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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)
i
„¦P e’iωP t a†2 ’ „¦— eiωP t a2 ,
HSS = (15.96)
P
2
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
resonator.
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)
P
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)
„¦P
= , (15.99)
a† (t) a† (t)

’κC /2
„¦P
dt
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:
κC
a (ω) + „¦P a† (’ω) + ξC (ω) ,
’iωa (ω) = ’ (15.100)
2
κC † †
’iωa† (’ω) = ’ a (ω) + „¦— a (ω) + ξC (’ω) , (15.101)
P
2
Nonclassical states of light

which have the solution

(κC /2 ’ iω) ξC (ω) + „¦P ξC (’ω)
a (ω) = ,
2 2
(κC /2 ’ iω) ’ |„¦P |
(15.102)

(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
κC
ω = ω± = ’i ± |„¦P | . (15.104)
2
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
2
|„¦P |
1

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
1
V (a (t)) = , (15.110)
4 (κC /2)2 ’ |„¦P |2

so that
2
|„¦P |
1 κC 1
’2iβ
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 |
1
VN (X) = ’ (15.113)
4 κC /2 + |„¦P |
and
|„¦P |
1
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,

d„¦
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

with
√ [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
1
b2,out (t) e’iβ + b† iβ
Xout (t) = 2,out (t) e ,
2 (15.120)
1 †
’iβ
’ b2,out (t) e iβ
Yout (t) = b2,out (t) e
2i
in the time domain, or
1
b2,out (ω) e’iβ + b† iβ
Xout (ω) = 2,out (’ω) e ,
2 (15.121)
1
b2,out (ω) e’iβ ’ b† iβ
Yout (ω) = 2,out (’ω) e
2i
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
where
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
1
VN (Xout (ω ) , Xout (ω )) = ’ 2πδ (ω + ω ’ 2ω0 ) ,
2 [κC /2 + |„¦P |]2 + (ω ’ ω0 )2
(15.125)
|„¦P | κ2
1
2πδ (ω + ω ’ 2ω0 ) .
VN (Yout (ω ) , Yout (ω )) =
2 [κC /2 ’ |„¦P |]2 + (ω ’ ω0 )2
(15.126)
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-
tion,

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
1
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
1
SN (ω) > ’ ; (15.130)
8
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
1
’;
SN (ω) (15.131)
4
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
light.
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-

PBS
MLO
Nd:YAG laser ω0 ω0
with output
second-harmonic crystal 2ω0
θLO
Pump
at 2ω0
χ(2) crystal
M1 ω0
Optical
parametric Idler
Signal
oscillator
(OPO)
LO
M2
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

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