ńņš. 19 |

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 deļ¬ning property, V aā , a = 0, is strictly conserved by dissipa-

tive scatteringā”i.e. coupling to a zero-temperature reservoirā”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)| ā’ 0 as t ā’ ā . (15.92)

Even this result depends on the detection of a quadrature that is rotating at the

optical frequency Ļ0 . Detector response times are large compared to optical periods,

so even the reduced squeezing shown by eqn (15.91) would be extremely diļ¬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 generationā—

15.2

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 eļ¬ect discussed in Section 13.3.2; but two important changes in the experimental

arrangement are shown in Fig. 15.1. The ļ¬rst is that the Ļ(2) crystal is cut so as

to produce collinear phase matching with degenerate pairs (Ļ1 = Ļ2 = Ļ0 = ĻP /2) of

photons, and also anti-reļ¬‚ection coated for both the ļ¬rst- and the second-harmonic

frequencies Ļ0 and ĻP = 2Ļ0 . In this conļ¬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 Ļ0 = ĻP /2 and, therefore, also to the pump frequency ĻP .

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 Ļ0 and Ļ2

Degenerate

Pump Ļ1

Ļ2

signal and idler

at Ļ2 at Ļ1 = Ļ2 = Ļ0 = Ļ2 /2

Ļ2

1 2

crystal

Ļ(2)

Fig. 15.1 A simpliļ¬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 crystalā”pumped by the strong laser beam at the second-

harmonic frequency 2Ļ0 ā”becomes an optical parametric ampliļ¬er1 (OPA) for the

degenerate photon pairs at the ļ¬rst-harmonic frequency Ļ0 .

This device can be understood at the classical level in the following way. The Ļ(2)

nonlinearity couples the two weak down-converted light beams to the strong-pump

laser beam, so that the weak light signals can be ampliļ¬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 ampliļ¬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 eā’iĻP t = Re |EP | eiĪøP eā’iĻP t , (15.93)

with the two weak beams via the Ļ(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 ampliļ¬cation process is ignored, the mutual-

reinforcement mechanism leads to an exponential growth of both of the weak beams.

With suļ¬cient feedback from the mirrors surrounding the crystal, this ampliļ¬erā”like

that of a laserā”can begin to oscillate, and thereby become an optical parametric

oscillator (OPO). When operated just below the threshold of oscillation, the optical

parametric ampliļ¬er emits strongly squeezed states of light.

The resonant enhancement at the degenerate signal and idler frequencies justiļ¬es

the use of the phenomenological model Hamiltonian,

HS = HS0 + HSS , (15.94)

HS0 = Ļ0 aā a , (15.95)

i

ā„¦P eā’iĻP t aā 2 ā’ ā„¦ā— eiĻP 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 deļ¬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 deļ¬ned by eqn

(15.30). The coupling frequency ā„¦P , which is proportional to the product Ļ(2) EP ,

characterizes the strength of the nonlinear interaction. The term ā„¦P aā 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 ā„¦P is proportional to EP = |EP | exp (iĪøP ), 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 ā˜parametricā™ ampliļ¬er was originally introduced in microwave engineering. The ā˜para-

meterā™ in the optical case is the pump wave amplitude, which is assumed to be unchanged by the

nonlinear interaction.

Theory of squeezed-light generationā—

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-ļ¬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 ĪŗC

a (t) + ā„¦P ei(2Ļ0 ā’ĻP )t aā (t) + Ī¾C (t) ,

a (t) = ā’ (15.97)

dt 2

where a (t) = a (t) exp (iĻ0 t) is the slowly-varying envelope operator, ĪŗC = Īŗ1 + Īŗ2

is the cavity damping rate, and Ī¾C (t) is the cavity noise operator deļ¬ned by eqn

(14.97). The explicit time dependence on the right side is eliminated by imposing the

resonance condition ĻP = 2Ļ0 on the cavity. The equation for the adjoint envelope

operator aā (t) is then

dā ĪŗC ā ā

a (t) + ā„¦ā— a (t) + Ī¾C (t) .

a (t) = ā’ (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:

ā’ĪŗC /2

d a (t) a (t)

ā„¦P

= , (15.99)

aā (t) aā (t)

ā—

ā’ĪŗC /2

ā„¦P

dt

where we have used Ī¾C (t) = 0. The 2 Ć— 2 matrix on the right side has eigenvalues

ĪĀ± = ā’ĪŗC /2 Ā± |ā„¦P |, so the general solution is a linear combination of special solutions

varying as exp [ĪĀ± (t ā’ t0 )]. Since ĪŗC > 0, the eigenvalue Īā’ always describes an

exponentially decaying solution. On the other hand, the eigenvalue Ī+ can describe

an exponentially growing solution if |ā„¦P | > ĪŗC /2.

At the threshold value |ā„¦P | = ĪŗC /2, the average a (t) of the slowly-varying enve-

ĪŗC , so that a (t) ā¼ exp (ā’iĻ0 t)

lope operator approaches a constant for times tā’t0

is oscillatory at large times. This describes the transition from optical parametric am-

pliļ¬cation to optical parametric oscillation. Operation above the oscillation threshold

would produce an exponentially rapid build-up of the intracavity ļ¬eld that would

quickly lead to a violation of the weak-ļ¬eld assumptions justifying the model Hamil-

tonian HSS in eqn (15.96). Dealing with pump ļ¬elds exceeding the threshold value

requires the inclusion of nonlinear eļ¬ects that would lead to gain saturation and thus

prevent runaway ampliļ¬cation. We avoid these complications by imposing the condi-

tion |ā„¦P | < ĪŗC /2. On the other hand, we will see presently that the largest squeezing

occurs for pump ļ¬elds just below the threshold value.

The coupled equations (15.97) and (15.98) for a (t) and aā (t) are a consequence of

the special form of the down-conversion Hamiltonian. Since the diļ¬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 diļ¬erential

equations are transformed into algebraic equations:

ĪŗC

a (Ļ) + ā„¦P aā (ā’Ļ) + Ī¾C (Ļ) ,

ā’iĻa (Ļ) = ā’ (15.100)

2

ĪŗC ā ā

ā’iĻaā (ā’Ļ) = ā’ a (Ļ) + ā„¦ā— a (Ļ) + Ī¾C (ā’Ļ) , (15.101)

P

2

Nonclassical states of light

which have the solution

ā

(ĪŗC /2 ā’ iĻ) Ī¾C (Ļ) + ā„¦P Ī¾C (ā’Ļ)

a (Ļ) = ,

2 2

(ĪŗC /2 ā’ iĻ) ā’ |ā„¦P |

(15.102)

ā

(ĪŗC /2 ā’ iĻ) Ī¾C (ā’Ļ) + ā„¦ā— Ī¾C (Ļ)

aā (ā’Ļ) = P

.

2 2

(ĪŗC /2 ā’ iĻ) ā’ |ā„¦P |

Combining the deļ¬nition (14.97) with the result (14.116) for the in-ļ¬elds in the fre-

quency domain gives

ā ā

Īŗ1 b1,Ļ+Ļ0 (t0 ) + Īŗ2 b2,Ļ+Ļ0 (t0 ) eiĻt0 .

Ī¾C (Ļ) = (15.103)

This shows that a (Ļ) and aā (Ļ) are entirely expressed in terms of the reservoir op-

erators at the initial time. The correlation functions of the intracavity ļ¬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 (Ļ) has two polesā”

determined by the roots of the denominator in eqn (15.102)ā”located at

ĪŗC

Ļ = ĻĀ± = ā’i Ā± |ā„¦P | . (15.104)

2

Since ĪŗC is positive, the pole at Ļ+ always remains in the lower half planeā”correspon-

ding to the exponentially damped solution of eqn (15.99)ā”but when the coupling

frequency exceeds the threshold value, |ā„¦P |crit = ĪŗC /2, the pole at Ļā’ inļ¬ltrates

into the upper half planeā”corresponding to the exponentially growing solution of eqn

(15.99). Thus the OPAā“OPO transition occurs at the same value for the operator

solution and the ensemble-averaged solution.

B Squeezing of the intracavity ļ¬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 aā (t) , a (t) and the variance V (a (t)),

which can in turn be expressed in terms of the Fourier transforms aā (Ļ) and a (Ļ).

For example, eqns (14.112) and (14.114) lead to

dĻ dĻ ā’i(Ļ +Ļ)t

V aā (t) , a (t) = V aā (ā’Ļ ) , a (Ļ) .

e (15.105)

2Ļ 2Ļ

Applying the relations

a (Ļ) = a (Ļ ā’ Ļ0 ) , aā (ā’Ļ ) = aā (ā’Ļ ā’ Ļ0 ) (15.106)

that follow from eqn (14.119), and the change of variables Ļ ā’ Ļ + Ļ0 , Ļ ā’ Ļ ā’ Ļ0

allows this to be expressed in terms of the slowly-varying operators a (Ļ):

dĻ dĻ ā’i(Ļ +Ļ)t

V aā (t) , a (t) = V aā (ā’Ļ ) , a (Ļ) .

e (15.107)

2Ļ 2Ļ

The solution (15.102) gives a (Ļ) and aā (Ļ) as linear combinations of the initial

reservoir creation and annihilation operators. In the experiment under consideration,

Theory of squeezed-light generationā—

there is no injected signal at the resonance frequency Ļ0 , and the incident pump ļ¬eld

at ĻP is treated classically. The Heisenberg-picture density operator can therefore be

treated as the vacuum for the initial reservoir ļ¬elds, i.e. ĻE = |0 0|, where

b1,ā„¦ (t0 ) |0 = b2,ā„¦ (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 deļ¬ned with

respect to the reservoir vacuum greatly simpliļ¬es the calculation. To begin with, ļ¬rst

calculating a (Ļ) |0 provides the happy result that many terms vanish. Once this is

done, the commutation relations (14.115) lead to

2

|ā„¦P |

1

ā

V a (t) , a (t) = . (15.109)

2 (ĪŗC /2)2 ā’ |ā„¦P |2

In the same way, the crucial variance V (a (t)) is found to be

ā„¦P Īŗ C

1

V (a (t)) = , (15.110)

4 (ĪŗC /2)2 ā’ |ā„¦P |2

so that

2

|ā„¦P |

1 ĪŗC 1

ā’2iĪ²

VN (XĪ² ) = 2 Re e ā„¦P + . (15.111)

2 4 (ĪŗC /2)2 ā’ |ā„¦P |2

8 (ĪŗC /2) ā’ |ā„¦P |

The minimum value of VN (X) is attained at the quadrature phase

ĪøP Ļ

ā’,

Ī²= (15.112)

2 2

where ĪøP is the phase of ā„¦P . For this choice of Ī²,

|ā„¦P |

1

VN (X) = ā’ (15.113)

4 ĪŗC /2 + |ā„¦P |

and

|ā„¦P |

1

VN (Y ) = . (15.114)

4 ĪŗC /2 ā’ |ā„¦P |

Keeping in mind the necessity of staying below the oscillation threshold, i.e. |ā„¦P | <

ĪŗC /2, we see that VN (X) > ā’1/8. The relation (15.16) then yields

1 1

< V (X) < ; (15.115)

8 4

in other words, the cavity ļ¬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 ĪŗC = Īŗ1 + Īŗ2 and not on Īŗ1 or Īŗ2 separately. This feature reļ¬‚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.

Ā¼ Nonclassical states of light

C Squeezing of the emitted light

The limits on cavity ļ¬eld squeezing are not the end of the story, since only the output

of the OPAā”i.e. the ļ¬eld emitted through one of the mirrorsā”can be experimentally

studied. We therefore consider a time t1 t0 when the light emittedā”say through

mirror M2ā”reaches a detector. The detected signal is represented by the out-ļ¬eld

operator b2,out (t) introduced in Section 14.3. We reproduce the deļ¬nition,

ā

dā„¦

b2,ā„¦ (t1 ) eā’iā„¦(tā’t1 ) ,

b2,out (t) = (15.116)

2Ļ

ā’ā

here, in order to emphasize the dependence of the output signal on the ļ¬nal value

b2,ā„¦ (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-ļ¬elds:

2 2

ā

bJ,out (Ļ) = PJL (Ļ) bL,in (Ļ) + CJL (Ļ) bL,in (ā’Ļ) , (15.117)

L=1 L=1

with

ā [ĪŗC /2 ā’ iĻ]

PJL (Ļ) = Ī“Jl ā’ ĪŗJ ĪŗL , (15.118)

2 2

[ĪŗC /2 ā’ iĻ] ā’ |ā„¦P |

ā ā„¦P

CJL (Ļ) = ā’ ĪŗJ ĪŗL . (15.119)

2 2

[ĪŗC /2 ā’ iĻ] ā’ |ā„¦P |

The M2-output quadratures are deļ¬ned by replacing a (t) with b2,out (t) in eqn (15.14)

to get

1

b2,out (t) eā’iĪ² + bā iĪ²

Xout (t) = 2,out (t) e ,

2 (15.120)

1 ā

ā’iĪ²

ā’ b2,out (t) e iĪ²

Yout (t) = b2,out (t) e

2i

in the time domain, or

1

b2,out (Ļ) eā’iĪ² + bā iĪ²

Xout (Ļ) = 2,out (ā’Ļ) e ,

2 (15.121)

1

b2,out (Ļ) eā’iĪ² ā’ bā iĪ²

Yout (Ļ) = 2,out (ā’Ļ) e

2i

in the frequency domain. The parameter Ī² is again chosen to satisfy eqn (15.112). The

normal-ordered variances for the output quadratures are

dĻ dĻ ā’i(Ļ +Ļ )t

VN (Xout (t)) = e VN (Xout (Ļ ) , Xout (Ļ )) , (15.122)

2Ļ 2Ļ

dĻ dĻ ā’i(Ļ +Ļ )t

VN (Yout (t)) = e VN (Yout (Ļ ) , Yout (Ļ )) , (15.123)

2Ļ 2Ļ

where

Theory of squeezed-light generationā— Ā½

VN (F, G) = : F G : ā’ : F : : G: (15.124)

is the joint normal-ordered variance. Calculations very similar to those for the

cavity quadratures lead to

|ā„¦P | Īŗ2

1

VN (Xout (Ļ ) , Xout (Ļ )) = ā’ 2ĻĪ“ (Ļ + Ļ ā’ 2Ļ0 ) ,

2 [ĪŗC /2 + |ā„¦P |]2 + (Ļ ā’ Ļ0 )2

(15.125)

|ā„¦P | Īŗ2

1

2ĻĪ“ (Ļ + Ļ ā’ 2Ļ0 ) .

VN (Yout (Ļ ) , Yout (Ļ )) =

2 [ĪŗC /2 ā’ |ā„¦P |]2 + (Ļ ā’ Ļ0 )2

(15.126)

The delta functions in the last two equations reļ¬‚ect the fact that the output ļ¬eld

b2,out (t)ā”by contrast to the discrete cavity mode described by a (t)ā”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,

dteiĻt VN (Xout (t) , Xout (0)) ,

SN (Ļ) =

dĻ

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

2Ļ

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

|ā„¦P | Īŗ2

1

SN (Ļ) = ā’ , (15.128)

2 [ĪŗC /2 + |ā„¦P |]2 + (Ļ ā’ Ļ0 )2

which has its minimum value for |ā„¦P | = ĪŗC /2 = (Īŗ1 + Īŗ2 ) /2 and Ļ = Ļ0 , i.e.

1 Īŗ2

SN (Ļ) > ā’ . (15.129)

4 Īŗ1 + Īŗ2

For a symmetrical cavityā”i.e. Īŗ1 = Īŗ2 ā”the degree of squeezing is bounded by

1

SN (Ļ) > ā’ ; (15.130)

8

therefore, the output ļ¬eld can at best be squeezed by 50%, just as for the intracavity

ļ¬eld. However, the degree of squeezing for the output ļ¬eld is not a symmetrical function

of Īŗ1 and Īŗ2 . For an extremely unsymmetrical cavityā”e.g. Īŗ1 Īŗ2 ā”we see that

1

ā’;

SN (Ļ) (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 ļ¬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

Ā¾ Nonclassical states of light

right-propagating waves. The left-propagating component of the intracavity mode is

partially reļ¬‚ected from the mirror M1 and then partially transmitted through the

mirror M2, together with the right-propagating component. Reļ¬‚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 ļ¬‚uctuations in the M1 reservoir that are partially transmitted through

the mirror M1 into the cavity. Interference between these ļ¬‚uctuations and the right-

propagating component of the intracavity mode is impossible, since the phases are

statistically independent. For a symmetrical cavity, Īŗ1 = Īŗ2 , 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 reļ¬‚ector at Ļ0 , i.e. Īŗ1 = 0, then

the vacuum ļ¬‚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 Ļ0 Ļ0

with output

second-harmonic crystal 2Ļ0

ĪøLO

Pump

at 2Ļ0

Ļ(2) crystal

M1 Ļ0

Optical

parametric Idler

Signal

oscillator

(OPO)

LO

M2

at Ļ0

Squeezed light output at Ļ0

Homodyne detector

Fig. 15.2 Simpliļ¬ed schematic of an experiment to generate squeezed light. ā˜PBSā™ stands

for ā˜polarizing beam splitterā™ and ā˜MLO ā™ is a mirror for the local oscillator (LO) beam at ĻLO .

(Adapted from Wu et al. (1986).)

Āæ

Experimental squeezed-light generation

ing a diode-laser-pumped, neodymium-doped, yttrium aluminum garnet (Nd:YAG)

crystalā”which produces an intense laser beam at the ļ¬rst-harmonic frequency Ļ0 ā”

and an intracavity, second-harmonic crystal (barium sodium niobate), which produces

a strong beam at the second-harmonic frequency 2Ļ0 . The solid lines represent beams

at the ļ¬rst harmonic, corresponding to a wavelength of 1.06 Āµm, and the dashed lines

represent beams at the second harmonic, corresponding to a wavelength of 0.53 Āµm.

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 ļ¬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 ĪøLO . 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 Ļ(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, Ļ0 , and are also collinear. The entrance mir-

ror M1 has an extremely high reļ¬‚ectivity at the ļ¬rst-harmonic frequency Ļ0 , but only

a moderately high reļ¬‚ectivity at the second-harmonic frequency 2Ļ0 . Thus M1 allows

the second-harmonic, pump light to enter the OPO, while also serving as one of the

reļ¬‚ecting surfaces deļ¬ning a resonant cavity for both the ļ¬rst- and second-harmonic

frequencies. This arrangement enhances the pump intensity inside the crystal.

By contrast, the exit mirror M2 has an extremely high reļ¬‚ectivity for the second-

harmonic frequency, but only a moderately high reļ¬‚ectivity at the ļ¬rst-harmonic fre-

quency. Thus the mirrors M1 and M2 form a resonatorā”for both the ļ¬rst- and second-

harmonic frequenciesā”but at the same time M2 allows the degenerate signal and idler

beamsā”at the ļ¬rst-harmonic frequency Ļ0 ā”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 diļ¬erential ampliļ¬er. Finally, the output of

the diļ¬erential ampliļ¬er is fed into a spectrum analyzer, as explained in Section 9.3.3.

It is important to emphasize that the extremely high reļ¬‚ectivity, for frequency Ļ0 ,

of the entrance mirror, M1, blocks out vacuum ļ¬‚uctuations from entering the system,

thereby preventing them from contributing unwanted vacuum ļ¬‚uctuation noise at this

frequency. As explained in Section 15.2-C, the asymmetry in the reļ¬‚ectivities of the

mirrors M1 and M2 at the ļ¬rst-harmonic frequency Ļ0 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 (Īø), of

the spectrum analyzer associated with the homodyne detector is plotted versus the

local oscillator phase Īø = ĪøLO , for a ļ¬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

8Īø

Īø Īø + Ļ Īø + Ļ Īø Īø + Ļ

Īø

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 levelā”which repre-

sents the noise from vacuum ļ¬‚uctuationsā”is indicated by the dashed line in this ļ¬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. Ļi = Ļ0 + āĻi and Ļs = Ļ0 + āĻs ,

with āĻi = ā’āĻs . 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 ĪøLO of the local oscillatorā”which can be adjusted

by the piezoelectric translator that controls the location of the mirror MLO ā”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 ampliļ¬cation proļ¬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 ļ¬eld. Any state, whether pure or mixed,

can be expressed in terms of number states. By deļ¬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 ) ā’ N

Q (|n ) ā” = ā’1 , (15.132)

N

where X = n |X| n . Thus the number states saturate the general inequality

Q (|ĪØ ) ā’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 identiļ¬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 (Ī±) 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 eļ¬ciency.

The ļ¬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 Ī» = 351 nm

enters a crystalā”potassium dihydrogen phosphate (KDP)ā”with a Ļ(2) nonlinearity.

Conjugate down-converted photon pairs are generated on opposite sides of the UV

Amp. & Counter

disc.

PMT

A

Interference

filter

Gated PDP

UV

counter 11/23+

filter

MCP

Argon-ion UV laser KDP

B Amp. & Counter

disc.

Lens Field Interference

lens filter

Fig. 15.4 Schematic of Hong and Mandelā™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 Aā”and the less-than-unity quantum

eļ¬ciency of counter B is accounted forā”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

Bā”conditioned on the detection of a signal photon at counter Aā”is plotted versus

the photon number n.

The data show that within small uncertainties (indicated by the cross-hatched

regions),

p(n) = Ī“n,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 oļ¬ in counter A,

one can, with almost complete certainty, predict that there is one and only one photon

within a well-deļ¬ned wave packet propagating in the idler channel. The Mandel Q-

parameter derived from these data, Q = ā’1.06 Ā± 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-deļ¬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 dotā”via the Coulomb blockade mechanismā”

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.

p(n)

1.0

Fig. 15.5 The derived probability p(n) for

the detection of n idler photons conditioned

0.5

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-

n

0 1 2 3 4

del (1986).)

Exercises

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 ļ¬rst method, a completed measurement of the ļ¬nal state of the atom after it

exits the cavity allows the experimenter to knowā”with certaintyā”whether the atom

has made a downwards transition inside the cavity. Combining this knowledge with

the conservation of energy determinesā”again with certaintyā”the number state of the

cavity ļ¬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 eļ¬ective Rabi frequency for an on-resonance, n-photon state is ā„¦n = 2g (n + 1),

where g is the coupling constant of the two-level atom with the cavity mode. The

ā

Rabi period is therefore Tn = 2Ļ/ā„¦n = Ļ/ g n + 1 . If the interaction time Tint of

the atom with the ļ¬eld satisļ¬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 unchangedā”i.e. trapped ā”if the condition

ā

n + 1gTint = kĻ (15.134)

is satisļ¬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 ļ¬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 diļ¬cult to extract them from the high-Q superconducting cavity for use in external

experiments.

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) suļ¬cient, as well as necessary? If not, what

are the suļ¬cient conditions?

(3) Explain why number states and coherent states are not squeezed states.

(4) Is the state |Ļ = cos Īø |0 + sin Īø |1 squeezed for any value of Īø? In other words,

for a given Īø, 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

|Ī¶, n = S (Ī¶) |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 statesā—

15.3

Use the properties of S (Ī¶) and D (Ī±) to derive the relations (15.52)ā“(15.54).

Photon statistics for the displaced squeezed stateā—

15.4

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 lightā—

15.5

(1) Carry out the calculations required to derive eqns (15.125) and (15.126).

(2) Use these results to derive eqn (15.128).

16

Linear optical ampliļ¬ersā—

Generally speaking, an optical ampliļ¬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 ampliļ¬ers to be studied.

To this end, we consider the special class of unitary inputā“output transformations

that can be expressed as follows. The annihilation operator for each output mode

is a linear combination, with c-number coeļ¬cients, of the creation and annihilation

operators for the input modes. Devices of this kind are called linear ampliļ¬ers. We

note in passing that linear ampliļ¬ers are quite diļ¬erent from laser oscillator-ampliļ¬ers,

which typically display the highly nonlinear phenomenon of saturation (Siegman, 1986,

Sec. 4.5).

For typical applications of linear, optical ampliļ¬ersā”e.g. optical communication or

the generation of nonclassical states of lightā”it is desirable to minimize the noise added

to the input signal by the ampliļ¬er. The ļ¬rst source of noise is the imperfect coupling

of the incident signal into the ampliļ¬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 eļ¬ectsā”such as noise in the associated electronic circuitsā”into the category

of technical noise. Since these eļ¬ects vary from device to device, we will concentrate

on the intrinsic quantum noise associated with the act of ampliļ¬cation itself.

In the present chapter we ļ¬rst discuss the general properties of linear ampliļ¬ers

and then describe several illustrative examples. In the ļ¬nal sections we present a

simpliļ¬ed version of a general theory of linear ampliļ¬ers due to Caves (1982), which

is an extension of the earlier work of Haus and Mullen (1962).

16.1 General properties of linear ampliļ¬ers

The degenerate optical parametric ampliļ¬er (OPA) studied in Section 15.2 is a linear

device, by virtue of the assumption that depletion of the pump ļ¬eld can be neglected. In

the application to squeezing, the input consists of vacuum ļ¬‚uctuationsā”represented

by b2,in (t)ā”entering the mirror M2, and the corresponding output is the squeezed

stateā”represented by b2,out (t)ā”emitted from M2. Both the input and the output

have the carrier frequency Ļ0 . Rather than extending this model to a general theory

of linear ampliļ¬ers that allows for multiple inputs and outputs and frequency shifts

Linear optical ampliļ¬ersā—

Ā¼Ā¼

between them, we choose to explain the basic ideas in the simplest possible context:

linear ampliļ¬ers with a single input ļ¬eld and a single output ļ¬eldā”denoted by bin (t)

and bout (t) respectivelyā”having a common carrier frequency.

We will also assume that the characteristic response frequency of the ampliļ¬er and

the bandwidth of the input ļ¬eld are both small compared to the carrier frequency. This

narrowband assumption justiļ¬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 deļ¬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 ampliļ¬er.

In the sampleā“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 ā˜internalā™ modes are ļ¬eld modes, e.g. vacuum ļ¬‚uctuations, that exist

in the space outside the physical ampliļ¬er.

The deļ¬nition of the ampliļ¬er is completed by specifying the Heisenberg-picture

density operator Ļ that describes the state of both the input ļ¬eld and the internal

modes of the ampliļ¬er. This is the same thing as specifying the initial value of the

SchrĀØdinger-picture density operator. Since we want to use the ampliļ¬er for a broad

o

range of input ļ¬elds, it is natural to require that the operating state of the ampliļ¬er is

independent of the incident ļ¬eld state. This condition is imposed by the factorizability

assumption

Ļ = Ļin Ļamp , (16.1)

where Ļin and Ļamp respectively describe the states of the input ļ¬eld and the ampliļ¬er.

In the generic states of interest for communications, the expectation value of the

input ļ¬eld does not vanish identically:

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

Situations for which bin (t) = 0 for all tā”e.g. injecting the vacuum state or a

squeezed-vacuum state into the ampliļ¬erā”are to be treated as special cases.

The identiļ¬cation of the measured values of the input and output ļ¬elds with the

expectation values bin (t) and bout (t) runs into the apparent diļ¬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 eļ¬ectively measure a

hermitian quadrature operator. For example, it is possible to measure one member of

the conjugate pair (XĪ²,in (t) , YĪ²,in (t)), where

1 ā’iĪ² ā

e bin (t) + eiĪ² bin (t) ,

XĪ²,in (t) =

2 (16.3)

1 ā’iĪ² ā

e bin (t) ā’ eiĪ² bin (t) .

YĪ²,in (t) =

2i

The quadrature angle Ī² is determined by the relative phase between the input signal

and the local oscillator employed in the detection scheme. The operational signiļ¬cance

Ā¼Ā½

General properties of linear ampliļ¬ers

of the complex expectation value bin (t) is established by carrying out measurements

of XĪ²,in (t) for several quadrature angles and using the relation

1 ā’iĪ² ā

= Re eā’iĪ² bin (t)

bin (t) + eiĪ² bin (t)

XĪ²,in (t) = e . (16.4)

2

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) ā’ bin (t) = eā’iĪø bin (t) , (16.5)

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

XĪ²,in (t) ā’ Xin (t) = XĪ²,in (t) cos Īø + YĪ²,in (t) sin Īø , (16.6)

YĪ²,in (t) ā’ Yin (t) = YĪ²,in (t) cos Īø ā’ XĪ²,in (t) sin Īø , (16.7)

represents a rotation through the angle Īø 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 ampliļ¬ers

From Section 14.1.1-C, we know that the noise properties of the input/output ļ¬elds are

described by the correlation functions of the ļ¬‚uctuation operators, Ī“bĪ³ (Ļ) ā” bĪ³ (Ļ) ā’

bĪ³ (Ļ) , where Ī³ = in, out. Thus the input/output noise correlation functions are

deļ¬ned by

1 ā ā

KĪ³ (Ļ, Ļ ) = Ī“bĪ³ (Ļ) Ī“bĪ³ (Ļ ) + Ī“bĪ³ (Ļ ) Ī“bĪ³ (Ļ) (Ī³ = in, out) . (16.8)

2

The deļ¬nitions (14.98) and (14.107) relating the input/output ļ¬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 satisļ¬ed.

(1) The noise in diļ¬erent frequencies is uncorrelated, i.e.

KĪ³ (Ļ, Ļ ) = NĪ³ (Ļ) 2ĻĪ“ (Ļ ā’ Ļ ) , (16.9)

where

1

ā

NĪ³ (Ļ) = Ī“bĪ³ (Ļ) Ī“bĪ³ (Ļ) + (16.10)

2

is the noise strength.

(2) The phases of the ļ¬‚uctuation operators are randomly distributed, so that

Ī“bĪ³ (Ļ) Ī“bĪ³ (Ļ ) = 0 . (16.11)

With this preparation, we are now ready to introduce an important division of the

family of linear ampliļ¬ers into two classes. A phase-insensitive ampliļ¬er is deļ¬ned

by the following conditions.

Linear optical ampliļ¬ersā—

Ā¼Ā¾

2

(i) The output ļ¬eld strength, bout (Ļ) , is invariant under phase transformations

of the input ļ¬eld.

(ii) If the input noise is phase insensitive, so is the output noise.

Condition (i) means that the only eļ¬ect of a phase shift in the input ļ¬eldā”i.e. a rota-

tion of the quadraturesā”is to produce a corresponding phase shift in the output ļ¬eld.

Condition (ii) means that the noise added by the ampliļ¬er is randomly distributed in

phase. An ampliļ¬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, ampliļ¬ers

can also be classiļ¬ed according to their physical conļ¬guration. In the degenerate OPA

the gain medium is enclosed in a resonant cavity, and the input ļ¬eld is coupled into

one of the cavity modes. The cavity mode in turn couples to an output mode to

produce the ampliļ¬ed signal. This conļ¬guration is called a regenerative ampliļ¬er,

which is yet another term borrowed from radio engineering. One way to understand

the regenerative ampliļ¬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 conļ¬guration, there are no

reļ¬‚ected wavesā”and therefore no multiple passes through the gain mediumā”so these

devices are called traveling-wave ampliļ¬ers.

16.2 Regenerative ampliļ¬ers

In this section we take advantage of the remarkable versatility of the spontaneous

down-conversion process to describe three regenerative ampliļ¬ers, two phase insensi-

tive and one phase sensitive.

16.2.1 Phase-insensitive ampliļ¬ers

A modiļ¬cation of the degenerate OPA design of Section 15.2 provides two examples

of phase-insensitive ampliļ¬ers. In the modiļ¬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 deļ¬ne a set of discrete cavity modes, and

the fundamental cavity modeā”which we will call the idlerā”is chosen to satisfy the

phase-matching condition Ļ0 = ĻP /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,ā„¦ (t).

Ā¼Āæ

Regenerative ampliļ¬ers

Fig. 16.1 Two examples of phase-insensitive optical ampliļ¬ers based on down-conversion in

a Ļ(2) crystal: (a) taking the signal-mode in- and out-ļ¬elds as the input and output of the

ampliļ¬er deļ¬nes a phase-preserving ampliļ¬er; (b) taking the signal-mode in-ļ¬eld as the input

and the out-ļ¬eld through mirror M2 as the output deļ¬nes a phase-conjugating ampliļ¬er.

The ļ¬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, sinceā”as we have seen in Chapter 13ā”the

atoms in the crystal are only virtually excited. This means that the eļ¬ect of the

atoms is completely accounted for by the signalā“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,ā„¦ (t)

and two noise reservoirs represented by the operators b1,ā„¦ (t) and b2,ā„¦ (t) describing

radiation entering and leaving the cavity through the mirrors.

Analyzing this model requires a slight modiļ¬cation of the method of in- and out-

ļ¬elds described in Section 14.3. The new feature requiring the modiļ¬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:

ā

D (Ļ)

vP (ā„¦) eā’iĻP t bā aā ā’ vP (ā„¦) eiĻP t absig,ā„¦ , (16.12)

ā—

sigā’idl

HSE =i dā„¦ sig,ā„¦

2Ļ

0

where vP (ā„¦) is the strength of the couplingā”induced by the nonlinear crystalā”

between the signal mode, the idler mode, and the pump ļ¬eld. The presence of the

products bā aā and absig,ā„¦ represents the fact that the signal and idler photons are

sig,ā„¦

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

modiļ¬ed Heisenberg equations

ā ā

2

D (Ļ) D (Ļ)

d ā

a (t) = dā„¦ vP (ā„¦) bsig,ā„¦ (t) + dā„¦ vm (ā„¦) bm,ā„¦ (t) ,

dt 2Ļ 2Ļ

0 0

m=1

(16.13)

D (Ļ)

d

vP (ā„¦) aā (t) ,

bsig,ā„¦ (t) = ā’i (ā„¦ ā’ Ļ0 ) bsig,ā„¦ (t) + (16.14)

dt 2Ļ

Linear optical ampliļ¬ersā—

Ā¼

where vm (ā„¦) describes the coupling of the idler to the noise modes, and a (t) =

a (t) exp (iĻ0 t), etc. The equations for the noise reservoir operators bm,ā„¦ (t) have the

generic form of eqn (14.89). The retarded and advanced solutions of eqn (16.14) for

the signal mode are respectively

t

bsig,ā„¦ (t) = bsig,ā„¦ (t0 ) eā’i(ā„¦ā’Ļ0 )(tā’t0 ) + vP (ā„¦) dt aā (t ) eā’i(ā„¦ā’Ļ0 )(tā’t ) (16.15)

t0

and

t1

ā’i(ā„¦ā’Ļ0 )(tā’t1 )

dt aā (t ) eā’i(ā„¦ā’Ļ0 )(tā’t ) .

ā’ vP (ā„¦)

bsig,ā„¦ (t) = bsig,ā„¦ (t1 ) e (16.16)

t

The corresponding results for the noise reservoir operators, bm,ā„¦ (t), are given by eqns

(14.94) and (14.105).

After substituting the retarded solutions for bsig,ā„¦ (t) and bm,ā„¦ (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

ā

vm (ā„¦) ā¼ Īŗm for the noise modes. The signal mode must be treated diļ¬erently, since

vP (ā„¦) is proportional to the classical pump ļ¬eld, which has a well-deļ¬ned phase ĪøP .

ā

In this case the broadband rule is vP (ā„¦) ā¼ gP exp (iĪøP ), where gP is positive.

The contributions from the noise reservoirs yield the expected loss term ā’ĪŗC 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 processā”in

which a signalā“idler pair is annihilated to create a pump photonā”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 ļ¬elds at the expense of the

(undepleted) classical pump ļ¬eld.

After carrying out these calculations, one ļ¬nds the retarded Langevin equation for

the idler mode:

ā ā

ā

d 1 ā

a (t) = ā’ (ĪŗC ā’ gP ) a (t)+ gP eiĪøP bsig,in (t)+ Īŗ1 b1,in (t)+ Īŗ2 b2,in (t) , (16.17)

dt 2

where ā

dā„¦

bsig,ā„¦ (t0 ) eā’i(ā„¦ā’Ļ0 )(tā’t0 )

bsig,in (t) = (16.18)

2Ļ

ā’ā

is the signal in-ļ¬eld, and the in-ļ¬elds for the mirrors are given by eqn (14.98). For

gP > ĪŗC , eqn (16.17) predicts an exponential growth of the idler ļ¬eld that would

violate the weak-ļ¬eld assumptions required for the model. Consequentlyā”just as in

the treatment of squeezing in Section 15.2-Aā”the pump ļ¬eld must be kept below the

threshold value (gP < ĪŗC ).

Ā¼

Regenerative ampliļ¬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 (Ļ), with the result

ā ā

ā ā

eiĪøP gP bsig,in (ā’Ļ) + Īŗ1 b1,in (Ļ) + Īŗ2 b2,in (Ļ)

a (Ļ) = . (16.19)

(ĪŗC ā’ gP ) ā’ iĻ

1

2

The inputā“output relation for the signal mode is obtained by equating the right sides

of eqns (16.15) and (16.16) and integrating over ā„¦ to get

ā

gP eiĪøP aā (t)

bsig,out (t) = bsig,in (t) + (16.20)

in the time domain, or

ā

gP eiĪøP aā (ā’Ļ)

bsig,out (Ļ) = bsig,in (Ļ) + (16.21)

in the frequency domain. The inputā“output relations for the mirror reservoirs are given

by the frequency-domain form of eqn (14.109):

ā

b1,out (Ļ) = b1,in (Ļ) ā’ Īŗ1 a (Ļ) , (16.22)

ā

b2,out (Ļ) = b2,in (Ļ) ā’ Īŗ2 a (Ļ) . (16.23)

A Phase-transmitting OPA

The ļ¬rst step in deļ¬ning an ampliļ¬er is to decide on the identity of the input and

output ļ¬elds. In other words: What is to be measured? For the ļ¬rst example, we choose

the in-ļ¬eld and out-ļ¬eld of the signal mode as the input and output ļ¬elds, i.e. bin (Ļ) =

bsig,in (Ļ) and bout (Ļ) = bsig,out (Ļ). The idler ļ¬eld and the two mirror reservoir in-

ļ¬elds are then internal modes of the ampliļ¬er. Substituting these identiļ¬cations and

the solution (16.19) into eqn (16.21) yields the ampliļ¬er inputā“output equation

bout (Ļ) = P (Ļ) bin (Ļ) + Ī· (Ļ) , (16.24)

where the coeļ¬cient

(ĪŗC + gP ) ā’ iĻ

1

2

P (Ļ) = (16.25)

(ĪŗC ā’ gP ) ā’ iĻ

1

2

has the symmetry property

P (Ļ) = P ā— (ā’Ļ) , (16.26)

and the operator

ā ā

gP eiĪøP Ī¾C (ā’Ļ)

Ī· (Ļ) = 1

2 (ĪŗC ā’ gP ) ā’ iĻ

ā

ā

ā

gP eiĪøP ā ā

=1 Īŗ1 b1,in (ā’Ļ) + Īŗ2 b2,in (ā’Ļ) (16.27)

2 (ĪŗC ā’ gP ) ā’ iĻ

is called the ampliļ¬er noise.

Linear optical ampliļ¬ersā—

Ā¼

This result shows that the noise added by the ampliļ¬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 ļ¬‚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 Ī· (Ļ) implies that the inputā“output

equation for the average ļ¬eld is

bout (Ļ) = P (Ļ) bin (Ļ) . (16.28)

Subtracting this equation from eqn (16.24) yields the inputā“output equation

Ī“bout (Ļ) = P (Ļ) Ī“bin (Ļ) + Ī· (Ļ) (16.29)

for the ļ¬‚uctuation operators.

The ļ¬rst step in the proof that this ampliļ¬er is phase insensitive is to use eqn

(16.28) to show that the eļ¬ect of a phase transformation applied to the input ļ¬eld is

bout (Ļ) = P (Ļ) bin (Ļ) = eiĪø bout (Ļ) . (16.30)

In other words, changes in the phase of the input signal are simply passed through the

ampliļ¬er. Ampliļ¬ers with this property are said to be phase transmitting. The ļ¬eld

2

strength bout (Ļ) is evidently unchanged by a phase transformation; therefore the

ampliļ¬er satisļ¬es condition (i) of Section 16.1.1.

Turning next to condition (ii), we note that the operators Ī“bin (Ļ) and Ī· (Ļ) are lin-

ear functions of the uncorrelated reservoir operators bsig,ā„¦ (t0 ) and bm,ā„¦ (t0 ) (m = 1, 2).

This feature combines with eqn (16.29) to give

Kout (Ļ, Ļ ) = P (Ļ) P ā— (Ļ ) Kin (Ļ, Ļ ) + Kamp (Ļ, Ļ ) , (16.31)

where

1

Ī· (Ļ) Ī· ā (Ļ ) + Ī· ā (Ļ ) Ī· (Ļ)

Kamp (Ļ, Ļ ) = (16.32)

2

is the ampliļ¬erā“noise correlation function. Since Ī· (Ļ) is a linear combination of the

mirror noise operators, the assumption that the mirror noise is phase insensitive guar-

antees that

Kamp (Ļ, Ļ ) = Namp (Ļ) 2ĻĪ“ (Ļ ā’ Ļ ) , (16.33)

where Namp (Ļ) is the ampliļ¬er noise strength. If the correlation function Kin (Ļ, Ļ )

satisļ¬es eqn (16.9), then eqns (16.31) and (16.33) guarantee that Kout (Ļ, Ļ ) does

also. The output noise strength is then given by

Nout (Ļ) = |P (Ļ)|2 Nin (Ļ) + Namp (Ļ) . (16.34)

It is also necessary to verify that the output noise satisļ¬es eqn (16.11), when the

input noise does. This is an immediate consequence of the phase insensitivity of the

ampliļ¬er noise and the inputā“output equation (16.29), which together yield

Ī“bout (Ļ) Ī“bout (Ļ ) = P (Ļ) P (Ļ ) Ī“bin (Ļ) Ī“bin (Ļ ) . (16.35)

Putting all this together shows that the ampliļ¬er is phase insensitive, since it satisļ¬es

conditions (i) and (ii) from Section 16.1.1.

Ā¼

Regenerative ampliļ¬ers

For this ampliļ¬er, it is reasonable to deļ¬ne the gain as the ratio of the output ļ¬eld

strength to the input ļ¬eld strength:

2

| bout (Ļ + Ļ0 ) |2

bout (Ļ)

G (Ļ) = = . (16.36)

2 2

| bin (Ļ + Ļ0 ) |

bin (Ļ)

Using eqn (16.28) yields the explicit expression

2 2

(ĪŗC + gP ) /4 + (Ļ ā’ Ļ0 )

G (Ļ ā’ Ļ0 ) = , (16.37)

2 2

(ĪŗC ā’ gP ) /4 + (Ļ ā’ Ļ0 )

which displays the expected peak in the gain at the resonance frequency Ļ0 . An alter-

native procedure is to deļ¬ne the gain in terms of the quadrature operators, and then

to showā”see Exercise 16.1ā”that the gain is the same for all quadratures.

B Phase-conjugating OPA

The crucial importance of the choice of input and output ļ¬elds is illustrated by using

the apparatus shown in Fig. 16.1 to deļ¬ne a quite diļ¬erent ampliļ¬er. In this version the

input ļ¬eld is still the signal-mode in-ļ¬eld bsig,in (Ļ), but the output ļ¬eld is the out-ļ¬eld

b2,out (Ļ) for the mirror M2. The internal modes are the same as before. The inputā“

output equation for this ampliļ¬erā”which is derived from eqn (16.23) by using the

solution (16.19) and the identiļ¬cations bin (Ļ) = bsig,in (Ļ) and bout (Ļ) = b2,out (Ļ)ā”

has the form

ā

bout (Ļ) = C (Ļ) eiĪøP bin (ā’Ļ) + Ī· (Ļ) . (16.38)

The coeļ¬cient C (Ļ) and the ampliļ¬er noise operator are respectively given by

ā

Īŗ2 g P

C (Ļ) = ā’ 1 (16.39)

(ĪŗC ā’ gP ) ā’ iĻ

2

and

ā

(Īŗ1 ā’ Īŗ2 ā’ gP ) ā’ iĻ

1

Īŗ1 Īŗ2

b2,in (Ļ) ā’

2

Ī· (Ļ) = b1,in (Ļ) . (16.40)

2 (ĪŗC ā’ gP ) ā’ iĻ 2 (ĪŗC ā’ gP ) ā’ iĻ

1 1

The important diļ¬erence from eqn (16.24) is that the output ļ¬eld depends on the

adjoint of the input ļ¬eld. Note that C (Ļ) has the same symmetry as P (Ļ):

C (Ļ) = C ā— (ā’Ļ) . (16.41)

The ensemble average of eqn (16.38) is

ā

bout (Ļ) = C (Ļ) bin (ā’Ļ) , (16.42)

so the phase transformation bin (Ļ) ā’ bin (Ļ) = exp (iĪø) bin (Ļ) results in

ā

bout (Ļ) = eā’iĪø C (Ļ) bin (ā’Ļ) = eā’iĪø bout (Ļ) . (16.43)

Instead of being passed through the ampliļ¬er unchanged, the phasor exp (iĪø) is re-

placed by its conjugate. A device with this property is called a phase-conjugating

ampliļ¬er.

Linear optical ampliļ¬ersā—

Ā¼

This ampliļ¬er nevertheless satisļ¬es condition (i) of Section 16.1.1, since

2 2

bout (Ļ) = bout (Ļ) . (16.44)

The argument used in Section 16.2.1-A to establish condition (ii) works equally well

here; therefore, the alternative design also deļ¬nes a phase-insensitive ampliļ¬er. The

form of the inputā“output relation in this case suggests that the gain should be deļ¬ned

as

2

bout (Ļ) Īŗ2 g P

= |C (Ļ)|2 = 1

G (Ļ) = . (16.45)

2

(ĪŗC ā’ gP )2 + Ļ 2

ā

bin (ā’Ļ) 4

16.2.2 Phase-sensitive OPA

In the design shown in Fig. 16.2 the ļ¬elds entering and leaving the cavity through

the mirror M1 are designated as the input and output ļ¬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 ļ¬eld b2,in (t) for the mirror M2 are the internal modes of the ampliļ¬er.

The inputā“output relation is obtained from eqn (15.117) by applying this identiļ¬cation

of the input and output ļ¬elds:

ā

bout (Ļ) = P (Ļ) bin (Ļ) + C (Ļ) eiĪøP bin (ā’Ļ) + Ī· (Ļ) . (16.46)

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

Īŗ1 (ĪŗC /2 ā’ iĻ)

P (Ļ) = 1 ā’ (16.47)

2 2

(ĪŗC /2 ā’ iĻ) ā’ |ā„¦P |

and

|ā„¦P | Īŗ1

C (Ļ) = ā’ . (16.48)

2 2

(ĪŗC /2 ā’ iĻ) ā’ |ā„¦P |

2, out

1, in

Pump

2, in

1, out

M2

M1

1 2

Fig. 16.2 A phase-sensitive ampliļ¬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.

Ā¼

Regenerative ampliļ¬ers

The functions P (Ļ) and C (Ļ) satisfy eqns (16.26) and (16.41) respectively. The am-

pliļ¬er noise operator,

ā

Īŗ1 Īŗ2 ā

Ī· (Ļ) = ā’ (ĪŗC /2 ā’ iĻ) b2,in (Ļ) + ā„¦P b2,in (ā’Ļ) , (16.49)

2 2

(ĪŗC /2 ā’ iĻ) ā’ |ā„¦P |

only depends on the reservoir operators associated with the mirror M2, so the ampliļ¬er

noise is entirely caused by vacuum ļ¬‚uctuations passing through the unused port at

M2.

According to eqn (16.46), the output ļ¬eld strength is

2

ā

2 2

2 2

= |P (Ļ)| + |C (Ļ)|

bout (Ļ) bin (Ļ) bin (ā’Ļ)

ā—

ā

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

We ļ¬rst test condition (i) of Section 16.1.1, by applying the phase transformation

(16.5) to the input ļ¬eld and evaluating the diļ¬erence between the transformed and

the original output intensities to get

2

2 2

ā’

Ī“ bout (Ļ) = bout (Ļ) bout (Ļ)

ā—

ā

e2iĪø ā’ 1 P (Ļ) C ā— (Ļ) bin (Ļ)

= 2 Re bin (ā’Ļ) .

(16.51)

Satisfying condition (i) would require the right side of this equation to vanish as an

identity in Īø. The generic assumption (16.2) combined with the explicit forms of the

functions P (Ļ) and C (Ļ) makes this impossible; therefore, the ampliļ¬er is phase

sensitive.

This feature is a consequence of the fact that P (Ļ) and C (Ļ) are both nonzero, so

ā

that the right side of eqn (16.46) depends jointly on bin (Ļ) and bin (ā’Ļ). 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 deļ¬nition of the gain for a phase-sensitive ampliļ¬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 ampliļ¬ed signal is propagating back-

wards toward the source of the input signal. Devices of this kind are sometimes called

reļ¬‚ection ampliļ¬ers. This is not a useful feature for communications applications;

therefore, it is necessary to reverse the direction of the ampliļ¬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 ampliļ¬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

noise.

Linear optical ampliļ¬ersā—

Ā½Ā¼

16.3 Traveling-wave ampliļ¬ers

The regenerative ampliļ¬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 conļ¬guration shown in Fig. 16.3, but this experimental simpliļ¬cation inevitably

comes at the expense of some theoretical complication.

The mirrors in the regenerative ampliļ¬ers perform two closely related functions.

The ļ¬rst is to guarantee that the ļ¬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 ļ¬eld is then entirely given by

the corresponding mode function; in eļ¬ect, the cavity is a zero-dimensional system.

The second functionā”which follows from the ļ¬rstā”is to justify the sampleā“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 simpliļ¬cations. Since

there are no discrete cavity modes, each of the continuum of external modes propagates

through the ampliļ¬er and interacts with the gain medium. Thus all ļ¬eld modes are

reservoir modes, and the sample consists of the atoms in the gain medium.

The interaction of the ļ¬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 ļ¬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 ampliļ¬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 positionā“space approach that

closely resembles the classical theory of parametric ampliļ¬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 conļ¬guration (Deutsch and Garrison,

1991b).

16.3.1 Laser ampliļ¬er

As a concrete example, we consider a sample composed of a collection of three-level

atomsā”with the level structure displayed in Fig. 16.4ā”which is made into a gain

Fig. 16.3 A black box schematic of a travel-

ing-wave ampliļ¬er. The shaded box indicates

the gain medium and the ļ¬elds at the two ports

are the input and output values of the signal.

The vacuum ļ¬‚uctuations entering port 2 are

not indicated, since they do not couple to the

signal.

Ā½Ā½

Traveling-wave ampliļ¬ers

3

Ļ32

Fig. 16.4 A three-level atom with a popula-

2

42 tion inversion between levels 1 and 2, main-

tained by an incoherent pump (dark double ar-

Signal

Ļ21 Ļ31 row) with rate RP . The solid arrow, the dashed

arrow, and the wavy arrows respectively rep-

resent the ampliļ¬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-ļ¬re modesā”i.e. ļ¬eld modes with

frequencies Ļ Ļ21 and propagation vectors, k, lying in a narrow cone around the

axis of the cylinderā”will be preferentially ampliļ¬ed.

This new feature requires a modiļ¬cation of the reservoir assignment used for the

pumping calculation. The noise reservoir previously associated with the spontaneous

emission 2 ā’ 1 is replaced by two reservoirs: (1) a noise reservoir associated with

spontaneous emission into modes with propagation vectors outside the end-ļ¬re cone;

and (2) a signal reservoir associated with the end-ļ¬re modes.

In the undepleted pump approximation, the back action of the atoms on the pump

ļ¬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

(n)

HS1 = ā’ S21 (t) d21 Ā· E(+) (rn , t) + HC , (16.52)

n

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-ļ¬re modes to the ļ¬eld

operator can be represented in terms of the slowly-varying envelope operators Ļs (r, t)

appearing in eqn (7.33). We will assume that the ampliļ¬er has been designed so that

only one polarization will be ampliļ¬ed; consequently, only one operator Ļ (r, t) will be

needed.

Turning next to the input signal, we recall that a paraxial state is characterized

by transverse and longitudinal length scales Ī = 1/ (Īøk0 ) and Ī = 1/ Īø2 k0 re-

spectively, where Īø is the opening angle of the paraxial ray bundle. The scale lengths

Ī and Ī correspond respectively to the spot size and Rayleigh range of a classical

Gaussian beam. We choose Īø so that Ī > 2RS and Ī LS , where RS and LS are

respectively the radius and length of the cylinder. This allows a further simpliļ¬cation

in which diļ¬raction is ignored and the envelope operator is approximated by

Linear optical ampliļ¬ersā—

Ā½Ā¾

1

Ļ (r, t) = ā Ļ (z, t) , (16.53)

Ļ

2

where Ļ = ĻRS . In this 1D approximation, the ļ¬eld expansion (7.33) and the commu-

tation relation (7.35) are respectively replaced by

Ļ0 (vg0 /c)

E(+) (r, t) = i e0 Ļ (z, t) ei(k0 zā’Ļ0 t) (16.54)

2 0 n0 Ļ

and

Ļ (z, t) , Ļā (z , t) = Ī“ (z ā’ z ) . (16.55)

The discretely distributed atoms and the continuous ļ¬eld are placed on a more

even footing by introducing the spatially coarse-grained operator density

1

Sqp (t) Ļ (z ā’ zn ) .

(n)

Sqp (z, t) = (16.56)

āz n

The averaging interval āz is chosen to satisfy the following two conditions. (1) A slab

with volume Ļāz contains many atoms. (2) The envelope operator Ļ (z, t) is essentially

constant over an interval of length āz. The function

Ļ (z ā’ zn ) = Īø (āz/2 ā’ z + zn ) Īø (z ā’ zn + āz/2) (16.57)

serves to conļ¬ne the n-sum to the atoms in a slab of thickness āz centered at z. The

atomic envelope operators are deļ¬ned by

Sqp (z, t) = S qp (z, t) eiĻqp t ei[Ļq (z,t)ā’Ļp (z,t)] , (16.58)

where the phases satisfy

Ļ2 (z, t) ā’ Ļ1 (z, t) = ā0 t ā’ k0 z . (16.59)

Using this notation, together with eqn (16.54), allows us to rewrite eqn (16.52) as

LS

HS1 = ā’i dz f S 21 (z, t) Ļ (z, t) ā’ HC , (16.60)

0

where

(vg0 /c) Ļ0 d21 Ā· e0

fā” (16.61)

2 0Ļ

is the coupling constant.

The total electromagnetic part of the Hamiltonian for this 1D model is, therefore,

ā

dzĻā (z, t) vg0 āz Ļ (z, t) + HS1 .

Hem = (16.62)

i

ā’ā

This leads to the corresponding Heisenberg equation

ā‚ ā‚

Ļ (z, t) = fā— S 12 (z, t) for 0

+ vg0 z LS , (16.63)

ā‚t ā‚z

ā‚ ā‚

+ vg0 Ļ (z, t) = 0 for z < 0 or z > LS (16.64)

ā‚t ā‚z

for the ļ¬eld.

Ā½Āæ

Traveling-wave ampliļ¬ers

The atomic operators are coupled to the reservoirs describing the incoherent pump

and spontaneous emission into oļ¬-axis modes; therefore, we insert eqn (16.60) into the

coarse-grained version of eqn (14.177) to ļ¬nd

d

S 12 (z, t) = [iā0 ā’ Ī“12 ] S 12 (z, t) ā’ f S 11 (z, t) ā’ S 22 (z, t) Ļ (z, t) + Ī¾12 (z, t) .

dt

(16.65)

The coarse-grained noise operator

1 (n)

Ī¾12 (t) Ļ (z ā’ zn )

Ī¾12 (z, t) = (16.66)

āz n

has the correlation function

ā

Ī¾12 (z, t) Ī¾12 (z , t ) = nat ĻC12,12 Ī“ (t ā’ t ) Ī“ (z ā’ z ) , (16.67)

where Ī“ (z ā’ 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 Ī“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) ā’ S 11 (z, t) Ī¾12 (z, t)

ńņš. 19 |