. 19
( 27)


With the results (15.78) and (15.91) in hand, we can now judge the robustness
of squeezed states. Let us begin by recalling that coherent states are regarded as
robust because the de¬ning property, V a† , a = 0, is strictly conserved by dissipa-
tive scattering”i.e. coupling to a zero-temperature reservoir”as well as by passage
through passive, linear devices. By contrast, dissipative scattering degrades the degree
of squeezing as well as the overall intensity of the squeezed input light, so that

|VN (X : t)| ’ 0 as t ’ ∞ . (15.92)

Even this result depends on the detection of a quadrature that is rotating at the
optical frequency ω0 . Detector response times are large compared to optical periods,
so even the reduced squeezing shown by eqn (15.91) would be extremely di¬cult to
detect. Passage through a linear optical device also degrades the degree of squeezing,
as shown by eqn (15.78). This combination of properties is the basis for the general
opinion that squeezed states are not robust.

Theory of squeezed-light generation—
The method used by Kimble and co-workers (Wu et al., 1986) to generate squeezed
states relies on the microscopic process responsible for the spontaneous down-conver-
sion e¬ect discussed in Section 13.3.2; but two important changes in the experimental
arrangement are shown in Fig. 15.1. The ¬rst is that the χ(2) crystal is cut so as
to produce collinear phase matching with degenerate pairs (ω1 = ω2 = ω0 = ωP /2) of
photons, and also anti-re¬‚ection coated for both the ¬rst- and the second-harmonic
frequencies ω0 and ωP = 2ω0 . In this con¬guration the down-converted photons have
identical frequencies and propagate in the same direction as the pump photons; in
other words, this is time-reversed second harmonic generation. The second change
is that the crystal is enclosed by a resonant cavity that is tuned to the degenerate
frequency ω0 = ωP /2 and, therefore, also to the pump frequency ωP .
The degeneracy conditions between the down-converted photons and the cavity
resonance frequency are maintained by a combination of temperature tuning for the
crystal and a servo control of the optical resonator length. This arrangement strongly

Mirrors for ω0 and ω2

Pump ω1
signal and idler
at ω2 at ω1 = ω2 = ω0 = ω2 /2

1 2

Fig. 15.1 A simpli¬ed schematic for the squeezed state generator employed in the experiment
of Kimble and co-workers (Wu et al., 1986).
Nonclassical states of light

favors the degenerate pairs over all other pairs of photons that are produced by down-
conversion. In this way, the crystal”pumped by the strong laser beam at the second-
harmonic frequency 2ω0 ”becomes an optical parametric ampli¬er1 (OPA) for the
degenerate photon pairs at the ¬rst-harmonic frequency ω0 .
This device can be understood at the classical level in the following way. The χ(2)
nonlinearity couples the two weak down-converted light beams to the strong-pump
laser beam, so that the weak light signals can be ampli¬ed by drawing energy from
the pump. The basic process is analogous to that of a child pumping a swing by
standing and squatting twice per period of the swing, thus increasing the amplitude
of the motion. This kind of ampli¬cation process depends on the timing (phase) of the
pumping motion relative to the timing (phase) of the swinging motion.
In the case of light beams, the mechanism for the transfer of energy from the pump
to the degenerate weak beams is the mixing of the strong-pump beam,
EP = Re EP e’iωP t = Re |EP | eiθP e’iωP t , (15.93)
with the two weak beams via the χ(2) nonlinearity. This leads to a mutual reinforce-
ment of the weak beams at the expense of the pump beam. If the depletion of the
strong-pump beam by the parametric ampli¬cation process is ignored, the mutual-
reinforcement mechanism leads to an exponential growth of both of the weak beams.
With su¬cient feedback from the mirrors surrounding the crystal, this ampli¬er”like
that of a laser”can begin to oscillate, and thereby become an optical parametric
oscillator (OPO). When operated just below the threshold of oscillation, the optical
parametric ampli¬er emits strongly squeezed states of light.
The resonant enhancement at the degenerate signal and idler frequencies justi¬es
the use of the phenomenological model Hamiltonian,
HS = HS0 + HSS , (15.94)
HS0 = ω0 a† a , (15.95)
„¦P e’iωP t a†2 ’ „¦— eiωP t a2 ,
HSS = (15.96)
for the sample shown in Fig. 15.1. The resonant mode associated with the annihilation
operator a is jointly de¬ned by the collinear phase-matching condition for the non-
linear crystal and by the boundary conditions at the two mirrors forming the optical
Note that HSS has exactly the form of the squeezing generator de¬ned by eqn
(15.30). The coupling frequency „¦P , which is proportional to the product χ(2) EP ,
characterizes the strength of the nonlinear interaction. The term „¦P a†2 describes the
down-conversion process in which a pump photon is converted into the degenerate
signal and idler photons. It is important to keep in mind that the complex coupling
parameter „¦P is proportional to EP = |EP | exp (iθP ), so that the parametric gain
depends on the phase of the pump wave. The consequences of this phase dependence
will be examined in the following sections.
1 The term ˜parametric™ ampli¬er was originally introduced in microwave engineering. The ˜para-
meter™ in the optical case is the pump wave amplitude, which is assumed to be unchanged by the
nonlinear interaction.
Theory of squeezed-light generation—

A The Langevin equations
The experimental signal in this case is provided by photons that escape the cavity, e.g.
through the mirror M2. In Section 14.3 this situation was described by means of in-
and out-¬elds for a general interaction HSS . In the present application, HSS is given
by eqn (15.96), and an explicit evaluation of the interaction term [a, HSS ] /i gives us
d κC
a (t) + „¦P ei(2ω0 ’ωP )t a† (t) + ξC (t) ,
a (t) = ’ (15.97)
dt 2
where a (t) = a (t) exp (iω0 t) is the slowly-varying envelope operator, κC = κ1 + κ2
is the cavity damping rate, and ξC (t) is the cavity noise operator de¬ned by eqn
(14.97). The explicit time dependence on the right side is eliminated by imposing the
resonance condition ωP = 2ω0 on the cavity. The equation for the adjoint envelope
operator a† (t) is then
d† κC † †
a (t) + „¦— a (t) + ξC (t) .
a (t) = ’ (15.98)
dt 2
Before considering the solution of the operator equations, it is instructive to write
the ensemble-averaged equations in matrix form:
’κC /2
d a (t) a (t)
= , (15.99)
a† (t) a† (t)

’κC /2
where we have used ξC (t) = 0. The 2 — 2 matrix on the right side has eigenvalues
Λ± = ’κC /2 ± |„¦P |, so the general solution is a linear combination of special solutions
varying as exp [Λ± (t ’ t0 )]. Since κC > 0, the eigenvalue Λ’ always describes an
exponentially decaying solution. On the other hand, the eigenvalue Λ+ can describe
an exponentially growing solution if |„¦P | > κC /2.
At the threshold value |„¦P | = κC /2, the average a (t) of the slowly-varying enve-
κC , so that a (t) ∼ exp (’iω0 t)
lope operator approaches a constant for times t’t0
is oscillatory at large times. This describes the transition from optical parametric am-
pli¬cation to optical parametric oscillation. Operation above the oscillation threshold
would produce an exponentially rapid build-up of the intracavity ¬eld that would
quickly lead to a violation of the weak-¬eld assumptions justifying the model Hamil-
tonian HSS in eqn (15.96). Dealing with pump ¬elds exceeding the threshold value
requires the inclusion of nonlinear e¬ects that would lead to gain saturation and thus
prevent runaway ampli¬cation. We avoid these complications by imposing the condi-
tion |„¦P | < κC /2. On the other hand, we will see presently that the largest squeezing
occurs for pump ¬elds just below the threshold value.
The coupled equations (15.97) and (15.98) for a (t) and a† (t) are a consequence of
the special form of the down-conversion Hamiltonian. Since the di¬erential equations
are linear, they can be solved by a variant of the Fourier transform technique used for
the empty-cavity problem in Section 14.3.3. In the frequency domain the di¬erential
equations are transformed into algebraic equations:
a (ω) + „¦P a† (’ω) + ξC (ω) ,
’iωa (ω) = ’ (15.100)
κC † †
’iωa† (’ω) = ’ a (ω) + „¦— a (ω) + ξC (’ω) , (15.101)
Nonclassical states of light

which have the solution

(κC /2 ’ iω) ξC (ω) + „¦P ξC (’ω)
a (ω) = ,
2 2
(κC /2 ’ iω) ’ |„¦P |

(κC /2 ’ iω) ξC (’ω) + „¦— ξC (ω)
a† (’ω) = P
2 2
(κC /2 ’ iω) ’ |„¦P |
Combining the de¬nition (14.97) with the result (14.116) for the in-¬elds in the fre-
quency domain gives
√ √
κ1 b1,ω+ω0 (t0 ) + κ2 b2,ω+ω0 (t0 ) eiωt0 .
ξC (ω) = (15.103)
This shows that a (ω) and a† (ω) are entirely expressed in terms of the reservoir op-
erators at the initial time. The correlation functions of the intracavity ¬eld a (t) are
therefore expressible in terms of the known statistical properties of the reservoirs.
Before turning to these calculations, we note that operator a (ω) has two poles”
determined by the roots of the denominator in eqn (15.102)”located at
ω = ω± = ’i ± |„¦P | . (15.104)
Since κC is positive, the pole at ω+ always remains in the lower half plane”correspon-
ding to the exponentially damped solution of eqn (15.99)”but when the coupling
frequency exceeds the threshold value, |„¦P |crit = κC /2, the pole at ω’ in¬ltrates
into the upper half plane”corresponding to the exponentially growing solution of eqn
(15.99). Thus the OPA“OPO transition occurs at the same value for the operator
solution and the ensemble-averaged solution.

B Squeezing of the intracavity ¬eld
As explained in Section 15.1.2, the properties of squeezed states are best exhibited
in terms of the normal-ordered variances VN (X) and VN (Y ) of conjugate pairs of
quadrature operators. According to eqns (15.17) and (15.18), these quantities can be
evaluated in terms of the joint variance V a† (t) , a (t) and the variance V (a (t)),
which can in turn be expressed in terms of the Fourier transforms a† (ω) and a (ω).
For example, eqns (14.112) and (14.114) lead to
dω dω ’i(ω +ω)t
V a† (t) , a (t) = V a† (’ω ) , a (ω) .
e (15.105)
2π 2π
Applying the relations
a (ω) = a (ω ’ ω0 ) , a† (’ω ) = a† (’ω ’ ω0 ) (15.106)
that follow from eqn (14.119), and the change of variables ω ’ ω + ω0 , ω ’ ω ’ ω0
allows this to be expressed in terms of the slowly-varying operators a (ω):
dω dω ’i(ω +ω)t
V a† (t) , a (t) = V a† (’ω ) , a (ω) .
e (15.107)
2π 2π
The solution (15.102) gives a (ω) and a† (ω) as linear combinations of the initial
reservoir creation and annihilation operators. In the experiment under consideration,
Theory of squeezed-light generation—

there is no injected signal at the resonance frequency ω0 , and the incident pump ¬eld
at ωP is treated classically. The Heisenberg-picture density operator can therefore be
treated as the vacuum for the initial reservoir ¬elds, i.e. ρE = |0 0|, where

b1,„¦ (t0 ) |0 = b2,„¦ (t0 ) |0 = 0 . (15.108)

This means that only antinormally-ordered products of the reservoir operators will
contribute to the right side of eqn (15.107). The fact that the variance is de¬ned with
respect to the reservoir vacuum greatly simpli¬es the calculation. To begin with, ¬rst
calculating a (ω) |0 provides the happy result that many terms vanish. Once this is
done, the commutation relations (14.115) lead to
|„¦P |

V a (t) , a (t) = . (15.109)
2 (κC /2)2 ’ |„¦P |2

In the same way, the crucial variance V (a (t)) is found to be
„¦P κ C
V (a (t)) = , (15.110)
4 (κC /2)2 ’ |„¦P |2

so that
|„¦P |
1 κC 1
VN (Xβ ) = 2 Re e „¦P + . (15.111)
2 4 (κC /2)2 ’ |„¦P |2
8 (κC /2) ’ |„¦P |

The minimum value of VN (X) is attained at the quadrature phase
θP π
β= (15.112)
2 2
where θP is the phase of „¦P . For this choice of β,
|„¦P |
VN (X) = ’ (15.113)
4 κC /2 + |„¦P |
|„¦P |
VN (Y ) = . (15.114)
4 κC /2 ’ |„¦P |
Keeping in mind the necessity of staying below the oscillation threshold, i.e. |„¦P | <
κC /2, we see that VN (X) > ’1/8. The relation (15.16) then yields
1 1
< V (X) < ; (15.115)
8 4
in other words, the cavity ¬eld cannot be squeezed by more than 50%. In this con-
nection, it is important to note that these results only depend on the symmetrical
combination κC = κ1 + κ2 and not on κ1 or κ2 separately. This feature re¬‚ects the
fact that the mode associated with a (t) is a standing wave that is jointly determined
by the boundary conditions at the two mirrors.
¼ Nonclassical states of light

C Squeezing of the emitted light
The limits on cavity ¬eld squeezing are not the end of the story, since only the output
of the OPA”i.e. the ¬eld emitted through one of the mirrors”can be experimentally
studied. We therefore consider a time t1 t0 when the light emitted”say through
mirror M2”reaches a detector. The detected signal is represented by the out-¬eld
operator b2,out (t) introduced in Section 14.3. We reproduce the de¬nition,

b2,„¦ (t1 ) e’i„¦(t’t1 ) ,
b2,out (t) = (15.116)


here, in order to emphasize the dependence of the output signal on the ¬nal value
b2,„¦ (t1 ) of the reservoir operator.
Combining the Fourier transforms of the scattering relations (14.109) with eqn
(15.102) produces the following relations between the in- and out-¬elds:
2 2

bJ,out (ω) = PJL (ω) bL,in (ω) + CJL (ω) bL,in (’ω) , (15.117)
L=1 L=1

√ [κC /2 ’ iω]
PJL (ω) = δJl ’ κJ κL , (15.118)
2 2
[κC /2 ’ iω] ’ |„¦P |
√ „¦P
CJL (ω) = ’ κJ κL . (15.119)
2 2
[κC /2 ’ iω] ’ |„¦P |
The M2-output quadratures are de¬ned by replacing a (t) with b2,out (t) in eqn (15.14)
to get
b2,out (t) e’iβ + b† iβ
Xout (t) = 2,out (t) e ,
2 (15.120)
1 †
’ b2,out (t) e iβ
Yout (t) = b2,out (t) e
in the time domain, or
b2,out (ω) e’iβ + b† iβ
Xout (ω) = 2,out (’ω) e ,
2 (15.121)
b2,out (ω) e’iβ ’ b† iβ
Yout (ω) = 2,out (’ω) e
in the frequency domain. The parameter β is again chosen to satisfy eqn (15.112). The
normal-ordered variances for the output quadratures are

dω dω ’i(ω +ω )t
VN (Xout (t)) = e VN (Xout (ω ) , Xout (ω )) , (15.122)
2π 2π

dω dω ’i(ω +ω )t
VN (Yout (t)) = e VN (Yout (ω ) , Yout (ω )) , (15.123)
2π 2π
Theory of squeezed-light generation— ½

VN (F, G) = : F G : ’ : F : : G: (15.124)
is the joint normal-ordered variance. Calculations very similar to those for the
cavity quadratures lead to

|„¦P | κ2
VN (Xout (ω ) , Xout (ω )) = ’ 2πδ (ω + ω ’ 2ω0 ) ,
2 [κC /2 + |„¦P |]2 + (ω ’ ω0 )2
|„¦P | κ2
2πδ (ω + ω ’ 2ω0 ) .
VN (Yout (ω ) , Yout (ω )) =
2 [κC /2 ’ |„¦P |]2 + (ω ’ ω0 )2
The delta functions in the last two equations re¬‚ect the fact that the output ¬eld
b2,out (t)”by contrast to the discrete cavity mode described by a (t)”lies in a contin-
uum of reservoir modes. In this situation, it is necessary to measure the time-dependent
correlation function VN (Xout (t) , Xout (0)), or rather the corresponding spectral func-

dteiωt VN (Xout (t) , Xout (0)) ,
SN (ω) =

= VN (Xout (ω) , Xout (ω )) . (15.127)

Using eqn (15.125) to carry out the remaining integral produces

|„¦P | κ2
SN (ω) = ’ , (15.128)
2 [κC /2 + |„¦P |]2 + (ω ’ ω0 )2

which has its minimum value for |„¦P | = κC /2 = (κ1 + κ2 ) /2 and ω = ω0 , i.e.
1 κ2
SN (ω) > ’ . (15.129)
4 κ1 + κ2
For a symmetrical cavity”i.e. κ1 = κ2 ”the degree of squeezing is bounded by
SN (ω) > ’ ; (15.130)
therefore, the output ¬eld can at best be squeezed by 50%, just as for the intracavity
¬eld. However, the degree of squeezing for the output ¬eld is not a symmetrical function
of κ1 and κ2 . For an extremely unsymmetrical cavity”e.g. κ1 κ2 ”we see that
SN (ω) (15.131)
in other words, the output light can be squeezed by almost 100%.
The surprising result that the emitted light can be more squeezed than the light in
the cavity demands some additional discussion. The ¬rst point to be noted is that the
intracavity mode associated with the operator a (t) is a standing wave. Thus photons
generated in the nonlinear crystal are emitted into an equal superposition of left- and
¾ Nonclassical states of light

right-propagating waves. The left-propagating component of the intracavity mode is
partially re¬‚ected from the mirror M1 and then partially transmitted through the
mirror M2, together with the right-propagating component. Re¬‚ection from the ideal
mirror M1 does not introduce any phase jitter between the two waves; therefore,
interference is possible between the two right-propagating waves emitted from the
mirror M2. This makes it possible to achieve squeezing in one quadrature of the emitted
In estimating the degree of squeezing that can be achieved, it is essential to account
for the vacuum ¬‚uctuations in the M1 reservoir that are partially transmitted through
the mirror M1 into the cavity. Interference between these ¬‚uctuations and the right-
propagating component of the intracavity mode is impossible, since the phases are
statistically independent. For a symmetrical cavity, κ1 = κ2 , the result is that the
squeezing of the output light can be no greater than the squeezing of the intracavity
light. On the other hand, if the mirror M1 is a perfect re¬‚ector at ω0 , i.e. κ1 = 0, then
the vacuum ¬‚uctuations in the M1 reservoir cannot enter the cavity. In this case it is
possible to approach 100% squeezing in the light emitted through the mirror M2.

15.3 Experimental squeezed-light generation
In Fig. 15.2, an experiment by Kimble and co-workers (Wu et al., 1986) to generate
squeezed light is sketched. The light source for this experiment is a ring laser contain-

Nd:YAG laser ω0 ω0
with output
second-harmonic crystal 2ω0
at 2ω0
χ(2) crystal
M1 ω0
parametric Idler
at ω0
Squeezed light output at ω0

Homodyne detector

Fig. 15.2 Simpli¬ed schematic of an experiment to generate squeezed light. ˜PBS™ stands
for ˜polarizing beam splitter™ and ˜MLO ™ is a mirror for the local oscillator (LO) beam at ωLO .
(Adapted from Wu et al. (1986).)
Experimental squeezed-light generation

ing a diode-laser-pumped, neodymium-doped, yttrium aluminum garnet (Nd:YAG)
crystal”which produces an intense laser beam at the ¬rst-harmonic frequency ω0 ”
and an intracavity, second-harmonic crystal (barium sodium niobate), which produces
a strong beam at the second-harmonic frequency 2ω0 . The solid lines represent beams
at the ¬rst harmonic, corresponding to a wavelength of 1.06 µm, and the dashed lines
represent beams at the second harmonic, corresponding to a wavelength of 0.53 µm.
The two outputs of the ring laser source are each linearly polarized along orthogo-
nal axes, so that the polarizing beam splitter (PBS) can easily separate them into two
beams. The ¬rst-harmonic beam is transmitted through the polarizing beam splitter
and then directed downward by the mirror MLO . This beam serves as the local oscilla-
tor (LO) for the homodyne detector, and the mirror MLO is mounted on a translation
stage so as to be able to adjust the LO phase θLO . The second-harmonic beam is
directed downward by the polarizing beam splitter, and it provides the pump beam of
the optical parametric oscillator (OPO).
The heart of the experiment is the OPO system, which is operated just below the
threshold of oscillation, where a maximum of squeezed-light generation occurs. The
OPO consists of a χ(2) crystal (lithium niobate doped with magnesium oxide), sur-
rounded by the two confocal mirrors M1 and M2. The crystal is cut so that the signal
and idler modes have the same frequency, ω0 , and are also collinear. The entrance mir-
ror M1 has an extremely high re¬‚ectivity at the ¬rst-harmonic frequency ω0 , but only
a moderately high re¬‚ectivity at the second-harmonic frequency 2ω0 . Thus M1 allows
the second-harmonic, pump light to enter the OPO, while also serving as one of the
re¬‚ecting surfaces de¬ning a resonant cavity for both the ¬rst- and second-harmonic
frequencies. This arrangement enhances the pump intensity inside the crystal.
By contrast, the exit mirror M2 has an extremely high re¬‚ectivity for the second-
harmonic frequency, but only a moderately high re¬‚ectivity at the ¬rst-harmonic fre-
quency. Thus the mirrors M1 and M2 form a resonator”for both the ¬rst- and second-
harmonic frequencies”but at the same time M2 allows the degenerate signal and idler
beams”at the ¬rst-harmonic frequency ω0 ”to escape toward the homodyne detector.
In Fig. 15.2, the left and right ports of the box indicating the homodyne detector
correspond to two ports of a central balanced beam splitter which respectively emit the
signal and local oscillator beams. The output ports of the beam splitter are followed
by two balanced photodetectors, and the detected outputs of the photodetectors are
then subtracted by means of a balanced di¬erential ampli¬er. Finally, the output of
the di¬erential ampli¬er is fed into a spectrum analyzer, as explained in Section 9.3.3.
It is important to emphasize that the extremely high re¬‚ectivity, for frequency ω0 ,
of the entrance mirror, M1, blocks out vacuum ¬‚uctuations from entering the system,
thereby preventing them from contributing unwanted vacuum ¬‚uctuation noise at this
frequency. As explained in Section 15.2-C, the asymmetry in the re¬‚ectivities of the
mirrors M1 and M2 at the ¬rst-harmonic frequency ω0 allows more squeezing of the
light to occur outside than inside the cavity.
The resulting data is shown in Fig. 15.3, where the output noise voltage, V (θ), of
the spectrum analyzer associated with the homodyne detector is plotted versus the
local oscillator phase θ = θLO , for a ¬xed intermediate frequency of 1.8 MHz.
The crucial comparison of this noise output is with the noise from the standard
Nonclassical states of light


θ θ + π θ + π θ θ + π

Fig. 15.3 Homodyne-detector, spectrum-analyzer output noise voltage (i.e. the rms noise
voltage at an intermediate frequency of 1.8 MHz) versus the local oscillator phase. (Repro-
duced from Wu et al. (1986).)

quantum limit (SQL), which is determined either by blocking the output of the OPO,
or by changing the temperature of the lithium niobate crystal so that the signal and
idler modes are detuned away from the cavity resonance. The SQL level”which repre-
sents the noise from vacuum ¬‚uctuations”is indicated by the dashed line in this ¬gure.
By inspection of these and similar data, the authors concluded that, in the absence
of linear attenuation, the light output from the OPO would have been squeezed by a
factor e2r > 10. This means the semiminor axis of the noise ellipse of the Gaussian
Wigner function in phase space would be more than ten times the semimajor axis.
Strictly speaking, this experiment in squeezed state generation and detection did
not involve exactly degenerate photon pairs, since the detected photons were symmet-
rically displaced from exact degeneracy by 1.8 MHz (within a bandwidth of 100 kHz).
The exact conservation of energy in parametric down-conversion guarantees that the
shifts in the two frequencies are anti-correlated, i.e. ωi = ω0 + ∆ωi and ωs = ω0 + ∆ωs ,
with ∆ωi = ’∆ωs . Thus the beat notes produced by interference of the upper and
lower sidebands with the local oscillator are exactly the same. Both sidebands are de-
tected in the balanced homodyne detector, but their phases are correlated in just such
a way that for one particular phase θLO of the local oscillator”which can be adjusted
by the piezoelectric translator that controls the location of the mirror MLO ”the sen-
sitive spots of homodyne detection coincide with the least noisy quadrature of the
squeezed light. This is true in spite of the fact that the two conjugate photons may
not be exactly degenerate in frequency, as long as they are inside the gain-narrowed
line width of the optical parametric ampli¬cation pro¬le just below threshold. The
noise analysis for this case of slightly nondegenerate parametric down-conversion can
be found in Kimble (1992).
Number states

15.4 Number states
We have seen in Section 2.1.2 that the number states provide a natural basis for the
Fock space of a single mode of the radiation ¬eld. Any state, whether pure or mixed,
can be expressed in terms of number states. By de¬nition, the variance of the number
operator vanishes for a number state |n ; so evaluating eqn (9.58) for the Mandel
Q-parameter of the number state |n gives

V (N ) ’ N
Q (|n ) ≡ = ’1 , (15.132)

where X = n |X| n . Thus the number states saturate the general inequality
Q (|Ψ ) ’1. Furthermore, every state with negative Q is nonclassical; consequently,
a pure number state is as nonclassical as it can be. Since this is true no matter how
large n is, the classical limit cannot be identi¬ed with the large-n limit. Further ev-
idence of the nonclassical nature of number states is provided by eqn (5.153), which
shows that the Wigner distribution W (±) for the single-photon number state |1 is
negative in a neighborhood of the origin in phase space.

15.4.1 Single-photon wave packets from SDC
States containing exactly one photon in a classical traveling-wave mode, e.g. a Gaussian
wave packet, are of particular interest in contemporary quantum optics. In the approx-
imate sense discussed in Section 7.8 the photon is localized within the wave packet.
With almost complete certainty, such a single-photon wave packet state would register
a single click when it falls on an ideal photodetector with unit quantum e¬ciency.
The ¬rst experiment demonstrating the existence of single-photon wave packet
states was performed by Hong and Mandel (1986). The single-photon state is formed
by one of the pair of photons emitted in spontaneous down-conversion, using the appa-
ratus shown in Fig. 15.4. An argon-ion UV laser beam at a wavelength of » = 351 nm
enters a crystal”potassium dihydrogen phosphate (KDP)”with a χ(2) nonlinearity.
Conjugate down-converted photon pairs are generated on opposite sides of the UV

Amp. & Counter
Gated PDP
counter 11/23+

Argon-ion UV laser KDP
B Amp. & Counter
Lens Field Interference
lens filter

Fig. 15.4 Schematic of Hong and Mandel™s experiment to generate and detect single-photon
wave packets. (Reproduced from Hong and Mandel (1986).)
Nonclassical states of light

beam wavelength at the signal and idler wavelengths of 746 nm and 659 nm, respec-
tively, and enter the photon counters A and B. Counter B is gated by the pulse derived
from counter A, for a counting time interval of 20 ns.
Whenever a click is registered by counter A”and the less-than-unity quantum
e¬ciency of counter B is accounted for”there is one and only one click at counter B.
This is shown in Fig. 15.5, in which the derived probability p(n) for a count at counter
B”conditioned on the detection of a signal photon at counter A”is plotted versus
the photon number n.
The data show that within small uncertainties (indicated by the cross-hatched
p(n) = δn,1 ; (15.133)
that is, the idler photons detected by B have been prepared in the single-photon
number state |n = 1 . In other words, the moment that the click goes o¬ in counter A,
one can, with almost complete certainty, predict that there is one and only one photon
within a well-de¬ned wave packet propagating in the idler channel. The Mandel Q-
parameter derived from these data, Q = ’1.06 ± 0.11, indicates that this state of light
is maximally nonclassical, as expected for a number state.

15.4.2 Single photons on demand
The spontaneous down-conversion events that yield the single-photon wave packet
states occur randomly, so there is no way to control the time of emission of the wave
packet from the nonlinear crystal. Recently, work has been done on a controlled pro-
duction process in which the time of emission of a well-de¬ned single-photon wave
packet is closely determined. Such a deterministic emission process for an individual
photon wave packet is called single photons on demand or a photon gun. One such
method involves quantum dots placed inside a high-Q cavity. When a single electron is
controllably injected into the quantum dot”via the Coulomb blockade mechanism”
the resonant enhancement of the rate of spontaneous emission by the high-Q cavity
produces an almost immediate emission of a single photon. Deterministic production
of single-photon states can be useful for quantum information processing and quantum
computation, since often the photons must be synchronized with the computer cycles
in a controllable manner.



Fig. 15.5 The derived probability p(n) for
the detection of n idler photons conditioned
on the detection of a single signal photon in
the 1986 experiment of Hong and Mandel. The
cross-hatched regions indicate the uncertain-
ties of p(n). (Reproduced from Hong and Man-
0 1 2 3 4
del (1986).)

15.4.3 Number states in a micromaser
Number states have been produced in a standing-wave mode inside a cavity, as opposed
to the traveling-wave packet described above. In the microwave region, number states
inside a microwave cavity have been produced by means of the micromaser described
in Section 12.3. This is accomplished by two methods described below.
In the ¬rst method, a completed measurement of the ¬nal state of the atom after it
exits the cavity allows the experimenter to know”with certainty”whether the atom
has made a downwards transition inside the cavity. Combining this knowledge with
the conservation of energy determines”again with certainty”the number state of the
cavity ¬eld.
In the second method, an exact integer number of photons is maintained inside
the cavity by means of a trapping state (Walther, 2003). According to eqn (12.21),
the e¬ective Rabi frequency for an on-resonance, n-photon state is „¦n = 2g (n + 1),
where g is the coupling constant of the two-level atom with the cavity mode. The

Rabi period is therefore Tn = 2π/„¦n = π/ g n + 1 . If the interaction time Tint of
the atom with the ¬eld satis¬es Tint = kTn , where k is an integer, then an atom that
enters the cavity in an excited state will leave in an excited state. Thus the number of
photons in the cavity will be unchanged”i.e. trapped ”if the condition

n + 1gTint = kπ (15.134)

is satis¬ed.
Trapping states are characterized by the number of photons remaining in the cav-
ity, and the number of Rabi cycles occurring during the passage of an atom through the
cavity. Thus the trapping state (n, k) = (1, 1) denotes a state in which a one-photon,
one-Rabi-oscillation trapped ¬eld state is maintained by a continuous stream of Ry-
dberg atoms prepared in the upper level. Experiments show that, under steady-state
excitation conditions, the one-photon cavity state is stable. Although this technique
produces number states of microwave photons in a beautifully simple and clean way, it
is di¬cult to extract them from the high-Q superconducting cavity for use in external

15.5 Exercises
15.1 Quadrature variances
(1) Use eqn (15.14) and the canonical commutation relations to calculate : X 2 : and
to derive eqns (15.17) and (15.18).
(2) Are the conditions (15.19) and (15.20) su¬cient, as well as necessary? If not, what
are the su¬cient conditions?
(3) Explain why number states and coherent states are not squeezed states.
(4) Is the state |ψ = cos θ |0 + sin θ |1 squeezed for any value of θ? In other words,
for a given θ, is there a quadrature X with VN (X) < 0?
Nonclassical states of light

15.2 Squeezed number state
Number states are not squeezed, but it is possible to squeeze a number state. Consider
|ζ, n = S (ζ) |n .
(1) Evaluate the Mandel Q-parameter for this state and comment on the result.
(2) What quadrature exhibits maximum squeezing?

Displaced squeezed states and squeezed coherent states—
Use the properties of S (ζ) and D (±) to derive the relations (15.52)“(15.54).

Photon statistics for the displaced squeezed state—
Carry out the integral in eqn (15.67) using polar coordinates and combine this with
the other results to get eqn (15.69).

Squeezing of emitted light—
(1) Carry out the calculations required to derive eqns (15.125) and (15.126).
(2) Use these results to derive eqn (15.128).
Linear optical ampli¬ers—

Generally speaking, an optical ampli¬er is any device that converts a set of input modes
into a set of output modes with increased intensity. The only absolutely necessary
condition is that the creation and annihilation operators for the two sets of modes
must be connected by a unitary transformation. Paradoxically, this level of generality
makes it impossible to draw any general conclusions; consequently, further progress
requires some restriction on the family of ampli¬ers to be studied.
To this end, we consider the special class of unitary input“output transformations
that can be expressed as follows. The annihilation operator for each output mode
is a linear combination, with c-number coe¬cients, of the creation and annihilation
operators for the input modes. Devices of this kind are called linear ampli¬ers. We
note in passing that linear ampli¬ers are quite di¬erent from laser oscillator-ampli¬ers,
which typically display the highly nonlinear phenomenon of saturation (Siegman, 1986,
Sec. 4.5).
For typical applications of linear, optical ampli¬ers”e.g. optical communication or
the generation of nonclassical states of light”it is desirable to minimize the noise added
to the input signal by the ampli¬er. The ¬rst source of noise is the imperfect coupling
of the incident signal into the ampli¬er. Some part of the incident radiation will be
scattered or absorbed, and this loss inevitably adds partition noise to the transmitted
signal. In the literature, this is called insertion-loss noise, and it is gathered together
with other e¬ects”such as noise in the associated electronic circuits”into the category
of technical noise. Since these e¬ects vary from device to device, we will concentrate
on the intrinsic quantum noise associated with the act of ampli¬cation itself.
In the present chapter we ¬rst discuss the general properties of linear ampli¬ers
and then describe several illustrative examples. In the ¬nal sections we present a
simpli¬ed version of a general theory of linear ampli¬ers due to Caves (1982), which
is an extension of the earlier work of Haus and Mullen (1962).

16.1 General properties of linear ampli¬ers
The degenerate optical parametric ampli¬er (OPA) studied in Section 15.2 is a linear
device, by virtue of the assumption that depletion of the pump ¬eld can be neglected. In
the application to squeezing, the input consists of vacuum ¬‚uctuations”represented
by b2,in (t)”entering the mirror M2, and the corresponding output is the squeezed
state”represented by b2,out (t)”emitted from M2. Both the input and the output
have the carrier frequency ω0 . Rather than extending this model to a general theory
of linear ampli¬ers that allows for multiple inputs and outputs and frequency shifts
Linear optical ampli¬ers—

between them, we choose to explain the basic ideas in the simplest possible context:
linear ampli¬ers with a single input ¬eld and a single output ¬eld”denoted by bin (t)
and bout (t) respectively”having a common carrier frequency.
We will also assume that the characteristic response frequency of the ampli¬er and
the bandwidth of the input ¬eld are both small compared to the carrier frequency. This
narrowband assumption justi¬es the use of the slowly-varying amplitude operators
introduced in Chapter 14, but it should be remembered that both the input and the
output are reservoir modes that do not have sharply de¬ned frequencies. Just as in the
calculation of the squeezing of the emitted light in Section 15.2, the input and output
are described by continuum modes.
All other modes involved in the analysis are called internal modes of the ampli¬er.
In the sample“reservoir language, the internal modes consist of the sample modes and
any reservoir modes other than the input and output. A peculiarity of this jargon is
that some of the ˜internal™ modes are ¬eld modes, e.g. vacuum ¬‚uctuations, that exist
in the space outside the physical ampli¬er.
The de¬nition of the ampli¬er is completed by specifying the Heisenberg-picture
density operator ρ that describes the state of both the input ¬eld and the internal
modes of the ampli¬er. This is the same thing as specifying the initial value of the
Schr¨dinger-picture density operator. Since we want to use the ampli¬er for a broad
range of input ¬elds, it is natural to require that the operating state of the ampli¬er is
independent of the incident ¬eld state. This condition is imposed by the factorizability
ρ = ρin ρamp , (16.1)
where ρin and ρamp respectively describe the states of the input ¬eld and the ampli¬er.
In the generic states of interest for communications, the expectation value of the
input ¬eld does not vanish identically:

bin (t) = Tr [ρin bin (t)] = 0 . (16.2)

Situations for which bin (t) = 0 for all t”e.g. injecting the vacuum state or a
squeezed-vacuum state into the ampli¬er”are to be treated as special cases.
The identi¬cation of the measured values of the input and output ¬elds with the
expectation values bin (t) and bout (t) runs into the apparent di¬culty that the
annihilation operators bin (t) and bout (t) do not represent measurable quantities. To
see why this is not really a problem, we recall the discussion in Section 9.3, which
showed that both heterodyne and homodyne detection schemes e¬ectively measure a
hermitian quadrature operator. For example, it is possible to measure one member of
the conjugate pair (Xβ,in (t) , Yβ,in (t)), where

1 ’iβ †
e bin (t) + eiβ bin (t) ,
Xβ,in (t) =
2 (16.3)
1 ’iβ †
e bin (t) ’ eiβ bin (t) .
Yβ,in (t) =
The quadrature angle β is determined by the relative phase between the input signal
and the local oscillator employed in the detection scheme. The operational signi¬cance
General properties of linear ampli¬ers

of the complex expectation value bin (t) is established by carrying out measurements
of Xβ,in (t) for several quadrature angles and using the relation
1 ’iβ †
= Re e’iβ bin (t)
bin (t) + eiβ bin (t)
Xβ,in (t) = e . (16.4)
With this reassuring thought in mind, we are free to use the algebraically simpler
approach based on the annihilation operators. An important example is provided by
the phase transformation,

bin (t) ’ bin (t) = e’iθ bin (t) , (16.5)

of the annihilation operator. The corresponding transformation for the quadratures,

Xβ,in (t) ’ Xin (t) = Xβ,in (t) cos θ + Yβ,in (t) sin θ , (16.6)
Yβ,in (t) ’ Yin (t) = Yβ,in (t) cos θ ’ Xβ,in (t) sin θ , (16.7)

represents a rotation through the angle θ in the (X, Y )-plane. As explained in Section
8.1, these transformations are experimentally realized by the use of phase shifters.

16.1.1 Phase properties of linear ampli¬ers
From Section 14.1.1-C, we know that the noise properties of the input/output ¬elds are
described by the correlation functions of the ¬‚uctuation operators, δbγ (ω) ≡ bγ (ω) ’
bγ (ω) , where γ = in, out. Thus the input/output noise correlation functions are
de¬ned by
1 † †
Kγ (ω, ω ) = δbγ (ω) δbγ (ω ) + δbγ (ω ) δbγ (ω) (γ = in, out) . (16.8)
The de¬nitions (14.98) and (14.107) relating the input/output ¬elds to the reservoir
operators allow us to apply the conditions (14.27) and (14.34) for phase-insensitive
noise. The input/output noise reservoir is phase insensitive if the following conditions
are satis¬ed.
(1) The noise in di¬erent frequencies is uncorrelated, i.e.

Kγ (ω, ω ) = Nγ (ω) 2πδ (ω ’ ω ) , (16.9)


Nγ (ω) = δbγ (ω) δbγ (ω) + (16.10)
is the noise strength.
(2) The phases of the ¬‚uctuation operators are randomly distributed, so that

δbγ (ω) δbγ (ω ) = 0 . (16.11)

With this preparation, we are now ready to introduce an important division of the
family of linear ampli¬ers into two classes. A phase-insensitive ampli¬er is de¬ned
by the following conditions.
Linear optical ampli¬ers—

(i) The output ¬eld strength, bout (ω) , is invariant under phase transformations
of the input ¬eld.
(ii) If the input noise is phase insensitive, so is the output noise.
Condition (i) means that the only e¬ect of a phase shift in the input ¬eld”i.e. a rota-
tion of the quadratures”is to produce a corresponding phase shift in the output ¬eld.
Condition (ii) means that the noise added by the ampli¬er is randomly distributed in
phase. An ampli¬er is said to be phase sensitive if it fails to satisfy either one of
these conditions.
In addition to the categories of phase sensitive and phase-insensitive, ampli¬ers
can also be classi¬ed according to their physical con¬guration. In the degenerate OPA
the gain medium is enclosed in a resonant cavity, and the input ¬eld is coupled into
one of the cavity modes. The cavity mode in turn couples to an output mode to
produce the ampli¬ed signal. This con¬guration is called a regenerative ampli¬er,
which is yet another term borrowed from radio engineering. One way to understand
the regenerative ampli¬er is to visualize the cavity mode as a traveling wave bouncing
back and forth between the two mirrors. These waves make many passes through the
gain medium before exiting through the output port.
The advantage of greater overall gain, due to multiple passes through the gain
medium, is balanced by the disadvantage that the useful gain bandwidth is restricted
to the bandwidth of the cavity. This restriction on the bandwidth can be avoided
by the simple expedient of removing the mirrors. In this con¬guration, there are no
re¬‚ected waves”and therefore no multiple passes through the gain medium”so these
devices are called traveling-wave ampli¬ers.

16.2 Regenerative ampli¬ers
In this section we take advantage of the remarkable versatility of the spontaneous
down-conversion process to describe three regenerative ampli¬ers, two phase insensi-
tive and one phase sensitive.

16.2.1 Phase-insensitive ampli¬ers
A modi¬cation of the degenerate OPA design of Section 15.2 provides two examples
of phase-insensitive ampli¬ers. In the modi¬ed design, shown in Fig. 16.1, the signal
and idler modes are frequency degenerate, but not copropagating. In the absence
of the mirrors M1 and M2, down-conversion of the pump radiation would produce
symmetrical cones of light around the pump direction, but this azimuthal symmetry
is broken by the presence of the cavity axis joining the two mirrors. This arrangement
picks out a single pair of conjugate modes: the idler and the signal.
The boundary conditions at the mirrors de¬ne a set of discrete cavity modes, and
the fundamental cavity mode”which we will call the idler”is chosen to satisfy the
phase-matching condition ω0 = ωP /2. The discrete idler mode is represented by a
single operator a (t). On the other hand, the signal mode is a traveling wave with
propagation direction determined by the phase-matching conditions in the nonlin-
ear crystal. Thus the signal mode is represented by a continuous family of operators
bsig,„¦ (t).
Regenerative ampli¬ers

Fig. 16.1 Two examples of phase-insensitive optical ampli¬ers based on down-conversion in
a χ(2) crystal: (a) taking the signal-mode in- and out-¬elds as the input and output of the
ampli¬er de¬nes a phase-preserving ampli¬er; (b) taking the signal-mode in-¬eld as the input
and the out-¬eld through mirror M2 as the output de¬nes a phase-conjugating ampli¬er.

The ¬rst step in dividing the world into sample and reservoirs is to identify the
sample. From the experimental point of view, the sample in this case evidently consists
of the atoms in the nonlinear crystal, combined with the idler mode in the cavity. The
theoretical description is a bit simpler, since”as we have seen in Chapter 13”the
atoms in the crystal are only virtually excited. This means that the e¬ect of the
atoms is completely accounted for by the signal“idler coupling constant; consequently,
the sample can be taken to consist of the idler mode alone. There are then three
environmental reservoirs: the signal reservoir represented by the operators bsig,„¦ (t)
and two noise reservoirs represented by the operators b1,„¦ (t) and b2,„¦ (t) describing
radiation entering and leaving the cavity through the mirrors.
Analyzing this model requires a slight modi¬cation of the method of in- and out-
¬elds described in Section 14.3. The new feature requiring the modi¬cation is the form
of the coupling between the idler (sample) mode and the signal (reservoir) mode. This
term in the interaction Hamiltonian HSE does not have the generic form of eqn (14.88);
instead, it is described by eqn (15.7). In a notation suited to the present discussion:

D (ω)
vP („¦) e’iωP t b† a† ’ vP („¦) eiωP t absig,„¦ , (16.12)

HSE =i d„¦ sig,„¦


where vP („¦) is the strength of the coupling”induced by the nonlinear crystal”
between the signal mode, the idler mode, and the pump ¬eld. The presence of the
products b† a† and absig,„¦ represents the fact that the signal and idler photons are
created and annihilated in pairs in down-conversion.
After including this new term in HSE , the procedures explained in Section 14.3 can
be applied to the present problem. The interaction term in eqn (16.12) leads to the
modi¬ed Heisenberg equations

∞ ∞
D (ω) D (ω)
d †
a (t) = d„¦ vP („¦) bsig,„¦ (t) + d„¦ vm („¦) bm,„¦ (t) ,
dt 2π 2π
0 0
D (ω)
vP („¦) a† (t) ,
bsig,„¦ (t) = ’i („¦ ’ ω0 ) bsig,„¦ (t) + (16.14)
dt 2π
Linear optical ampli¬ers—

where vm („¦) describes the coupling of the idler to the noise modes, and a (t) =
a (t) exp (iω0 t), etc. The equations for the noise reservoir operators bm,„¦ (t) have the
generic form of eqn (14.89). The retarded and advanced solutions of eqn (16.14) for
the signal mode are respectively
bsig,„¦ (t) = bsig,„¦ (t0 ) e’i(„¦’ω0 )(t’t0 ) + vP („¦) dt a† (t ) e’i(„¦’ω0 )(t’t ) (16.15)

’i(„¦’ω0 )(t’t1 )
dt a† (t ) e’i(„¦’ω0 )(t’t ) .
’ vP („¦)
bsig,„¦ (t) = bsig,„¦ (t1 ) e (16.16)

The corresponding results for the noise reservoir operators, bm,„¦ (t), are given by eqns
(14.94) and (14.105).
After substituting the retarded solutions for bsig,„¦ (t) and bm,„¦ (t) into the equation
of motion (16.13), we impose the Markov approximation by assuming that the idler
mode is coupled to a broad band of excitations in the two mirror reservoirs and in
the signal reservoir. The general discussion in Section 14.3 yields the broadband rule

vm („¦) ∼ κm for the noise modes. The signal mode must be treated di¬erently, since
vP („¦) is proportional to the classical pump ¬eld, which has a well-de¬ned phase θP .

In this case the broadband rule is vP („¦) ∼ gP exp (iθP ), where gP is positive.
The contributions from the noise reservoirs yield the expected loss term ’κC a (t) /2,
but the contribution from the signal reservoir instead produces a gain term +gP a (t) /2.
This new feature is another consequence of the fact that the down-conversion mech-
anism creates and annihilates the signal and idler photons in pairs. Emission of a
photon into the continuum signal reservoir can never be reversed; therefore, the asso-
ciated idler photon can also never be lost. On the other hand, the inverse process”in
which a signal“idler pair is annihilated to create a pump photon”does not contribute
in the approximation of constant pump strength. Consequently, in the linear approx-
imation the coupling of the signal and idler modes through down-conversion leads
to an increase in the strength of both signal and idler ¬elds at the expense of the
(undepleted) classical pump ¬eld.
After carrying out these calculations, one ¬nds the retarded Langevin equation for
the idler mode:
√ √

d 1 †
a (t) = ’ (κC ’ gP ) a (t)+ gP eiθP bsig,in (t)+ κ1 b1,in (t)+ κ2 b2,in (t) , (16.17)
dt 2
where ∞
bsig,„¦ (t0 ) e’i(„¦’ω0 )(t’t0 )
bsig,in (t) = (16.18)


is the signal in-¬eld, and the in-¬elds for the mirrors are given by eqn (14.98). For
gP > κC , eqn (16.17) predicts an exponential growth of the idler ¬eld that would
violate the weak-¬eld assumptions required for the model. Consequently”just as in
the treatment of squeezing in Section 15.2-A”the pump ¬eld must be kept below the
threshold value (gP < κC ).
Regenerative ampli¬ers

We now imitate the empty-cavity analysis of Section 14.3.3 by transforming eqn
(16.17) to the frequency domain and solving for a (ω), with the result
√ √
√ †
eiθP gP bsig,in (’ω) + κ1 b1,in (ω) + κ2 b2,in (ω)
a (ω) = . (16.19)
(κC ’ gP ) ’ iω

The input“output relation for the signal mode is obtained by equating the right sides
of eqns (16.15) and (16.16) and integrating over „¦ to get

gP eiθP a† (t)
bsig,out (t) = bsig,in (t) + (16.20)

in the time domain, or

gP eiθP a† (’ω)
bsig,out (ω) = bsig,in (ω) + (16.21)

in the frequency domain. The input“output relations for the mirror reservoirs are given
by the frequency-domain form of eqn (14.109):

b1,out (ω) = b1,in (ω) ’ κ1 a (ω) , (16.22)

b2,out (ω) = b2,in (ω) ’ κ2 a (ω) . (16.23)

A Phase-transmitting OPA
The ¬rst step in de¬ning an ampli¬er is to decide on the identity of the input and
output ¬elds. In other words: What is to be measured? For the ¬rst example, we choose
the in-¬eld and out-¬eld of the signal mode as the input and output ¬elds, i.e. bin (ω) =
bsig,in (ω) and bout (ω) = bsig,out (ω). The idler ¬eld and the two mirror reservoir in-
¬elds are then internal modes of the ampli¬er. Substituting these identi¬cations and
the solution (16.19) into eqn (16.21) yields the ampli¬er input“output equation

bout (ω) = P (ω) bin (ω) + · (ω) , (16.24)

where the coe¬cient
(κC + gP ) ’ iω
P (ω) = (16.25)
(κC ’ gP ) ’ iω

has the symmetry property
P (ω) = P — (’ω) , (16.26)
and the operator
√ †
gP eiθP ξC (’ω)
· (ω) = 1
2 (κC ’ gP ) ’ iω

gP eiθP † †
=1 κ1 b1,in (’ω) + κ2 b2,in (’ω) (16.27)
2 (κC ’ gP ) ’ iω

is called the ampli¬er noise.
Linear optical ampli¬ers—

This result shows that the noise added by the ampli¬er is entirely due to the noise
reservoirs associated with the mirrors. The absence of noise added by the atoms in
the nonlinear crystal is a consequence of the fact that the excitations of the atoms
are purely virtual. In most applications, only vacuum ¬‚uctuations enter through M1
and M2, but the following calculations are valid in the more general situation that
both mirrors are coupled to any phase-insensitive noise reservoirs. In particular, the
vanishing ensemble average of the noise operator · (ω) implies that the input“output
equation for the average ¬eld is
bout (ω) = P (ω) bin (ω) . (16.28)
Subtracting this equation from eqn (16.24) yields the input“output equation
δbout (ω) = P (ω) δbin (ω) + · (ω) (16.29)
for the ¬‚uctuation operators.
The ¬rst step in the proof that this ampli¬er is phase insensitive is to use eqn
(16.28) to show that the e¬ect of a phase transformation applied to the input ¬eld is

bout (ω) = P (ω) bin (ω) = eiθ bout (ω) . (16.30)
In other words, changes in the phase of the input signal are simply passed through the
ampli¬er. Ampli¬ers with this property are said to be phase transmitting. The ¬eld
strength bout (ω) is evidently unchanged by a phase transformation; therefore the
ampli¬er satis¬es condition (i) of Section 16.1.1.
Turning next to condition (ii), we note that the operators δbin (ω) and · (ω) are lin-
ear functions of the uncorrelated reservoir operators bsig,„¦ (t0 ) and bm,„¦ (t0 ) (m = 1, 2).
This feature combines with eqn (16.29) to give
Kout (ω, ω ) = P (ω) P — (ω ) Kin (ω, ω ) + Kamp (ω, ω ) , (16.31)
· (ω) · † (ω ) + · † (ω ) · (ω)
Kamp (ω, ω ) = (16.32)
is the ampli¬er“noise correlation function. Since · (ω) is a linear combination of the
mirror noise operators, the assumption that the mirror noise is phase insensitive guar-
antees that
Kamp (ω, ω ) = Namp (ω) 2πδ (ω ’ ω ) , (16.33)
where Namp (ω) is the ampli¬er noise strength. If the correlation function Kin (ω, ω )
satis¬es eqn (16.9), then eqns (16.31) and (16.33) guarantee that Kout (ω, ω ) does
also. The output noise strength is then given by
Nout (ω) = |P (ω)|2 Nin (ω) + Namp (ω) . (16.34)
It is also necessary to verify that the output noise satis¬es eqn (16.11), when the
input noise does. This is an immediate consequence of the phase insensitivity of the
ampli¬er noise and the input“output equation (16.29), which together yield
δbout (ω) δbout (ω ) = P (ω) P (ω ) δbin (ω) δbin (ω ) . (16.35)
Putting all this together shows that the ampli¬er is phase insensitive, since it satis¬es
conditions (i) and (ii) from Section 16.1.1.
Regenerative ampli¬ers

For this ampli¬er, it is reasonable to de¬ne the gain as the ratio of the output ¬eld
strength to the input ¬eld strength:
| bout (ω + ω0 ) |2
bout (ω)
G (ω) = = . (16.36)
2 2
| bin (ω + ω0 ) |
bin (ω)
Using eqn (16.28) yields the explicit expression
2 2
(κC + gP ) /4 + (ω ’ ω0 )
G (ω ’ ω0 ) = , (16.37)
2 2
(κC ’ gP ) /4 + (ω ’ ω0 )
which displays the expected peak in the gain at the resonance frequency ω0 . An alter-
native procedure is to de¬ne the gain in terms of the quadrature operators, and then
to show”see Exercise 16.1”that the gain is the same for all quadratures.

B Phase-conjugating OPA
The crucial importance of the choice of input and output ¬elds is illustrated by using
the apparatus shown in Fig. 16.1 to de¬ne a quite di¬erent ampli¬er. In this version the
input ¬eld is still the signal-mode in-¬eld bsig,in (ω), but the output ¬eld is the out-¬eld
b2,out (ω) for the mirror M2. The internal modes are the same as before. The input“
output equation for this ampli¬er”which is derived from eqn (16.23) by using the
solution (16.19) and the identi¬cations bin (ω) = bsig,in (ω) and bout (ω) = b2,out (ω)”
has the form

bout (ω) = C (ω) eiθP bin (’ω) + · (ω) . (16.38)
The coe¬cient C (ω) and the ampli¬er noise operator are respectively given by

κ2 g P
C (ω) = ’ 1 (16.39)
(κC ’ gP ) ’ iω


(κ1 ’ κ2 ’ gP ) ’ iω
κ1 κ2
b2,in (ω) ’
· (ω) = b1,in (ω) . (16.40)
2 (κC ’ gP ) ’ iω 2 (κC ’ gP ) ’ iω
1 1

The important di¬erence from eqn (16.24) is that the output ¬eld depends on the
adjoint of the input ¬eld. Note that C (ω) has the same symmetry as P (ω):
C (ω) = C — (’ω) . (16.41)
The ensemble average of eqn (16.38) is

bout (ω) = C (ω) bin (’ω) , (16.42)

so the phase transformation bin (ω) ’ bin (ω) = exp (iθ) bin (ω) results in

bout (ω) = e’iθ C (ω) bin (’ω) = e’iθ bout (ω) . (16.43)

Instead of being passed through the ampli¬er unchanged, the phasor exp (iθ) is re-
placed by its conjugate. A device with this property is called a phase-conjugating
Linear optical ampli¬ers—

This ampli¬er nevertheless satis¬es condition (i) of Section 16.1.1, since
2 2
bout (ω) = bout (ω) . (16.44)

The argument used in Section 16.2.1-A to establish condition (ii) works equally well
here; therefore, the alternative design also de¬nes a phase-insensitive ampli¬er. The
form of the input“output relation in this case suggests that the gain should be de¬ned
bout (ω) κ2 g P
= |C (ω)|2 = 1
G (ω) = . (16.45)
(κC ’ gP )2 + ω 2

bin (’ω) 4

16.2.2 Phase-sensitive OPA
In the design shown in Fig. 16.2 the ¬elds entering and leaving the cavity through
the mirror M1 are designated as the input and output ¬elds respectively, i.e. bin (t) =
b1,in (t) and bout (t) = b1,out (t). The degenerate signal and idler modes of the cavity
and the input ¬eld b2,in (t) for the mirror M2 are the internal modes of the ampli¬er.
The input“output relation is obtained from eqn (15.117) by applying this identi¬cation
of the input and output ¬elds:

bout (ω) = P (ω) bin (ω) + C (ω) eiθP bin (’ω) + · (ω) . (16.46)

The phase-transmitting and phase-conjugating coe¬cients are respectively

κ1 (κC /2 ’ iω)
P (ω) = 1 ’ (16.47)
2 2
(κC /2 ’ iω) ’ |„¦P |

|„¦P | κ1
C (ω) = ’ . (16.48)
2 2
(κC /2 ’ iω) ’ |„¦P |

2, out
1, in

2, in
1, out

1 2

Fig. 16.2 A phase-sensitive ampli¬er based on the degenerate OPA. The heavy solid arrow
represents the classical pump; the thin solid arrows represent the input and output modes
for the mirror M1; and the dashed arrows represent the input and output for the mirror M2.
Regenerative ampli¬ers

The functions P (ω) and C (ω) satisfy eqns (16.26) and (16.41) respectively. The am-
pli¬er noise operator,

κ1 κ2 †
· (ω) = ’ (κC /2 ’ iω) b2,in (ω) + „¦P b2,in (’ω) , (16.49)
2 2
(κC /2 ’ iω) ’ |„¦P |

only depends on the reservoir operators associated with the mirror M2, so the ampli¬er
noise is entirely caused by vacuum ¬‚uctuations passing through the unused port at
According to eqn (16.46), the output ¬eld strength is


2 2
2 2
= |P (ω)| + |C (ω)|
bout (ω) bin (ω) bin (’ω)

+ 2 Re P (ω) C — (ω) bin (ω) bin (’ω) . (16.50)

We ¬rst test condition (i) of Section 16.1.1, by applying the phase transformation
(16.5) to the input ¬eld and evaluating the di¬erence between the transformed and
the original output intensities to get
2 2

δ bout (ω) = bout (ω) bout (ω)

e2iθ ’ 1 P (ω) C — (ω) bin (ω)
= 2 Re bin (’ω) .

Satisfying condition (i) would require the right side of this equation to vanish as an
identity in θ. The generic assumption (16.2) combined with the explicit forms of the
functions P (ω) and C (ω) makes this impossible; therefore, the ampli¬er is phase
This feature is a consequence of the fact that P (ω) and C (ω) are both nonzero, so

that the right side of eqn (16.46) depends jointly on bin (ω) and bin (’ω). A straight-
forward calculation shows that condition (ii) of Section 16.1.1 is also violated, even
for the simple case that the reservoir for the mirror M2 is the vacuum. Choosing an
appropriate de¬nition of the gain for a phase-sensitive ampli¬er is a bit trickier than
for the phase-insensitive cases, so this step will be postponed to the general treatment
in Section 16.4.
The alert reader will have noticed that the ampli¬ed signal is propagating back-
wards toward the source of the input signal. Devices of this kind are sometimes called
re¬‚ection ampli¬ers. This is not a useful feature for communications applications;
therefore, it is necessary to reverse the direction of the ampli¬er output so that it
propagates in the same direction as the input signal. Mirrors will not do for this task,
since they would interfere with the input. One solution is to redirect the ampli¬er
output by using an optical circulator, as described in Section 8.6. This device will
redirect the output signal, but it will not interfere with the input signal or add further
Linear optical ampli¬ers—

16.3 Traveling-wave ampli¬ers
The regenerative ampli¬ers discussed above enhance the nonlinear interaction for a
relatively weak cw pump beam by means of the resonant cavity formed by the mirrors
M1 and M2. This approach has the disadvantage of restricting the useful bandwidth to
that of the cavity. An alternative method is to remove the mirrors M1 and M2 to get
the con¬guration shown in Fig. 16.3, but this experimental simpli¬cation inevitably
comes at the expense of some theoretical complication.
The mirrors in the regenerative ampli¬ers perform two closely related functions.
The ¬rst is to guarantee that the ¬eld inside the cavity is a superposition of a discrete
set of cavity modes. In practice, the design parameters are chosen so that only one
cavity mode is excited. The position dependence of the ¬eld is then entirely given by
the corresponding mode function; in e¬ect, the cavity is a zero-dimensional system.
The second function”which follows from the ¬rst”is to justify the sample“reservoir
model that treats the discrete modes inside the cavity and the continuum of reservoir
modes outside the cavity as kinematically-independent degrees of freedom.
Removing the mirrors eliminates both of these conceptual simpli¬cations. Since
there are no discrete cavity modes, each of the continuum of external modes propagates
through the ampli¬er and interacts with the gain medium. Thus all ¬eld modes are
reservoir modes, and the sample consists of the atoms in the gain medium.
The interaction of the ¬eld with the gain medium could be treated by generalizing
the scattering description of passive, linear devices developed in Section 8.2, but this
approach would be quite complicated in the present application. The fact that the
sample occupies a ¬xed interval, say 0 z LS , along the propagation (z) axis
violates translation invariance and therefore conservation of momentum. Consequently,
the scattering matrix for the ampli¬er connects each incident plane wave, exp (ikz),
to a continuum of scattered waves exp (ik z).
We will avoid this complication by employing a position“space approach that
closely resembles the classical theory of parametric ampli¬cation (Yariv, 1989, Chap.
17). This technique can also be regarded as the Heisenberg-picture version of a method
developed to treat squeezing in a traveling-wave con¬guration (Deutsch and Garrison,

16.3.1 Laser ampli¬er
As a concrete example, we consider a sample composed of a collection of three-level
atoms”with the level structure displayed in Fig. 16.4”which is made into a gain

Fig. 16.3 A black box schematic of a travel-
ing-wave ampli¬er. The shaded box indicates
the gain medium and the ¬elds at the two ports
are the input and output values of the signal.
The vacuum ¬‚uctuations entering port 2 are
not indicated, since they do not couple to the
Traveling-wave ampli¬ers

Fig. 16.4 A three-level atom with a popula-
42 tion inversion between levels 1 and 2, main-
tained by an incoherent pump (dark double ar-
ω21 ω31 row) with rate RP . The solid arrow, the dashed
arrow, and the wavy arrows respectively rep-
resent the ampli¬ed signal transition, a nonra-
1 diative decay, and spontaneous emission.

medium by maintaining a population inversion between levels 1 and 2 through the
use of the incoherent pumping mechanism described in Section 14.5. By virtue of
the cylindrical shape of the gain medium, the end-¬re modes”i.e. ¬eld modes with
frequencies ω ω21 and propagation vectors, k, lying in a narrow cone around the
axis of the cylinder”will be preferentially ampli¬ed.
This new feature requires a modi¬cation of the reservoir assignment used for the
pumping calculation. The noise reservoir previously associated with the spontaneous
emission 2 ’ 1 is replaced by two reservoirs: (1) a noise reservoir associated with
spontaneous emission into modes with propagation vectors outside the end-¬re cone;
and (2) a signal reservoir associated with the end-¬re modes.
In the undepleted pump approximation, the back action of the atoms on the pump
¬eld can be ignored. This certainly cannot be done for the interaction with the signal
reservoir; after all, the action of the gain medium on the signal is the whole purpose of
the device. Thus the coupling of the entire collection of atoms to the signal reservoir
must be included by using the interaction Hamiltonian
HS1 = ’ S21 (t) d21 · E(+) (rn , t) + HC , (16.52)

where the sum runs over the atoms in the sample and the coordinate, rn , of the nth
atom is treated classically.
The description of the signal reservoir given above amounts to the assumption that
the Heisenberg-picture density operator for the input signal is a paraxial state with
respect to the z-axis; consequently, the contribution of the end-¬re modes to the ¬eld
operator can be represented in terms of the slowly-varying envelope operators φs (r, t)
appearing in eqn (7.33). We will assume that the ampli¬er has been designed so that
only one polarization will be ampli¬ed; consequently, only one operator φ (r, t) will be
Turning next to the input signal, we recall that a paraxial state is characterized
by transverse and longitudinal length scales Λ = 1/ (θk0 ) and Λ = 1/ θ2 k0 re-
spectively, where θ is the opening angle of the paraxial ray bundle. The scale lengths
Λ and Λ correspond respectively to the spot size and Rayleigh range of a classical
Gaussian beam. We choose θ so that Λ > 2RS and Λ LS , where RS and LS are
respectively the radius and length of the cylinder. This allows a further simpli¬cation
in which di¬raction is ignored and the envelope operator is approximated by
Linear optical ampli¬ers—

φ (r, t) = √ φ (z, t) , (16.53)
where σ = πRS . In this 1D approximation, the ¬eld expansion (7.33) and the commu-
tation relation (7.35) are respectively replaced by

ω0 (vg0 /c)
E(+) (r, t) = i e0 φ (z, t) ei(k0 z’ω0 t) (16.54)
2 0 n0 σ
φ (z, t) , φ† (z , t) = δ (z ’ z ) . (16.55)
The discretely distributed atoms and the continuous ¬eld are placed on a more
even footing by introducing the spatially coarse-grained operator density
Sqp (t) χ (z ’ zn ) .
Sqp (z, t) = (16.56)
∆z n
The averaging interval ∆z is chosen to satisfy the following two conditions. (1) A slab
with volume σ∆z contains many atoms. (2) The envelope operator φ (z, t) is essentially
constant over an interval of length ∆z. The function
χ (z ’ zn ) = θ (∆z/2 ’ z + zn ) θ (z ’ zn + ∆z/2) (16.57)
serves to con¬ne the n-sum to the atoms in a slab of thickness ∆z centered at z. The
atomic envelope operators are de¬ned by
Sqp (z, t) = S qp (z, t) eiωqp t ei[ψq (z,t)’ψp (z,t)] , (16.58)
where the phases satisfy
ψ2 (z, t) ’ ψ1 (z, t) = ∆0 t ’ k0 z . (16.59)
Using this notation, together with eqn (16.54), allows us to rewrite eqn (16.52) as
HS1 = ’i dz f S 21 (z, t) φ (z, t) ’ HC , (16.60)
(vg0 /c) ω0 d21 · e0
f≡ (16.61)
2 0σ
is the coupling constant.
The total electromagnetic part of the Hamiltonian for this 1D model is, therefore,

dzφ† (z, t) vg0 ∇z φ (z, t) + HS1 .
Hem = (16.62)
This leads to the corresponding Heisenberg equation
‚ ‚
φ (z, t) = f— S 12 (z, t) for 0
+ vg0 z LS , (16.63)
‚t ‚z
‚ ‚
+ vg0 φ (z, t) = 0 for z < 0 or z > LS (16.64)
‚t ‚z
for the ¬eld.
Traveling-wave ampli¬ers

The atomic operators are coupled to the reservoirs describing the incoherent pump
and spontaneous emission into o¬-axis modes; therefore, we insert eqn (16.60) into the
coarse-grained version of eqn (14.177) to ¬nd
S 12 (z, t) = [i∆0 ’ “12 ] S 12 (z, t) ’ f S 11 (z, t) ’ S 22 (z, t) φ (z, t) + ξ12 (z, t) .
The coarse-grained noise operator
1 (n)
ξ12 (t) χ (z ’ zn )
ξ12 (z, t) = (16.66)
∆z n

has the correlation function

ξ12 (z, t) ξ12 (z , t ) = nat σC12,12 δ (t ’ t ) δ (z ’ z ) , (16.67)

where δ (z ’ z ) is a coarse-grained delta function, nat is the density of atoms, and
C12,12 is an element of the noise correlation matrix discussed in Section 14.6.2.
In the strong-pump limit, the dephasing rate “12 = (w21 + RP ) /2 is large com-
pared to the other terms in eqn (16.65); therefore, applying the adiabatic elimination
rule (11.187) provides the approximate solution

S 22 (z, t) ’ S 11 (z, t) ξ12 (z, t)


. 19
( 27)