VOLUME 86, NUMBER 13 PHYSICAL REVIEW LETTERS 26 MARCH 2001
Entanglement, Interference, and Measurement
in a Degenerate Parametric Oscillator
Hua Deng, Daniel Erenso, Reeta Vyas, and Surendra Singh
Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701
(Received 25 August 2000)
Quantum dynamical equations of motion for homodyne detection of the degenerate optical parametric
oscillator are solved exactly. Nonclassical photon statistics are shown to be a consequence of interfer-
ence of probability amplitudes, entanglement of photon pairs from such an oscillator, and the role of
measurement in quantum evolution.
DOI: 10.1103/PhysRevLett.86.2770 PACS numbers: 42.50.Dv, 42.50.Ar, 42.65.Ky
Fluctuations of photon beams reflect the quantum dy-
namics of photoemissive sources. In quantum mechanics,
probabilities for observed events are derived from an un-
derlying wave function that can interfere and collapse as it
evolves. A consequence of this is that quantum mechan-
ics can lead to correlations between observed events which
a classical stochastic theory may not. Examples of these
nonclassical correlations include squeezing, antibunching,
and violations of Bell’s inequalities [1,2].
The subthreshold degenerate parametric oscillator
(DPO) has played a central role in the study of nonclassi-
cal photon correlations, particularly, squeezing [1,2]. The
DPO radiates a highly bunched light beam that exhibits a
large degree of squeezing. Interestingly, the squeezed and
highly bunched light from the DPO when combined with a
coherent light field, as in homodyne detection, is expected
to display a rich variety of nonclassical photon correlations
including antibunching [3]. It is intriguing that a highly
bunched entangled photon beam from the DPO when
mixed with a coherent field will exhibit correlations similar
to those exhibited by a single-atom resonance fluorescence
in free space or in cavity quantum electrodynamics (QED)
[4,5]. Antibunching of light emitted by a single two-level
atomic system can be eventually traced to the atomic dead
time that a two-level atom cannot emit a second photon
immediately after the emission of a first photon. The situ-
ation is not so simple for homodyne detection of the light
from the DPO because there is no obvious mechanism
for a dead time. By solving the equations of motion for
homodyne detection exactly, we show that nonclassical
photon correlations in homodyne detection of the DPO are
a consequence of the interference of probability ampli-
tudes, entangled nature of photon pairs generated by the
DPO, and measurement. These are the features that most
distinguish quantum mechanics from classical mechanics.
An outline of the experimental setup for homodyne de-
tection of the DPO light is shown in Fig. 1. The DPO and
local oscillator (LO) fields are combined by a beam splitter
to produce the source field at the detector. The field from
the DPO is governed by the interaction Hamiltonian for a
phase matched DPO driven by a classical injected signal
of amplitude ´ [6]:
ˆ
H
i ¯ hk´
2
ˆ a
y2
d
2 ˆ a
2
d
1
ˆ
H
loss
. (1)
Here k is the mode-coupling constant and ˆ a
d
and ˆ a
y
d
are
the annihilation and creation operators, respectively, for
the DPO.
ˆ
H
loss
describes the loss suffered by the DPO
field. The combination k´ can be chosen to be real by a
suitable definition of phases.
The equation of motion for the density matrix ˆ r
d
of the
DPO field is then
ˆ r
d
k´
2
ˆ a
y2
d
2 ˆ a
2
d
,ˆ r
d
1g2ˆ a
d
ˆ r
d
ˆ a
y
d
2 ˆ a
y
d
ˆ a
d
ˆ r
d
2 ˆ r
d
ˆ a
y
d
ˆ a
d
, (2)
where 2g is the cavity decay rate. The steady-state solution
to this equation in positive-P representation is given by
[6,7]
ˆ r
d
ss
1
p
2¯ n
d
p
ZZ
dx dy
jx y j
y j x
3 exp
∑
2xy 2
g
k´
x
2
1 y
2
∏
, (3)
where 2` , x , y ,` are both real variables and jx is a
coherent state of ˆ a
d
with ˆ a
d
jx x jx . From this expres-
sion for the density matrix, the steady-state expectation
value of an operator
ˆ
O can be calculated as
ˆ
O
ss
Tr
ˆ
O ˆ r
ss
. This leads to the following expectation values
FIG. 1. Schematic experimental setup for the homodyne de-
tection of the light from a degenerate parametric oscillator.
2770 0031-9007 01 86(13) 2770(4)$15.00 © 2001 The American Physical Society