5

Chapter 5
Quantum Phenomena in Simple Systems in Nonlinear Optics












Abstract In this chapter we will analyse some simple processes in nonlinear optics where analytic solutions are possible. This will serve to illustrate how the formalism developed in the preceding chapters may be applied. In addition, the simple exam- ples chosen illustrate many of the quantum phenomena studied in more complex systems in later chapters.
This chapter will serve as an introduction to how quantum phenomena such as
photon antibunching, squeezing and violation of certain classical inequalities may occur in nonlinear optical systems. In addition, we include an introduction to quan- tum limits to amplification.



5.1 Single-Mode Quantum Statistics


A single-mode field is the simplest example of a quantum field. However, a num- ber of quantum features such as photon antibunching and squeezing may occur in a single-mode field. To illustrate these phenomena we consider the degenerate para- metric amplifier which displays interesting quantum behaviour.



5.1.1 Degenerate Parametric Amplifier


One of the simplest interactions in nonlinear optics is where a photon of frequency 2? splits into two photons each with frequency ?. This process known as para-
metric down conversion may occur in a medium with a second-order nonlinear sus- ceptibility ? (2). A detailed discussion on nonlinear optical interactions is left until
Chap. 9.
We shall make use of the process of parametric down conversion to describe a parametric amplifier. In a parametric amplifier a signal at frequency ? is amplified by pumping a crystal with a ? (2) nonlinearity at frequency 2?. We consider a simple


73


model where the pump mode at frequency 2? is classical and the signal mode at frequency ? is described by the annihilation operator a. The Hamiltonian describing
the interaction is
H = k?a†a ? ik ? .a2e2i?t ? a†2e?2i?t . , (5.1) where ? is a constant proportional to the second-order nonlinear susceptibility and
the amplitude of the pump. If we work in the interaction picture we have the time-
independent Hamiltonian


HI = ?ik 2 .a

The Heisenberg equations of motion are


? a†2.


. (5.2)




da 1
=
dt ik


[a, HI]= ? a†,

da† dt

= .a†, HI . = ? a . (5.3)
ik


The interaction picture can be viewed equivalently as transforming to a frame rotat- ing at frequency ?.
These equations have the solution
a (t)= a (0) cosh ?t + a† (0) sinh ?t , (5.4) which has the form of a generator of the squeezing transformation, see (2.60). As
such we expect the light produced by parametric amplification to be squeezed. This can immediately be seen by introducing the two quadrature phase amplitudes
a ? a†

X1 = a + a†, X2 =
i

which diagonalize (5.2 and 5.3)

(5.5, 5.6)


dX1 = +? X , dX2 =


? X . (5.7, 5.8)

dt 1

dt ? 2


These equations demonstrate that the parametric amplifier is a phase-sensitive am- plifier which amplifies one quadrature and attenuates the other:
X1 (t)= e?t X1 (0) , (5.9)
X2 (t)= e??t X2 (0) . (5.10)

The parametric amplifier also reduces the noise in the X2 quadrature and increases the noise in the X1 quadrature. The variances V (Xi,t) satisfy the relations
V (X1,t)= e2?tV (X1, 0) , (5.11)
V (X2,t)= e?2?tV (X2, 0) . (5.12)


5.1 Single-Mode Quantum Statistics 75
For initial vacuum or coherent states V (Xi, 0)= 1, hence
V (X1,t)= e2?t ,
V (X2,t)= e?2?t ,

(5.13)
and the product of the variances satisfies the minimum uncertainty relation V (X1) V (X2)= 1. Thus the deamplified quadrature has less quantum noise than the vacuum level. The amount of squeezing or noise reduction is proportional to the strength of the nonlinearity, the amplitude of the pump and the interaction time.



Photon Statistics


We shall next consider the photon statistics of the light produced by the parametric amplifier. First we analyse the light produced from an initial vacuum state. The intensity correlation function g(2)(0) in this case is

† †
g(2) (0)= (a (t ) a (t ) a (t ) a (t ))
(a (t) a (t))
cosh 2?t

= 1 +

sinh2 ?t

. (5.14)


This indicates that the squeezed light generated from an initial vacuum exhibits photon bunching (g(2)(0) > 1). This is expected for a squeezed vacuum which must contain correlated pairs of photons.
For an initial coherent state |?) we find the mean photon number

(a (t) a (t)) = |?|

(cosh 2?t + cos 2? sinh 2?t)+ sinh2

?t , (5.15)

where we have used ? = |?|ei? , and the intensity correlation function


g(2) (0) ? 1 +


2


1
|?|2 e?2?t

2


.e?2?t ? 1. , ? =

?
4 , (5.16)

where |?|

is large compared with sinh

?t and sinh ?t cosh ?t.

Thus under these conditions the photon statistics of the output light is anti-
bunched. We see that a parametric amplifier evolving from an initial coherent state
|i|?|) evolves towards an amplitude squeezed state with a coherent amplitude of
|?|e??t . This reduction in amplitude is due to the dynamic contraction in the X2
direction described by (5.10) (Fig. 5.1).


Fig. 5.1 Schematic represen- tation of the evolution of an initial coherent state undergo- ing parametric amplification















Wigner Function

The full photon statistics of the light generated in parametric amplification may be calculated via a quasi-probability distribution. While we could choose to calculate the P function we would find that it would become singular due to the quantum correlations which build up during the amplification process. Therefore we shall calculate the Wigner distribution which is a nonsingular positive function for this problem.
The Wigner function describing the state of the parametric oscillator at any time
t may now be calculated via the symmetrically ordered characteristic function,

? (?,t)= Tr ,

(0) e?a†(t)???a(t), . (5.17)


Let us take the initial state to be the coherent state ? (0)= |?0)(?0 |. Then substitut-
ing (5.4) into (5.17) we find

. ?|2 .

? (?,t)= exp

??? (t) ? ???0 (t) ? |

cosh 2?t + 1 .?2 + ??2. sinh 2?t ,



where


This may be written as

(5.18)

?0 (t)= ?0 cosh ?t + ?? sinh ?t . (5.19)

? (?,t)= exp .?T • ?? (t)+ 1 ?T??. , (5.20)
0 2
?T = (?, ???) , (5.21)
?T ?
0 (t)= (?0 (t) , ?0 (t)) (5.22)



and


1 . sinh 2?t cosh 2?t .

? = 2

cosh 2?t sinh 2?t

. (5.23)

The Wigner function is then given by the Fourier transform of ? (?, t), see (4.33). Using (4.36) the result is


2
W (?,t)=
?

. 1
exp
2

.
[? ? ?0 (t)]T C?1 [? ? ?0 (t)]



, (5.24)



where ?T = (?, ?? ). This is a two variable Gaussian with mean ?0(t) and covari- ance matrix C? = ?. In terms of the real variables x1 = ? + ??. x2 = ?i(? ? ??)
(corresponding to the quadrature phase operators), the Wigner function becomes
1 . 1 1 .

W (x1, x2)= 2? exp

? 2 [x ? x0 (t)]C?

[x ? x0 (t)]

, (5.25)


where



Cx =


. e2?t 0 .
0 e?2?t



. (5.26)


Thus the Wigner function is a two-dimensional Gaussian with the variance in fluctu- ations in the quadratures X1 and X2 given by the major and minor axes of the elliptic contours.



Two-Mode Quantum Correlations


In two-mode systems there are a richer variety of quantum phenomena since there exists the possibility of quantum correlations between the modes. These correlations may give rise to two mode squeezing such, as described by (2.85). There may also exist intensity and phase correlations between the modes. A simple system which displays many of the above features is the non-degenerate parametric amplifier [1].



Non-degenerate Parametric Amplifier


The non-degenerate parametric amplifier is a simple generalization of the degen- erate parametric amplifier considered in the previous section. In this case the clas-
sical pump mode at frequency 2?1 interacts in a nonlinear optical medium with
two modes at frequency ?1 and ?2. These frequencies sum to the pump frequency, 2? = ?1 + ?2. It is conventional to designate one mode as the signal and the other
as the idler.
The Hamiltonian describing this system is
H = k?1a†a1 + k?2a†a2 + ik? .a†a†e?2i?t ? a1a2e2i?t . , (5.27)
1 2 1 2



where a1(a2) is the annihilation operator for the signal (idler) mode. The coupling constant ? is proportional to the second-order susceptibility of the medium and to
the amplitude of the pump.
The Heisenberg equations of motion in the interaction picture are

da1 = ? a† , (5.28)
dt 2
da†

The solutions to these equations are

2 = ? a1 . (5.29)
dt


a1 (t)= a1 cosh ?t + a† sinh ?t , (5.30)
a2 (t)= a2 cosh ?t + a† sinh ?t . (5.31)

If the system starts in an initial coherent stat |?1), |?2), the mean photon number in mode one after time t is
. .
(n1 (t)) = (?1 , ?2 .a† (t) a1 (t). ?1, ?2)
. 1 .
2 2

= |?1 cosh ?t + ?? sinh ?t|

+ sinh

?t . (5.32)


The last term in this equation represents the amplification of vacuum fluctuations since if the system initially starts in the vacuum (?1 = ?2 = 0) a number of photons given by sinh2 ?t will be generated after a time t.
The intensity correlation functions of this system exhibit interesting quantum fea- tures. With a two-mode system we may consider cross correlations between the two modes. We shall show that quantum correlations may exist which violate classical inequalities.
Consider the moment (a†a1a†a2). We may express this moment in terms of the
1 2
Glauber–Sudarshan P function as follows:


(a†a1a†a2) =


d2?1


d2?2 |?1 |2 |?2|2 P (?1, ?2 ) . (5.33)

1 2

If a positive P function exists the right-hand side of this equation is the classical intensity correlation function for two fields with the fluctuating complex amplitudes
?1 and ?2. It follows from the Ho¨ lder inequality that

.¸
d2?1 d2?2 |?1 |2 |?2|2 P (?1, ?2 ) ?

.1/2
d2?1 d2?2 |?1 |4 P (?1, ?2 )

.¸ .1/2
× d2?1d2?2 |?2|4 P (?1, ?2 )



. (5.34)


Re-expressed in terms of operators this inequality implies
(a†a1a†a2)? [((a†)2 a2)((a†)2a2)]1/2 , (5.35)
1 2 1 1 2 2


a result known as the Cauchy–Schwarz inequality. If the two modes are symmetric as for the non-degenerate parametric amplifier this inequality implies
(a†a1a†a2)? ((a†)2a2) . (5.36)
1 2 1 1

Because we have assumed a positive P function this is a weak inequality and there exists certain quantum fields which will violate it.
It is more usual to express the Cauchy–Schwarz inequality in terms of the second-
order intensity correlation functions defined for a single-mode field in (3.63). The two-mode intensity correlation function is defined by

† †

g(2)

a1 a1 a2 a2

12 (0)= (

) . (5.37)

(a†a1)(a†a2)
1 2

This definition together with

† †

g(2)

ai ai ai ai

i (0)= (

)
(a†ai)

(5.38)

i

enables one to write the Cauchy–Schwarz inequality as


[g(2)

(2)

(2)

12 (0)]2 ? g1 (0)g2 (0) . (5.39)

A stronger inequality may be derived for quantum fields when a Glauber–Sudarshan P representation does not exist. The appropriate inequality for two non-commuting operators is, see (3.26),

2 † 2 † 2

(a†a1a†a2)

? ((a a1) )((a a2) ) . (5.40)

1 2 1 2

For symmetrical systems this implies
(a†a1a†a2)? ((a†)2a2) + (a†a1) (5.41)
1 2 1 1 1

or

g(2)

(2)

1

12 (0) ? g1 (0)+

(a†a1)

. (5.42)

We now show that the non-degenerate parametric amplifier if initially in the ground state leads to a maximum violation of the Cauchy–Schwarz inequality (5.39), as is consistent with the inequality (5.42). That is, the correlations built up in the parametric amplifier are the maximum allowed by quantum mechanics.
In this system the following conservation law is easily seen to hold,
n1 (t) ? n2 (t)= n1 (0) ? n2 (0) , (5.43)
where ni(t) ? a†(t)ai(t). This conservation law has been exploited to give squeezing in the photon number difference in a parametric amplifier as will be described in


Chap. 8. Using this relation the intensity correlation function may be written
(n1 (t) n2 (t)) = (n1 (t)2) + (n1 (t) [n2 (0) ? n1 (0)]) . (5.44)

If the system is initially in the vacuum state the last term is zero, thus
(n1 (t) n2 (t)) = (a† (t) a† (t) a1 (t) a1 (t)) + (a† (t) a1 (t)) , (5.45)
1 1 1

which corresponds to the maximum violation of the Cauchy–Schwarz inequality allowed by quantum mechanics.
Thus the non-degenerate parametric amplifier exhibits quantum mechanical cor-
relations which violate certain classical inequalities. These quantum correlations may be further exploited to give squeezing and states similar to those discussed in the EPR paradox, as will be described in the following subsections.



Squeezing


In the interaction picture, the unitary operator for time evolution of the non- degenerate parametric amplifier is
U (t)= exp .?t .a†a2 ? a1a2.. (5.46) Comparison with (2.87) shows that U (t) is the unitary two-mode squeezing opera-
tor, S(G) with G = ??t. We will define the squeezing with respect to the quadrature phase amplitudes of the field at the local oscillator frequency ?LO and phase refer-
ence ? [2].
To see this explicitly consider the positive frequency components of the field that
results for a super-position of the signal and a local oscillator field in a coherent state with (E(?)(t)) = |?|e?i(?LOt+?). The positive frequency components of the
superposed field is then well approximated by

T (t)= E(?) (t)+ |?| e?i(?LOt+? ) (5.47) The photo-current when such a field is directed to a detector is then proportional to

(?)

(+)

i(t) ? (ET (t)ET (t)). We now define the average homodyne detection signal by
subtracting off the known local oscillator intensity and normalising by |?|,


s (t)= (E(?) (t) ei(? +?LOt)+E


(+)

(t)e?i(?+?LOt)


) (5.48)


The noise in the signal will then be determined by the variance in the operator
sˆ(t)= E(?)(t)ei(?+?LOt) + E(+)(t)e?i(?+?LOt) .
We can now make a change of variable for the frequencies of the signal and idler fields by writing ?1 = ? ? ?, ?2 = ? + ? with ? > 0. This change of variable



anticipates a homodyne detection scheme. If we mix these two modes with a local oscillator at half the pump frequency, i.e. at ?LO = ?, the resulting signal will have Fourier components at frequencies ±?.
In the new frequency variables, the total field of the signal plus idler in the Heisenberg picture is the sum of two modes ? ± ? symmetrically displaced about
the local oscillator,


1
E (t)= ?2

.
a1 (t) e


?i(?+?)t


+ a2 (t) e


?i(?+?)t

+ h.c .


where ai(t) is the solution given in (5.30 and 5.31), h.c means hermitian conjugate,
and the factor 1/?2 has been inserted to give a convenient definition of the vacuum.
This may be written as
E (t)= X? (t, ?) cos (?t + ?) ? X? +?/2 (t, ?) sin (?t + ?) (5.49) and the quadrature phase operators are defined as

1
X? (t, ?)= ?2


a1 (t) ei?


+ a† (t) e

?i? .

ei?t

+ h.c .

i ..


i? †

?i?.

i?t .

X?+?/2 (t, ?)= ?2

a1 (t) e

? a2 (t) e

e + h.c.



In this form, as ? > 0, we can distinguish the positive and negative frequency com-
ponents of the quadrature phase operators with respect to the local oscillator fre-
quency.
If the system starts in the vacuum state, the homodyne detection signal at ? = 0
(the DC signal) will have a variance given by

X? (t, ? = 0)= cosh 2?t + cos 2? sinh 2?t (5.50)


Thus for ? = 0, we find that



V (X0 (t, ? = 0)) = e2?t (5.51)
V .X?/2 (t, ? = 0). = e?2?t (5.52)

Changing the phase of the local oscillator by ?/2 enables one to move from en- hanced to diminished noise in the homodyne signal. We note that the squeezing in the non-degenerate parametric amplifier is due to the development of quantum cor- relations between the signal and idler mode. The individual signal and idler modes are not squeezed as is easily verified.


Quadrature Correlations and the Einstein–Podolsky–Rosen Paradox


The non-degenerate parametric amplifier can also be used to prepare states similar to those discussed in the Einstein–Podolsky–Rosen (EPR) paradox [3]. In the original treatment two systems are prepared in a correlated state. One of two canonically conjugate variables is measured on one system and the correlation is such that the value for a physical variable in the second system may be inferred with certainty.
To see how this behaviour is manifested in the non-degenerate parametric am- plifier we first define two sets, one for each mode, of canonically conjugate vari- ables, i.e.,

i = aiei?

+ a†e?i?

(i = 1, 2) . (5.53)

The variables X ? and X ? +?/2 obey the commutation relation
i i
X ? ? +?/2.

i , Xi

= ?2i (5.54)


and are thus directly analogous to the position and momentum operators discussed in the original EPR paper.
To measure the degree of correlation between the two modes in terms of these
operators, we consider the quantity

V (? , ? ) ? 1 ((X ? ? X ? )2) . (5.55)
2 1 2
If V (? , ? )= 0 then X ? is perfectly correlated with X ? . This means a measurement
1 2
of X ? can be used to infer a value of X ? with certainty. To appreciate why such a
1 2
correlation should occur in the non-degenerate parametric amplifier, we can write
the interaction Hamiltonian directly in terms of the defined canonical variables,

HI = ? 2k? sin (? + ? ) .


1 X ? ? X


? +?/2X

? +?/2.

X ? 2
? 2k? cos (? + ? ) .


1 2
X + X ? X


. . (5.56)

? +?/2 ?
1 2

? +?/2
1 2


The Heisenberg equation of motion for X ? is then


X? ? X ?

? +?/2 .

1 = ?4? .

2 cos (? + ? ) ? X2

sin (? + ? )

(5.57)


and we see that X ? is coupled solely to X ? when ? + ? = 0.
1 2
Direct calculation of V (? , ? ) using the solutions in (5.30 and 5.31) gives
V (? , ? )= cosh 2?t ? sinh 2?t cos (? + ? ) . (5.58) When ? + ? = 0, V (? , ? )= e?2?t and, for long times, V (? , ? ) becomes increas- ingly small reflecting the build up of correlation between the two variables for this


case. Initially, of course, the two systems are uncorrelated and V (? , ? ) = 1. As V (? , ? ) tends to zero the system becomes correlated in the sense of the EPR para- dox. As time proceeds a measurement of X ? yields an increasingly certain value of X ? . However one could equally have measured X ? ??/2. Thus certain values for
2 1
two noncommuting observables X ? , X ? +?/2 may be obtained without in anyway
2 2
disturbing system 2. This outcome constitutes the centre of the EPR argument.
Of course, in reality no measurement enables a perfect inference to be made. To quantify the extent of the apparent paradox, we can define the variances Vinf(X ? ) and Vinf(X ? +?/2) which determine the error in inferring X ? and X ? +?/2 from di-
2 2 1
rect measurements on X ? and X ? ??/2. In the case of direct measurements made on
2 1
(X ? , X ? +?/2
2 2 ) quantum mechanics would suggest

X ? .V .X

? +?/2

.
? 4 .

2 2

However the variances in the inferred values are not constrained. Thus whenever
Vinf(X ? )Vinf(X ? +?/2) < 4, we can claim an EPR correlation paradoxically less than
2 2
expected by direct measurement on the same state. This result seems to contradict
the uncertainty principle. That this is not the case is seen as follows. In the standard uncertainty principle the variances are calculated with respect to the same state. However in the inference uncertainty product the variances are not calculated in the
same state. That is to say Vinf(X ? +?/2) is calculated on the conditional state given a
result for a measurement of X ? +?/2, however Vinf(X ? ) is calculated on the different
2 2
conditional state given a result for the measurement of X ? .
Ou et al. [4] performed an experimental test of these for the parametric amplifier. Using their quadrature normalization, the inferred variances indicate a paradoxical
result if

Vinf .

X ? +?/2

.
? 2 .

The experimental result for the lowest value of the product was 0.7 ± 0.01.



Wigner Function


The full quantum correlations present in the parametric amplifier may be repre- sented using a quasi-probability distribution. If both modes of the amplifier are ini- tially in the vacuum state no Glauber P function for the total system exists at any time. However, a Wigner function may be obtained. We shall proceed to derive the Wigner function for the parametric amplifier.
We may define a two mode characteristic function by a simple generalization of
the single-mode definition. For both modes initially in the vacuum state this may be expressed as

. . .

? (?2, ?2,t)= (0| exp .

a† (t) ? ??a1 (t)

exp

?2a† (t) ? ??a2 (t)

|0)

?1 1 1 2 2
. 1 . 2 1 2.

= exp ? 2

?1 (t) |

? 2 ?2 (t) |

. (5.59)

where

?1 (t)= ?1 cosh ?t ? ?? sinh ?t,
?2 (t)= ?2 cosh ?t ? ?? sinh ?t,

The Wigner function is then given by

1 ¸ 2 ¸ 2

? ? ? ?

W (?1, ?2 ,t)= ? 4
4
=
? 2

d ?1
exp .

d ?2 exp (?1 ?1 ??1?1 ) exp (?2 ?2 ? ?2?2 ) ? (?1, ?2,t)

2|?1 cosh ?t ? ?? sinh ?t|2

? 2|?2 cosh ?t ? ?? sinh ?t|2) . (5.60)

This distribution may be written in terms of the uncoupled c-number variables

?1 = ?1 + ??,
?2 = ?1 ? ??.

In these new variables the Wigner function is


4

. 1 . |?1|2

|?2|2 ..

W (?1, ?2)= ?2 exp ? 2

e2?t + e?2?t

, (5.61)



in which form it is particularly easy to see that squeezing occurs in a linear combi- nation of the two modes. The variances in the two quadratures being given by e?2?t and e2?t , respectively. It is interesting to note that even though the state produced
contains non-classical correlations the Wigner function always remains positive.



Reduced Density Operator


When a two component system is in a pure state the reduced state of each component system, determined by a partial trace operation, will be a mixed state. An interesting feature of the non-degenerate parametric amplifier is that the reduced state of each mode is a thermal state, if each mode starts from the vacuum.
To demonstrate this result we first show the high degree of correlation between
the photon number in each mode. The state of the total system at time t is

|? (t)) = exp .?t . a

? a1a2.. |0) (5.62)


We now make use of the disentangling theorem [5]

. † † .

. † † .

e? a1a2?a1a2

= e?a1a2 e

?g a1a1+a2a2+1

e??a1a2 , (5.63)



where


? = tanh ? ,
g = ln(cosh ? ).



Thus


† † ?

|? (t)) = e?ge?a1a2 |0) = (cosh ?t)?1 ? (tanh ?t)n |n, n) , (5.64)
n=1
where |n, n)? |n)1 |n)2. As photons are created in pairs there is perfect correlation between the photon number in each mode. The reduced state of either mode is then
easily seen to be
?
?i (t)= Tr j {|? (t))(? (t) |} = (cosh ?t)?2 ? (tanh ?t)2n |n)i(n| i ?= j . (5.65)
n=0
This is a thermal state with mean n¯ = sinh2 ?t, having strong analogies with the Hawking effect associated with the thermal evaporation of black holes.
Suppose, however, that a photodetector with quantum efficiency ? has counted
m photons in mode b. What is the state of mode a conditioned on this result? Such
a conditional state for mode a is referred to as the selected state as it is selected from an ensemble of systems each with different values for the number of photons counted in mode b. We shall now describe how the conditional state of mode a may be calculated.
In Chap. 3 we saw that the probability to detect m photons from a field with the photon-number distribution P(n) and detector efficiency ? is


?
P? (m)= ?
n=m

. n .
m


(1 ? ?)n?m


?mP1 (n) , (5.66)


where P1(n) is the photon number distribution for the field. This equation may be written as
P? (m) ? Tr .? ?† (m) ?? (m). (5.67)
and the operator ? on mode b is defined by


?
?? (m)= ?
n=m

. n .1/2
m


(1 ? ?)(n


?m)/2


?m/2


|n ? m)b(n| . (5.68)


Note that when ? ? 1 this operator approaches the projection operator |0)b(m|. This is quite different to the projection operator |m)b(m| that a naive application of
the von Neumann projection postulate would indicate for photon counting measure-
ments, and reflects the fact that real photon-counting measurements are destructive,




Fig. 5.2 Photon number distribution for mode a given that 5 photons are counted in mode b of a parametric amplifier. (a) ? = 0.9. (b) ? = 0.6



i.e. photons are absorbed upon detection. The conditional state of mode a is then given by

? (m) = .P? (m).?1 Trb ,

(m)

?† (m)

. (5.69)

?? ? m

This equation is a generalisation of the usual projection postulate. In the case of the correlated two mode state in (5.65), the conditional state of mode a becomes





with

?
? (m) = .P? (m).?1 ?
n=m

. n .
m


?m (1 ? ?)n?m |n)(n| , (5.70)

P? (m)= (1 + n¯)?1 (??)m [1 ? ? (1 ? ?)](m+1) , (5.71)
where ? = tanh2 ?t, n¯ = sinh2 ?t. Equation (5.70) represents a state with at least m quanta. In Fig. 5.2 we show the photon number distribution P(m)(n) for this con-
ditional state. As one would expect, when ? ? 1, this approaches a number state
|m). It should be noted, however, that the conditional state computed above refers
to a situation in which the counting is done after the interaction which produces
the correlated state, is turned off. In a cavity configuration, however, it is likely that photon counting is proceeding simultaneously with the process of parametric ampli- fication. In that case one must proceed a little differently, however the overall result is much the same, i.e., mode a is left with at least m quanta. The details of this more complicated calculation will be found in the paper by Holmes et al. [6].


Quantum Limits to Amplification


The non-degenerate parametric amplifier exemplifies many features of general lin- ear amplification. One such feature is the limit placed on the amplifier gain if the output is to be squeezed. To see how this limit arises, and to see how it might be overcome, we write the solutions (5.30 and 5.31) in the form

5.3 Quantum Limits to Amplification 87

X ? 1/2 ?

1/2 ?

1, OUT = G

X1, IN + (G ? 1)

X2, IN , (5.72)

where G = cosh2 ?t and the quadrature phase operators are defined in (5.53). The subscript IN denotes operators defined at t = 0 and the subscripts 1 and 2 refer to the signal and idler modes, respectively. In (5.72) the first term describes the amplification of the quadrature and the second term the noise added by the amplifier. The variances obey the equation

V .X ?

. = GV .

. + (G ? 1)V .

. . (5.73)

?
1, OUT ,

X ? IN

The maximum gain consistent with any squeezing at the output is

GMAX =

. ? .
2, IN

. (5.74)

X ? . + V .X ? .

1, IN

2, IN


If the idler mode is in the vacuum state, V (X ?

2
GMAX =


)= 1 then


, (5.75)

. ? .
1, IN

which gives a maximum gain of 2 for a highly squeezed state at the signal input. For higher values of the gain the squeezing at the signal output is lost due to contamina- tion from the amplification of vacuum fluctuation in the idler input.
Greater gains may be achieved while still retaining the squeezing in the output

signal if the input to the idler mode, is squeezed (V (X ?

) < 1).

If we define the total noise in the signal as the sum of the noise in the two quadra- tures

N = Var ,

, + Var ,X

, (5.76)


then

? ? +?/2
1



where

NOUT = G (NIN + A) , (5.77)

. 1 . .

, ? ,

, ? +?/2,.

A = 1 ? G
. 1 .

Var

X2, IN

+ Var

X2, IN

? 2 1 ? G

. (5.78)


This is in agreement with a general theorem for the noise added by a linear am-
plifier [7]. The minimum added noise A = 2(1 ? 1/G) occurs when Var(X ? )=
2, IN )= 1, that is, when the idler is in a coherent or vacuum state.


5.4 Amplitude Squeezed State with Poisson Photon Number Statistics


Finally we consider a simple nonlinear optical model which produces an amplitude squeezed state which has Poissonian photon number statistics [8, 9]. The model describes a quantised field undergoing a self-interaction via the Kerr effect. The Kerr effect is a nonlinear process involving the third-order nonlinear polarisability of a nonlinear medium. The field undergoes an intensity dependent phase shift, and thus we regard the medium as having a refractive index proportional to the intensity of the field.
Quantum mechanically the Kerr effect may be described by the effective Hamil-

tonian

H = k 2 .a

†.2 a2


, (5.79)

where ? is proportional to the third-order nonlinear susceptibility. The Heisenberg
equation of motion for the annihilation operator is

da
= i? a†aa . (5.80)
dt

As a†a, the photon number operator, is a constant of motion the photon number statistics is time invariant. The solution is then
a (t)= e?i?ta†aa (0) . (5.81) Assume the initial state is a coherent state with real amplitude ?. The mean ampli-
tude at a later time is then
(a (? )) = ? exp .??2 (1 ? cos? ) ? i?2 sin ? . , (5.82) where we have defined ? = ?t. Typically ? ? 1 and then

2 2 2
(a (? ))? ?e?i? ? ?? ? /2 . (5.83)

This result displays two effects. Firstly, there is a rotation of the mean amplitude by ?2? ; the expected nonlinear phase shift. Secondly, there is a decay of the am-
plitude which goes quadratically with time. This decay is due to the fact that the Kerr effect transforms intensity fluctuations in the initial coherent state into phase fluctuations (Fig. 5.3). In effect, the initial coherent state error circle undergoes a rotational shearing while the area remains constant.
Inspection of Fig. 5.3 suggests that, at least for short times, this system is likely to produce a squeezed state with reduced amplitude fluctuations. This is indeed the
case. For short times (? ? 1) and large intensities (?2 ± 1) one finds the minimum
variance of the in-phase quadrature approaches the value

V (X1)min = 0.4 . (5.84)

5.4 Amplitude Squeezed State with Poisson Photon Number Statistics 89


Fig. 5.3 Contour of the Q-function (at a height of
0.3) for the state of a single mode, prepared in a coherent
state with ? = 2.0, evolved
with a Kerr nonlinearity for
? = 0.25. The equivalent con-
tour for the initial coherent
state is shown as dashed


3


2


1


X2 0


-1

-2

-3
-3 -2 -1 0 1 2 3
X1



This occurs at the value


? 0.55 . (5.85)
?

This short time behaviour is evident in Fig. 5.4a. Thus even though the photon statis- tics is at all times Poissonian, for short times the field is amplitude squeezed.
We now consider mixing the output of the nonlinear process with a coherent field on a beam splitter of low reflectivity. The output field is now given by

?
a0 = T


e?i? a†a

a + ?


? , (5.86)


where ? is the coherent amplitude of the mixing field, and T and R are, respec- tively, the transmitivity and reflectivity of the beam splitter. We assume T ? 1 with
?R? ? ? , that is the mixing field is very strong. In this limit we have
a0 = e?i? a aa + ? . (5.87)

We now can choose ? so as to minimise the photon number noise at the output. This requires ? to be ?? /2 out-of-phase with the coherent excitation of the input. As ? increases, the ratio of the number variance to number mean decreases to a minimum at ? = 1/2(n0)?2/3 (for optimal ? ), and then increases. The minimum
photon number variance is [10]

V (n0)= (n0 1/3 2


, (5.88)

where (n0) = (n) + |? | . This is smaller than the similar result for a squeezed state,
2/3
which has a minimum value of (n) .


Fig. 5.4 A plot of the variance in the in-phase quadrature for the Kerr interaction, versus the dimensionless interaction
time ? , with an initial coher-
ent state of amplitude ? = 4.0.
(a) Short time behaviour. (b)
Long time behaviour



















Were the rotational shearing to continue (as one might expect from a classical model) the variance in the in-phase and out-of-phase quadratures would saturate at
the value 2?2 + 1. This would be the variance for a number state with the photon
number equal to ?2. That this does not happen is evident in Fig. 5.4b. Indeed, from (5.82) it is clear that for ? = 2? the mean amplitude returns to the initial value. A
similar result holds for the variances (Exercise 5.6). This is an example of a quantum recurrence and arises from the discrete nature of the photon number distribution for
a quantised field. The details are left for Exercise 5.6. In fact at ? = ? the system
evolves to a coherent superposition of coherent states:


1
|? (? = ?))| = ?2

.ei?/4


|i?) + e


?i?/4

.
|? i?)


. (5.89)


This phenomenon would be very difficult to observe experimentally as typical val- ues of ? would require absurdly large interaction times, which in practice means
extremely large interaction lengths. In Chap. 15 we will show that dissipation also makes the observation of such a coherent superposition state unlikely in a Kerr medium.

References 91

Exercises


Derive the Wigner and P functions for the reduced density operator of the signal mode for the non-degenerate parametric amplifier.
Show that, n1 ? n2, the difference in the number of photons in the signal and
idler mode is a constant for the parametric amplifier.
The Hamiltonian for the frequency up-converter is
H = k?1a†a1 + k?2a†a2 + k? .ei?t a†a2 + e?i?t a1a†. ,
1 2 1 2

where ? = ?2 ? ?1. Show that n1 + n2, the sum of the number of photons in
the signal and idler modes, is a constant.
Show that the process of parametric frquency upconversion is noiseless, that is a coherent state remains coherent.
Take the initial state for the frequency upconverter to be |N, N). Express the
density operator at time t as the tensor product of number states. Hint: Use the
disentangling theorem, see (5.63). What is the reduced density operator for a single mode?
5.6(a) If the initial state for the Kerr-effect model is a coherent state with real mean amplitude, calculate the variances for the in-phase and out-of-phase
quadratures. Show that at ?t = ? the field exhibits amplitude squeezing for
small values of the amplitude.
(b) Show that at ?t = ? the state may be written in the form

1 .ei?/4
?2


|? i?) + e


?i?/4

|i?. .