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Reduction to the Fock state via a degenerate four-wave mixing
1 October 1996
OPTiCIS COMMUNICATIONS Optics Communications 130 (1996) 365-376
Full length article
Reduction to the Fock state via a degenerate four-wave mixing Masashi Ban Advanced Research Laboratory, Hitachi Ltd., Hatoyama, Saitama 350-03, Japan
Received 19 December 1995; revised version received 4 March 1996; accepted 22 March 1996
Abstract A scheme to generate the Fock state via a degenerate four-wave mixing and partial measurement is proposed. A conditional state of the signal mode of a degenerate four-wave mixer will be shown to reduce to the Fock state when the outcome of the measurement on the reference mode satisfies a certain condition. The photon number of the Fock state is distributed when the measurement is repeated many times. Conditioning for the quantum state of the signal mode is carried out by the photon counting measurement on the reference mode whose input state is assumed to be the vacuum. The phase coherence of the conditional signal state is also considered. Furthermore conditioning for the signal mode by means of heterodyne and homodyne detections are briefly considered. Keywords: Four-wave mixing; Fock state; Quantum nondemolition measurement
1. Introduction Nonclassical states of photons, exhibiting the quadrature squeezing and the sub-Poissonian photon statistics, are of great importance in high precision measurement and quantum communication. One of the most typical and simple nonclassical states is the Fock state, that is, the eigenstate of photon number. The scheme for generating the Fock state by means of quantum nondemolition measurements [l-3] has been recently proposed by several authors [ 4-101. In this scheme, a beam of Rydberg atoms (which have two or three energy levels) is used to measure nondestructively the photon number in a microcavity. The interaction between the cavity photon and the atom is assumed to be described by the Jaynes-Cummings Hamiltonian [ 11,121. The quantum nondemolition measurement of photon number in the cavity by means of the Raman-coupled model [ 13,141 has been also proposed [ 151. These measurements can produce the Fock state of the cavity field. In this paper, the Fock-state generation via a degenerate four-wave mixing and partial measurement is considered. In this scheme, the photon number of the signal mode can be measured nondestructively by a photon counting measurement on the reference mode of the four-wave mixer. It will be shown that when the outcome of the measurement satisfies a certain condition, the conditional state of the signal mode of the degenerate four-wave mixer reduces to the Fock state. The photon number of the Fock state is distributed when the experiment is repeated many times. Conditioning for the signal state of the four-wave mixer is carried out by means of the photon counting measurement on the reference mode whose input state is assumed to be the vacuum. Furthermore, a decay of phase coherence of the conditional state of the signal mode caused by the 0030-4018/96/$12.00 Copyright @ 1996 Elsevier Science B.V. All rights reserved. PII SOO30-40 18 (96) 00263-5
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reduction to the Fock state is investigated. As an example of the initial state of the signal mode, the Fock state, coherent state and superposition of two Fock states are considered. Finally, the heterodyne and homodyne detection measurements on the reference mode are briefly considered as conditioning for the signal mode.
2. Conditional
state of a degenerate
2.1. Model of a degenerate four-wave
four-wave
mixer
mixer
The model of a degenerate four-wave mixer considered in this paper is explained. The one mode of the four-wave mixer is highly excited so that it can be treated classically, and the two other modes are degenerate. Thus the Hamiltonian of the degenerate four-wave mixer [ 161 can be written as
(1) where ii (6) and &t (6t) are the bosonic annihilation and creation operators, satisfying the canonical commutation relations, [ 8, at] = 1 and [h, St ] = 1, and eeeiot stands for the classical pump field. In the following, the mode represented by (a, at) is referred to as the signal mode and the mode represented by (&, 6t) is referred to as the reference mode. It is further assumed, for simplicity, that WB = w and E is real. Thus the equation of motion for the statistical operator p(t) of the four-wave mixer in the interaction picture is given by
i@(t) = -i[Ei,,,iii(t)],
(2)
I?i~~i,,~fi(S+&+)y
(3)
where we set ,u = he and A = Bt& Note that the photon number of the signal mode is conserved since [A, Ei] = 0. 2.2. Conditional
in this model
state of the signal mode
Suppose that the statistical m(O) = @ii/;,, ‘8
operator of the initial state I@(O) of the degenerate
lO)bb(Ol~
four-wave
mixer is given by (4)
where I@i” is the initial statistical operator of the signal mode and ]O)b is the vacuum state of the reference mode, &]O)b = 0. Then in the interaction picture, the statistical operator at time t becomes
*(t> =~(t>(@iiri, @lo)bb(o1)@(t),
(5)
D(t) =exp[-it@(S+L+)].
(6)
After an interaction time r of the four-wave mixing, the photon number of the reference mode is observed by means of an ideal photodetector with unit quantum efficiency [ 171. When the photodetector registers m photons, the statistical operator pa< t, m) of the signal mode reduces to (7) where we set g, = ( ,LL~)~ and Tr, (Trb) stands for the trace operation over the Hilbert space of the signal (reference) mode, i, is the unit operator of the signal mode, and 1m)b is the Fock state of the reference mode. The statistical operator tiO( r, m) represents the conditional signal state of the degenerate four-wave mixer. In
M. Ban/Optics ConznzunicationsI30 (1996) 365-376
this case, the conditioning other hand, the probability
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is carried out by the photon counting measurement on the reference mode. On the P,,f(m) that the photodetector for the reference mode registers m photons is given
by P,,f(m)
= -$cTr[fi’“‘exp(
which is normalized
-g7A2)&],
as crZ0
(8)
P,& m) = TrIPi” = 1. This probability
the conditional signal state @a (7, m) . When the outcome of the photon counting the signal mode is given by @A(r) = exp[ -ig7(A+
ii-ii
the relative frequency
is not referred to, the statistical
of obtaining
operator
@L(r) of
- A_)2] VVin,
where we have defined the operations A+A = aA,
measurement
represents
(9)
A+ and A_ as
= AA,
(10)
for an arbitrary operator d of the signal mode. Such a measurement is called a non-selective measurement. This result indicates that the statistical operator I@:(r) describes the process which is equivalent to a pure dephasing process due to the quadratic coupling between the signal mode and the thermal reservoir [ 181. The pure dephasing process is governed by the quantum master equation, ;w(t) where
K
=
-K[i2,
[fi,fi((t)]],
(11)
is the phase diffusion
2.3. Statistics
constant
in the photon counting
due to the thermal reservoir.
measurement
We briefly consider the statistics of the outcomes of the photon counting measurement on the reference mode. To this end, we first calculate a characteristic function defined by (eiAm)rer 2 C,“, eiAmPrer(m). Using Eq. (8), we can easily obtain (exp(ihm)),,f
= (exp[gAe
iA -
l)fi2])in,
(12)
where we set (A)i, = Tr, [ AL%‘iin].In particular,
the 1st and 2nd moments
(13)
+&f=gT(fi2)in, (m2)rt2f=&(fi2)in
(
+ grin.
In the limit g7 + co, the following lim
[ (m2jref
g,--=
are given by
-
bEf1
relation
is established
1’2= [ (Fi4)in- (fi2)fn]‘i2
(n&f
14)
(s2)in
’
(15)
It is reasonable to consider that the photon number registered by the photodetector is proportional to the parameter g, when g, is sufficiently large. So we introduce a normalized random variable as ti = m/gT. In this case we obtain the characteristic function, (exp(iA*)),,f
= (exp[g,(ei”lg’
- 1)A2])i,.
(16)
Thus we find the relation (eiAfi’)ref= (eiAA’)in for g, >> 1. This shows that the statistics of the normalized random variable Yszis equivalent to that of squared photon number n2 with the probability (nl@i”jn) when the parameter g, is sufficiently large.
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2.4. Reduction
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(1996) 365-376
to Fock state
Now we will show that the conditional signal state given by Eq. (7) reduces to the Fock state Iii) (ii1 in the limit gr -+ cc under a certain condition, where the non-negative integer fi is determined by the outcome of the photon counting measurement on the reference mode. To this end, we suppose that the photon number m registered by the photodetector is proportional to the parameter g7 when the parameter g, is sufficiently large, and we set m = g,.fz, where Cr is a positive real number. In this case we obtain from Eq. (7))
(17) We first calculate be expressed as
the matrix element
(kl@dT,m> II) = where the function
of this statistical
operator with the Fock states. The matrix element
exp{-$g7[f~l(k)+ ftin<~>l}(kIiYinIZ)
can
(18)
C~exp[-gT.fa(k)l (kl%nlk) ’
f3rz(n) is given by
j-+$(n) = n2 - *Inn”.
(19)
It should be noted that ffi (x) is a convex function which takes the minimum value at x = A. Thus, the function J”,?~ (n) takes the minimum value at integer n = A which is determined by the parameter fi and is close to 6. If &% is an integer, we obtain ii = &. When riz = 0, A becomes the minimum integer that satisfies the relation (filIintl&) # 0. We assume here that 2 is uniquely determined and that (fiIPiTinlii) # 0. According to this consideration, we rewrite Eq. (18) into
ew[-&V~t(k)l
(Jw~(~Jd0= (A,Wi”lfii) + czk o(k+fi) where we set Af,T,(k> = fsz(k) the relation,
lim g,-00
- f,~(ti).
exp[-3g~Af~~(Z>l(kliYinlZ)
exp[-gAf~L(k)l
S’mce fITI
(kjiYin/k)’
(20)
> f,Tz(f?.) for k # A and (A/lVt”l&) # 0, we can obtain
(kl@a(T, m>II) = 6kfi&S3
(21)
or equivalently lim Qn(9-,m) g,-03
= lS)(Al.
(22)
Thus, we have found that the conditional signal state of the degenerate four-wave mixer reduces to the Fock state in the limit g, + co. The photon number of the Fock state is determined by the outcome of the photon counting measurement on the reference mode, but we cannot assign an arbitrary integer value for ii, independent of the result of the photon counting measurement. The integer fi is not always uniquely determined. If there are integers ~11,n2,. . . , np such that fSz(ni) = fe1(n2) = . . . = ffiL(np) = min, the conditional signal state becomes lim TVO(7-,m) = gT-00
cT=l c”,,
In this case, the conditional
CT=,
\n.i)(njlwi\fyinlndbkI (njl@in/nj)
’
signal state does not reduce to the Fock state.
(23)
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130 (1996) 365-376
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It is found from the result in Section 2.3 that when the parameter g, is sufficiently large, the statistics of (n]@‘i,,]n). Thus, it the parameter 6~ is equivalent to that of squared photon number n2 with the probability is considered that when we use the initial state of the signal mode in which the width of its photon number probability is sufficiently narrow, the distribution of the Fock states becomes narrow around the state Ifi). When we would like to obtain the Fock state with a given photon number, we have to repeat the experiment until such a Fock state is obtained. However, if we chose an appropriate initial state of the signal mode, we can reduce the number of repetitions of the experiment. When we do not refer to the outcome of the photon counting measurement, we find from Eq. (9) that (24) Thus the statistical 2.5. Continuous
operator of the signal mode becomes
measurement
diagonal
with respect to the Fock states.
of photon number
We have assumed that the photon counting measurement obeys the projection postulate [ 191 and we have used the projection operator lm)bb(ml to describe the measurement on the reference mode. This is valid in the idealized situation. The real photon counting measurement is a time continuous measurement and does not obey the projection postulate. When we use the continuous measurement of the photon number of the reference mode, the result obtained above will be modified. Using the Srinivas-Davies model of the photon counting process [20-221 for the reference mode, we consider the state reduction of the signal mode. This model is formulated by assuming that the influence of a photodetector on a measured system is characterized by the quantum Markov process [23]. We briefly explain the photon counting mode proposed by Srinivas and Davies [20]. When one photon of the reference mode is registered by the photodetector at time t, the statistical operator of the system changes as follows: @7(t) -
biir(t)bt Tr,Trb[6+&@(t)]
On the other hand, when the photodetector the statistical operator becomes lV(t + t’) =
(25)
’ does not register any photon of the reference
mode during time t’,
exp(-&&t’6+&)W(t)exp(-$ct’S+6> Tr,Trb[exp(-&g+6)I%‘(t)]
(26)
’
where .$ is a parameter that characterizes the interaction between the reference mode and the photodetector. When we begin the photon counting measurement on the reference mode at time r and m photons of the reference mode are registered during time t, the statistical operator becomes W(r(t,T,rn) =
exp(-~&&+&)PlV(r)(6+)n~exp(-$$tF;+&) Tr,Trb[exp(-&6+6)&‘l@(r)(&+)nZ]
(27)
’
It should be noted that the effective quantum efficiency of the photodetector is given by yt [20,21] and that the projection postulate for the photon counting measurement is recovered in To investigate the state reduction of the signal mode, we substitute Eq. (5) into Eq. (27) trace over the Hilbert space of the reference mode. Then we obtain the statistical operator of
C~~P(k>Tra[exp(-g,A2)A2(k+“)~i”]
’
= 1 - exp(-&) the limit yr ---) 1. and we take the the signal mode,
(28)
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where P(n) P(n)
is a Poissonian
distribution
Here let us introduce k=g,Zi(k),
functions
130 (19963 365-376
given by
-7+)lnexp[-gAl
=-$&Al
Communications
-x)1.
(29)
fi( k) and fEt(k) (n) as
ffiL(k)(n) =n2-Cz(k)lnn2.
(30)
Since for real value X, f,rz(k) (x) is a convex function which takes the minimum value at x = m, the function f,r,tk,(n) takes the minimum value at some integer ii(k) which depends on k. We assume that the integer Z(k) is uniquely determined and (n(k) Il%‘ii/;,,lii( k)) # 0 for all k. Then the same argument in terms which we derived Eq. (22) gives us the statistical operator for g, >> 1, I@&v)
=~P(k),l(k+m))(R(k+m)(. k=o
(31)
Thus it is found that the statistical operator eLi( t, r, m) reduces to the Fock state ]ff( m)) (E(m) 1for g7( 1 -rt) < 1. This is possible if the effective quantum efficiency of the photodetector is nearly equal to unity (rt M 1). On the other hand, when g7( 1 - rf) >> 1, we obtain the broad distribution of the Fock states. Therefore the reduction of the signal state to the Fock state by the photon counting measurement on the reference mode occurs under the condition, 1 Kg,<<
(1 -Yt)_r,
(32)
which is satisfied when we can perform the photon counting following sections, we assume that this condition is satisfied.
measurement
for a sufficiently
long time. In the
2.6. Decay of the phase coherence When the conditional signal state of the degenerate four-wave mixer approaches the Fock state, the phase coherence vanishes. To investigate a decay of the phase coherence, we calculate the phase probability and the phase dispersion of the conditional state of the signal mode. The phase probability PgT(4; m) and the phase dispersion D,(m) [24,25] are given, respectively, by Ps,(&m)
=
(4@o(~~m)14)~
Dg,(m)
= 1s -7r
dqSe-i4PgV(&m)
where 14) is the eigenstate
(33) 2
?T
,
(34)
of the Susskind-Glogower
phase operator
[26-281
of the signal mode, (35)
Thus, substituting
Eq. (17) into Eqs. (33) and (34),
we can obtain
p (~;m> = JJ&C~exp{-&,[_fr72(k) +fr~t(~)l)e~‘~-“~(kl~nl~) 87 2~C~exp[-g,f~~(k)l(kI~“Ik) ’ D,(m)
= 1-
C~exp{-&,[f~r(k)
+fmtk+l)l)(k+
C~exp[-g,f~~tk)l(kI~i”Ik)
(36)
ll%nIk) 2 ’
(37)
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130 (1996) 365-376
371
where the function fti (k) is given by Eq. ( 19). It is easy to see from these equations that lim,,,, PgT(4; m) = (2~)~’ and limg,+oo Dg,( m) = 1. Thus, as expected, the phase in the conditional signal state becomes completely uncertain in the limit g, 4 co. When the photon number registered by the photodetector is not referred to, the phase probability Pi7 (4) and the phase dispersion Di, can be calculated from Eq. (9),
(38)
Thus we also find the relations, limsT,, Pi?<#) = (2~)~’ and lims,_+cr, Di, = 1. It should be noted that the phase probability PLT(4) is equivalent to that obtained in the pure phase relaxation [29], where the phase probability distribution satisfies the Fokker-Planck equation with respect to the phase variable. 2.7. Several initiaE states of the signal mode 2.7.1. Fock state When the initial state of the signal mode is the Fock state /nin)r it is easy to see from Eq. (7) that the conditional signal state becomes fiIa(r, m) = Inin) (ni,\ for any value of the parameter g7. This shows that the conditional signal state is equal to the initial Fock state and does not depend on the outcome of the photon counting measurement on the reference mode. If the photon number registered by the photodetectors is not referred to, the state of the signal mode also becomes @L(r) = Ini,) (ni” 1, w h ere we have used Eq. (9). Thus the degenerate four-wave mixer makes the Fock state invariant. On the other hand, the probability P,f( m) that the photodetector registers m photons of the reference mode is given by
which gives (m),,r = g&.
In this case, rii = n,?, and fi = &z = nin are obtained.
2.7.2. Coherent state Let us now consider a coherent state Icz) as the initial state of the signal mode. To investigate the photon statistics of the conditional signal state, we calculate the average photon number (A),(m) and the Mandel Q-factor Q,(m) [ 301 by means of Eq. ( 17) and we obtain (41) where the quantity
Fg. (m; k) is given by
(42)
Q,(m) = - 1 which means that the conditional signal and we set CY= Iale- i4. It is easy to see that lim,,,, state becomes the Fock state in the limit g, --f co. From Eq. (37), the phase dispersion in the conditional signal state is calculated as
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Table 1 The values of (iI)g,(m),
Q,(m)
and D,(m)
ST 0.0 5.0 1.0 5.0 5.0 0.1 0.5 1.o 3.0 5.0
x x x x
10-d 10-3 10-3 10-Z
D,(m) = 1 - exp( -gr)
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130 (1996) 345-376
for several values of g7 with 1~11~= 6, where we assign A = 9, for simplicity
($8, Cm)
Q, (ml
6.000000 5.986125 5.957943 5.751930 4.574189 4.078494 3.311263 3.151679 3.005801 3.000161
0.0 -2.254746 -3.002306 -8.467228 -4.119755 -5.699732 -8.611952 -9.335847 -9.977465 -9.999430
C~II &,tn)G,(n
Dg,(?n)
x x x x x x x x x
+ 1) expl-g&n
4.588578 x 1O-2 5.355189 x lo-* 5.407979 x 10-2 5.822228 x 1O-2 9.962435 x lo-* 1.420634 x IO-’ 4.253367 x 10-l 6.743202 x 10-l 9.897015 x 10-l 9.991107 x 10-t
IO-* 1O-2 lo-* 10-l 10-l IO-’ 10-l IO-’ 10-l
+ l)]
* ’
CFZ~&(n)*exp(-g,n*)
(43)
and D,(m) for several values of g, are with d,,(n) = (n!)-‘/2n~r”lal”. The values of (A),(m), Q,(m), given in Table 1. Since we assign * = 9, we obtain the Fock state 13), that is, lim,,,, fia(r, m) = 13) (31. It is found from the table that the conditional signal state approaches the Fock state rapidly as g, becomes large. When we do not refer to the outcome of the photon counting measurement on the reference mode, the state of the signal mode is given by (44) Thus the signal state in the limit g7 + co becomes
(45)
which is nothing
but the mixed coherent
state.
2.7.3. Superposition of the two Fock states For a coherent initial state, the relation (fil@in/ii) f 0 is satisfied for any non-negative integer ii. Thus we now consider here a superposition state I&n) of the two Fock states as the initial state of the signal mode, I&> = cos(@/2)Ini)
+
(46)
sin(O/2)b2),
where we assume that FQ > nt > 0. In this case, we obtain the relation (fflmi”j6) = 0 unless ii = nt or n2. Substituting G’i” = [@in)(@in1 into Eq. ( 17), we can get the conditional signal state,
iir,(T,rn) = 1 + cm 0 n:&le--g,“; 2Z(7)
h)hl+
1-
~0s
2Z(7-)
e 26,fie-6,nf %
b2)(72*1
sin B
_(,,,,>o~~,-~~(n:+n2?)/*
+22(r) where the quantity
(I~*)(4
+
Mn2l)
3
(47)
Z (7) is given by (48)
M. Ban/Optics Communications 130 (1996) 365-376
Therefore
in the limit g, --+ 00, we find the reduction
of the conditional
373
signal state to the Fock state 1~~1)or
InA (49) where riz, is a critical value given by 72: - n: (50)
mc = ln(n2/nt)2’
If fi = tic, the conditional signal state becomes equal to the initial superposition state. To consider the decay of the phase coherence of the conditional signal state, we calculate the phase dispersion from Eq. (37), Dgr(m)
Hg,(m)
(
= I-
2
sin6
l+cosB+H,,(m)(l--cos8)
where the quantity
1
(51)
’
Hg, (m) is given by (52)
Thus, we can see the decay of the phase coherence
of the conditional
signal state.
3. Heterodyne and homodyne detections 3.1. Heterodyne
detection
We now consider the conditional signal state of the degenerate four-wave mixer, obtained by means of the heterodyne detection for the reference mode. The heterodyne detection is equivalent to the simultaneous measurement of two quadrature operators, &, = (6 + 6t ) /1/2 and pb = -i( 6 - &t )/A. The measurement outcome is expressed as a complex number cr*. It is shown that the heterodyne detection is described by means of the nonorthogonal projection operator ~-‘]cz*)bb(a*I, where Icr*)b is the coherent state of the reference mode, &Icx*)~ = a” Icv*)~. The quantum mechanical heterodyne detection was investigated in detail by several authors [ 31-331. When we obtain the complex number LY*as the measurement outcome, the statistical operator @a (7, (u) of the conditional signal state is calculated from Eqs. (5) and (6))
=
exp [ - t$gTA2- igi’2aA] @in exp [ - ig,A2 + ig,l/2 a* A] Tr,{exp[
The probability Pref(a*)
-g7A2 - igi’2( a - a*)21 Pi”}
P,r( a) that the measurement = ~~exp[-(Rea)2
-
(Ima
outcome -
cz* is obtained
(53)
’ becomes
g~/2n)2](nlkiiri,ln),
(54)
Go which is normalized as J d2cu P,f(cu*) the complex number z.
= 1, where Re( z) and Im( z) stand for the real and imaginary
parts of
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We first consider the statistics of the outcome end, we calculate the characteristic function, (e
Alx+AzY)ref= e-_(A:+A:)/4
of the heterodyne
detection
for the reference
mode. To this
~exp(-Azfif/‘n)(n/~i”l~),
(55)
n=O where we set x = Rea (X)ref =
0,
and y = -1mcu. In particular
we obtain the average values and fluctuations,
(Ax)~ = ;,
(Y)ref = -g:‘2(+qi”>
(56)
(AY>~ = i + g7[(fi2)in -
(fi)i?,ly
(57)
where Ax and Ay stand for the fluctuations of the real and imaginary parts of the complex number a*. Thus we find from Eq. (57) that the relation Ay/l (~)~~fl = [ (A2)i,, - (A)z] ‘/2/(A)i, is obtained in the limit g, --+ CXJ. Furthermore,
since
((~*)~~f = -igb’2(ii)in,
it is reasonable
to consider
that the measurement
outcome
CX*is
proportional to the parameter g:‘” when g, is sufficiently large. Thus we set a* = -igi’2[*, where 5 is a complex variable. Next we show that the conditional signal state given by Eq. (53) reduces to the Fock state in the limit g, + 00. We first calculate the matrix element of Eq. (53) with the Fock states, (58) where the function ff (k) is given by fs (k) = (k - 5) 2. Let A be a non-negative integer which minimizes the real part of the function Re ft (k), that is, mink20 Re fs (k) = Re f.c( A).We assume here that A is uniquely determined and that (EllVi”lA) # 0 is satisfied. Then we rewrite I$. (58) into
=
bw~(~~4I0
exp[-&Afdk)l (nl~inl_)
+
c”
k-0(k+a)
where we set A fc( k) = lim (klijl,(T, g,-+m
exP[-~g,Afg*(Z)l(kl~iVi,lz) exp[-g,ReAfS(k)l(kI~,lk)'
(59)
fc( k) - f,t( iti>. Since Afs( k) > 0 for k f A, we can obtain the following result:
a) II) = &+&A,
(60)
which means that lim,,, !@0(7,~) = Ifi)(fil. Th us we see the reduction of the conditional signal state to the Fock state. If A is not uniquely determined, however, the reduction to the Fock state does not occur. 3.2. Homodyne
detection
Finally we carry out the homodyne detection for the reference mode to obtain the conditional signal state. The ideal homodyne detection measures the quadrature operator given by $ = (he-‘4 + &tei4) /a [ 34,351, where 4 is the phase of the local oscillator. Thus the homodyne detection for the reference mode is described by the projection operator lq)bb(ql, where jq)b is the eigenstate of the quadrature operator 6 of the reference mode. We assume that 4 = 0, for simplicity. When the measurement outcome is a real number q, the statistical operator pQ( 7, q) of the conditional signal state is obtained from Eqs. (5) and (6)) . wa(T’q)=
TrbL.@(T)(f, TraTrb[*(7)(h
‘8 \q)bb(q()l @ jq)bb(ql)l
Eq. (61) shows that the conditional state. Thus the homodyne detection
=eXp[-i(2g,)‘/2qfi]lVi~exp[i(2g,)’/2qA].
(61)
signal state is given by the phase-shift transformation of the initial signal of the reference mode does not induce the reduction of the conditional
M. Ban/Optics Communications 130 (1996) 365-376
signal state to the Fock state. The probability that the value 4 is obtained Gaussian distribution, that is, Pdq) = s--l exp( -q2).
375
as the measurement
outcome
is the
4. Summary In this paper, we have investigated the properties of the conditional signal state of the degenerate four-wave mixers. The conditioning for the signal state is performed by the photon counting, heterodyne and homodyne detection measurements on the reference mode. We have found that by carrying out the photon counting and heterodyne detection measurements, the conditional signal state reduces to the Fock state under a certain condition. The photon number of the Fock state is determined by the measurement outcome of the reference mode. These results show that the measurement carried out on the reference mode nondestructively measures the photon number of the signal mode. In fact, the measurement setup we considered satisfies the following relations:
[A(t),%s)l
=o,
[fiint,Fi(t)l =03
[fiii,tv~b(t)l + 07
(62)
where ri( t) is the photon-number operator of the signal mode at time t and 6, stands for the measured physical quantity of the reference mode. Thus we find that the condition for the quantum nondemolition measurement of the photon number is fulfilled. Since the homodyne detection for the reference mode does not satisfy the last relation in Es. (62), we cannot obtain any information of the signal mode. Furthermore we have considered the phase probability and the phase dispersion in the conditional signal state. The phase dispersion increases and the phase becomes completely uncertain in the conditional signal state in the limit g, -+ 00. This behaviour is consistent with the reduction to the Fock state. If the photon numbers registered by the photodetectors are not referred to, the state of the signal mode is equivalent to that of the pure phase relaxation process caused by the interaction with the thermal reservoir.
References [ 11 [2] [3] [4] [S] [6] [7] [8] [9]
[lo] [ 111 [ 121 [ 131 [ 141 [IS] [ 161 [ 171 [ 181 [ 191 [20] [21] [22] [23]
V.B. Braginsky, Y.I. Vorontsov and F.Y. Khalili, Sov. Phys. JETP 46 (1977) 705. W.G. Unruh, Phys. Rev. D 18 (1978) 1764. V.B. Braginsky and F.Y. Khalili, Quantum Measurement (Cambridge University Press, 1992). M. Brune, S. Haroche, V. Leferve, J.M. Raimond and N. Zagury. Phys. Rev. Lett. 65 (1990) 976. M.J. Holland, D.F. Walls and P. Zoller, Phys. Rev. Lett. 67 (1991) 1716. R. &hack, A. Breitenbach and A. Schenzle, Phys. Rev. A 45 (1992) 3260. D. Ivanov and T.A.B. Kennedy, Phys. Rev. A 47 (1993) 566. J.I. Cirac, R. Blatt, A.S. Parkins and P Zoller, Phys. Rev. Lett. 70 (1993) 762. A.S. Parkins, P. Matte and P. Zoller, Phys. Rev. Lett. 71 ( 1993) 3095. U. Herzog and Th. Richter, J. Mod. Optics 41 (1994) 553. E.T. Jaynes and F.W. Cummings, Proc. IEEE 51 (1963) 89. B.W. Shore and P.L. Knight, J. Mod. Optics 40 (1993) 1195. S.J.D. Phoenix and PL. Knight, .I. Opt. Sot. Am. B 7 (1990) 116. L. SchoendortT and H. Risken, Phys. Rev. A 41 (1990) 5 147. M.. Ueda, N. Imoto, H. Nagaoka and T. Ogawa, Phys. Rev. A 46 (1992) 2859. J.G. Milbum and D.F. Walls, Phys. Rev. A 30 (1984) 56. M. Ban, Phys. Rev. A 49 (1994) 5078. D.F. Walls, M.J. Collet and G.J. Milbum, Phys. Rev. D 32 (1985) 3208. J. von Neumann, Mathematical Foundation of Quantum Mechanics (Princeton University Press, Princeton, M.D. Srinivas and E.B. Davies, Optica Acta 28 (1981) 981. C.A. Holmes, G.J. Milbum and D.F. Walls, Phys. Rev. A 39 (1989) 2439. M. Ban, Optics Comm. 117 (1995) 477. E.B. Davies, Quantum Theory of Open Systems (Academic Press, New York, 1976).
1955).
376 [24]
[25] [26] [27] [28] [29] [30] [31] [32] [ 331 1341 [35]
M. Ban/Optics
Comnzunications
130 (1996) 365-376
A. Bandilla, H. Paul and H. Ritze, Quantum Optics 3 (1991) 267. Z. Hradil, Quantum Optics 4 (1992) 93. L. Susskind and J. Glogower, Physics 1 ( 1964) 49. P. Cam&hers and M.M. Nieto, Rev. Mod. Phys. 49 (1968) 41 I. R. Lynchi, Phys. Rep. 256 (1995) 367. M. Ban, Found. Phys. L.&t. 5 ( 1992) 297. L. Mandel, Optics Lett. 4 ( 1979) 205. HP Yuen and J.H. Shapiro, IEEE Trans. Inform. Theory IT-24 (1978) 657. J.H. Shapiro, HP. Yuen and J.A. Machado Mata, IEEE Trans. Inform. Theory IT-25 (1979) 179. HP Yuen and J.H. Shapiro, IEEE Trans. Inform. Theory IT-26 (1980) 78. P Busch, M. Grabowski and PJ. Lahti, Operational Quantum Mechanics (Springer-Verlag, Berlin, 1995), and references C. Bendjaballah, Introduction to Photon Communication (Springer-Verlag, Berlin, 1995), and references therein.
therei
OPTiCIS COMMUNICATIONS Optics Communications 130 (1996) 365-376
Full length article
Reduction to the Fock state via a degenerate four-wave mixing Masashi Ban Advanced Research Laboratory, Hitachi Ltd., Hatoyama, Saitama 350-03, Japan
Received 19 December 1995; revised version received 4 March 1996; accepted 22 March 1996
Abstract A scheme to generate the Fock state via a degenerate four-wave mixing and partial measurement is proposed. A conditional state of the signal mode of a degenerate four-wave mixer will be shown to reduce to the Fock state when the outcome of the measurement on the reference mode satisfies a certain condition. The photon number of the Fock state is distributed when the measurement is repeated many times. Conditioning for the quantum state of the signal mode is carried out by the photon counting measurement on the reference mode whose input state is assumed to be the vacuum. The phase coherence of the conditional signal state is also considered. Furthermore conditioning for the signal mode by means of heterodyne and homodyne detections are briefly considered. Keywords: Four-wave mixing; Fock state; Quantum nondemolition measurement
1. Introduction Nonclassical states of photons, exhibiting the quadrature squeezing and the sub-Poissonian photon statistics, are of great importance in high precision measurement and quantum communication. One of the most typical and simple nonclassical states is the Fock state, that is, the eigenstate of photon number. The scheme for generating the Fock state by means of quantum nondemolition measurements [l-3] has been recently proposed by several authors [ 4-101. In this scheme, a beam of Rydberg atoms (which have two or three energy levels) is used to measure nondestructively the photon number in a microcavity. The interaction between the cavity photon and the atom is assumed to be described by the Jaynes-Cummings Hamiltonian [ 11,121. The quantum nondemolition measurement of photon number in the cavity by means of the Raman-coupled model [ 13,141 has been also proposed [ 151. These measurements can produce the Fock state of the cavity field. In this paper, the Fock-state generation via a degenerate four-wave mixing and partial measurement is considered. In this scheme, the photon number of the signal mode can be measured nondestructively by a photon counting measurement on the reference mode of the four-wave mixer. It will be shown that when the outcome of the measurement satisfies a certain condition, the conditional state of the signal mode of the degenerate four-wave mixer reduces to the Fock state. The photon number of the Fock state is distributed when the experiment is repeated many times. Conditioning for the signal state of the four-wave mixer is carried out by means of the photon counting measurement on the reference mode whose input state is assumed to be the vacuum. Furthermore, a decay of phase coherence of the conditional state of the signal mode caused by the 0030-4018/96/$12.00 Copyright @ 1996 Elsevier Science B.V. All rights reserved. PII SOO30-40 18 (96) 00263-5
366
M. Bnn/Oyfics Commuaicntions130 (1996) 365-376
reduction to the Fock state is investigated. As an example of the initial state of the signal mode, the Fock state, coherent state and superposition of two Fock states are considered. Finally, the heterodyne and homodyne detection measurements on the reference mode are briefly considered as conditioning for the signal mode.
2. Conditional
state of a degenerate
2.1. Model of a degenerate four-wave
four-wave
mixer
mixer
The model of a degenerate four-wave mixer considered in this paper is explained. The one mode of the four-wave mixer is highly excited so that it can be treated classically, and the two other modes are degenerate. Thus the Hamiltonian of the degenerate four-wave mixer [ 161 can be written as
(1) where ii (6) and &t (6t) are the bosonic annihilation and creation operators, satisfying the canonical commutation relations, [ 8, at] = 1 and [h, St ] = 1, and eeeiot stands for the classical pump field. In the following, the mode represented by (a, at) is referred to as the signal mode and the mode represented by (&, 6t) is referred to as the reference mode. It is further assumed, for simplicity, that WB = w and E is real. Thus the equation of motion for the statistical operator p(t) of the four-wave mixer in the interaction picture is given by
i@(t) = -i[Ei,,,iii(t)],
(2)
I?i~~i,,~fi(S+&+)y
(3)
where we set ,u = he and A = Bt& Note that the photon number of the signal mode is conserved since [A, Ei] = 0. 2.2. Conditional
in this model
state of the signal mode
Suppose that the statistical m(O) = @ii/;,, ‘8
operator of the initial state I@(O) of the degenerate
lO)bb(Ol~
four-wave
mixer is given by (4)
where I@i” is the initial statistical operator of the signal mode and ]O)b is the vacuum state of the reference mode, &]O)b = 0. Then in the interaction picture, the statistical operator at time t becomes
*(t> =~(t>(@iiri, @lo)bb(o1)@(t),
(5)
D(t) =exp[-it@(S+L+)].
(6)
After an interaction time r of the four-wave mixing, the photon number of the reference mode is observed by means of an ideal photodetector with unit quantum efficiency [ 171. When the photodetector registers m photons, the statistical operator pa< t, m) of the signal mode reduces to (7) where we set g, = ( ,LL~)~ and Tr, (Trb) stands for the trace operation over the Hilbert space of the signal (reference) mode, i, is the unit operator of the signal mode, and 1m)b is the Fock state of the reference mode. The statistical operator tiO( r, m) represents the conditional signal state of the degenerate four-wave mixer. In
M. Ban/Optics ConznzunicationsI30 (1996) 365-376
this case, the conditioning other hand, the probability
367
is carried out by the photon counting measurement on the reference mode. On the P,,f(m) that the photodetector for the reference mode registers m photons is given
by P,,f(m)
= -$cTr[fi’“‘exp(
which is normalized
-g7A2)&],
as crZ0
(8)
P,& m) = TrIPi” = 1. This probability
the conditional signal state @a (7, m) . When the outcome of the photon counting the signal mode is given by @A(r) = exp[ -ig7(A+
ii-ii
the relative frequency
is not referred to, the statistical
of obtaining
operator
@L(r) of
- A_)2] VVin,
where we have defined the operations A+A = aA,
measurement
represents
(9)
A+ and A_ as
= AA,
(10)
for an arbitrary operator d of the signal mode. Such a measurement is called a non-selective measurement. This result indicates that the statistical operator I@:(r) describes the process which is equivalent to a pure dephasing process due to the quadratic coupling between the signal mode and the thermal reservoir [ 181. The pure dephasing process is governed by the quantum master equation, ;w(t) where
K
=
-K[i2,
[fi,fi((t)]],
(11)
is the phase diffusion
2.3. Statistics
constant
in the photon counting
due to the thermal reservoir.
measurement
We briefly consider the statistics of the outcomes of the photon counting measurement on the reference mode. To this end, we first calculate a characteristic function defined by (eiAm)rer 2 C,“, eiAmPrer(m). Using Eq. (8), we can easily obtain (exp(ihm)),,f
= (exp[gAe
iA -
l)fi2])in,
(12)
where we set (A)i, = Tr, [ AL%‘iin].In particular,
the 1st and 2nd moments
(13)
+&f=gT(fi2)in, (m2)rt2f=&(fi2)in
(
+ grin.
In the limit g7 + co, the following lim
[ (m2jref
g,--=
are given by
-
bEf1
relation
is established
1’2= [ (Fi4)in- (fi2)fn]‘i2
(n&f
14)
(s2)in
’
(15)
It is reasonable to consider that the photon number registered by the photodetector is proportional to the parameter g, when g, is sufficiently large. So we introduce a normalized random variable as ti = m/gT. In this case we obtain the characteristic function, (exp(iA*)),,f
= (exp[g,(ei”lg’
- 1)A2])i,.
(16)
Thus we find the relation (eiAfi’)ref= (eiAA’)in for g, >> 1. This shows that the statistics of the normalized random variable Yszis equivalent to that of squared photon number n2 with the probability (nl@i”jn) when the parameter g, is sufficiently large.
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2.4. Reduction
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I30
(1996) 365-376
to Fock state
Now we will show that the conditional signal state given by Eq. (7) reduces to the Fock state Iii) (ii1 in the limit gr -+ cc under a certain condition, where the non-negative integer fi is determined by the outcome of the photon counting measurement on the reference mode. To this end, we suppose that the photon number m registered by the photodetector is proportional to the parameter g7 when the parameter g, is sufficiently large, and we set m = g,.fz, where Cr is a positive real number. In this case we obtain from Eq. (7))
(17) We first calculate be expressed as
the matrix element
(kl@dT,m> II) = where the function
of this statistical
operator with the Fock states. The matrix element
exp{-$g7[f~l(k)+ ftin<~>l}(kIiYinIZ)
can
(18)
C~exp[-gT.fa(k)l (kl%nlk) ’
f3rz(n) is given by
j-+$(n) = n2 - *Inn”.
(19)
It should be noted that ffi (x) is a convex function which takes the minimum value at x = A. Thus, the function J”,?~ (n) takes the minimum value at integer n = A which is determined by the parameter fi and is close to 6. If &% is an integer, we obtain ii = &. When riz = 0, A becomes the minimum integer that satisfies the relation (filIintl&) # 0. We assume here that 2 is uniquely determined and that (fiIPiTinlii) # 0. According to this consideration, we rewrite Eq. (18) into
ew[-&V~t(k)l
(Jw~(~Jd0= (A,Wi”lfii) + czk o(k+fi) where we set Af,T,(k> = fsz(k) the relation,
lim g,-00
- f,~(ti).
exp[-3g~Af~~(Z>l(kliYinlZ)
exp[-gAf~L(k)l
S’mce fITI
(kjiYin/k)’
(20)
> f,Tz(f?.) for k # A and (A/lVt”l&) # 0, we can obtain
(kl@a(T, m>II) = 6kfi&S3
(21)
or equivalently lim Qn(9-,m) g,-03
= lS)(Al.
(22)
Thus, we have found that the conditional signal state of the degenerate four-wave mixer reduces to the Fock state in the limit g, + co. The photon number of the Fock state is determined by the outcome of the photon counting measurement on the reference mode, but we cannot assign an arbitrary integer value for ii, independent of the result of the photon counting measurement. The integer fi is not always uniquely determined. If there are integers ~11,n2,. . . , np such that fSz(ni) = fe1(n2) = . . . = ffiL(np) = min, the conditional signal state becomes lim TVO(7-,m) = gT-00
cT=l c”,,
In this case, the conditional
CT=,
\n.i)(njlwi\fyinlndbkI (njl@in/nj)
’
signal state does not reduce to the Fock state.
(23)
M. Ban/Optics
Comnzunications
130 (1996) 365-376
369
It is found from the result in Section 2.3 that when the parameter g, is sufficiently large, the statistics of (n]@‘i,,]n). Thus, it the parameter 6~ is equivalent to that of squared photon number n2 with the probability is considered that when we use the initial state of the signal mode in which the width of its photon number probability is sufficiently narrow, the distribution of the Fock states becomes narrow around the state Ifi). When we would like to obtain the Fock state with a given photon number, we have to repeat the experiment until such a Fock state is obtained. However, if we chose an appropriate initial state of the signal mode, we can reduce the number of repetitions of the experiment. When we do not refer to the outcome of the photon counting measurement, we find from Eq. (9) that (24) Thus the statistical 2.5. Continuous
operator of the signal mode becomes
measurement
diagonal
with respect to the Fock states.
of photon number
We have assumed that the photon counting measurement obeys the projection postulate [ 191 and we have used the projection operator lm)bb(ml to describe the measurement on the reference mode. This is valid in the idealized situation. The real photon counting measurement is a time continuous measurement and does not obey the projection postulate. When we use the continuous measurement of the photon number of the reference mode, the result obtained above will be modified. Using the Srinivas-Davies model of the photon counting process [20-221 for the reference mode, we consider the state reduction of the signal mode. This model is formulated by assuming that the influence of a photodetector on a measured system is characterized by the quantum Markov process [23]. We briefly explain the photon counting mode proposed by Srinivas and Davies [20]. When one photon of the reference mode is registered by the photodetector at time t, the statistical operator of the system changes as follows: @7(t) -
biir(t)bt Tr,Trb[6+&@(t)]
On the other hand, when the photodetector the statistical operator becomes lV(t + t’) =
(25)
’ does not register any photon of the reference
mode during time t’,
exp(-&&t’6+&)W(t)exp(-$ct’S+6> Tr,Trb[exp(-&g+6)I%‘(t)]
(26)
’
where .$ is a parameter that characterizes the interaction between the reference mode and the photodetector. When we begin the photon counting measurement on the reference mode at time r and m photons of the reference mode are registered during time t, the statistical operator becomes W(r(t,T,rn) =
exp(-~&&+&)PlV(r)(6+)n~exp(-$$tF;+&) Tr,Trb[exp(-&6+6)&‘l@(r)(&+)nZ]
(27)
’
It should be noted that the effective quantum efficiency of the photodetector is given by yt [20,21] and that the projection postulate for the photon counting measurement is recovered in To investigate the state reduction of the signal mode, we substitute Eq. (5) into Eq. (27) trace over the Hilbert space of the reference mode. Then we obtain the statistical operator of
C~~P(k>Tra[exp(-g,A2)A2(k+“)~i”]
’
= 1 - exp(-&) the limit yr ---) 1. and we take the the signal mode,
(28)
370
M. Ban/Optics
where P(n) P(n)
is a Poissonian
distribution
Here let us introduce k=g,Zi(k),
functions
130 (19963 365-376
given by
-7+)lnexp[-gAl
=-$&Al
Communications
-x)1.
(29)
fi( k) and fEt(k) (n) as
ffiL(k)(n) =n2-Cz(k)lnn2.
(30)
Since for real value X, f,rz(k) (x) is a convex function which takes the minimum value at x = m, the function f,r,tk,(n) takes the minimum value at some integer ii(k) which depends on k. We assume that the integer Z(k) is uniquely determined and (n(k) Il%‘ii/;,,lii( k)) # 0 for all k. Then the same argument in terms which we derived Eq. (22) gives us the statistical operator for g, >> 1, I@&v)
=~P(k),l(k+m))(R(k+m)(. k=o
(31)
Thus it is found that the statistical operator eLi( t, r, m) reduces to the Fock state ]ff( m)) (E(m) 1for g7( 1 -rt) < 1. This is possible if the effective quantum efficiency of the photodetector is nearly equal to unity (rt M 1). On the other hand, when g7( 1 - rf) >> 1, we obtain the broad distribution of the Fock states. Therefore the reduction of the signal state to the Fock state by the photon counting measurement on the reference mode occurs under the condition, 1 Kg,<<
(1 -Yt)_r,
(32)
which is satisfied when we can perform the photon counting following sections, we assume that this condition is satisfied.
measurement
for a sufficiently
long time. In the
2.6. Decay of the phase coherence When the conditional signal state of the degenerate four-wave mixer approaches the Fock state, the phase coherence vanishes. To investigate a decay of the phase coherence, we calculate the phase probability and the phase dispersion of the conditional state of the signal mode. The phase probability PgT(4; m) and the phase dispersion D,(m) [24,25] are given, respectively, by Ps,(&m)
=
(4@o(~~m)14)~
Dg,(m)
= 1s -7r
dqSe-i4PgV(&m)
where 14) is the eigenstate
(33) 2
?T
,
(34)
of the Susskind-Glogower
phase operator
[26-281
of the signal mode, (35)
Thus, substituting
Eq. (17) into Eqs. (33) and (34),
we can obtain
p (~;m> = JJ&C~exp{-&,[_fr72(k) +fr~t(~)l)e~‘~-“~(kl~nl~) 87 2~C~exp[-g,f~~(k)l(kI~“Ik) ’ D,(m)
= 1-
C~exp{-&,[f~r(k)
+fmtk+l)l)(k+
C~exp[-g,f~~tk)l(kI~i”Ik)
(36)
ll%nIk) 2 ’
(37)
M. Ban/Optics
Communications
130 (1996) 365-376
371
where the function fti (k) is given by Eq. ( 19). It is easy to see from these equations that lim,,,, PgT(4; m) = (2~)~’ and limg,+oo Dg,( m) = 1. Thus, as expected, the phase in the conditional signal state becomes completely uncertain in the limit g, 4 co. When the photon number registered by the photodetector is not referred to, the phase probability Pi7 (4) and the phase dispersion Di, can be calculated from Eq. (9),
(38)
Thus we also find the relations, limsT,, Pi?<#) = (2~)~’ and lims,_+cr, Di, = 1. It should be noted that the phase probability PLT(4) is equivalent to that obtained in the pure phase relaxation [29], where the phase probability distribution satisfies the Fokker-Planck equation with respect to the phase variable. 2.7. Several initiaE states of the signal mode 2.7.1. Fock state When the initial state of the signal mode is the Fock state /nin)r it is easy to see from Eq. (7) that the conditional signal state becomes fiIa(r, m) = Inin) (ni,\ for any value of the parameter g7. This shows that the conditional signal state is equal to the initial Fock state and does not depend on the outcome of the photon counting measurement on the reference mode. If the photon number registered by the photodetectors is not referred to, the state of the signal mode also becomes @L(r) = Ini,) (ni” 1, w h ere we have used Eq. (9). Thus the degenerate four-wave mixer makes the Fock state invariant. On the other hand, the probability P,f( m) that the photodetector registers m photons of the reference mode is given by
which gives (m),,r = g&.
In this case, rii = n,?, and fi = &z = nin are obtained.
2.7.2. Coherent state Let us now consider a coherent state Icz) as the initial state of the signal mode. To investigate the photon statistics of the conditional signal state, we calculate the average photon number (A),(m) and the Mandel Q-factor Q,(m) [ 301 by means of Eq. ( 17) and we obtain (41) where the quantity
Fg. (m; k) is given by
(42)
Q,(m) = - 1 which means that the conditional signal and we set CY= Iale- i4. It is easy to see that lim,,,, state becomes the Fock state in the limit g, --f co. From Eq. (37), the phase dispersion in the conditional signal state is calculated as
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M. Ban/Optics
Table 1 The values of (iI)g,(m),
Q,(m)
and D,(m)
ST 0.0 5.0 1.0 5.0 5.0 0.1 0.5 1.o 3.0 5.0
x x x x
10-d 10-3 10-3 10-Z
D,(m) = 1 - exp( -gr)
Communications
130 (1996) 345-376
for several values of g7 with 1~11~= 6, where we assign A = 9, for simplicity
($8, Cm)
Q, (ml
6.000000 5.986125 5.957943 5.751930 4.574189 4.078494 3.311263 3.151679 3.005801 3.000161
0.0 -2.254746 -3.002306 -8.467228 -4.119755 -5.699732 -8.611952 -9.335847 -9.977465 -9.999430
C~II &,tn)G,(n
Dg,(?n)
x x x x x x x x x
+ 1) expl-g&n
4.588578 x 1O-2 5.355189 x lo-* 5.407979 x 10-2 5.822228 x 1O-2 9.962435 x lo-* 1.420634 x IO-’ 4.253367 x 10-l 6.743202 x 10-l 9.897015 x 10-l 9.991107 x 10-t
IO-* 1O-2 lo-* 10-l 10-l IO-’ 10-l IO-’ 10-l
+ l)]
* ’
CFZ~&(n)*exp(-g,n*)
(43)
and D,(m) for several values of g, are with d,,(n) = (n!)-‘/2n~r”lal”. The values of (A),(m), Q,(m), given in Table 1. Since we assign * = 9, we obtain the Fock state 13), that is, lim,,,, fia(r, m) = 13) (31. It is found from the table that the conditional signal state approaches the Fock state rapidly as g, becomes large. When we do not refer to the outcome of the photon counting measurement on the reference mode, the state of the signal mode is given by (44) Thus the signal state in the limit g7 + co becomes
(45)
which is nothing
but the mixed coherent
state.
2.7.3. Superposition of the two Fock states For a coherent initial state, the relation (fil@in/ii) f 0 is satisfied for any non-negative integer ii. Thus we now consider here a superposition state I&n) of the two Fock states as the initial state of the signal mode, I&> = cos(@/2)Ini)
+
(46)
sin(O/2)b2),
where we assume that FQ > nt > 0. In this case, we obtain the relation (fflmi”j6) = 0 unless ii = nt or n2. Substituting G’i” = [@in)(@in1 into Eq. ( 17), we can get the conditional signal state,
iir,(T,rn) = 1 + cm 0 n:&le--g,“; 2Z(7)
h)hl+
1-
~0s
2Z(7-)
e 26,fie-6,nf %
b2)(72*1
sin B
_(,,,,>o~~,-~~(n:+n2?)/*
+22(r) where the quantity
(I~*)(4
+
Mn2l)
3
(47)
Z (7) is given by (48)
M. Ban/Optics Communications 130 (1996) 365-376
Therefore
in the limit g, --+ 00, we find the reduction
of the conditional
373
signal state to the Fock state 1~~1)or
InA (49) where riz, is a critical value given by 72: - n: (50)
mc = ln(n2/nt)2’
If fi = tic, the conditional signal state becomes equal to the initial superposition state. To consider the decay of the phase coherence of the conditional signal state, we calculate the phase dispersion from Eq. (37), Dgr(m)
Hg,(m)
(
= I-
2
sin6
l+cosB+H,,(m)(l--cos8)
where the quantity
1
(51)
’
Hg, (m) is given by (52)
Thus, we can see the decay of the phase coherence
of the conditional
signal state.
3. Heterodyne and homodyne detections 3.1. Heterodyne
detection
We now consider the conditional signal state of the degenerate four-wave mixer, obtained by means of the heterodyne detection for the reference mode. The heterodyne detection is equivalent to the simultaneous measurement of two quadrature operators, &, = (6 + 6t ) /1/2 and pb = -i( 6 - &t )/A. The measurement outcome is expressed as a complex number cr*. It is shown that the heterodyne detection is described by means of the nonorthogonal projection operator ~-‘]cz*)bb(a*I, where Icr*)b is the coherent state of the reference mode, &Icx*)~ = a” Icv*)~. The quantum mechanical heterodyne detection was investigated in detail by several authors [ 31-331. When we obtain the complex number LY*as the measurement outcome, the statistical operator @a (7, (u) of the conditional signal state is calculated from Eqs. (5) and (6))
=
exp [ - t$gTA2- igi’2aA] @in exp [ - ig,A2 + ig,l/2 a* A] Tr,{exp[
The probability Pref(a*)
-g7A2 - igi’2( a - a*)21 Pi”}
P,r( a) that the measurement = ~~exp[-(Rea)2
-
(Ima
outcome -
cz* is obtained
(53)
’ becomes
g~/2n)2](nlkiiri,ln),
(54)
Go which is normalized as J d2cu P,f(cu*) the complex number z.
= 1, where Re( z) and Im( z) stand for the real and imaginary
parts of
374
M. Ban/Optics Communications I30 (1996) 365-376
We first consider the statistics of the outcome end, we calculate the characteristic function, (e
Alx+AzY)ref= e-_(A:+A:)/4
of the heterodyne
detection
for the reference
mode. To this
~exp(-Azfif/‘n)(n/~i”l~),
(55)
n=O where we set x = Rea (X)ref =
0,
and y = -1mcu. In particular
we obtain the average values and fluctuations,
(Ax)~ = ;,
(Y)ref = -g:‘2(+qi”>
(56)
(AY>~ = i + g7[(fi2)in -
(fi)i?,ly
(57)
where Ax and Ay stand for the fluctuations of the real and imaginary parts of the complex number a*. Thus we find from Eq. (57) that the relation Ay/l (~)~~fl = [ (A2)i,, - (A)z] ‘/2/(A)i, is obtained in the limit g, --+ CXJ. Furthermore,
since
((~*)~~f = -igb’2(ii)in,
it is reasonable
to consider
that the measurement
outcome
CX*is
proportional to the parameter g:‘” when g, is sufficiently large. Thus we set a* = -igi’2[*, where 5 is a complex variable. Next we show that the conditional signal state given by Eq. (53) reduces to the Fock state in the limit g, + 00. We first calculate the matrix element of Eq. (53) with the Fock states, (58) where the function ff (k) is given by fs (k) = (k - 5) 2. Let A be a non-negative integer which minimizes the real part of the function Re ft (k), that is, mink20 Re fs (k) = Re f.c( A).We assume here that A is uniquely determined and that (EllVi”lA) # 0 is satisfied. Then we rewrite I$. (58) into
=
bw~(~~4I0
exp[-&Afdk)l (nl~inl_)
+
c”
k-0(k+a)
where we set A fc( k) = lim (klijl,(T, g,-+m
exP[-~g,Afg*(Z)l(kl~iVi,lz) exp[-g,ReAfS(k)l(kI~,lk)'
(59)
fc( k) - f,t( iti>. Since Afs( k) > 0 for k f A, we can obtain the following result:
a) II) = &+&A,
(60)
which means that lim,,, !@0(7,~) = Ifi)(fil. Th us we see the reduction of the conditional signal state to the Fock state. If A is not uniquely determined, however, the reduction to the Fock state does not occur. 3.2. Homodyne
detection
Finally we carry out the homodyne detection for the reference mode to obtain the conditional signal state. The ideal homodyne detection measures the quadrature operator given by $ = (he-‘4 + &tei4) /a [ 34,351, where 4 is the phase of the local oscillator. Thus the homodyne detection for the reference mode is described by the projection operator lq)bb(ql, where jq)b is the eigenstate of the quadrature operator 6 of the reference mode. We assume that 4 = 0, for simplicity. When the measurement outcome is a real number q, the statistical operator pQ( 7, q) of the conditional signal state is obtained from Eqs. (5) and (6)) . wa(T’q)=
TrbL.@(T)(f, TraTrb[*(7)(h
‘8 \q)bb(q()l @ jq)bb(ql)l
Eq. (61) shows that the conditional state. Thus the homodyne detection
=eXp[-i(2g,)‘/2qfi]lVi~exp[i(2g,)’/2qA].
(61)
signal state is given by the phase-shift transformation of the initial signal of the reference mode does not induce the reduction of the conditional
M. Ban/Optics Communications 130 (1996) 365-376
signal state to the Fock state. The probability that the value 4 is obtained Gaussian distribution, that is, Pdq) = s--l exp( -q2).
375
as the measurement
outcome
is the
4. Summary In this paper, we have investigated the properties of the conditional signal state of the degenerate four-wave mixers. The conditioning for the signal state is performed by the photon counting, heterodyne and homodyne detection measurements on the reference mode. We have found that by carrying out the photon counting and heterodyne detection measurements, the conditional signal state reduces to the Fock state under a certain condition. The photon number of the Fock state is determined by the measurement outcome of the reference mode. These results show that the measurement carried out on the reference mode nondestructively measures the photon number of the signal mode. In fact, the measurement setup we considered satisfies the following relations:
[A(t),%s)l
=o,
[fiint,Fi(t)l =03
[fiii,tv~b(t)l + 07
(62)
where ri( t) is the photon-number operator of the signal mode at time t and 6, stands for the measured physical quantity of the reference mode. Thus we find that the condition for the quantum nondemolition measurement of the photon number is fulfilled. Since the homodyne detection for the reference mode does not satisfy the last relation in Es. (62), we cannot obtain any information of the signal mode. Furthermore we have considered the phase probability and the phase dispersion in the conditional signal state. The phase dispersion increases and the phase becomes completely uncertain in the conditional signal state in the limit g, -+ 00. This behaviour is consistent with the reduction to the Fock state. If the photon numbers registered by the photodetectors are not referred to, the state of the signal mode is equivalent to that of the pure phase relaxation process caused by the interaction with the thermal reservoir.
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therei