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Role of particle-number statistics in interference of independent Bose fields
Toru Kawakubo and Katsuji Yamamoto
Department of Nuclear Engineering, Kyoto University, Kyoto 606-8501, Japan
(Dated: November 7, 2011)
We elucidate generally the interference of independent Bose fields in view of the conditional
probability for the particle number measurements, and clarify its relation to the source number
statistics. Despite lack of intrinsic phases, the interference phase can be inferred from the particle
number registered at one detector by using the classical mean fields. If the conditional number
distributions for the other detectors, given the outcome of the first detector, exhibit sufficiently
narrow peaks around the values specified by the estimated phases, the mean field description is
valid in a single run of interference. The widths in the conditional distribution are determined by
the number statistics of the sources, among which notable scaling behavior is found depending on
the detector configurations with the boundary at the Poissonian. The mean field description is found
to be applicable to Poissonian and sub-Poissonian sources, whereas for super-Poissonian sources it
is likely invalidated with the rather broad conditional distribution.
PACS numbers: 42.50.Ar, 42.50.St, 03.65.Ta
Interference is often considered as a signature of su-
perposition in quantum systems. In particular, interfer-
ence in many-body systems as a macroscopic quantum
effect has been attracting many interests. In usual ex-
periments, two states originating from a common source
are subject to interfere, namely, each particle interferes
with itself. However, in many-boson systems includ-
ing lasers [1, 2] and atomic Bose-Einstein condensates
(BECs) [3], interference between independently prepared
particles has also been observed. Such interference is of-
ten explained with the spontaneous symmetry breaking
for the relative phase, which gives nonvanishing expecta-
tion values of the field operators or mean fields. In BECs,
a U(1) symmetry is relevant for the global phase rotation
of atomic wavefunctions, the breakdown of which relies
on a nonphysical interaction [4, 5]. In optical systems,
a U(1) symmetry also arises from lack of an absolute
phase reference, which is ensured by the effective photon-
number conservation in optical processes [6–8]. The U(1)
symmetry breaking hence seems problematic in the ab-
sence of real mechanism.
The interference observed for independent sources un-
der the U(1) symmetry has been attributed to the back-
action of particle detection on the systems, which causes
localization of the relative phase in a single run [6, 7, 9].
Another approach to the interference is to calculate the
correlation functions of the particle numbers measured
by the different detectors, which show the spatial mod-
ulation. By evaluating the statistical moments of the
Fourier components of the spatial modulation up to the
fourth order, the plane-wave interference of atomic BECs
is predicted in a single run with a random phase [4, 10].
This analysis exploits the nature of the plane-wave mode
functions. Generally, some common understanding will
be presented for the interference appearing under vari-
ous configurations, which is based on the probability the-
ory on quantum measurement. Moreover, there will be
some intimate relationship between the interference and
the particle-number statistics of sources, by considering
the fact that the interference is observed so far for lasers
(Poissonian states) and BECs (sub-Poissonian states).
In this paper, we investigate the interference of in-
dependent Bose fields under general configurations for
sources and detectors, and clarify its relation to the
source statistics. We examine the joint probabilities of
the particle numbers registered by the detectors directly,
rather than the correlations, to see the interference in a
single run. The outcome at one detector provides infor-
mation about the relative phase, despite lack of intrin-
sic phases due to the U(1) symmetry. This information
appears in the conditional distributions for the particle
numbers at the other detectors, which are derived from
the joint probabilities with the given outcome of the first
detector. The relative phase is estimated by applying the
mean field description to the measurement outcome. If
the conditional distribution has sufficiently narrow peaks
around the values predicted by the estimated phases, it
is almost certain that the outcome at the second detector
takes a value close to one of the mean field predictions.
Hence, the conditional distribution provides a quantita-
tive criterion for the validity of the mean field descrip-
tion. The mean field description is found to be appli-
cable to Poissonian and sub-Poissonian sources, whereas
for super-Poissonian sources it is likely invalidated with
the rather broad conditional distribution.
We consider a system of noninteracting Bose parti-
cles, photons or cold atoms, where two independent
sources are contained. The positive-frequency field oper-
ator ψˆ(x, t) is given generally in terms of the annihilation
operators aˆl for a complete set of mode functions {φl}:
ψˆ(x, t) = ∑l aˆlφl(x, t), where the time evolution of the
free field is represented in the mode functions φl(x, t),
which is determined in practice by expanding ψˆ alterna-
tively in terms of the plane-wave modes. In order to de-
scribe an interference experiment, the mode functions are

2chosen suitably to provide the two independent sources as
aˆ1 ≡ aˆ and aˆ2 ≡ bˆ. For example, in interference between
two wavepackets of light the wavevector distributions are
localized around the central wavevectors of the respective
sources. In the case of two atomic BECs [3], the initial
mode functions φl(x, 0) are divided into two groups con-
sisting of the eigenstates of the respective one-particle
Hamiltonians with harmonic traps. In the following we
assume for simplicity that all the particles are populated
in the two source modes (l = 1, 2), while the other modes
(l ≥ 3) are in the vacuum states. (This will be almost
valid in typical interference experiments.) Then, the den-
sity matrix for the sources is given by ρˆ = ρˆa⊗ ρˆb, where
each source state, respecting the U(1) symmetry, is given
with the particle-number statistics ps(N) [7] as
ρˆs =
∞
∑
N=0
ps(N)|N〉〈N | (s = a, b). (1)
In the photon measurement for optical interference ex-
periments, a commonly used photodetector records the
number of photoelectrons emitted from the detector sur-
face during a time interval T . The time and surface inte-
grated photon-flux operator for the photoelectron emis-
sion at the detector m is given [11] by
Iˆm = ηm
∫ T
0
dt
∫
Sm
dxdy ψˆ†(x, t)ψˆ(x, t), (2)
where ηm is the quantum efficiency, and the z axis is
taken normal to the detector surface Sm. The band-
width ∆ω of the incident radiation is assumed to be
small enough compared with the central frequency ω0.
The photon-flux operators in Eq. (2) are specifically ex-
pressed as bilinear forms of the mode operators, Iˆm =
∑
ll′ R
(m)
ll′ aˆ
†
l aˆl′ , with the Hermitian matrices R(m) ob-
tained from Eq. (2) by substitution ψˆ†ψˆ → φ∗l φl′ . For the
detection of cold atoms, we may take a resonant inter-
action between the atomic internal levels and the probe
light, which transfers the information of the atomic den-
sity to ψˆ†ψˆ of the probe light [12]. Hence, the detection
of cold atoms is treated in the same way as the photon
number detection.
The joint probabilities of the photon counts n1, . . . , nM
by the M detectors (1 ≤M ≤Mend), which characterize
the full statistics of interference, are given by
P (n1, . . . , nM ) =
〈
:
M
∏
m=1
1
nm!
(Iˆm)nme−Iˆm :
〉
, (3)
where :: stands for normal ordering [11]. The flux oper-
ators are presented explicitly as
Iˆm = R(m)aa aˆ†aˆ+ R
(m)
bb bˆ†bˆ+R
(m)
ab aˆ†bˆ+R
(m)
ba bˆ†aˆ. (4)
Here, it should be noted that the terms involving the
vacuum modes (l ≥ 3) are dropped in Iˆm since they pro-
vide null contributions to Eq. (3) as the normal-ordered
expectation values. The mean particle number measured
at each detector is given by
〈nm〉 = 〈Iˆm〉 = R(m)aa N¯a +R
(m)
bb N¯b. (5)
Here, N¯s = Tr[ρˆssˆ†sˆ] are the mean particle numbers ini-
tially contained in the sources, which are assumed to be
large enough to produce 〈nm〉 ≫ 1 for high accuracy
statistics. The coefficients R(m)aa and R(m)bb indicate the
probabilities for each particle from the respective sources
to fall into the detector m. They may represent the reso-
lution of interference. Specifically, R(m)ss ∝ 1/Mend → 0,
but keeping R(m)ss N¯s ≫ 1 for 〈nm〉 ≫ 1, when the par-
ticles are measured by almost continuously distributed
many detectors, resulting in a fine interference pattern,
e.g., spatial interference fringes [1, 3].
In the above sense, as seen in Eq. (5), a change of R(m)ss
(or resolution) for the detectors may be viewed alterna-
tively as an modification of the source statistics. Here,
consider scaling of the detector matrices (by removing
several detectors and changing the quantum efficiencies),
R˜(m)(q;M) = R(m)/q (m = 1, . . . ,M) (6)
with R˜(m)(q;M) = 0 (m > M), and define the binomial
distribution
BNN ′(q) ≡
(N
N ′
)
qN ′(1− q)N−N ′ (0 ≤ N ′ ≤ N). (7)
In evaluating the joint probabilities, 〈:(Iˆ1)k1 · · · (IˆM )kM :〉
contained in Eq. (3) are calculated for a Fock state
|Na, Nb〉 with the normal-ordered expectation values
〈(bˆ†)kb(aˆ†)ka aˆka bˆkb〉 = [Na!/(Na−ka)!]× [Nb!/(Nb−kb)!]
(ka+kb = k1 + · · ·+kM ), which are multiplied by qkaqkb
under the scaling. Then, by considering the relation
qk[N !/(N − k)!] =
∑N
N ′=k BNN ′(q)[N ′!/(N ′ − k)!], the ef-
fects of this scaling can be renormalized to the source
statistics without changing the calculations in Eq. (3) as
p˜s(N ; q) =
∞
∑
N ′=N
ps(N ′)BN
′
N (q), (8)
which is also normalized as the original ps(N). Hence,
the number statistics of the sources may be replaced with
the effective ones in Eq. (8) for any scaling of q, repro-
ducing the same joint probability for the measurement
by the M detectors (namely the M -detector model):
{R˜(m)(q;M), p˜s(N ; q)} → P (n1, . . . , nM ). (9)
This may be viewed as a renormalization transformation
among the number statistics. It indicates universal re-
lation for various interference phenomena, ranging from
two-mode homodyne detection (M = 2) to measurement
of spacial fringes (M = Mend ≫ 1). According to Eq. (8),
the mean ˜¯Ns and variance V˜s for the effective statistics

3are given in terms of the original ones as ˜¯Ns = qN¯s and
V˜s = q2Vs+(1−q)qN¯s. Then, for a sub-Poissonian distri-
bution (Vs < N¯s), the effective one is still sub-Poissonian
(V˜s < ˜¯Ns) as
V˜s/ ˜¯Ns = q(Vs/N¯s) + 1− q. (10)
The Poissonian form is preserved under the renormaliza-
tion up to the scaling of mean as ˜¯Ns = qN¯s. On the
other hand, for a super-Poissonian distribution the effec-
tive one is still super-Poissonian.
We now examine the validity of the mean field descrip-
tion for interference phenomena, where the field opera-
tors are replaced with c-numbers as aˆ → α and bˆ → β
(expectation values for coherent states |α, β〉). Specifi-
cally, we have
n¯m = 〈α, β|Iˆm|α, β〉
= 〈nm〉+ 2|R(m)ab |N¯1/2a N¯
1/2
b cos(δab + θm), (11)
where N¯a = |α|2, N¯b = |β|2, δab = argα − arg β,
θm = argR(m)ab , and 〈nm〉 is the same as Eq. (5) for
the U(1)-invariant sources. The set of {n¯m} exhibits the
interference pattern with the cosine term in Eq. (11),
which oscillates with θm depending on the detector loca-
tion. The mean field description is, however, not directly
applicable to the U(1)-invariant sources in Eq. (1) with
〈aˆ†bˆ〉 = 0, eliminating the cosine term in Eq. (11). Never-
theless, by experiments and theoretical calculations the
interference fringes are observed in a single run with a
random relative phase for Poissonian sources (laser fields
[1]) and sub-Poissonian sources (optical number states
[6, 7] and BECs [3, 4, 10]).
We hence consider the relationship between the in-
terference phenomena and the source number statistics.
Specifically, we examine the validity of the mean field
description by inspecting the joint probability P (n1, n2)
for any pair of detectors, say 1 and 2, depending on the
source statistics. Given the outcome n1 at detector 1, the
mean field description in Eq. (11) provides an estimate
for the relative phase, generally with two possibilities δ±ab
due to the cosine. Then, the outcome n2 at detector 2 is
inferred with the estimated phases:
n¯1 = n1 → δ±ab(n1) → n¯2[δ±ab(n1)]. (12)
If the actual count n2 is close to one of n¯2[δ±ab(n1)], fixing
the estimation of δab, we find that the interference occurs
as described by the mean (classical) fields. This criterion
for the interference can be checked readily by calculat-
ing the conditional distribution Pc(n2|n1) from P (n1, n2)
with given n1. If Pc(n2|n1) has sufficiently narrow peaks
at n¯2[δ±ab(n1)], the second outcome n2 should be close to
either of the peaks with high probability. Specifically,
the width of the peak should be no greater than that
of the Poisson distribution e−n¯2(n¯2)n2/n2!, which is the
shot noise level for the coherent states |α, β〉. Here, we
conjecture that sub-Poissonian sources lead to the narrow
peaks, showing the interference pattern. It is pointed out
[13] that wavepackets emitted from a cavity maintain a
pronounced relative phase coherence when the intracav-
ity field has a narrow number distribution. Light beams
from such sub-Poissonian cavities will exhibit the inter-
ference. This phase coherence of each source is essen-
tial to fix the interference phase in the number measure-
ments.
Consider first the case of fine detector resolution with
|R(m)| ≪ 1 in the usual measurement of spatial inter-
ference fringes. This case can be treated by scaling
as the two-detector model with R˜(1,2) = R(1,2)/q ∼ 1
and q → 0, which provides the same P (n1, n2) with
the effective statistics in Eq. (9). Then, as seen in Eq.
(10), the effective statistics of sub-Poissonian sources ap-
proach the Poissonian for q → 0. Hence, by using any
sub-Poissonian sources, essentially the same result is ob-
tained for the interference fringes as the Poissonian case,
where the mean field description is valid as numerically
confirmed in the following. This is not the case for
super-Poissonian sources. For R(m) = qR˜(m) → 0 with
R˜(m) ∼ 1 fixed, the large N¯s = ˜¯Ns/q ∝ 1/|R(m)|, which
is required to produce 〈nm〉 ≫ 1, may derive even the
larger Vs, e.g., Vs ∝ N¯2s , for a super-Poissonian source,
giving a nonzero q(Vs/N¯s) for q → 0 in Eq. (10).
In order to examine the validity of the mean field
description for general R(m)ss , we have calculated nu-
merically Pc(n2|n1) by using Eq. (3) for some typi-
cal sources. The detector matrices are chosen for in-
stance as R(1)aa = R(2)bb = 0.6R, R
(1)
bb = R
(2)
aa = 0.4R,
|R(1,2)ab |2 = R
(1,2)
aa R(1,2)bb , giving the maximum interference
term in Eq. (11), with the relative phase θ2 − θ1 = 0.9pi.
The first outcome is set as n1 = 118, which corresponds
to δ±ab(n1) + θ1 = ∓1.39 and n¯2[δ±ab(n1)] ≈ 53, 113.
Due to limitation on the numerical calculation, RN¯a =
RN¯b = 100 are taken, giving 〈n1〉 = 〈n2〉 = 100 with
R(1,2)aa +R(1,2)bb = R, and consistently 〈n1〉+ 〈n2〉 = 200 ≈
118 + (53 + 113)/2. The scaling for the effective statis-
tics is also used by taking R/q = R˜ = 0.867 to calcu-
late Pc(n2|n1) for the increasing N¯a,b = 100/R with the
smaller R, after it is checked numerically for q ∼ 0.5
with N¯a,b = 25/R. A bound on θ2 − θ1 may appear for
the increasing R from the condition n¯1 + n¯2 ≤ N¯a + N¯a
(= 200/R) due to the unitarity or the total number con-
servation, e.g., 0.9pi ≤ θ2 − θ1 ≤ pi for R = 0.867. This
is clearly seen in the familiar two-mode homodyne detec-
tion, where Iˆ1,2 = (aˆ†± bˆ†)(aˆ± bˆ)/2 with ei(θ2−θ1) = −1.
The results for number states |N/R,N/R〉 with N =
100 and some values of R are shown in Fig. 1. The
case of Poissonian source is also plotted for comparison,
corresponding to R → 0, where the Poisson distribu-
tion ∝ (n¯2)n2/n2! for n2 is confirmed around the peaks
(though rather broad due to not so large 〈n1,2〉 = 100).

4 0
0.01
0.02
0.03
0.04
0 20 40 60 80 100 120 140 160
P c
(n 2
|n 1
)
n2
Poissonian
R = 0.2
R = 0.5
R = 0.8
FIG. 1. (Color online) Conditional distribution for number
states |N/R,N/R〉 with N = 100 and some values of R. The
Poissonian case corresponds to R → 0. The mean field values
n¯2[δ±ab(n1)] ≈ 53, 113 for n1 = 118 are indicated with vertical
dotted lines.
0
0.005
0.01
0.015
0 20 40 60 80 100 120 140 160
P c
(n 2
|n 1
)
n2
Poissonian
Q = 0.01
Q = 0.1
Q = 1
FIG. 2. (Color online) Conditional distribution for super-
Poissonian sources with P(α) in Eq. (13). The mean field
values are shown the same as in Fig. 1.
The peaks agree with n¯2[δ±ab(n1)] ≈ 53, 113 (vertical dot-
ted lines), and exhibit the narrower widths than the Pois-
sonian case. Therefore, the mean field description is valid
for these sub-Poissonian number states and also their ef-
fective statistics, i.e., the binomial distributions in Eq.
(7). Here, the limit R → 1 becomes unphysical with the
dominating n¯2[δ+ab(n1)] to give n1 + n2 ≈ 118 + 113 >
200(R = 1), violating the unitarity.
We have also considered a super-Poissonian source
with a U(1)-invariant P-representation as
P(α) ∝ (|α|2/QN¯)1/Q−1 exp(−|α|2/QN¯), (13)
where Q = (V − N¯)/N¯2 with Q > 0. The limit Q → 0
corresponds to the Poissonian, whereas Q = 1 to the
thermal state. The conditional distribution is shown in
Fig. 2, which does not depend on R in this case with
RN¯a,b (=100) fixed. The increasing Q broadens the dis-
tribution, eventually washing out the peaks. We have
further examined the single-photon-added thermal state
[14]. This nonclassical super-Poissonian state has the
variance smaller than the thermal case. Despite this fact,
for the small R (RN¯a,b fixed), the conditional distribu-
tion becomes flatter than that for the thermal sources.
These results indicate that the behavior of interference
is rather complicated for super-Poissonian sources, likely
invalidating the mean field description.
To conclude, in view of the conditional probability
for the number measurements, we have elucidated the
common mechanism for the interference of independent
Bose fields under various situations, ranging from two-
mode homodyne interference to spacial fringes. The in-
terference is determined by the source number statistics,
among which the scaling behavior is present depending
on the detector characteristics with the boundary at the
Poissonian. For sub-Poissonian and Poissonian sources
the interference pattern appears in a single run, consis-
tently with the mean field description, whereas this is not
the case for super-Poissonian sources. It will be a chal-
lenge for future experiments to confirm the role of source
statistics with the scaling behavior, by preparing various
source states and detector configurations.
T. K. was supported by the JSPS Grant No. 22.1355.
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