K

where X (t ) and Y (t ) are sample operators that depend on O J (t ) for t < t <

t. Equations (14.209) and (14.210) are special cases of the quantum regression

theorem ¬rst proved by Lax (1963). We will study the general version in Chapter 18.

Quantum noise and dissipation

Photon bunching—

14.8

We mentioned in Section 10.1.1 that the Hanbury Brown“Twiss e¬ect can be measured

by coincidence counting. As explained in Section 9.2.4, the coincidence-count rate is

proportional to the second-order correlation function

G(2) (r , t , r, t; r , t , r, t) = E (’) (r , t ) E (’) (r, t) E (+) (r, t) E (+) (r , t ) , (14.211)

where r and r are the locations of the detectors, t = t + „ , and the ¬elds are all

projected on a common polarization vector. By placing suitable ¬lters in front of the

detectors, we can con¬ne our attention to a single mode, so that G(2) is proportional

to the correlation function

C („ ) = a† (t + „ ) a† (t) a (t) a (t + „ ) = a† (t) N (t + „ ) a (t) , (14.212)

where N (t) = a† (t) a (t) is the mode number operator in the Heisenberg picture.

The quantum regression theorem can be applied to the evaluation of C („ ) by using

the Langevin equation for a (t) to derive the di¬erential equation

d N (t)

= ’κ N (t) + ξ † (t) a (t) + a† (t) ξ (t) (14.213)

dt

for the average photon number. It is shown in Exercise 14.2 that

ξ † (t) a (t) + a† (t) ξ (t) = n0 κ , (14.214)

so that the equation for N (t) can be rewritten as

d δN (t)

= ’κ δN (t) , (14.215)

dt

where δN (t) = N (t) ’ n0 . The solution,

δN (t) = e’κ(t’t0 ) δN (t0 ) , (14.216)

of this equation is a special case of the linear regression equation (14.206), with the

Green function G („ ) = exp (’κ„ ). According to the quantum regression theorem

(14.210), the correlation function a† (t) δN (t + „ ) a (t) obeys the same regression

law, so

a† (t) δN (t + „ ) a (t) = e’κ„ a† (t) δN (t) a (t) , (14.217)

and

C („ ) = e’κ„ a†2 (t) a2 (t) + 1 ’ e’κ„ n0 N (t) . (14.218)

For large times, κ (t ’ t0 ) 1, eqn (14.216) shows that N (t) ≈ n0 . The remaining

†2 2

expectation value a (t) a (t) can be calculated by using the solution (14.65) for

Resonance ¬‚uorescence—

a (t). In the same large-time limit, the initial-value term in eqn (14.65) can be dropped

to get the asymptotic result

⎡ ¤

4

t t

κ

dt4 exp ⎣’ (t ’ tj )¦ ξ † (t1 ) ξ † (t2 ) ξ (t3 ) ξ (t4 ) .

a†2 (t) a2 (t) = dt1 · · ·

2 j=1

t0 t0

(14.219)

For a thermal noise distribution,

ρE = exp ’β „¦ν N ν , (14.220)

ν

the discussion in Section 14.2.2 shows that

ξ † (t1 ) ξ † (t2 ) ξ (t3 ) ξ (t4 ) = (n0 κ) {δ (t1 ’ t3 ) δ (t2 ’ t4 ) + δ (t1 ’ t4 ) δ (t2 ’ t3 )} .

2

(14.221)

Substituting this result into eqn (14.219) and carrying out the integrals yields

a†2 (t) a2 (t) = 2n2 for κ (t ’ t0 ) 1. (14.222)

0

The correlation function C („ ) is then given by

C („ ) = n2 1 + e’κ„ , (14.223)

0

which shows that the coincidence rate is largest at „ = 0. In other words, photon

detections are more likely to occur at small rather than large time separations, as

shown explicitly by eqn (14.223) which yields

C (0) = 2C (∞) . (14.224)

This e¬ect is called photon bunching; it represents the quantum aspect of the Han-

bury Brown“Twiss e¬ect. For a contrasting situation, consider an experiment in which

the thermal light is replaced by light from a laser operated well above threshold. There

are no cavity walls and consequently no external reservoir, so the operator a (t) evolves

freely as a exp (’iω0 t). The density operator for the ¬eld is a coherent state |± ±|,

so that

C („ ) = ± a†2 a2 ± = |±|4 . (14.225)

In this case, the coincidence rate is independent of the delay time „ ; photon bunching

is completely absent.

Resonance ¬‚uorescence—

14.9

When an atom is exposed to a strong, plane-wave ¬eld that is nearly resonant with

an atomic transition, some of the incident light will be inelastically scattered into

all directions. This phenomenon, which is called resonance ¬‚uorescence, has been

studied experimentally and theoretically for over a century. Early experiments (Wood,

Quantum noise and dissipation

1904, 1912; Dunoyer, 1912) provided support for Bohr™s model of the atom, and af-

ter the advent of a quantum theory for light the e¬ects were explained theoretically

(Weisskopf, 1931).

In the ideal case of scattering from an isolated atom at rest, the theory predicts

(Mollow, 1969) a three-peaked spectrum (the Mollow triplet) for the scattered radi-

ation. After the invention of the laser and the development of atomic beam techniques,

it became possible to approximate this ideal situation. The ¬rst experimental veri¬ca-

tions of the Mollow triplet were obtained by crossing an atomic beam with a laser beam

at right angles, and observing the resulting ¬‚uorescent emission (Schuda et al., 1974;

Wu et al., 1975; Hartig et al., 1976). This experimental technique was later re¬ned by

reducing the atomic beam current”so that at most one atom is in the interaction re-

gion at any given time”and by employing counter-propagating laser beams to reduce

the Doppler broadening due to atomic motion transverse to the beam direction. These

improvements cannot, however, eliminate the transit broadening ∆ωtran ∼ 1/Ttran

caused by the ¬nite transit time Ttran for an atom crossing the laser beam. In more

recent experiments (Schubert et al., 1995; Stalgies et al., 1996) the ideal case is al-

most exactly realized by observing resonance ¬‚uorescence from a laser-cooled ion in

an electrodynamic trap.

In the interests of simplicity, we will only consider the case of resonance ¬‚uorescence

from a two-level atom. The previous discussion of Rabi oscillations, in Section 11.3.2,

neglected spontaneous emission, but a theory of resonance ¬‚uorescence must include

both the classical driving ¬eld and the quantized radiation ¬eld. This can be done by

using the result” obtained in Section 11.3.1”that the e¬ective Hamiltonian is the

sum of the semiclassical Hamiltonian for the atom in the presence of the laser ¬eld

and the radiation Hamiltonian describing the interaction with the quantized radiation

¬eld. In the present case, this yields the e¬ective Schr¨dinger-picture Hamiltonian

o

HW = HS0 + VS (t) + HE + HSE , (14.226)

where

ω21

HS0 = σz , (14.227)

2

VS (t) = „¦L e’iωL t σ+ + „¦— eiωL t σ’ , (14.228)

L

ωk a† aks ,

HE = (14.229)

ks

ks

and

vks σ’ a† ’ σ+ aks .

HSE = i (14.230)

ks

ks

The explicit time dependence of VS (t) comes from the semiclassical treatment of the

laser ¬eld. Since we are dealing with a single atom, the location of the atom can be

chosen as the origin of coordinates.

Resonance ¬‚uorescence—

The quantity to be measured is the counting rate for photons of polarization e at

a detector located at r. According to eqn (9.33),

w(1) (t) = S G(1) (r, t; r, t)

= S Tr ρe— · E(’) (r, t) e · E(+) (r, t) , (14.231)

where S is the sensitivity factor for the detector, and the Heisenberg-picture density

operator,

ρW = ρatom |±0 ±0 | , (14.232)

is the product of the density operator for the coherent state |±0 describing the laser

¬eld and the initial density operator ρatom for the atom. Our ¬rst objective is to show

that the counting rate can be expressed in terms of atomic correlation functions.

14.9.1 The counting rate

The discussion in Section 11.3, in particular eqn (11.149), shows that the density oper-

ator ρ in eqn (14.232) is the vacuum state for the ¬‚uorescent modes; consequently, the

only di¬erence between the problem at hand and the spontaneous emission calculation

in Section 11.2.1 is the e¬ect of the laser ¬eld on the atom. Furthermore, the operator

aks (t) commutes with Hsc (t), so the atom“laser coupling does not change the form of

the Heisenberg equation for aks (t). Consequently, we can still use the formal solution

(11.51) and the argument contained in eqns (11.52)“(11.66). The new feature is that

the de¬nition (11.63) of the slowly-varying envelope operators for the atom must be

replaced by

σ ’ (t) = eiωL t σ’ (t) , (14.233)

in order to eliminate the explicit time dependence in VS (t). This is permissible, because

of the near-resonance assumption |δ| ω21 , where δ = ω21 ’ ωL is the detuning. For

a detector in the radiation zone, the counting rate is therefore given by

w(1) (t) = S TrW ρW e · Erad (r, t) e— · Erad (r, t) ,

(’) (+)

(14.234)

where

kL [(d— — r) — r] eikL r ’iωL t

2

(+)

σ ’ (t ’ r/c) ,

Erad (r, t) = e (14.235)

4π 0 r

and kL = ωL /c. Combining the last two equations gives us the desired result

w(1) (t) = R σ + (t ’ r/c) σ ’ (t ’ r/c) , (14.236)

S

where X = TrS (ρatom X), and the rate

S

2

S 2

kL

|(d— — r) — r|

2

R= 2 (14.237)

r 4π 0

carries all the information on the angular distribution of the radiation.

¼ Quantum noise and dissipation

14.9.2 Langevin equations for the atom

The result (14.236) has eliminated any direct reference to the radiation ¬eld; therefore,

we are free to treat the ¬‚uorescent ¬eld modes as a reservoir and the atom”under the

in¬‚uence of the laser ¬eld”as the sample. Elimination of the ¬eld operators by means

of the formal solution (11.51) and the Markov approximation yields the Langevin

equations

dσ + (t)

= ’ (“ ’ iδ) σ + (t) ’ i„¦— σ z (t) + ξ+ , (14.238)

L

dt

dσ z (t)

= ’w [1 + σ z (t)] + 2i„¦— σ ’ (t) ’ 2i„¦L σ + (t) + ξz , (14.239)

L

dt

where “ = “12 is the dipole dephasing rate, w = w21 is the spontaneous decay rate,

and the noise operators are de¬ned in Section 14.4.

We begin with the averaged Langevin equations,

d σ + (t)

= ’ (“ ’ iδ) σ + (t) ’ i„¦— σ z (t) , (14.240)

L

dt

d σ z (t)

= ’w [1 + σ z (t) ] + 2i„¦— σ ’ (t) ’ 2i„¦L σ + (t) , (14.241)

L

dt

and note that the averaged atomic operators approach steady-state values, σ + ss and

σ z ss , for times t max (1/“, 1/w). These values are determined by setting the time

derivatives to zero and solving the resulting algebraic equations, to get

1

=’

σz , (14.242)

ss 2

1 + |„¦L | /„¦2

sat

„¦—

= ’i L

σ+ σz , (14.243)

“ ’ iδ

ss ss

where

w (“2 + δ 2 )

„¦sat = (14.244)

4“

is the saturation value for the Rabi frequency. For |„¦L | „¦sat , σ z ss ≈ 0, which

means that the two levels are equally populated. In the same limit, one ¬nds

’ 0,

σ+ (14.245)

ss

i.e. the average dipole moment goes to zero for large laser intensities. This e¬ect is

2

called bleaching. The ratio |„¦L | /„¦2 is often expressed as

sat

2

|„¦L | IL

= , (14.246)

„¦2 Isat

sat

where IL is the laser intensity and

2

δ 2 + “2

3 0 cw

Isat = (14.247)

2

8“ |d|

is the saturation intensity.

Resonance ¬‚uorescence— ½

The fact that the population di¬erence σ z ss and the dipole moment σ + ss are

independent of time raises a question: What happened to the Rabi oscillations of the

atom? The answer is that they are still present, but concealed by the ensemble average

de¬ned by the initial density operator. This can be seen more explicitly by applying

the long-time averaging procedure

T

1

(» = +, z, ’)

σ» = lim dt σ » (t) (14.248)

∞ T ’∞ T 0

to eqns (14.240) and (14.241). It is easy to show that the average of the left side

vanishes in both equations, so that the time averages σ » ∞ satisfy the same equations

as the steady-state solutions σ » ss . Thus the steady-state solutions are equivalent to

a long-time average over the Rabi oscillations. This result is conceptually similar to

the famous ergodic theorem in statistical mechanics (Chandler, 1987, Chap. 3).

Since the distance r to the detector is ¬xed, we can use the retarded time tr = t’r/c

instead of t. With this understanding, the total number of counts in the interval

(tr0 , tr0 + T ) is

tr0 +T

N (T ) = R dtr σ + (tr ) σ ’ (tr ) , (14.249)

tr0

and the Pauli-matrix identity,

1

σ + (tr ) σ ’ (tr ) = [1 + σ z (tr )] = S22 (tr ) , (14.250)

2

allows this to be written in the equivalent form

tr0 +T

N (T ) = R dtr S22 (tr ) . (14.251)

tr0

For su¬ciently large tr0 the average in eqn (14.251) can be replaced by the stationary

value, so that

2

RT |„¦L | /„¦2 sat

N (T ) = RT S22 ss = . (14.252)

2

2 1 + |„¦L | /„¦2sat

This result tells us the total number of counts, but it does not distinguish between

the coherent contribution due to Rabi oscillations of the atomic dipole and the inco-

herent contribution arising from quantum noise, i.e. spontaneous emission. In order to

bring out this feature, we introduce the ¬‚uctuation operators

δσ z (tr ) = σ z (tr ) ’ σ z (tr ) , δσ ± (tr ) = σ ± (tr ) ’ σ ± (tr ) , (14.253)

and rewrite eqn (14.249) as

N (T ) = Ncoh (T ) + Ninc (T ) , (14.254)

with

tr0 +T

Ncoh (T ) = R dtr σ + (tr ) σ ’ (tr ) (14.255)

tr0

¾ Quantum noise and dissipation

and

tr0 +T

Ninc (T ) = R dtr δσ + (tr ) δσ ’ (tr ) . (14.256)

tr0

The coherent contribution is what one would predict from forced oscillations of a

classical dipole with magnitude | σ ’ (tr ) |, and the incoherent contribution depends

on the strength of the quantum ¬‚uctuation operators δσ + (tr ) and δσ ’ (tr ). In the

limit of large tr0 the coherent contribution is obtained by substituting the asymptotic

result (14.243) into eqn (14.256), with the result

|„¦L |2 /„¦2

w sat

Ncoh (T ) = RT . (14.257)

2

4“ 2

1 + |„¦L | /„¦2

sat

The incoherent contribution can be evaluated directly from eqn (14.256), but it is

easier to use eqns (14.252) and (14.254) to get

2

RT IL 1 ’ (w/2“) + |„¦L | /„¦sat

2

Ninc (T ) = . (14.258)

2

2 Isat 2

1 + |„¦L | /„¦sat

2

In the high intensity limit, the laser ¬eld should become more classical, and one might

expect that the coherent contribution would dominate the counting rate. Examination

of the results shows exactly the opposite; Ncoh (T ) ’ 0 and Ninc (T ) ’ RT /2. This

apparent paradox is resolved by the bleaching of the average dipole moment”shown in

eqn (14.245)”and the fact that half the atoms are in the excited state and consequently

available for spontaneous emission.

14.9.3 The ¬‚uorescence spectrum

Spectral data for ¬‚uorescent emission are acquired by using one of the narrowband

counting techniques discussed in Section 9.1.2-C. It is safe to assume that the ¬eld

correlation functions approximately satisfy time-translation invariance for times tr

much larger than the decay times for the sample; therefore, we can immediately use

the result (9.45) for the spectral density to get

d„ e’iω„ G(1) (r, „ + tr ; r, tr ) .

S (ω, tr ) = SG(1) (r, ω) = S (14.259)

Substituting the solution (14.235) for the radiation ¬eld into this expression yields