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INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS
J. Opt. B: Quantum Semiclass. Opt. 3 (2001) 21–28
www.iop.org/Journals/ob
PII: S1464-4266(01)15224-9
Amplification of Schrödinger-cat state:
distinguishability and interference in
phase space
R Filip and J Peřina
Department of Optics, Palacký University, 17 Listopadu 50, 772 07 Olomouc, Czech Republic
Received 5 July 2000, in final form 12 January 2001
Abstract
Amplification and cloning of the Schrödinger-cat state (even coherent-state
superposition) in the optical nondegenerate parametric amplifier are
analysed. In the long-time limit, distinguishability and interference in the
marginal probability distributions of the Schrödinger-cat state cannot be
preserved simultaneously during amplification. The mutual ratio between
distinguishability and interference can be controlled by the squeezed
vacuum state injected in the idler mode. Thus only one of either quantum
interference of indistinguishable states or distinguished states without
interference can be obtained on the output of the amplifier. The purity of the
initial state in the signal mode disappears during the amplification for both
cases, even if the interference is preserved. Thus the interference in the
marginal probability distribution is connected with indistinguishability in
the phase space rather than with purity of the state. On the other hand, the
nonclassical oscillations in the photon-number distribution vanish
simultaneously, as the state is more mixed, irrespective of the existence of
the interference.
Keywords: Schrödinger-cat state, phase space, superposition
1. Introduction
Complementarity is a fundamental notion of quantum theory.
It says that any quantum system has at least two properties
that cannot be simultaneously known. When one applies
the complementarity principle to the one-photon interference
experiment in the Mach–Zehnder interferometer, it can be
found that it is impossible to obtain the complete whichpath information for the interfering paths of a photon and
simultaneously to observe the interference effect in a single
experiment. The visibility of interference fringes will be zero if
we know exactly which path the photon goes through, whereas
no knowledge of the photon path will give rise to maximal
visibility [1, 2].
The quantum state superpositions of the two almost
orthogonal coherent states (Schrödinger-cat states) can exhibit
simultaneously the interference and distinguishability of the
component states in the marginal probability distribution
of complementary variables [3]. A distinct feature is the
occurrence of interference between different amplitudes in
the phase space, in contrast to the interference between
real paths in the interferometer. The atom-amplifier with
1464-4266/01/020021+08$30.00
© 2001 IOP Publishing Ltd
phase-selective diffusion [4–7] and nondegenerate parametric
amplification [8–11] were suggested to amplify an initial
Schrödinger-cat state with small amplitude up to a macroscopic
level.
In this paper, the Schrödinger-cat state evolution is
analysed in the nondegenerate parametric amplifier and the
relation between the distinguishability and interference is
discussed for state superposition with a larger photon number.
Adjusting the squeezing in the idler mode, amplification and
copying of either the interference or distinguishable states can
be obtained in the large-gain limit. Preservation of both the
properties together is not possible. This can be considered as a
simple phase-space analogy of the well known correspondence
between distinguishability and interference for the one-photon
Mach–Zehnder experiment and can be expressed in the form of
a simple relation. The interference in the marginal probability
distribution [3] can be preserved during the amplification,
whereas the purity of state and the nonclassical oscillations in
the photon-number distribution of the Schrödinger-cat state [6]
are vanishing. Thus chaotic light exhibiting interference in the
marginal probability distribution can be obtained on the output
of a parametric amplifier.
Printed in the UK
21
R Filip and J Peřina
Figure 1. Evolution of the Wigner function of the Schrödinger-cat state in the signal (left) and idler (right) modes in the parametric
amplifier injected by the squeezed light in idler mode; for kt = 0.5, ξ(0) = 2: (a), (b) for r = 2, (c), (d) for r = −2.
2. Evolution of the Schrödinger-cat state
Thus for r > 0, the first quadrature is squeezed, the other
having enhanced fluctuations.
The optical parametric amplifier can be considered as a
nondegenerate parametric process (NOPA) in the classical
pump approximation [12, 13]. Generally, the nondegenerate
amplifier exhibits dissipations, but in the strong-amplification
and small-thermal-noise limit they can be neglected.
Thus the approximate evolution is the same as for
a unitary amplifier and can be simply described after
elimination of free oscillations by the evolution operator
ÛN (t):
ÛN (t) = exp(kt (Â+S Â+I − ÂS ÂI )),
(1)
We consider the superposition of two coherent states
|ξS (0) and | − ξS (0), ξS (0) ∈ R,
where ÂS and ÂI are the annihilation operators of the
signal and idler modes, k is a real positive amplification
constant and the product kt is adimensional. The idler
mode is stimulated by the squeezed vacuum state |0, r, r ∈
R [12], which is produced from the initial vacuum state by
the degenerate parametric process (DOPA) in the classical
pump approximation. The preparation of the squeezed
vacuum state can be described by the unitary transformation
ÛD :
ÛD = exp(r(Â+2 − Â2 )).
(2)
The squeezing parameter r (−∞ < r < ∞) determines
the orientation and rate of squeezing of the vacuum state.
For this squeezed vacuum state in the idler mode, we obtain
for initial variance of quadrature fluctuations the following
relation:
(Q̂I (0))2 =
22
1
4
exp(−2r),
(P̂I (0))2 =
1
4
exp(2r).
(3)
1
|ψS (0) = √ (|ξS (0) + | − ξS (0)),
N
(4)
N = 2(1 + exp(−2ξS2 (0))),
injected in the signal mode. For ξ 2, the coherent
state superposition exhibits the well known Schrödinger-cat
behaviour and the states in the superposition are almost
orthogonal. If we are interested in statistics of the signal
(or idler) mode separately, a time evolution of the coherent
state superposition (4) can be described by the Wigner
quasidistribution in the phase space
1
P2
exp −
bS,I
π N aS,I bS,I
2
(Q − ξS,I (t))
(Q + ξS,I (t))2
× exp −
+ exp −
aS,I
aS,I
2
2
ξS,I (t)
Q
+2 exp(−2ξS2 (0)) exp
−
bS,I
aS,I
2ξS,I (t)P
× cos
,
(5)
bS,I
WS,I (Q, P , t) =
where Q and P are the phase-space variables corresponding to
the quadrature operators Q̂ = 21 (Â + Â+ ) and P̂ = 2i1 (Â − Â+ ),
Amplification of Schrödinger-cat state: distinguishability and interference in phase space
Figure 2. Density matrix elements p(Q, Q ) in the coordinate representation in dependence on the initial squeezing r in the idler mode, for
ξ(0) = 2: (a) kt = 0, (b) kt = 0.5, r = 0, (c) kt = 0.5, r = 2, (d) kt = 0.5, r = −2.
and
ξS (t) = cosh(kt)ξS (0),
ξI (t) = sinh(kt)ξS (0),
aS,I = BS,I + CS,I − 21 ,
bS,I = BS,I − CS,I − 21 ,
BS,I (t) = cosh2 (kt)BS,I (0) + sinh2 (kt)(BI,S (0) − 1),
∗
CS,I (t) = cosh2 (kt)CS,I (0) + sinh2 (kt)CI,S
(0),
BS (0) = 1,
BI (0) = cosh2 r,
CI (0) =
− 21
(6)
CS (0) = 0,
sinh 2r.
The Wigner function of the Schrödinger-cat state exhibits
negative values, which are an exhibition of the nonclassical
character of quantum interference. The nonclassical character
vanishes during the amplification and the Wigner function
becomes positive, as can be seen in figure 1. For positive r, the
destruction of the nonclassical character is more pronounced
than for negative r. In the signal mode, the distinguishable
marginal peaks without an interference term shift away in
the Schrödinger-cat state, as can be seen in figure 1(a),
in contrast to the negative-r case in figure 1(c), when the
interference term is preserved and the marginal peaks are
indistinguishable. In the idler mode, the copy of the signal
Wigner function is generated from the squeezed-vacuum state,
but the nonclassical character of the Schrödinger-cat state
cannot be cloned into the idler mode, as can be seen in
figures 1(b) and (d).
The density operator for the signal or idler mode in the
coordinate representation may be expressed as the Fourier
transform of the Wigner function W (Q, P )
∞
Q + Q
pi (Q, Q ) = Q|ρ̂|Q i =
,P
Wi
2
−∞
× exp(iP (Q − Q )) dP ,
(7)
where i = S, I and |Q, |Q are the eigenstates of the
quadrature operator Q̂. The analogous relation for the density
operator in the momentum representation P |ρ̂|P i can be
found by the exchange of a particular quadrature. From the
solution (5), the density matrix elements can be found in the
following form:
Q − Q 2
1
exp −
bi
pi (Q, Q ) = √
N π ai
2
( Q+Q
− ξ(t))2
2
× exp −
ai
Q+Q
( 2 + ξ(t))2
+ exp −
ai
( Q+Q
)2
ξ(t)2
2
+ exp(−2ξ(0)2 ) exp
−
bi
ai
ξ(t) Q − Q 2
× exp −
+
bi
bi
2
−ξ(t) Q − Q 2
+ exp −
.
(8)
+
bi
bi
2
The occurrence of off-diagonal terms in the density matrix
Q|ρ̂|Q in the coordinate representation is an exhibition
of interference, as can be seen for initial state superposition
in figure 2(a). In figure 2(b), the off-diagonal peaks are
suppressed, whereas the amplification is not pronounced. The
suppression is much more stimulated by the vacuum noise
injected in the idler mode than the proper amplification. For
the squeezed vacuum in the Q-quadrature, which is injected
in the idler mode, the diagonalization in the |Q-basis is
more pronounced on the same timescale, as can be seen in
figure 2(c). On the other hand, if the noise is put in the Qquadrature in such a way that the marginal diagonal peaks will
23
R Filip and J Peřina
Figure 3. Evolution of the relation between the interference R and distinguishability D parameters for the signal (a), (b) and idler (c), (d)
modes in dependence on the initial squeezing r of the idler mode, for ξ(0) = 2.
be indistinguishable then, simultaneously, the diagonalization
does not occur in the |Q-basis. It can be achieved by injecting
the squeezed vacuum noise in the P -quadrature in the idler
mode.
3. Distinguishability and interference
The marginal probability distributions pi (Q) and pi (P ) can
be obtained from equation (7) by taking only the diagonal
elements. The marginal distribution for P -quadrature
2
ξi (t)P
, (9)
pi (P ) = P |ρ̂|P i = i (P ) 1 + Ri cos 2
N
bi
where
1
P2
i (P ) = √
,
exp −
bi
πbi
2 ξ (t)
2
Ri = exp(−2ξS (0)) exp i
bi
where
i (Q) = √
(10)
can exhibit the interference effect, which reflects the coherent
state superposition. Equation (9) is an analogue of the
interference formula for partial coherence, where the function
i (P ), arising from coherent state noise, modulates the
interference fringes by the Gaussian envelope. Instead of using
the visibility of interference fringes, the relative peak-to-peak
ratio R between the maximum of density p(P ) for the coherent
and incoherent state superposition,
pcoh (P ) − pinc (P ) ,
(11)
R=
pinc (P )
P =0
24
where 0 R 1, is used to measure the interference [14].
The parameter R can be treated as the peak-to-peak ratio
between the interference and the mixed part of the Wigner
function.
On the other hand, if ξ(0) is sufficiently large, the marginal
distribution for Q-quadrature
(Q − ξi (t))2
1
exp −
pi (Q) = Q|ρ̂|Qi = √
N π ai
ai
2 (Q + ξi (t))
2
+ exp −
+ exp(−2ξS2 (0))i (Q),
ai
N
(12)
1
Q2
exp −
ai
πai
(13)
will consist essentially of two well separated Gaussian peaks.
The last term in equation (12) is a contribution of quantum
interference, which reduces the distinguishability of peaks for
small ξS (0). Distinguishability of the symmetrically posed
Gaussian peaks can be described by the ratio between peak
√
distance ξi (t) from the origin and the half-peak width ai .
Thus a distinguishability parameter D is introduced in an
analogous manner to the interference parameter R,
Di = exp(−2ξS2 (0)) exp
ξi2 (t)
ai
(14)
and can be connected with the relative peak-to-peak ratio
between density p(Q) in the origin of phase space for the
Amplification of Schrödinger-cat state: distinguishability and interference in phase space
Figure 4. Evolution of the marginal probability distribution p(Q) (a), (c), (e) and p(P ) (b), (d), (f ) of the Schrödinger-cat state in the signal
mode, for ξ(0) = 2: (a), (b) r = 0, (c), (d) r = 2, (e), (f ) r = −2.
coherent and incoherent state superposition
pcoh (Q) − pinc (Q) D=
,
p (Q)
inc
(15)
Q=0
where 0 D 1. For the signal and idler modes particularly,
the interference and distinguishability parameters R and D can
be found in the following explicit forms:
2 cosh2 (kt)ξS2 (0)
2
RS = exp(−2ξS (0)) exp
,
(1 + exp(2r)) sinh2 (kt) + 1
2 sinh2 (kt)ξS2 (0)
2
RI = exp(−2ξS (0)) exp
,
(1 + exp(2r)) cosh2 (kt) − 1
2 cosh2 (kt)ξS2 (0)
2
DS = exp(−2ξS (0)) exp
,
(1 + exp(−2r)) sinh2 (kt) + 1
2 sinh2 (kt)ξS2 (0)
2
DI = exp(−2ξS (0)) exp
.
(1 + exp(−2r)) cosh2 (kt) − 1
(16)
The marginal probability distributions in the P -quadrature
give information about interference, whereas the marginal
distribution for Q-quadrature contains information about
distinguishability of the coherent states in the superposition.
The marginal probability distributions can be connected with
results for the homodyne detection [3, 6] on the output of
the amplifier and thus the parameters Di and Ri can be
determined directly from the measurement. By adjusting
the local-oscillator phase, an experimentalist can verify that
the state has distinguishable components and it is a coherent
superposition rather than a statistical mixture.
The amplifier exhibits the nonexponential suppression of
interference, which can be controlled by the input squeezed
vacuum in the idler mode, as can be seen in figure 3. The
effect is stimulated by the change of noise in the quadratures
of signal mode: an enhanced noise in a particular quadrature
in the idler mode induces a proportional increase of noise
in the same quadrature of the signal mode. The evolution
of the marginal probability distribution can be observed in
figures 4 and 5. It can be found that quantum superposition
orientated in one quadrature is very sensitive to fluctuations
in the complementary quadrature. Thus the interference in
the signal mode is strongly suppressed if the idler mode
exhibits enhanced noise in the direction perpendicular to
25
R Filip and J Peřina
Figure 5. Evolution of the marginal probability distribution p(Q) (a), (c), (e) and p(P ) (b), (d), (f ) of the squeezed state in the idler mode,
for ξ(0) = 2: (a), (b): r = 0, (c), (d): r = 2, (e), (f ): r = −2.
the maximal separation of the superposition in the phase
space (for positive r), as can be seen in figure 3(a). In the
opposite case, slowing down of the decay can be observed if
the noise in the idler mode is squeezed in the direction of
maximal separation (negative r). For sufficiently negative
r, the interference is preserved. At the same time, the
peaks in the marginal probability distribution p(Q) become
indistinguishable and information about amplitude states in
the superposition disappears, as can be seen in figure 3(b). In
the course of time, in the idler mode we can find a copy of
the state in the signal mode: for positive r, the distinguished
states without interference are cloned, whereas for negative r
the interference of indistinguishable states can be cloned, as
can be seen in figures 3(c) and (d). Thus the interference can
be preserved during amplification and cloning, but the states
are indistinguishable. Only one of either distinguishability or
interference can be preserved and therefore the Schrödingercat state cannot be sufficiently amplified or cloned with
preservation of its essential properties.
The relations between distinguishability and interference
can be simply described by the parameters D and R in the form
26
of the following inequalities:
exp(−4ξS2 (0)) RI DI exp(−2ξS2 (0)) RS DS 1. (17)
For the signal (idler) mode, the right- (left-) hand equality
in equation (17) occurs at the initial time, whereas the left(right-) hand equality is approximately satisfied in the largegain limit G 1, G = cosh(kt). Thus, in the large-gain limit,
the relations are the same for both the signal and idler modes.
From the relation (17), it is evident that the interference and
distinguishability are inversely proportional, which is more
pronounced for large ξS (0). If the product Ri Di is treated
as a measure of the incomplementarity, i.e. the simultaneous
interference and distinguishability, then it can be concluded
that this property vanishes in the signal mode and is not
cloned in the idler mode. The ratio between interference and
distinguishability is equal to unity at the initial time for both
the modes. In the large-gain limit, it is determined only by the
parameters ξS (0) and r in the following form:
D
sinh 2r
≈ exp 2ξS2 (0)
.
(18)
R
1 + cosh 2r
Amplification of Schrödinger-cat state: distinguishability and interference in phase space
0,07
0,4
(a)
0,3
0,05
0,04
0,2
p(n)
p(n)
(b)
0,06
0,03
0,02
0,1
0,01
0,0
0
2
4
6
8
0,00
10 12 14 16 18 20
0
2
4
6
8
n
10 12 14 16 18 20
n
0,04
(c )
0,10
(d)
0,08
0,02
p(n)
p(n)
0,03
0,06
0,04
0,01
0,02
0,00
0
2
4
6
8
10 12 14 16 18 20
n
0,00
0
2
4
6
8
10 12 14 16 18 20
n
Figure 6. Evolution of the photon-number probability distribution p(n) of the Schrödinger-cat state in the signal mode in dependence on the
initial squeezing r in the idler mode, for ξ(0) = 2: (a) kt = 0, (b) kt = 1, r = 0, (c) kt = 1, r = 2, (d) kt = 1, r = −2.
Figure 7. Purity parameter of the signal mode for the
Schrödinger-cat state evolution in the parametric amplifier in
dependence on the initial squeezing r in the idler mode and on
normalized time kt, ξ(0) = 2.
In the limit |r| → ∞, the ratio exhibits the simple form
D/R ≈ exp(±2ξS2 (0)), where the argument is positive for
r > 0 and negative for r < 0. Thus for a sufficiently positive r,
the distinguishability is more pronounced than the interference,
whereas for sufficiently negative r the states in superposition
are not distinguishable and thus the interference is preserved on
the output of the amplifier. The value of the ratio is determined
by the initial extension ξS (0) of the Schrödinger-cat state.
Another insight into the sensitivity of the quantum
interference to the amplification process can be observed in
the photon-number distribution. This can be obtained from
the Wigner function by the relation
p(n) =
Wn (Q, P )W (Q, P ) dQ dP ,
(19)
n
2
2
2
2
Wn (Q, P )=2(−1) exp(−2(Q +P ))Ln (4(Q +P )),
where Wn is the Wigner function of the number state |n
and Ln is the Laguerre polynomial of order n. The photonnumber distribution of the Schrödinger-cat state exhibits
nonclassical oscillations, while the statistical mixture has a
Poissonian distribution [6]. The oscillations can be considered
as an exhibition of the high-order interference and they
disappear in the long-time limit if the system is under the
influence of amplification. This long-time effect occurs
irrespective of the squeezed state injected into the idler
mode, as can be found in figure 6. Thus the high-order
coherence is lost, whereas second-order coherence can be
preserved for sufficiently negative r. Thus this signature
of the nonclassical character of the Schrödinger-cat state is
suppressed, irrespective of the preserving interference in the
marginal probability distribution. A different behaviour of the
photon-number distribution for positive and negative r arises
from amplification of the distinguishable or indistinguishable
coherent states.
The purity parameter P of the state can be expressed by
means of the Wigner function W (Q, P ),
∞ ∞
Pi = Tr{ρ̂i2 } = π
Wi2 (Q, P ) dQ dP ,
(20)
−∞
−∞
where i = S, I. It is a minimum for a maximally mixed state
and is equal to unity when the state is pure. From solution (5),
the purity parameter can be expressed in the following form:
2ξ 2 (t)
1
Pi = 2 √
1 + exp − i
ai
N ai bi
2 ξ 2 (t)
(t)
ξ
exp − i
+4 exp(−2ξS2 (0)) exp i
2bi
2ai
2 2ξ
(t)
i
+ exp(−4ξS2 (0)) 1 + exp
.
(21)
bi
If the initial state of the total system is pure, the behaviour
of the purity parameter is identical for the signal and idler
modes. For the signal mode, it only slightly depends on the
27
R Filip and J Peřina
squeezing parameter r, in contrast with the peak-to-peak ratio
RS , as can be seen in figure 7. In the long-time limit, the purity
converges to zero for both positive and negative r. The purity
parameter (21) can be approximated by the simple expression
if the second and third terms are neglected and thus, for the
signal mode, it is connected with interference parameter RS
PS ≈
2(1 + RS2 )
N 2 cosh2 r sinh2 (2kt) + 1
.
(22)
Thus, irrespective of the preservation or vanishing of the
interference parameter RS , the purity parameter PS decreases
and the signal mode is more mixed.
Thus even a
strongly mixed state can exhibit interference in the marginal
probability distribution, but with indistinguishable states in the
superposition.
4. Conclusions
The amplifier can be considered as part of a simple
measurement device mapping a state with strong nonclassical
properties to a state with a classical analogue.
The
relation (17) between the distinguishability and interference
for the Schrödinger-cat state is an illustration of the principle
of complementarity in this specific measurement process.
The interference and distinguishability of the Schrödingercat state in the marginal probability distributions can be
controlled during amplification by the squeezing of the vacuum
state injected in the idler mode. The relation (17) between
interference and distinguishability excludes pronounced
amplification and cloning of the Schrödinger-cat state with
preservation of both the properties. This can be considered
as a simple analogue of the relation between interference and
distinguishability as for one-photon behaviour in the Mach–
Zehnder interferometer, but here it is obtained for a state with
a larger photon number.
From the analysis, the quantum and classical interference
can be distinguished during the amplification. The quantum
28
interference exhibits itself in the negative values of the
Wigner function or in the higher-order interference in the
photon-number distribution. As a measure of the quantum
interference, the purity parameter or product of interference
and distinguishability parameters Ri Di can be used. The
classical interference exhibits itself in the marginal probability
distribution and can be measured by the peak-to-peak ratio
Ri . The classical interference can be preserved during the
amplification, if the interfering states are indistinguishable,
whereas the signatures of the quantum interference vanish.
Acknowledgments
RF would like to thank Miloslav Dušek for very interesting and
useful discussions and Jaroslav Řeháček and Ladislav Mišta
for some suggestions. This research was supported by the
grant no VS96028, the research project CEZ: 314/98 ‘Wave
and particle optics’ of the Czech Ministry of Education and an
internal grant of Palacký University.
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