purpose, enlarge the sample to include
all the modes (ks) of the radiation ¬eld; but keep things simple by assuming that all
modes are coupled to a single reservoir with the same value of κ.
(1) Generalize the single-mode treatment by writing down the Langevin equation for
aks . Give the expression for the noise operator, ξk (t), and show that

ξk (t) ξk (t ) = κn (ωk ) δ (t ’ t ) ,

where n (ωk ) is the average number of reservoir excitations at the mode frequency
ωk .
(2) Apply the result in part (3) of Exercise 14.2 to ¬nd limt’∞ Nks (t) = n (ωk ).
What is the physical meaning of the limit t ’ ∞?
(3) Finally, use the general result (2.177) to argue that the photon distribution in the
cavity asymptotically relaxes to a blackbody distribution.

14.4 Noise operators for the two-level atom
By following the derivation of the Langevin equations (14.148)“(14.150) show that the
noise operators are

ξ22 (t) = ’ w21 b† (t) S 12 (t) + HC = ’ξ11 (t) ,
in


√ √

w22 c† (t) ’ w11 c† (t) S 12 (t)
ξ12 (t) = S 22 (t) ’ S 11 (t) w21 bin (t) + 2,in 1,in
√ √
’ S 12 (t) { w22 c2,in (t) ’ w11 c1,in (t)} .
Inverted-oscillator reservoir—
14.5
A gain medium enclosed in a resonant cavity has been modeled (Gardiner, 1991, Sec.
7.2.1) by the interaction of the cavity mode a (t) of Section 14.3 with an inverted-
oscillator reservoir described by the Hamiltonian

d„¦
„¦c† c„¦ ,
HIO = ’ „¦

0

where c„¦ , c† = 2πδ („¦ ’ „¦ ).
„¦
(1) Express the energy-raising and energy-lowering operators for the reservoir in terms
of c„¦ and c† .
„¦
(2) In addition to the two terms in eqn (14.88), the interaction Hamiltonian HSE now
has a third term, HS,IO describing the interaction with the inverted oscillators. In
the resonant wave approximation, show that HS,IO must have the form

d„¦
χ („¦) c„¦ a ’ a† c† .
HS,IO = i „¦

0
Quantum noise and dissipation

(3) Using the discussion in Section 14.3 as a guide, derive the Langevin equation
d 1
a (t) = (g ’ κC ) a (t) + ξ (t) ,
dt 2
and give expressions for the gain g and the noise operator ξ (t).
14.6 Langevin equations for incoherent pumping
Use the (N = 3)-case of eqns (14.159) and (14.160), without the phase-changing terms,
to derive the full set of Langevin equations for the three-level atom of Fig. 14.2.
14.7 Bloch equations for incoherent pumping
Consider the case of pure pumping, i.e. HS1 = 0.
(1) Derive the c-number Bloch equations by averaging eqns (14.174)“(14.177).
(2) Find the steady-state solutions for the populations.
14.8 Noise correlation coe¬cients
Consider the reduced Langevin equations (14.174)“(14.177), with HS1 = 0.
(1) How many independent coe¬cients Cqp,lk (q, p, k, l = 1, 2, 3) are there?
(2) Use the Einstein relation and the steady-state populations to calculate the inde-
pendent coe¬cients in the limit w32 ’ ∞.
Generalized transition operators—
14.9
The two important simplifying assumptions HSS ∼ 0 and eqn (14.182) were made
for the sole purpose of ensuring the linearity of the Heisenberg equations of motion,
which is essential for the relatively simple arguments establishing the nonanticipating
property (14.192) and the quantum regression theorem (14.209). Both of these as-
sumptions can be eliminated by a special choice of the sample operators. To this end,
de¬ne the stationary states, |¦A , of the full sample Hamiltonian HS = HS0 + HSS by
HS |¦A = µA |¦A , and for simplicity™s sake assume that A is a discrete label.
(1) Explain why the use of the |¦A s renders the assumption HSS ∼ 0 unnecessary.
(2) Show that the generalized transition operators SAB = |¦A ¦B | satisfy the
following:
(a) [SAB , HS ] = ’ ωAB SAB , with ωAB = (µA ’ µB ) / ;
(b) SAB SCD = δBC SAD ;
(c) [SAB , SCD ] = δBC SAD ’ δAD SCB ;
(d) X = A B ¦A |X| ¦B SAB , for any sample operator X.
(3) For an external ¬eld acting on the sample through VS (t), derive eqn (14.182) by
showing that
1
S AB (t) , VS (t) = i „¦AB,CD (t) S CD (t) .
i
CD

Give the explicit expression for „¦AB,CD (t) in terms of the matrix elements of
VS (t).
Exercises

Mollow triplet—
14.10
Use eqn (14.268) for a numerical evaluation of Sinc /R as a function of ∆/“. Assume
resonance (δ = 0) √ pure radiative decay (w = 2“), and consider two cases: |„¦L | =
and
5“ and |„¦L | = “/ 2. In each case, plot the numerical evaluation of eqn (14.278) and
the numerical evaluation of eqn (14.281) against ∆/“.
15
Nonclassical states of light

In Section 5.6.3 we de¬ned a classical state for a single mode of the electromagnetic
¬eld by the requirement that the Glauber“Sudarshan P (±)-function is everywhere
non-negative. When this condition is satis¬ed P (±) may be regarded as a probability
distribution for the classical ¬eld amplitude ±. Advances in experimental techniques
have resulted in the controlled generation of nonclassical states of the ¬eld, for which
P (±) is not a true probability density. In this chapter, we study the nonclassical states
that have received the most attention: squeezed states and number states.

15.1 Squeezed states
In the correspondence-principle limit, a coherent state of light approaches a noiseless
classical electromagnetic ¬eld as closely as allowed by the uncertainty principle for the
radiation oscillators. This might lead one to expect that a coherent state would describe
a light beam with the minimum possible quantum noise. On theoretical grounds, it
has long been known that this is not the case, and in recent years states with noise
levels below the standard quantum limit”known as squeezed states”have been
demonstrated experimentally.

15.1.1 Squeezed states for a radiation oscillator
As an introduction to the ideas involved, let us begin by considering a single ¬eld mode
which is described by the operators q and p for the corresponding radiation oscillator.
In Section 5.1 we saw that the coherent states are minimum-uncertainty states, with

∆q0 = /2ω , ∆p0 = ω/2 , ∆q0 ∆p0 = /2 . (15.1)

The simplest example is the vacuum state, which is described, in the momentum
representation, by
P2
2 ’1/4
¦0 (P ) = 2π∆p0 exp . (15.2)
4∆p2 0
Suppose that the radiation oscillator is prepared in the initial state,

P2
2 ’1/4
exp ’
ψ (P, 0) = 2π∆p , (15.3)
4∆p2
which is called a squeezed vacuum state if ∆p < ∆p0 . This wave function cannot
be a stationary state of the oscillator; instead, it is a superposition over the whole
family of energy eigenstates:
½
Squeezed states


ψ (P, 0) = Cn ¦n (P ) , (15.4)
n=0

where ¦n is the nth excited state (H¦n = n ω¦n ), and we have, as usual, subtracted
the zero-point energy. The excited state ¦n (P ) is an n-photon state, so we have
reached the paradoxical sounding conclusion that the squeezed vacuum contains many
photons.
The energy eigenvalues are n ω, so the initial state ψ (P, 0) evolves into

Cn ¦n (P ) e’inωt .
ψ (P, t) = (15.5)
n=0

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