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. 18
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>>

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

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

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—

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