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 = ’ „¦

2π

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 „¦

2π

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