(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

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 = ’ „¦

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

By virtue of the equal spacing of the energy levels”a unique property of the harmonic

oscillator”the wave function is periodic, with period T = 2π/ω. This in turn implies

that the time-dependent width,

2

ψ (t) |P 2 | ψ (t) ’ ψ (t) |P | ψ (t) ,

∆p (t) = (15.6)

will exhibit the same periodicity. In other words, ψ (P, t) is a breathing Gaussian wave

packet which expands in size”as measured by ∆p (t)”from its minimum initial value

to a maximum size half a period later, and then contracts back to its initial size. This

periodic cycling from minimum to maximum spread repeats inde¬nitely. We recall

from eqns (2.99) and (2.100) that the operators p and q respectively correspond to

the electric and magnetic ¬elds. According to Section 2.5 this means that the variance

in the electric ¬eld for the squeezed vacuum state (15.3) is smaller than the vacuum-

¬‚uctuation variance.

The Hamiltonian for a radiation oscillator is unchanged by the (unitary) parity

transformation p ’ ’p, q ’ ’q on the operators p and q; therefore the energy

eigenstates, e.g. the momentum-space eigenfunctions ¦n (P ), are also eigenstates of

parity:

¦n (P ) ’ (’1)n ¦n (P ) for P ’ ’P .

An immediate consequence of this fact is that an initial state having de¬nite parity,

i.e. a superposition of eigenstates which all have the same parity, will evolve into

a state with the same parity at all times. Inspection of eqn (15.3) shows that this

initial Gaussian state is an even function of P ; consequently, the coe¬cients Cn in the

expansion (15.5) must vanish for all odd integers n. In other words, the evolution of the

squeezed vacuum state can only involve even-parity eigenfunctions for the radiation

oscillators. Since these eigenfunctions represent number states, an equivalent statement

is that only even integer number states can be involved in the production and the time

evolution of a squeezed vacuum state. Thus we arrive at the important conclusion that

the simplest elementary process leading to such a state is photon pair production.

For production of photons in pairs one needs to look to nonlinear optical inter-

actions, such as those provided by χ(2) and χ(3) media. The ¬rst experiment demon-

strating a squeezed state of light was performed by Slusher et al. (1985), who used

four-wave mixing in an atomic-vapor medium with a χ(3) nonlinearity. More strongly

¾ Nonclassical states of light

squeezed states of light were subsequently generated in χ(2) crystals by Kimble and co-

workers (Wu et al., 1986). In both cases the internal interaction in the sample induced

by the external classical ¬eld has the form

HSS = i„¦P a†2 ’ HC , (15.7)

for some c-number, phenomenological coupling constant „¦P . Long before these exper-

iments were performed, squeezed states were discovered theoretically by Stoler (1970),

in a study of minimum-uncertainty wave packets that are unitarily equivalent to co-

herent states. Yuen (1976) introduced squeezed states into quantum optics through

the notion of two-photon coherent states. He also made the important observation

that squeezed states would lead to the possibility of quantum noise reduction. Caves

(1981) studied squeezed states in the context of possible improvements in the fun-

damental sensitivity of gravitational-wave detectors based on optical interferometers

that use squeezed light.

But how are squeezed states of light to be detected? If there is a synchronous

experimental method to measure p (t), i.e. the electric ¬eld, just at the integer multiples

of the period”when the p-noise, ∆p (t), is at a minimum”it is plausible that one can

observe p-noise that is less than the standard quantum limit. The price we pay for

reduced p-noise at integer multiples of the period (t = 0, T, 2T, . . .) is an increased

p-noise at odd multiples of a half-period (t = T /2, 3T /2, 5T /2, . . .). This increase must

be such that the product of the alternating deviations, e.g. ∆p (T ) ∆p (3T /2), remains

larger than /2. An equivalent argument is based on the fact that q = p, so that

™

—¦

the deviation in displacement, ∆q (t), is 90 out of phase (in quadrature) with ∆p (t).

Consequently ∆q (t) is a maximum when ∆p (t) is a minimum, and the uncertainty

relation is maintained at all times. A synchronous measurement method is provided

by balanced homodyne detection, as discussed in Section 9.3.3. This kind of detection

scheme has blind spots precisely at those times when the p-noise is at a maximum, and

sensitive spots at the intermediate times when the p-noise is at a minimum. In this

way, the signal-to-noise ratio of a synchronous measurement scheme for the electric

¬eld can, in principle, be increased over the prediction of the standard quantum limit

associated with a coherent state.

The theory required to describe the generation of squeezed states is signi¬cantly

more complex than the discussion showing that coherent states are generated by clas-

sical currents. For this reason, we will follow the historical sequence outlined above,

by ¬rst studying the formal properties of squeezed states. This background is quite

useful for the analysis of experiments, even in the absence of a detailed model of the

source. In subsequent sections we will present the theory of squeezed-light generation,

and ¬nally describe an actual experiment.

15.1.2 General properties of squeezed states

A Quadrature operators

In place of eqn (15.3) we could equally well consider a squeezed vacuum for which

the deviation in the magnetic ¬eld (i.e. ∆q (t)) periodically achieves minimum values

less than the vacuum ¬‚uctuation value B0 . This can be done by using the coordinate

¿

Squeezed states

representation, and replacing P by Q everywhere in the discussion. More generally,

there is no reason to restrict attention to purely electric or purely magnetic ¬‚uctua-

tions; we could, instead, decide to measure any linear combination of the two. For this

discussion, let us ¬rst introduce the dimensionless canonical operators

1† q ω

X0 ≡ a +a = = q,

2 2∆q0 2

(15.8)

i† p 1

Y0 ≡ a ’a = = p,

2 2∆p0 2ω

which satisfy the commutation relation

i

[X0 , Y0 ] = . (15.9)

2

Comparing this to the canonical relations [q, p] = i and ∆q∆p /2 shows that the

corresponding uncertainty product is

1

∆X0 ∆Y0 . (15.10)

4

The solution a (t) = a exp (’iωt) of the free-¬eld Heisenberg equations yields the

time evolution of X0 and Y0 :

X0 (t) = X0 cos (ωt) + Y0 sin (ωt) ,

(15.11)

Y0 (t) = ’X0 sin (ωt) + Y0 cos (ωt) ,

which describes a rotation in the phase plane. It is often useful to generalize the

conventional choice, t = 0, of the reference time to t = t0 , so that the annihilation

operator is given by

a (t) = ae’iω(t’t0 ) = ae’iβ e’iωt , (15.12)

where β = ’ωt0 . In a mechanical context, choosing t0 amounts to setting a clock; but

in the optical context, the preceding equation shows that choosing the reference time t0

is equivalent to choosing the reference phase β. In the homodyne detection experiments

to be described later on, the phase β can be controlled by means of changes in the

relative phase between a local oscillator beam and the squeezed light which is being

measured. With this choice of reference phase, the time evolution of the magnetic and

electric ¬elds is given by

X0 (t) = X cos (ωt) + Y sin (ωt) ,

(15.13)

Y0 (t) = ’X sin (ωt) + Y cos (ωt) ,

where

1

ae’iβ + a† eiβ = X0 cos (β) + Y0 sin (β) ,

X=

2 (15.14)

1 ’iβ † iβ

’a e = ’X0 sin (β) + Y0 cos (β) .

Y= ae

2i

These are the same quadrature operators introduced in the analysis of heterodyne

and homodyne detection in Section 9.3; they are related to the canonical operators

Nonclassical states of light

by a rotation through the angle β in the phase plane. The cases considered previously

correspond to β = ’π/2 and β = 0 for the electric and magnetic ¬elds respectively.

For any value of β, the quadrature operators satisfy eqns (15.9) and (15.10). Conse-

quently, for any coherent state |± ”in particular for the vacuum state”the variances

of the quadrature operators are

1

V (X) = V (Y ) = , (15.15)

4

and the uncertainty product ∆X∆Y = 1/4 has the minimum possible value at all

times. Fig. 5.1 shows that the phase space portrait of the coherent state in the dimen-

sionless variables (X0 , Y0 ) consists of a circular quantum fuzzball, which surrounds

the tip of the coherent-state phasor ±. The rotation to X and Y amounts to choos-

ing the X-axis along the phasor. The isotropic quantum fuzzball corresponds to a

quasi-probability distribution which has the form of an isotropic Gaussian in phase

space.

A state ρ is said to be squeezed along the quadrature X, if the variance

V (X) = X 2 ’ X 2 satis¬es V (X) < 1/4, where Z = Tr (ρZ), for any operator Z.

This condition can be expressed more conveniently in terms of the normal-ordered

2

variance VN (X) ≡ : X 2 : ’ : X : , where : Z : is the normal-ordering operation

de¬ned by eqn (2.107). Since X is a linear function of the creation and annihilation

operators, : X : = X, but : X 2 : = X 2 . An explicit calculation leads to the relation

1

VN (X) = V (X) ’ . (15.16)

4

With this notation, the squeezing condition becomes VN (X) < 0 and perfect squeez-

ing, i.e. V (X) = 0, corresponds to VN (X) = ’1/4.

The straightforward calculation suggested in Exercise 15.1 establishes the relations

1 1

Re e’2iβ V (a) + V a† , a ,

VN (X) = (15.17)

2 2

1 1

VN (Y ) = ’ Re e’2iβ V (a) + V a† , a , (15.18)

2 2

between the normal quadrature variances and variances of the annihilation opera-

tors. The quantity V a† , a = a† a ’ a† a is an example of the joint vari-

ance, V (F, G) = F G ’ F G , introduced in Section 5.1.1. It is easy to see that

V a† , a 0; therefore, necessary conditions for squeezing along X or Y are

Re e’2iβ V (a) < 0 (15.19)

and

Re e’2iβ V (a) > 0 (15.20)

respectively. Thus a state for which V (a) = 0 is not squeezed along any quadra-

ture. This fact excludes both number states and coherent states from the category of

squeezed states.

Squeezed states

B The squeezing operator

As an aid to understanding how single-mode squeezing is generated by the interaction

Hamiltonian (15.7), let us recall the argument used in Section 5.4.1 to guess the form

of the displacement operator that generates coherent states from the vacuum. The

Hamiltonian Hint describing the interaction of a classical current with a single mode

of the radiation ¬eld is linear in the creation and annihilation operators. For the mode

exactly in resonance with a purely sinusoidal current, the time evolution of the state

vector in the interaction picture is represented by the unitary operator exp (’itHint / ),

which leads to the form D (±) = exp ±a† ’ ±— a for the displacement operator.

By analogy with this argument, the quadratic interaction Hamiltonian (15.7) sug-

gests that the squeezing operator should be de¬ned by

—2

a ’ζa†2 )

1

S (ζ) = e 2 (ζ , (15.21)

where the c-number ζ = r exp (2iφ) is called the complex squeeze parameter. The

modulus r = |ζ| describes the amount of squeezing, and the phase 2φ determines the

angle of the squeezing axis in phase space.

The unitary squeezing operator applied to a pure state |Ψ de¬nes the squeezing

transformation,

|Ψ (ζ) = S (ζ) |Ψ , (15.22)

for states. It is also useful to de¬ne squeezed operators by

X (ζ) = S (ζ) XS † (ζ) , (15.23)

so that expectation values are preserved, i.e.

Ψ (ζ) |X (ζ)| Ψ (ζ) = Ψ |X| Ψ . (15.24)

Applying eqn (15.23) to the density operator describing a mixed state, as well as to

the observable X, shows that mixed-state expectation values are also preserved:

Tr [ρ (ζ) X (ζ)] = Tr [ρX] . (15.25)

The ¬rst example is the squeezed vacuum state ψ (P, 0) in eqn (15.3). With the

correct choice for ζ this can be expressed as

ψ (P, 0) = P |S (ζ)| 0 . (15.26)

In the limit of weak squeezing, i.e. |ζ| 1, the operator in eqn (15.22) can be expanded

to get

1 —2

ζ a ’ ζa†2 |0 + · · ·

S (ζ) |0 = |0 +

2

1 †2

= |0 ’ ζa |0 + · · · . (15.27)

2

The ¬rst-order term on the right side is the output state for the degenerate case of

the down-conversion process discussed in Section 13.3.2. Thus down-conversion rep-

resents incipient single-mode squeezing. The transformation of a single pump photon

Nonclassical states of light

of frequency ωP into a pair of photons, each with frequency ω0 = ωP /2, is the source

of the photons in the squeezed vacuum S (ζ) |0 . The general case of nondegenerate

down-conversion can similarly serve as the source of a two-mode squeezed state. In

this case, the nonlocal phenomena associated with entangled states would play an

important role.

For general squeezed states, the features of experimental interest are expressible

in terms of variances of the quadrature operators or other observables, such as the

number operator. For example, the variance, V (X), of X in the squeezed state is

2

V (X) = Tr ρ (ζ) X 2 ’ (Tr [ρ (ζ) X]) . (15.28)

The easiest way to evaluate these expressions is to use the relation (15.23) between the

original operators X and their squeezed versions X (ζ). Since all observables can be

expressed in terms of the creation and annihilation operators it is su¬cient to consider

a (ζ) = S (ζ) aS † (ζ) . (15.29)

The ¬rst step in evaluating the right side of this equation is to de¬ne the squeezing

generator K (ζ) by

i

K (ζ) = ’ ζ — a2 ’ ζa†2 , (15.30)

2

so that S (ζ) = exp [iK (ζ)]. The second step is to imitate eqn (5.49) by introducing

the interpolating operators

c („ ) = ei„ K(ζ) ae’i„ K(ζ) , (15.31)

where „ is a real variable in the interval (0, 1). The interpolation formula has the form

of a time evolution with Hamiltonian K, so the interpolating operators satisfy the

Heisenberg-like equations

d

i c („ ) = c („ ) , K , (15.32)

d„

where

i

K = ’ ζ — c2 („ ) ’ ζc†2 („ ) . (15.33)

2

If we identify ζ with ’2i„¦P , then K has the general form (15.7). This means that

we will be able to use the results obtained here to treat the model for squeezed-state

generation to be given in Section 15.2.

The explicit form (15.33), together with the canonical commutation relation

c („ ) , c† („ ) = 1, yields a pair of ¬rst-order equations for c and c† :

dc†

dc

= ζc† , = ζ—c , (15.34)

d„ d„

and eliminating c† produces a single second-order equation:

d2 c 2

= |ζ| c . (15.35)

2

d„

Since |ζ| = r2 is real and positive the fundamental solutions are e±r„ and the general

2

solution is c („ ) = C+ er„ +C’ e’r„ . Substituting this form into either of the ¬rst-order

Squeezed states

equations yields one relation between C+ and C’ , and the initial condition c (0) = a

gives another. The solution of this pair of algebraic equations provides the expression

a (ζ) = µa + νa† for the squeezed annihilation operator, where the coe¬cients

µ = cosh (r) , ν = e2iφ sinh (r) , (15.36)

2

satisfy the identity µ2 ’ |ν| = 1. The relation between a and a (ζ) is another example

of the Bogoliubov transformation. The inverse transformation,

a = µa (ζ) ’ νa† (ζ) , (15.37)

will be useful in subsequent calculations.

Let us ¬rst apply eqn (15.37) to express the quadrature operators, de¬ned by eqn

(15.14), as

1

cosh (r) ’ e’2i(φ’β) sinh (r) a (ζ) e’iβ + HC ,

X=

2 (15.38)

i

cosh (r) + e’2i(φ’β) sinh (r) a (ζ) e’iβ ’ HC .

Y=

2

For the quadrature angle β = φ this simpli¬es to X = e’r X (ζ) and Y = er Y (ζ), so

that

V (X) = V e’r X (ζ) = e’2r V (X (ζ)) = e’2r V0 (X) ,

(15.39)

V (Y ) = V (er Y (ζ)) = e2r V (Y (ζ)) = e2r V0 (Y ) ,

i.e. the X-quadrature is squeezed and the Y -quadrature is stretched, relative to the

variances V0 in the original state. The alternative choice β = φ ’ π/2 reverses the roles

of X and Y . For either choice, the deviations in the squeezed state satisfy

V (X) = e±r ∆0 X , ∆Y = V (Y ) = e“r ∆0 Y ,

∆X = (15.40)

which shows that the uncertainty product is unchanged by squeezing. In particular, if

|Ψ is a minimum-uncertainty state, then so is the squeezed state |Ψ (ζ) , i.e.

1

∆X∆Y = ∆0 X∆0 Y = . (15.41)

4

We now turn to the question of the classical versus nonclassical nature of squeezed

states. Suppose that ρ (ζ) is squeezed along X. The P -representation (5.168) can be

used to express the variance as

d2 ± 2

P (±) ± (X ’ X ) ±

V (X) =

π

d2 ± 2

P (±) ± X 2 ± ’ 2 X ± |X| ± + X

= , (15.42)

π

where P (±) is the P -function representing the squeezed state ρ (ζ) and X =

Tr [ρ (ζ) X]. The coherent-state expectation values can be evaluated by ¬rst using eqn

Nonclassical states of light

(15.14) and the commutation relations to express X 2 in normal-ordered form. After a

little further algebra one ¬nds that the normal-ordered variance is

2

±e’iβ + ±— eiβ

d2 ±

’X

VN (X) = P (±) . (15.43)

π 2

Now let us suppose that the squeezed state ρ (ζ) is classical, i.e. P (±) 0, then the

last result shows that VN (X) > 0. Since this contradicts the assumption that ρ (ζ) is

squeezed along X, we conclude that all squeezed states are nonclassical.

Multimode squeezed states—

15.1.3

A description of multimode squeezed states can be constructed by imitating the treat-

ment of multimode coherent states in Section 5.5.1. The single-mode squeezing oper-

ator can be applied to any member of a complete set of modes, e.g. the plane waves

of a box-quantized description; consequently, the simplest de¬nition of a multimode

squeezed state is

Ψ ζ = S ζ |Ψ , (15.44)

where

1 —2

ζks aks ’ ζks a†2

Sζ= exp (15.45)

ks

2

ks

is the multimode squeezing operator. Since the individual squeezing generators com-

mute, the de¬nition of S ζ can also be expressed as

1

ζks a2 ’ ζks a†2

—

S ζ = exp . (15.46)

ks ks

2

ks

Special squeezed states—

15.1.4

Coherent states are minimum-uncertainty states, so eqn (15.41) implies that the squee-

zed coherent states,

|ζ; ± ≡ S (ζ) |± = S (ζ) D (±) |0 , (15.47)

are also minimum-uncertainty states. In this notation, the squeezed vacuum state dis-

cussed previously is denoted by |ζ; 0 . The squeezed vacuum is generated by injecting

pump radiation into a nonlinear medium with an e¬ective interaction given by eqn

(15.7), and the more general squeezed coherent state can be obtained by simultane-

ously injecting the pump beam and the output of a laser matching the squeezed mode.

Furthermore, the squeezed coherent states are eigenstates of the transformed operator

a (ζ), since

a (ζ) |ζ; ± = S (ζ) a |± = ± |ζ; ± . (15.48)

The state |ζ; ± is therefore an analogue of the coherent state |± , but it is generated by

creating and annihilating pairs of photons. The squeezed coherent states are therefore

the two-photon coherent states introduced by Yuen.

Squeezed states

For a ¬xed value of the squeezing parameter ζ, the squeezed coherent states have

the same orthogonality and completeness properties as the coherent states. The or-

thogonality property follows from the unitary relation (15.47), which shows that the

inner product of two squeezed coherent states is

ζ; β |ζ; ± = β S † (ζ) S (ζ) ± = β |± . (15.49)

The resolution of the identity follows in the same way, since combining eqn (15.47)

with eqn (5.69) gives us

d2 ± d2 ±

|± ±| S † (ζ) = 1 .

|ζ; ± ζ; ±| = S (ζ) (15.50)

π π

An alternative family of states is de¬ned by the displaced squeezed states

|±; ζ ≡ D (±) |ζ = D (±) S (ζ) |0 , (15.51)

which are constructed by displacing a squeezed vacuum state. An idealized physical

model for this is to inject the output of a squeezed vacuum generator into a laser

ampli¬er for the squeezed mode. The squeezed vacuum is the simplest example of a

squeezed state, so the displaced squeezed states are also called ideal squeezed states

(Caves, 1981).

The states |ζ; ± and |±; ζ are quite di¬erent, since the operators S (ζ) and D (±)

do not commute. For this reason it is important to remember that ζ is the squeezing

parameter and ± is the displacement parameter. Despite their di¬erences, these two

states are both normalized, so there must be a unitary transformation connecting

them. Indeed it is not di¬cult to show that they are related by

|ζ; ± = |±’ ; ζ (15.52)

and

|±; ζ = |ζ; ±+ , (15.53)

where

±± = µ± ± ν±—

= ± cosh r ± ±— e2iφ sinh r . (15.54)

According to eqn (15.53) the displaced squeezed state |±; ζ is also an eigenvector of

a (ζ),

a (ζ) |±; ζ = ±+ |±; ζ , (15.55)

but the eigenvalue is ±+ rather than ±.

The relation (15.53) allows us to transfer the orthogonality and completeness re-

lations for squeezed coherent states to the displaced squeezed states. Applying eqn

(15.53) to eqns (15.49) and (15.50) yields

β; ζ |±; ζ = ζ; β+ |ζ; ±+ = β+ |±+ , (15.56)

¼ Nonclassical states of light

and

d2 β d2 β

|β’ ; ζ β’ ; ζ| = |ζ; β ζ; β| = 1 . (15.57)

π π

The general result (15.39) shows that squeezing any minimum-uncertainty state

produces the quadrature variances V (X) = e’r /4 and V (Y ) = er /4. For the case of

the squeezed coherent state |ζ; ± , with ± = |±| eiθ , the quadrature averages are given

by

ζ; ± |X| ζ; ± = |±| e’r cos (θ ’ φ) ,

(15.58)

ζ; ± |Y | ζ; ± = |±| er sin (θ ’ φ) .

For the special choice θ = φ one ¬nds ζ; ± |Y | ζ; ± = 0 and

ζ; ± |X| ζ; ± = |±| e’r , (15.59)

so the squeezed quadrature X represents the amplitude of the coherent state. Con-

sequently this process is called amplitude squeezing. This example has led to the

frequent use of the names amplitude quadrature and phase quadrature for X

and Y respectively.

Of course, the roles of X and Y can always be changed by making a di¬erent phase

choice. If we choose θ ’ φ = π/2, then ζ; ± |X| ζ; ± = 0 and ζ; ± |Y | ζ; ± = |±| er .

The amplitude of the coherent state is now carried by the stretched quadrature Y ,

and the squeezed quadrature X is conjugate to Y . Roughly speaking, the operator

conjugate to the amplitude is related to the phase; consequently, this process is called

phase squeezing.

Photon-counting statistics—

15.1.5

The variances and averages of the quadrature operators were used in the interpretation

of the homodyne detection scheme discussed in Section 9.3.3, but photon-counting

experiments are related to the average and variance of the photon number operator.

For the special squeezed states de¬ned by eqns (15.47) and (15.51), the most direct

way to calculate these quantities is ¬rst to use eqn (15.37) to express the operators N

and N 2 in terms of the transformed operators a (ζ) and a† (ζ), and then to rearrange

these expressions in normal-ordered form with respect to a (ζ) and a† (ζ). Finally, the

eigenvalue equations (15.48) and (15.55), together with their adjoints, can be used to

get the expectation values of N and N 2 as explicit functions of ζ and ±.

By virtue of the relation (15.52), it is enough to consider the expectation values

for the displaced squeezed state |±; ζ . Using eqn (15.37) produces the expression

N = a† a = µa† (ζ) ’ ν — a (ζ) µa (ζ) ’ νa† (ζ)

= µ2 + |ν|2 a† (ζ) a (ζ) ’ ν — µa (ζ)2 ’ νµa† (ζ) + |ν|2 (15.60)

for the number operator, so eqn (15.55) and its adjoint yield

N = ±; ζ |N | ±; ζ = µ±— ’ ν — ±+ µ±+ ’ ν±— + |ν|

2

+ +

2 2 2

= |±| + |ν| = |±| + sinh2 (r) . (15.61)

To get the ¬nal result we have used the solution ± = µ±+ ’ ν±— of eqn (15.54).

+

½

Squeezed states

For the calculation of N 2 , we ¬rst use the commutation relations to establish the

identity N 2 = a†2 a2 + N , which leads to

N 2 = a†2 a2 + N . (15.62)

The next step is to use eqn (15.37) to derive the normal-ordered expression”with

respect to the squeezed operators a (ζ) and a† (ζ)”for a2 :

a2 = µ2 a (ζ) ’ 2µνa† (ζ) a (ζ) + ν 2 a† (ζ) ’ µν .

2

(15.63)

This can be used in turn to derive the normal-ordered form for a†2 a2 and thus to

evaluate a†2 a2 in the same way as N . This calculation is straightforward but

rather lengthy. A somewhat more compact method is to use the completeness relation

(15.57) to get

a†2 a2 = ±; ζ a†2 a2 ±; ζ

d2 β

†2

|β’ ; ζ β’ ; ζ| a2 |±; ζ

= ±; ζ| a

π

d2 β 2

β’ ; ζ a2 ±; ζ

= . (15.64)

π

Applying the eigenvalue equation (15.55) to |±; ζ and the adjoint equation to β’ ; ζ|

produces a (ζ) |±; ζ = ±+ |±; ζ and β’ ; ζ| a† (ζ) = β’ ; ζ| β — , so the matrix element

in the integrand is given by

β’ ; ζ a2 ±; ζ = f (β — ) β’ ; ζ |±; ζ = f (β — ) β |±+ , (15.65)

where

f (β — ) = µ2 ±2 ’ 2µνβ — ±+ + ν 2 β —2 ’ µν . (15.66)

+

Substituting this result in eqn (15.64) and using the explicit formula (5.58) for the

inner product leaves us with

d2 β

†2 2

|f (β — )|2 e’|β’±+ |

2

aa =

π

d2 β 2 ’|β|2

f β — + ±—

= e , (15.67)

+

π

where the last line was obtained by the change of integration variables β ’ β + ±+ .

This rather elaborate preparation would be useless if the remaining integrals could

not be easily evaluated. Fortunately, the integrals can be readily done in polar co-

ordinates, β = b exp (i‘), as can be seen in Exercise 15.4. After a certain amount of

algebra, one ¬nds

a†2 a2 = |±| + µ2 |ν| ’ µ ±2 ν — + CC + 4 |±| |ν| + 2 |ν| .

4 2 2 2 4

(15.68)

Combining this result with eqns (15.36), (15.62), and (9.58) leads to the general ex-

pression

sinh2 r cosh 2r + 2 |±|2 sinh r [sinh r ’ cosh r cos (θ ’ φ)]

Q= (15.69)

2

|±| + sinh2 r

for the Mandel Q parameter.

¾ Nonclassical states of light

The Q parameter is positive (super-Poissonian statistics) for cos (θ ’ φ) 0, but

it can be negative (sub-Poissonian statistics) if cos (θ ’ φ) > 0. In the case θ = φ

we have amplitude squeezing (see eqn (15.59) for the squeezed quadrature X), so the

general result becomes

1 ’ e’2r

2

sinh2 r cosh 2r ’ |±|

Q= . (15.70)

2

|±| + sinh2 r

In the strong-¬eld limit |±| exp (4r), Q becomes

Q ≈ ’ 1 ’ e’2r . (15.71)

1), then Q ≈ ’1, i.e. there is negligible noise

If we also assume strong squeezing (r

in photon number. Consequently, amplitude squeezed states are also called number

squeezed states. This terminology is rather misleading, since eqn (15.19) shows that

a squeezed state can never be a number state.

Are squeezed states robust?—

15.1.6

In Section 8.4.3 we saw that a coherent state |±1 incident on a beam splitter is

scattered into a two-mode coherent state |±1 , ±2 , where ±1 = t ±1 and ±2 = r ±1 . A

similar result would be found for any passive, linear optical element. An even more

impressive feature appears in Section 18.5.2, where it is shown that an initial coherent

state |±0 coupled to a zero-temperature reservoir evolves into the coherent state

±0 e’“t/2 e’iω0 t . In other words, the de¬ning statistical property, V a† , a = 0, of

the coherent state is unchanged by this form of dissipation. Only the amplitude of the

parameter ±0 is reduced. For these reasons the coherent state is regarded as robust.

The situation for squeezed states turns out to be a bit more subtle.

Let us ¬rst consider an experiment in which light in a squeezed state enters through

port 1 of a beam splitter, as shown in Fig. 8.2. The input state |Ψ is the vacuum for

the mode entering through the unused port 2, i.e.

a2 |Ψ = 0 , (15.72)

but it is squeezed along a quadrature

1

a1 e’iβ + a† eiβ

X1 = (15.73)

1

2

of the incident mode 1, i.e. VN (X1 ) < 0. According to eqn (8.62) the scattered oper-

ators a1 and a2 are related to the incident operators a1 and a2 by

a 2 = r a1 + t a 2 ,

(15.74)

a 1 = t a 1 + r a2 ,

2 2

where |t| +|r| = 1. We choose the phases of r and t so that the transmission coe¬cient

t is real and the re¬‚ection coe¬cient r is purely imaginary.

¿

Squeezed states

The question to be investigated is whether there is squeezing along any output

quadrature. We begin by examining general quadratures

1

a1 e’iβ1 + a1† eiβ1

X1 = (15.75)

2

and

1

a2 e’iβ2 + a2† eiβ2

X2 = (15.76)

2

for the transmitted and re¬‚ected modes respectively. Applying eqns (15.17), (15.72),

and (15.74) to the X1 -quadrature leads to

1 1

Re V a1 e’iβ1 + V a1† , a1

VN (X1 ) =

2 2

1 12

= Re t2 V a1 e’iβ1 + |t| V a† , a1

1

2 2

t2

= t VN (X1 ) + Re ei• ’ 1 V a1 e’iβ

2

, (15.77)

2

where • = 2 (β ’ β1 ). Squeezing along X1 means that VN (X1 ) < 0, but the second

term depends on the value of β1 . The simplest choice”β1 = β”leads to

VN (X1 ) = t2 VN (X1 ) , (15.78)

which shows that squeezing along X1 implies squeezing along X1 for the quadrature

angle β1 = β. As might be expected, the inescapable partition noise at the beam

splitter reduces the amount of squeezing by the intensity transmission coe¬cient t2 <

1. This particular choice of output quadrature does answer the squeezing question,

but it does not necessarily yield the largest degree of squeezing.

A similar argument applied to X2 begins with

1 1

+ V a2† , a2 ,

Re V a2 e’iβ2

VN (X2 ) = (15.79)

2 2

2

but the relation r2 = ’ |r| produces

Re V a2 e’iβ2 = Re r2 V a1 e’iβ2 = ’ |r| Re V a1 e’iβ2

2

. (15.80)

The ¬nal result in this case is

|r|2

ei• + 1 V a1 e’iβ

2

V (X2 ) = |r| VN (X1 ) ’ Re , (15.81)

2

where • = 2 (β ’ β2 ). For the re¬‚ected mode, the choice β2 = β ’ π/2 (• = π) shows

reduced squeezing along X2 . Alternatively, we can use the relation

X2 |β2 =β’π/2 = ’ Y2 |β2 =β (15.82)

to say that squeezing occurs along the conjugate quadrature Y2 for β2 = β.

Nonclassical states of light

We next consider the evolution of a squeezed state coupled to a zero-temperature

reservoir. For the quadrature

1

ae’iβ + a† eiβ ,

Xβ = (15.83)

2

eqn (15.43) gives us

2

±e’iβ + ±— eiβ

d2 ±

P (±, ±— ; t) ’ Xβ ; t

VN (Xβ ; t) = , (15.84)

π 2

where

±e’iβ + ±— eiβ

d2 ± —

Xβ ; t = P (±, ± ; t) . (15.85)

π 2

The assumption that the state is initially squeezed along Xβ means that

2

±e’iβ + ±— eiβ

d2 ±

P0 (±, ±— ) ’ Xβ

VN (Xβ ; 0) = < 0, (15.86)

0

π 2

where P0 (±, ±— ) = P (±, ±— ; t = 0). Anticipating the general solution (18.88) for dis-

sipation by interaction with a zero-temperature reservoir leads to

d2 ±

P0 e(“/2+iω0 )t ±, e(“/2’iω0 )t ±— e“t

VN (Xβ ; t) =

π

2

±e’iβ + ±— eiβ

— ’ Xβ ; t , (15.87)

2

and

d2 ±

P0 e(“/2+iω0 )t ±, e(“/2’iω0 )t ±— e“t

Xβ ; t =

π

±e’iβ + ±— eiβ

— . (15.88)

2

Our next step is to make the change of integration variables ± ’ ± exp [’ (“/2 + iω0 ) t]

in the last two equations. For eqn (15.88) the result is

±e’i(β+ω0 t) + ±— e’i(β+ω0 t)

d2 ±

’“t/2

P0 (±, ±— )

Xβ ; t = e

π 2

= e’“t/2 Xβ+ω0 t , (15.89)

0

and a similar calculation starting with eqn (15.87) yields

VN (Xβ ; t) = e’“t VN (Xβ+ω0 t ; 0) . (15.90)

Just as in the case of the beam splitter, we are free to choose new quadratures to

investigate, in this case at di¬erent times. At time t we take advantage of this freedom

to let β ’ β ’ ω0 t, so that

VN (Xβ’ω0 t ; t) = e’“t VN (Xβ ; 0) < 0 . (15.91)

Thus at any time t, there is a squeezed quadrature”with the amount of squeezing

reduced by exp (’“t)”but the required quadrature angle rotates with frequency ω0 .

Theory of squeezed-light generation—