PHYSICAL
REVIEW A
VOLUME
48,
NUMBER
6
DECEMBER
1993
Phase-difference
operator
A.
Luis and L.L.
Sanchez-Soto
Departamento
de
Optica,
Facultad de
Ciencias
Fisicas,
Universidad
Complutense, P80$0
Madrid,
Spain
(Received
15
March
1993)
We introduce
a
unitary
operator representing
the exponential
of the
phase
difference between
two modes
of the electromagnetic
field. The eigenvalue
spectrum
has a
discrete
character that is
fully
analyzed.
We
relate this
operator
with a
suitable
polar
decomposition
of the
Stokes
parameters
of the
field,
obtaining
a natural
classical limit.
The
cases of
weakly
and
highly
excited states are
considered, discussing
to what
extent it is
possible
to talk
about the
phase
for a
single-mode
field.
This
operator
is
applied
to some interesting
two-mode
fields.
PACS
number(s):
42.50.
Dv,
03.
65.
—
w
I. INTRODUCTION
The
problem
of
a correct definition in
quantum
me-
chanics
of the
phase
variable has a
long
history
and
has
provoked
many
discussions
[1].
Given
the relevance of
such
a
variable,
there have been
numerous
attempts
to
solve
this
problem
and.
certainly
very
interesting
progress
has been done in
the last
years
[2
—
4].
A
few
experiments
[5
—
7]
have been
also
reported
in
which
phase
Huctuations for a monomode
laser were
mea-
sured,
and
attempts
have
been made to
test some of the
definitions
[8
—
10,
although
unfortunately
no
clear
con-
clusion
emerged
[11].
However,
most of this work has been
devoted to the
properties
of the
phase
operator
for a
one-mode
quantum
Geld
or,
equivalently,
for a
single
harmonic
oscillator. In
this
case,
the absence of
a
proper
phase
operator
is
usu-
ally
ascribed to the
semiboundedness of the
eigenvalue
spectrum
of the
number
operator.
On the
other
hand,
from a
practical
point
of
view
an
absolute
phase
has no
meaning
and
all measurements
must be made
relative
to
the
phase
of
a reference sys-
tem.
Therefore,
it seems that
the most
proper
way
to deal
with the
phase
should
be as a
phase
difference between
the state considered and some
reference
phase
state.
One
could think that this
phase-difference
operator
should be
just
the difference of
phase
operators
for
each field as it
were a
position
or momentum
difference.
But, given
the
periodic
character of this variable,
this statement must
be taken with
care.
It is worth
mentioning
that the variable
canonically
conjugated
to the
phase-difFerence
operator
is
the
num-
ber
difFerence,
that is not
bounded
from
beloiv.
So,
it is
reasonable to
expect
the
existence of a
phase-difference
operator
free from the
problems arising
in the
one-mode
case. This
is
to
say,
it is
possible
to have
a
definite
phase
difference between two independent
oscillators for a
fixed
number of
quanta
in the
two
systems.
Moreover, if
a
phase
measurement must be
thought
of
as
a measurement
of the
phase
difFerence,
this
phase-
difference
operator
could
give
to what extent
and in
which terms we
can talk
about the
phase
of
a
one-
dimensional
system.
The aim of the
present
paper
is
just
to show how to
introduce a
well-behaved
phase-difference
operator
and
to
study
its main
properties.
II.
PHASE-DIFFERENCE OPERATOR
A.
Polar decomposition
of the
amplitudes
of two
oscillators
aia
=
Ei2/&i(N2
+
1),
(2.
1)
where
aq
and
a2
are the
annihilation
operators
for both
modes. We shall see that
Eq.
(2.
1)
has
unitary
solu-
tions for
Ei2,
that
is,
Ei2
—
—
exp(i@i2), @i2
being
the
Hermitian relative-phase
operator.
As in the
one-dimensional
case,
the
polar
decomposi-
tion does not
completely
define the
exponential
of
phase.
In
our case the matrix elements
(ni,
O~Ei2iO,
n2)
are
un-
defined and thus
Ei2
cannot
be
uniquely
determined
by
the
unitarity requirement.
We must
impose
then further
conditions,
the most
adequate
being
the
commutation
relations.
For a classical
harmonic
oscillator
the action
(
j)
and
phase
(P)
variables
verify
the fundamental Poisson
Our
purpose
here
is to find an
operator
Ei2
exponential
of
the
phase
difference
between two
independent
oscilla-
tors like
two modes of the electromagnetic
field.
Let
Ei
and
E2
be the
Susskind-Glogower
phase
op-
erators
[12]
for the
two
oscillators. Given the classical
expression
exp'(Pi
—
P2)
=
exp(i/i)
exp(
—
i/2),
we
may
be
tempted
to introduce a relative
exponential
phase
op-
erator as
EqEz,
that
commutes with the
total-number
operator
N
=
Ni+
N2,
Ni
and
N2
being
the number op-
erators
for each mode.
Unfortunately,
EiE2
annihilates
the state
~O,
n)
=
~0)
I3
~n)
in the
subspace
with fixed
total
number of
quanta
n,
and this
precludes
the
exis-
tence of eigenstates
for this
operator.
This
is an obvious
consequence
of the lack of
unitarity
of
Ei
and
E2.
In order to
avoid this
problem,
let us
try
a
polar
de-
composition
of the
complex
amplitudes
of
the two modes
as
1050-2947/93/48(6)/4702(7)/$06.
00
4702 1993
The
American Physical Society
PHASE-DIFFERENCE
OPERATOR
4703
bracket
b
&}=1
so,
for
two
independent
oscillators
we have
(2.
2)
in fact
X
commutes
with all
the
operators
in
(2.
6).
The total Hilbert
space
of
the
problem
'Ri
(3'R2
can
be
expressed
then as
direct sum of the
subspaces invariant
under
these
angular
momentum
operators,
(2.
3)
n=O
(2.
9)
((j1
—
j2)/2
0
»}
=
1,
[u,
v]
++
ih(u,
v
},
(2
4)
we
have that
the
quantum counterpart
of I)'112 commutes
with the total
number
operator.
Thus
the
commuta-
tor associated with
the second Poisson bracket
should be
verified on
the
finite-dimensional
spaces
with fixed total
number
n,
but
this
is not
possible.
The
quantum
translation
of
(2.
3)
in
terms of
the
ex-
ponential
of
phase
difference
is
[E12,
K1
+ 1V2]
=
0,
(2.
5a)
where
ItI12
is
the classical
phase
difference.
With
the
stan-
dard
prescription
that
operator
commutators are related
to classical Poisson brackets
via
ln1, n2)
=
lZ
=
(n1+
n2)/2,
m
=
(n1
—
n2)/2)
.
(2.
10)
From
Eq.
(2.
6)
one
gets
immediately
that
J+
——
J~
+
iJ„=
aia2,
Z'
JJJ
aia2&
t
(2.
11)
and
then
Eq.
(2.
1)
can be
recast
as
J
=
E»QJ+
J
(2.
12)
Each
Q
having
fixed total number
n is
spanned
by
the
2j
+
1
=
n
+
1
vectors
l
j,
m),
simultaneous
eigenstates
of 3
and J
.
The
number
eigenstates
ln1, n2)
in
'R
correspond
to the
l
j,
m)
basis as follows:
[E12 (~1
~2)/2]
—
@12
(2.
5b)
J.
=-'
a,
-.
+a,
-.
,
2
J„=
—
a2ai
—
aia2
2
t
t
2
(2.
6)
satisfy
the commutation
relations
for
the
Lie
algebra
of
the
three-dimensional
rotation
group SU(2):
the second one
can
be
recognized
as
the
analogous
of
the
well-known
Lerner
criterion
[13].
It should be noted that there is a
nonunitary
solution
for
Ei2
verifying
simultaneously these
two commutation
relations,
namely
the Susskind-Glogower phase-difference
operator
EiE&.
The lack of
unitarity
of
this
solution
is a reminiscence of
the
incompatibility
in the
quantum
translation of
(2.
3).
Recently
Ban
[14]
and
Hradil
[15]
have introduced
a
unitary exponential
of
phase
operator
verifying
(2.5b)
but
neither
(2.5a)
nor a
polar
decompo-
sition.
Therefore,
we
shall consider
the
polar
decomposi-
tion
(2.
1)
together
with the condition
(2.
5a),
which leads
to a
unique
(up
to an
arbitrary
global
phase)
unitary
solution for
(2.1).
Instead
of
solving
Eq.
(2.
1)
directly,
we can take
ad-
vantage
of the fact that the
operators
[16]
Since the
operator
Ei~
commutes with the total
nurn-
ber
N,
we
may
rather
study
its
restriction
to
each
sub-
space
'8
.
Calling
E1z
this
restriction,
Eq.
(2.12)
can
be
easily
solved
obtaining
the
unitary
SU(2)
exponential
of
phase
operator
[17
—
19]
m=-
j+1
l~
m
—
1)(j
ml+"'""'~
l~ j)(~
—
jl
(2.
13)
@(~)
l
y(n)
)
iP("}
l
y(n)
)
(2.14)
with r
=
0,
.
.
.
,
n.
These states can
be
expressed
in
the
number basis
as
)
being
an
arbitrary
phase.
Note
that
the crucial
ex-
tra term in
Eq.
(2.
13),
which establishes the
unitarity
of
Eiz,
is
precisely
based
on the finite number of states.
For
the
harmonic-oscillator
operators
such
decomposition
is
unattainable because
the
number states
are extended to
infinity
and no extra term
projecting
the
upper
state
to
ground
state exists
(unless
one
truncates,
as in the
Pegg-
Barnett
approach,
the infinite ladder
of the
oscillator
and
creates
an
upper
state). Therefore,
in each
subspace
'R
there
are
n+
1 orthonormal states
verifying
that
[Jk,
Jl]
=
Xekl
J
(2.
7)
(2.
15)
The
Casimir invariant
for this
group
can
be
put
into
the
form
where,
by
taking
the same
2'
window in each
subspace,
we have
2
i2
)
(2.
8)
~(~}
~
7I
r
n+
1
(2.16)
4704
A.
LUIS
AND
L. L. SANCHEZ-SOTO
The
expression
for
E&2
on
the whole
space
is
just
made
of
infinity
many
copies
of the
SU(2)
exponential
of
phase
operator
can
be
defined
[24]:
So
=
aya1
+
a2a2
(2.
17)
)
~
E(~)
) )
ly(n))
i+&"&
(y(n)
l
n=o
n=o
v'=0
oo
n n
)
) )
lk
k)
i(k
—
k'+1)P("i
n+
1
n=o
~=0
k,
A.
"=0
x (k', n
—
A,
"l,
S~
—
—
a&a2
+
a2aq
S2
—
—
i
a2ag
—
a~
a2
S3
—
—
a~
ay
—
a2a2
(2.
19)
which
is
very
reminiscent
of
the
operator
introduced
by
Levy-Leblond
[20]
in a
diff'erent
way.
Since
E~2
is
unitary,
it
defines
a
Hermitian
phase-
difFerence
operator
)
~
)
ly(n))
y(n)
(y(n)
l
n=o ~=0
(2.18)
and
we have
E12
—
—
exp(i@12).
This
operator difFers
from other
approaches,
mainly
because
it
cannot
be
obtained
from
a
previous
construc-
tion
of
phase
operators for
the
individual
oscillators
in-
volved.
Unlike in
the
Susskind-Glogower
approach,
Eq2
is
unitary.
On the other
hand
412
has
discrete
eigenvalues
(for
each
subspace
'R
there
are
n+
1
uniformly
distributed
in the
interval
[0,
2vr]),
in
comparison
with
other
for-
malisms
[21,
22]
that
give
continuous
spectrum
in
the
in-
terval
[0,
47r].
The
situation
presents
the
same
qualitative features
as
in
the
Carruthers
and
Nieto definition
of
phase
difFer-
ence
by
means of
relative
sine
and
cosine
operators
[1].
However,
in
this
approach
of
Carruthers
and
Nieto,
the
eigenvalues of the
phase
difFerence
are
in
the
upper
half
circle
for
the
cosine
case,
and on the
right
one
in the sine
case.
In
the limit
of
high
n this
spectrum
becomes
dense,
as
Inight
be
expected.
But,
the
case
with n
=
0
is
es-
pecially
relevant, since the
state
l0, 0)
is an
eigenstate of
Eq2.
The
unitarity
requirement makes
the
corresponding
eigenvalue
to
be
an
arbitrary
phase.
In the next
section
we
shall
try
to
understand
this
be-
havior
(in
both
limits
of small
and
high
n)
by
means
of a
simple
relative
phase-sensitive
arrangement
for the
electromagnetic
field.
Note
that,
apart
from a
factor
of
2,
the
operators
S;
(i
=
1, 2,
3)
coincide
with
the
operators
(2.
6)
of an
angu-
lar
momentum, while
So
represents the total
number
¹
The
noncommutability
of
the Stokes
operators
precludes
the
simultaneous
measurement
of
the
physical quantities
represented
by
these
operators. The
S, may
be viewed
as the
generators
of
a
group
of
transformations
locally
isomorphic to
the
three-dimensional
rotation
group
and
which
leave
the
operator
So
invariant.
This
may
be
con-
sidered
as
the
sound
basis for
the
close
relation
between
Stokes
parameters
and the
Poincare
sphere
introduced
in the
classical
description of
light.
Note
that
the eigen-
states
of
So
belong
to
spaces
which
under
the
action
of
rotations
transform
according to some
irreducible
repre-
sentation
of this
group.
In order
to show
that the
operators
(2.
19)
are the
ana-
log
of
the
classical
Stokes
parameters,
let
us
compute
their
mean
values
for
a
two-mode
coherent
state
n1
n2
Qni!
n2!
(2.
20)
that
has
special
significance in
describing the
classical
limit
of the
system. It is
easy
to
get
[25]
Bp
=
(~1
~2l~pl~i
~2)
=
1~ii'+
I~21'
»
=
(~i ~2l~il~i
~2)
=
2l~ill~2I
cos(@1
—
@2)
(2.
21)
82
—
—
((Xl,
A2lS2
i&1,
o.
2)
=
—
2lnillo.
2l
sin($1
—
p2),
Bs
=
(~i
~2
1
~s
I
~i
~2)
=
I
~il'
—
1~2
I'
B.
Stokes
parameters
In order to
gain
physical
insight
into
the
previous
polar
decomposition, we
shall relate it
to the
Stokes
parame-
ters.
It
is well
known
that in
classical
optics
to
character-
ize the
polarization
ellipse
of
two
orthogonal oscillations
of the same
frequency,
three
independent
quantities
are
necessary: the two
amplitudes
and
the
phase
difFerence.
For
practical
purposes,
it is
customary to
characterize
the
resultant
oscillation
by
the
Stokes
parameters,
which
are
directly measurable
quantities
[23].
In the
quantum
treatment
of the two-mode
field
con-
sidered
here,
the
following
Hermitian
Stokes
operators
1
2
0!
y
0!
Bp + Bs 81
+
282
(
Bi
—
182
Bp
—
Bs
~le
—
i(4'i
—
4'~)
)
(2.
22)
where
n;
=
ln;l
exp(iP,
),
P,
being
the
classical
phase
of
the
state
It is
evi.
dent
that
(2.
21)
are
exactly the
Stokes
parameters
for
two
classical
oscillations
of
amplitudes
ln,
l
and
phases
P,
.
As
discussed
in
Ref.
[26],
for
a
given
state
of the
two-
mode
field, the
classical
polarization
properties
can
be
described
by
the
coherency
matrix
that,
in terms
of
the
Stokes
parameters, can
be
written
as
48
PHASE-DIFFERENCE
OPERATOR
4705
If we denote
s~
=
(si
6
is2)/2
then
the classical
phase
difference between the two modes
is
unambiguously
ob-
tained
as
s+
—
—
e
'
~' ~'
~o.
i~~o.
2~
=
e
'l~'
~'
gs
s+. (2.
23)
The
quantum analog
of
the
separation
of
the
complex
am-
plitude
into
a real
part
and a
phase
factor
is
just
(2.
12),
which seems to be
a
natural
way
to
characterize the
phase
difference with a
clear
counterpart
in the
classical limit.
III.
LIMIT
OF
WEAKLY
EXCITED STATES
We
shall
try,
in first
place,
to
justify
the
discrete
char-
acter of the
phase
difference in the
quantum case,
whose
efFects will be more
evident
in the limit of
small number
of
photons.
Perhaps,
the
simplest arrangement
sensitive
to the
rel-
ative
phase
is a
homodyne
detection
[27,
28]
schematized
in
Fig.
1.
The
beam
splitter couples
the
input
modes
1
and
2
transforming
them into
the
output
modes
1,
2
whose
photon
numbers
N»,
N2
are
measured.
Fluctua-
tions in the
modes therefore
get
coupled,
which causes
the
appearance
of
intriguing
behaviors.
For a
linear,
lossless,
and
passive
beam
splitter
the
number
operators
at
the
output
can be
expressed.
in
terms
of the
operators
J
and
J,
and
so
the action of the
beam
splitter
can be visualized as the
process
of
measuring
the
rotations of J
[29].
The parameters
of such an action
depends
on the
particular
choice for the transmission
and
reflection
coefI»cients
of
the beam
splitter.
The lossless beam
splitter
conserves the total
energy
in the
pair
of
modes and therefore
we
have
states to
be a
product
of
number states
~ni, n2)
with
n
=
ni
+
n2.
Due
to
(3.
1)
the
output
state will
be
an
eigenstate
of
¹ As it
is
well
known,
the
photon
num-
bers at
the
output
are no
longer
sharply
determined. The
beam
splitter
coupling
has
brought
about
noise
in
the
photon
number of each
mode,
although
the
total
photon
number
in
both modes
is
invariant and free noise
[30].
The state
~ni, n2)
transforms
into
a
highly
correlated
superposition
of states with n
total number
of
photons
in the two
modes, namely
the states
~n,
0),
n
—
1,
1),
~n—
2,
2),
..
.
,
~0, n),
and
then
the number of
possible
out-
comes
in
the measurement of
N»
and
N2
are
just
n
+
1.
Since we have considered the incident states with
well-
defined
amplitudes,
the number of
outcomes
can
only
depend
on the number
of
possible
values
for the
phase
difference,
so we can observe
just
n+1
values
which is pre-
cisely
the same number
predicted
by
the
phase-di6'erence
operator
for a state in a
Q
subspace,
as it is the
case.
Although
there are reasons for
requiring
the
phase
dif-
ference between
number-state
fields to be
completely
ran-
dom
[21],
in our
approach
only
in the limit of
ni
or
n2
high
enough
is
it to
be
expected
that the measured
phase
difference is
uniformly
distributed over the interval
[0,
27r],
and this is in
agreement
with
the theoretical and
experimental
results
of
Noh, Fougeres,
and Mandel
[11].
Moreover,
when the incident state is
~0,
0),
we have
clearly
just only
one outcome. This is consistent
with
taking
it
as
a
phase-difference
eigenstate.
This
strange
situation could be understood
considering
its field
fluc-
tuations
due
only
to
the
ones
of
the
phase
sum. This
is
equivalent
to
ascribe
the field
fluctuations
of the
vac-
uum in the
one-mode
case to the
phase
and not to the
amplitude.
N1
+
N2
—
N1
+
N2.
(3.
1)
IV. LIMIT OF
HIGHLY EXCITED STATES:
PHASE
FOR
A
ONE-DIMENSIONAL SYSTEM
Since
N»
and
N2
are linear
combinations
of J and
J,
the
fluctuations
in the
output
number of
photons
are due to
fluctuations in
the
amplitudes
and in
the
phase
d.
ifference
of the
incoming states,
but not in their
phase
sum.
To focus on
the behavior
of the
phase
difference,
we
shall consider
incident fields with
nonfluctuating
ampli-
tudes.
So,
for
definiteness,
we consider
the initial
photon
Actually,
the
more extended
use of the
arrangement
discussed in the
preceding
section
makes use
of a
very
in-
tense
state
of
well-defined
phase
(for
example
a
coherent
state of
high
mean number of
photons)
in one of the
inci-
dent
modes,
say
2. This scheme can be used to measure
the
properties
of the field
in
mode
1,
and
in
our
case,
its
phase
properties.
Our aim is to
study
the behavior
of
the phase-difference
operator
by
means
of
a
suitable
approximation
for
a
high
number
of
photons.
Since
we
expect
a continuous
char-
acter,
in this limit
we
can
approximate
the r sum in
ex-
pressions
like
(2.
17)
and
(2.
18)
by
an
integral.
For
defi-
niteness,
we shall consider
the
operator
E»2,
and we
have
'(e+I
—
k')
y,
C
i(X+A:
—
A.
")
„' ",
n+
1
(4
1)
that will
be
replaced
by
2
1
2
7t
po+27t-
i(1+k
—
k')Q
(4.
2)
FIG.
1. Outline of
the beam
splitter
geometry
used
in
a
phase-sensitive
measurement of the
electromagnetic
field.
While
the
integral
gives
bg+k
k
0,
the sum
gives
zero
4706
A. LUIS
AND L. L.
SANCHEZ-SOTO
48
Op+2m
Pp+2~
dP
l0+
$,
0)
e'
~(0+
$,
0l,
(4 3)
where
l0+
$,
0)
denotes
a
two-mode
Susskind-Glogower
phase
state
I
unless
+"
&" is an
integer
m,
,
taking
in this
case the
value
e'
The approximation
in
replacing
(4.
1)
by
(4.
2)
is
tanta-
mount
to considering
that the
contribution
when m
g
0
is
negligible.
Looking
at
(2.17)
this
will be the case
when
the
spread
of
the
photon
number
distribution
is small
compared
with
the mean
value of
the total
photon
num-
ber
n,
assumed
high
enough.
This can
be verified,
for
example,
when
just
only
one
mode is in
a
coherent
state
of
high
mean
photon
number and
the other
one
involves
a
small
number
of
photons
in comparison
with
the
co-
herent
one,
but
otherwise arbitrary.
This
condition is
verified
as
well
when the
two
modes are
intense
coherent
states.
With all
this in
mind,
this replacement
gives
an
ap-
proximate
expression
for
E~2
in the
form
1
(for
example,
by
means of
the
arrangement
discussed
in
Sec.
III),
and the
results obtained
from
4i2
can be
inter-
preted
as
information about the
phase
in mode 1.
Taking
P2(0)
to be an
arbitrarily
narrow
function of
0,
centered
for
simplicity
on
the
value
Po,
we have
from
(4.
7)
(f(C'»))
=
d4
P.
(4)f(4
—
0o).
(4.
8)
V. PHASE
DIFFERENCE
PROPERTIES OF
SOME
TWO-MODE
FIELD
STATES
What
we
get
is
nothing
but
the
Pegg-Barnett
phase
ap-
proach
for the
one-mode
case,
which
is
often
considered
as
giving
the
expected
results,
even for
the vacuum.
El-
linas
[18]
also derived similar results
in
an
elegant
way
performing
a
polar
decomposition
of the
SU(2)
algebra
and
taking
the
group
contraction in the limit
j
~
oo.
Finally,
we must stress
(according
with
the comments
made
in
the
Introduction)
that this
procedure
cannot
be understood as
giving
a
phase
operator
for the
one-
mode
problem.
In
fact,
the
Pegg-Barnett formalism,
re-
produced
here in the limit
of a
practical
observation of
phase,
does not
give
such
operator
in the infinite Hilbert
space.
I0+&
0)
=
2
).
n1,
n2
—
—
0
in'(9+/)
in28l
i
)
2
(4.
4)
A.
Two-mode
coherent states
Therefore,
in
this
limit we
recover
the
equivalent
version
of
the
Susskind-Glogower
phase-difference operator:
@12
—
(Ei
@2
)
(4.
5)
and we
lost the unitarity
of the operator.
Incidentally,
we
note
that,
for an arbitrary
function
f,
we
have
f(&»)
W
f(@i&2).
(4.
6)
As
we can
see,
in this
limit
we
get
expressions
for
the
phase
difference resembling
the
ones from
other
ap-
proaches
starting
from a
one-mode
analysis.
To
discuss
this
resemblance we
use
(4.
3)
to
obtain the mean
value of
a
periodic
function of
phase
difference
f(@i2)
on a
state
P,
(0)
=
ln,
l
exp
—
l
'
(0
—
@,
)
(ln;l'
2
(5.
1)
Among
other
basic
properties,
coherent states
have a
special
significance describing
the
classical limit of a
sys-
tem and
they
are
the
prototype
for
the radiation
emitted
by
a classical current source
[31].
Here we are
going
to
study
their
phase-difference
prop-
erties in
the limit of
high
excitation,
so
we are in the
con-
ditions of
application
of
(4.
7)
to the
two-mode
coherent
state
(2.
20).
In the
limit
of
large
coherent
amplitudes
the
P,
(0)
func-
tions can be
approximated
by
the
Gaussian
distributions
l3]
14)
=
ldi
42)
(&If(c'»)
I&)
=
Op+2~
@p+2vr
d4
Pi(&+ 0)
xP2(0)f(p),
(4.
7)
(f(C'»))
=
deaf(&)P(&)
(5.
2)
where
n;
=
ln,
le'~*,
and we
can
extend to koo the limits
of
integration
in
(4.
7)
without
significant error.
Then,
we
have
where
P(0)
=
l(0lvj)l
is
the
phase probability
distri-
bution function of
the
Pegg-Barnett
formalism for
the
one-mode
case.
To
some
extent,
expression
(4.
7)
is
an
expected result,
since it is valid
for
high
photon
numbers,
and all
phase
approaches
coincide in
this
limit.
However,
in the
two-
mode
case,
this limit can
be reached when
only
one of
the modes is
highly
excited,
while the other one can
be
one of few
photons.
If
we consider in mode 2
a state whose
phase
distribu-
tion
P2(0)
is narrow
enough,
we are in the
proper
condi-
tions for the observation of
the
phase properties
of
mode
P(&)
=
1
2
We
finally
get
where
P(P)
is
the
Gaussian
distribution
(5.
3)
