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Vol.
7,
No.
2/February
1990/J.
Opt.
Soc.
Am.
B
211
Photon amplification
by
parametric
downconversion
Z.
Y.
Ou,
L.
J.
Wang,
and
L.
Mandel
Department
of
Physics
and
Astronomy,
University
of
Rochester,
Rochester,
New
York
14627
Received
August
14,
1989;
accepted
October
26,
1989
We
investigate
the
theory
of
the
process
in which
an
idler
photon
emitted
spontaneously
from
a
nonlinear
crystal
in
the
process
of
parametric
downconversion
serves as
idler
input
to
a
second
downconverter.
The
signal
photon
from
this
second
crystal
is
then
detected
in
coincidence
with
the
signal
photon
from
the
first
crystal,
which
provides
the
time
reference.
We
show
that
the first
idler
photon
can
induce
a
stimulated
downconversion in
the
second
crystal,
which serves
as
a
photon
amplifier,
and
we
calculate
the
ratio
of
the
stimulated to the
spontaneous
emission
probability.
INTRODUCTION
The
problem
of amplifying,
or
cloning,
a
photon
with
a
light
amplifier
has
been
discussed
several
times
in
recent
years,'-'o
particularly
in connection
with
attempts
to
violate
the
uncertainty
principle.
Needless
to
say,
when all
aspects
of
the
amplification
process,
including
spontaneous
emis-
sion by
the
amplifier, are
taken
into
account,
there
is
no
violation.
Nevertheless,
the
theory
of
the
process
remains
of
interest,
if
only
as
a
guide
to
different
experimental
possi-
bilities
for
studying
photon
cloning.
In
what
follows
we
consider
the
process of
parametric
downconversion
in
a
nonlinear medium
as
the
basis
for
a
photon
amplifier. Although
spontaneous
downconversion
has
been
studied
both
experimentallyll-'
4
and
theoretical-
ly,'
5
-
24
as
has
the
parametric
oscillator,
there
appears
to
have
been
no
attempt
so
far
to
apply
it
to
the
problem
of
cloning
a
photon.
We
show
below
that
by combining
spontaneous
downconversion
with amplification,
and
using
the
output
of
one
nonlinear
crystal
as
the
input
to
a
second
one
acting
as
an
amplifier,
one
should
be
able
to
study
the
photon-cloning
process.
We
analyze
the
theory
of
the
process
and
show
that
the
process
lends
itself
to
an
experimental
investigation
of
amplification of
single
photons.
Consider
the
experiment
shown
schematically
in
Fig.
1. A
coherent
pump
beam
from
a
laser
is
used
to
optically
pump
two
nonlinear
crystals
NL1
and
NL2
with
a
second-order
nonlinear
susceptibility.
Some
of
the
incident photons
fall-
ing
upon
NL1
fission
into
simultaneous
signal
and
idler
photons
by
parametric
downconversion.
The
signal
photon
(S)
falls
upon detector
D1,
which
provides an
electrical
out-
put
signaling
the
appearance
of
a
photon
(i,)
in
the
idler
channel.
The
latter
photon
serves
as
input
to
a
second
downconverter
NL2,
whose
signal
output
2
falls
upon
a
second
detector
D
2.
By
studying simultaneous detections
by
D,
and
D
2
in
coincidence
with
the
arrangement
in
Fig.
1,
when
the
i
photons
are
blocked
and
unblocked
in
turn,
one
can
investigate
the
response
of NL2
acting
as
an amplifier
to
an
incident photon,
while
D,
provides
the
time
reference.
THEORY
We
shall
make
use
of
the
formalism
developed
in
Ref.
24.
We
treat
the
two
coherent
pump
fields falling
upon
NL1
and
NL2
classically
and
represent them
by
complex
analytic
signals
V
exp(-iwot),
V
2
exp(-icoot),
expressed
in
units
such
that
l
V12
is
in
photons per
second.
The
parametric
interac-
tion
in
each
downconverter
results
in
the
following
unitary
time-evolution
operators
in
the
interaction
picture
24:
Orj(t
+
r,
t)
=
1
+
77jVjcO
Z
Z
(',
")
sin
/2
+
-
)T
X
'exp[i('
+ "
-
)
X
(t
+
l/
2
r)]a&Sjt(cW/)a.it(,,)
+ . . .
(i
=
1, 2).
(1)
If
is
short
compared with
the
average
time
interval
be-
tween
two downconversions,
then
the
probability
of two
or
more
successive
downconversions
is
small
compared
with
that
for
one,
and
we
may
terminate the
expansion
as
in Eq.
(1).
j(w',
co")
is
a
spectral
function
that
is
symmetric
in
co',
cw",
is
peaked
at
'
=
1/2wo
=
co",
and
has
a
substantial
band-
width
Ace,
whose
reciprocal characterizes
the
length
of
each
downconverted
photon
wave
packet.
tqj
is
the
dimensionless
coupling
constant,
which
depends
on
the
nonlinear
suscepti-
bility,
whose
square
modulus
gives
the
fraction
of
pump
photons
that
is
downconverted.
zj&t(w)
and
&tjt(w)
are
pho-
ton-creation operators
for
signal
and
idler
photons
in
crystal
j (
=
1,
2),
and
bw
is
the
mode
spacing.
As
w
-
0,
sums
over
co
convert
into
integrals.
In
particular,
the
function
q5(w,
0
-
co)
is
normalized
so
that
24
2ir6
2
10(w,
o -)JI
2r
d(w,
wo
-W)1
2
=
1. (2)
Let
us
suppose
that
modes
s,
i
are
in
the
vacuum
state
at
time
t
=
0
when
the
pump
is
turned
on.
After
a
time
t,
which
0740-3224/90/020211-04$02.00
©
1990
Optical Society of
America
Ou
et
al.
