Quantum non-local effects with Bose-Einstein condensates
Quantum non-local effects with Bose-Einstein condensates
F. Laloë a and W. J. Mullin b
Laboratoire Kastler Brossel, ENS, UPMC, CNRS; 24 rue Lhomond, 75005 Paris, France
Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003 USA
We study theoretically the properties of two Bose-Einstein condensates in different spin states,
represented by a double Fock state. Individual measurements of the spins of the particles are per-
formed in transverse directions, giving access to the relative phase of the condensates. Initially, this
phase is completely undefined, and the first measurements provide random results. But a fixed value
of this phase rapidly emerges under the effect of the successive quantum measurements, giving rise
to a quasi-classical situation where all spins have parallel transverse orientations. If the number of
measurements reaches its maximum (the number of particles), quantum effects show up again, giving
rise to violations of Bell type inequalities. The violation of BCHSH inequalities with an arbitrarily
large number of spins may be comparable (or even equal) to that obtained with two spins.
PACS numbers: 03.65.Ta,03.65.Ud,03.75.Gg,03.75.Mn
The notion of non-locality in quantum mechanics
(QM) takes its roots in a chain of two theorems, the
EPR (Einstein Podolsky Rosen) theorem [1] and its log-
ical continuation, the Bell theorem. The EPR theorem
starts from three assumptions (Einstein realism, locality,
the predictions of quantum mechanics concerning some
perfect correlations are correct) and proves that QM is
incomplete: additional quantities, traditionally named λ,
are necessary to complete the description of physical re-
ality. The Bell theorem [2, 3] then proves that, if λ exists,
the predictions of QM concerning other imperfect corre-
lations cannot always be correct. The ensemble of the
three assumptions: Einstein realism, locality, all predic-
tions of QM are correct, is therefore self-contradictory;
if Einstein realism is valid, QM is non-local. Bohr [4]
rejected Einstein realism because, in his view, the no-
tion of physical reality could not correctly be applied to
microscopic quantum systems, defined independently of
the measurement apparatuses. Indeed, since EPR con-
sider a system of two microscopic particles, which can be
“seen” only with the help of measurement apparatuses,
the notion of their independent physical reality is open
to discussion.
Nevertheless, it has been pointed out recently [5, 6]
that the EPR theorem also applies to macroscopic sys-
tems, namely Bose-Einstein (BE) condensates in two dif-
ferent internal states. The λ introduced by EPR then cor-
responds to the relative phase of the condensates, i.e. to
macroscopic transverse spin orientations, physical quan-
tities at a human scale; it then seems more difficult to
deny the existence of their reality, even in the absence
of measurement devices. This gives even more strength
to the EPR argument and weakens Bohr’s refutation. It
is then natural to ask whether the Bell theorem can be
transposed to this stronger case.
The purpose of this article is to show that it can. We
consider an ensemble of N+ particles in a state defined by
an orbital state u and a spin state +, and N− particles
in the same state with spin orientation −. The whole
system is described quantum mechanically by a double
Fock state, that is, a “double BE condensate”:
| Φ > =
(au,+)
]N+ [
(au,−)
| vac. > (1)
where au,+ and au,− are the destruction operators asso-
ciated with the two populated single-particle states and
|vac. > is the vacuum state. We introduce a sequence
of transverse spin measurements that leads to quantum
predictions violating the so called BCHSH [7, 8] Bell in-
equality. This is reminiscent of the work of Mermin [9],
who finds exponential violations of local realist inequal-
ities with N -particle spin states that are maximally en-
tangled. By contrast, here we consider the simplest way
in which many bosons can be put in two different in-
ternal levels, with a N -particle state containing only the
minimal possible correlations, those due to statistics. We
find violations of inequalities that are the same order of
magnitude as with the usual singlet spin state and may
actually saturate the Cirel’son bound [10].
Double Fock states are experimentally more accessi-
ble and much less sensitive to dissipation and decoher-
ence than maximally entangled states [11]. Considering
a system in a double Fock state, we assume that a se-
ries of rapid spin measurements can be performed and
described by the usual QM postulate of measurement,
without worrying about decoherence between the mea-
surements, thermal effects, etc.
The operators associated with the local density of par-
ticles and spins can be expressed as functions of the
two fields operators Ψ±(r) associated with the two in-
ternal states ± as: n(r) = Ψ†+(r)Ψ+(r) + Ψ
−(r)Ψ−(r),
σz(r) = Ψ
+(r)Ψ+(r)−Ψ
−(r)Ψ−(r), while the spin com-
ponent in the direction of plane xOy making an angle ϕ
with Ox is: σϕ(r) = e
+(r)Ψ−(r)+ e
−(r)Ψ+(r).
Consider now a measurement of this component per-
formed at point r and providing result η = ±1. The
http://arxiv.org/abs/0704.0386v4
corresponding projector is:
Pη=±1(r, ϕ) =
[n(r) + η σϕ(r)] (2)
and, because the measurements are supposed to be per-
formed at different points (ensuring that these projectors
all commute) the probability P(η1, η2, ...ηN ) for a series
of results ηi± 1 for spin measurements at points ri along
directions ϕi can be written as:
< Φ | Pη1(r1, ϕ1)× Pη2(r2, ϕ2)× ....PηN (rN , ϕN ) | Φ >
We now substitute the expression of σϕ(r) into (2) and
(3), exactly as in the calculation of ref. [5], but with one
difference: here we do not assume that the number of
measurements is much smaller than N±, but equal to
its maximum value N = N+ + N−. In the product of
projectors appearing in (3), because all r’s are different,
commutation allows us to push all the field operators to
the right, all their conjugates to the left; one can then
easily see that each Ψ±(r) acting on | Φ > can be re-
placed by u(r) × au,± , and similarly for the Hermitian
conjugates. With our initial state, a non-zero result can
be obtained only if exactly N+ operators au,+ appear in
the term considered, and N− operators au,−; a similar
condition exists for the Hermitian conjugate operators.
To express these conditions, we introduce two additional
variables. As in [5], the first variable λ ensures an equal
number of creation and destruction operators in the in-
ternal states ± through the mathematical identity:
einλ = δn,0 (4)
The second variable Λ expresses in a similar way that the
difference between the number of destruction operators in
states + and − is exactly N+−N−, through the integral:
e−inΛ ei(N+−N−)Λ = δn,N+−N− (5)
The introduction of the corresponding exponentials into
the product of projectors (2) in (3) provides the expres-
sion (c.c. means complex conjugate):
|u(rj)|2
eiΛ + e−iΛ + ηj
ei(λ−ϕj+Λ) + c.c.
where, after integration over λ and Λ, the only surviving
terms are all associated with the same matrix element in
state | Φ > (that of the product of N+ operators a†u,+
and N− operators a
u,− followed by the same sequence
of destruction operators, providing the constant result
N+!N−!). We can thus write the probability as:
P(η1, η2, ...ηN ) ∼
ei(N+−N−)Λ
|u(rj)|2
eiΛ + e−iΛ + ηj
ei(λ−ϕj+Λ) + c.c.
or, by using Λ parity and changing one integration variable (λ′ = λ+ Λ), as:
P(η1, η2, ...ηN ) =
cos [(N+ −N−)Λ]
{cos (Λ) + ηj cos (λ′ − ϕj)} (8)
The normalization coefficient CN is readily obtained by writing that the sum of probabilities of all possible sequences
of η’s is 1 (this step requires discussion; we come back to this point at the end of this article):
cos [(N+ −N−)Λ] [cos (Λ)]N (9)
Finally, we generalize (8) to any number of measurements M < N . A sequence of M measurements can always be
completed by additional N −M measurements, leading to probability (8). We can therefore take the sum of (8) over
all possible results of the additional N −M measurements to obtain the probability for any M as:
P(η1, η2, ...ηM ) =
cos [(N+ −N−)Λ] [cosΛ]N−M
{cos (Λ) + ηj cos (λ′ − ϕj)} (10)
The Λ integral can be replaced by twice the integral between ±π/2 (a change of Λ into π −Λ multiplies the function
by (−1)N+−N−+N−M+M = 1). If M ≪ N , the large power of cosΛ in the first integral concentrates its contribution
around Λ ≃ 0, so that a good approximation is Λ = 0. We then recover the results of refs [5, 6], with a single integral
over λ defining the relative phase of the condensates (Anderson phase), initially completely undetermined, so that
the first spin measurement provides a completely random result. But the phase rapidly emerges under the effect of a
few measurements, and remains constant [12, 13, 14]; it takes a different value for each realization of the experiment,
as if it was revealing the pre-existing value of a classical quantity. Moreover, when cosΛ is replaced by 1, each factor
of the product over j remains positive (or zero), leading to a result similar to that of stochastic local realist theories;
the Bell inequalities can then be obtained. However, when N − M is small or even vanishes, cosΛ can take values
that are smaller than 1 and the factors may become negative, opening the possibility of violations. In a sense, the
additional variable Λ controls the amount of quantum effects in the series of measurements.
We now discuss when these standard QM predictions violate Bell inequalities. We need the value of the quantum
average of the product of results, that is the sum of η1, η2, ...ηM × P(η1, η2, ...ηM ) over all possible values of the η’s,
which according to (10) is given by:
E(ϕ1, ϕ2, ..ϕM ) =
cos [(N+ −N−)Λ] [cosΛ]N−M
Consider a thought experiment where two condensates
in different spin states (two eigenstates of the Oz spin
component) overlap in two remote regions of space A
and B , with two experimentalists Alice and Bob; they
measure the spins of the particles in arbitrary transverse
directions (perpendicular to Oz) at points of space where
the orbital wave functions of the two condensates are
equal. All measurements performed by Alice are made
along a single direction ϕa, which plays here the usual
role of the “setting” a, while all those performed by Bob
are made along angle ϕb. We assume that Alice retains
just the product A of all her measurements, while Bob
retains only the product B of his; A and B are both ±1.
We now assume two possible orientations ϕa and ϕ
for Alice, two possible orientations ϕb and ϕ
b for Bob.
Within deterministic local realism, for each realization of
the experiment, it is possible to define two numbers A,
A′, both equal to ±1, associated with the two possible
products of results η that Alice will observe, depending
of her choice of orientation; the same is obviously true
for Bob, introducing B and B′. Within stochastic local
realism [8, 15], A and A′ are the difference of probabilities
associated with Alice observing +1 or −1, i.e. numbers
that have values between +1 and −1. In both cases, the
following inequalities (BCHSH) are obeyed:
− 2 ≤ AB +AB′ ± (A′B − A′B′) ≤ 2 (12)
In standard quantum mechanics, of course, “unper-
formed experiments have no results” [16], and several of
the numbers appearing in (12) are undefined; only two
of them can be defined after the experiment has been
performed with a given choice of the orientations. Thus,
while one can calculate from (11) the quantum average
value < Q > of the sum of products of results appearing
in (12), there is no special reason why < Q > should be
limited between +2 and −2. Situations where the limit
is exceeded are called “quantum non-local”.
We have seen that the most interesting situations oc-
cur when the cosines do not introduce their peaking effect
around Λ = 0, i.e. when N+ = N− and M has its maxi-
mum value N . Then, for a given N , the only remaining
choice is how the number of measurements is shared be-
tween Na measurements for Alice and Nb for Bob.
Assume first that Na = 1 (Alice makes one measure-
ment) and therefore Nb = N − 1 (Bob makes all the oth-
ers). Since we assume that N+ = N− and M = N , the
Λ integral in (11) disappears, and the λ integral contains
only the product of cos (λ′ − ϕa) by the (N − 1)th power
of cos (λ′ − ϕb), which is straightforward and provides
cos (ϕa − ϕb) times the normalization integral CN . The
quantum average associated with the product AB is thus
merely equal to cos (ϕa − ϕb), exactly as the usual case
of two spins in a singlet state. Then it is well-known that,
when the angles form a “fan” [17] spaced by χ = π/4,
a strong violation of (12) occurs, by a factor
2, sat-
urating the Cirel’son bound [10]. A similar calculation
can be performed when Alice makes 2 measurements and
Bob N − 2, and shows that the quantum average is
now equal to 1
1 + 1
+ (1 − 1
) cos 2 (ϕa − ϕb)
no longer independent of N. If N = 4, the maximum
of < Q > is 2.28 < 2
2, and rises to 2.41 as N → ∞.
An expression for the generalization of the quantum av-
erage to any number P and N − P of measurements by
Alice and Bob, respectively, is (with χ = ϕa − ϕb):
E(χ) =
{P/2}
P !(N − 2k)!
k!(P − 2k)!(N
− k)!
sin2k χ cosP−2k χ
where {P/2} is the integer part of P/2. The maximum of
< Q > can then be found using a numerical Mathematica
routine. Results are shown for several values of P in Fig.
1. The angles maximizing the quantum Bell quantity
always occur in the fan shape, although the basic angle
χ changes with P and N. All of the curves where P is
held fixed have a finite < Q > limit with increasing N ,
and the optimum values of the angles approach constants.
For the curve P = N/2, the limit is 2.32 when N → ∞,
and the fan opening decreases as 1/
10080604020
       P
FIG. 1: The maximum of the quantum average < Q > for
Alice doing P experiments and Bob N − P , as a function of
the total number of particles N . The usual Bell situation is
obtained for N = 2, P = 1. Local realist theories predict an
upper limit of 2; large violations of this limit are obtained,
even with macroscopic systems (N → ∞). If P = 1, the
violation saturates the Cirel’son limit for any N .
We can also study cases where the number of measure-
ments is M < N : if Bob makes all his measurements,
but ignores one or two of them (independently of the or-
der of the measurements), when he correlates his results
with Alice, the BCHSH inequality is never violated. All
measurements have to be taken into account to obtain
violations. Furthermore, if the number of particles in the
two condensates are not equal, no violation occurs either.
Finally, it is possible to consider cases where we gener-
alize the angles considered: experimenter Carole makes
measurements at ϕc and ϕ
c, and David at ϕd and ϕ
We then find that a maximization of < Q > reduces to
the cases already studied, where the new angles collapse
to the previous angles ϕa, · · · , ϕ′b.
For the sake of simplicity, we have not yet discussed
some important issues that underlie our calculations. One
is related to the so called “sample bias loophole” (or
“detection/efficiency loophole”) and to the normalization
condition (9), which assumes that one spin is detected at
each point of measurement. A more detailed study (see
second ref. [5]) should include the integration of each
r in a small detection volume and the possibility that
no particle is detected in it. This is a well-known dif-
ficulty, which already appears in the usual two-photon
experiments [8], where most photons are missed by the
detectors. If this loophole still raises a real experimen-
tal challenge, the difficulty can be resolved in theory by
assuming the presence of additional spin-independent de-
tectors [2, 8], which ensure the detection of one particle
in each detector and create appropriate initial conditions
(see for instance [18] for a description of an experiment
with veto detectors). We postpone this discussion to an-
other article [19]. A second issue deals with the definition
of the local realist quantities A, B, etc. For two conden-
sates, we have a slightly different situation than in the
usual EPR situation: the local realist reasoning leads to
the existence of a well-defined phase λ between the con-
densates [5], not necessarily to deterministic properties
of the individual particles. Fortunately, Bell inequalities
can also be derived within stochastic local realist theories
[3, 8] (see also for instance [9] or appendix I of [15]), and
this difference is not a problem [19].
In conclusion, strong violations of local realism may
occur for large quantum systems, even if the state is a
simple double Fock state with equal populations; within
present experimental techniques, this seems reachable
with N ∼ 10 or 20. We have assumed that the mea-
sured quantity is the product of many microscopic mea-
surements, not their sum, which would be macroscopic; a
product of results remains sensitive to the last measure-
ment, even after a long sequence of others. Curiously, for
very few measurements only the results are quantum, for
many measurements they can be interpreted in terms of
a classical phase, but become again strongly quantum
when the maximum number of measurements is reached,
a sort of revival of quantum-ness of the system.
Laboratoire Kastler Brossel is “UMR 8552 du CNRS,
de l’ENS, et de l’Université Pierre et Marie Curie”.
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