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)

2π

’∞

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) =

dω

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

2π

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