Transcript Interference between Bose-Einstein condensates and quantum non-locality

Symposium to celebrate the 60th birthdays of
Jean-Paul Blaizot and Larry McLerran
Quantum field theory in extreme environments
IPhT, CEA, Saclay
Interference between Bose-Einstein condensates:
classical phase, quantum angle
(quantum non-local effects)
BE condensates are interesting for many reasons, but also provide
particularly illustrative examples of the relation between spontaneous
symmetry breaking and quantum measurement; there is also an unexpected
relation with so called « hidden variables ».
Franck Laloë, LKB/ENS, 23/04/2009
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Outline of the talk
• 1. Spontaneous symmetry breaking, Anderson’s
classical phase, quantum gases.
• 2. Orthodox quantum calculation (without
symmetry breaking); Leggett and Sols.
• 3. EPR (Einstein-Podolsky-Rosen) argument;
the classical phase is a macroscopic EPR
“element of reality” (hidden variable).
• 4. Quantum angle, quantum non-local effects
(Bell inequalities), populations oscillations.
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1. Spontaneous symmetry breaking (Anderson)
• When a system of bosons undergoes the superfluid
transition (BEC), spontaneous symmetry breaking takes
place; the order parameter <Y> takes a non-zero value. This
introduces a (complex) classical field with a phase.
Similar to ferromagnetic transition.
• Powerful idea: explains superfluid currents, vortex
quantization, etc.
• Violation of the conservation of the number of particles
• Anderson’s question: “When two superfluids that have
never seen each other before overlap, to they have a (relative)
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phase?
Dilute quantum gases
• These ideas were introduced in condensed
matter physics
• In dilute quantum gases of bosons, well below
the Bose-Einstein (BE) transition temperature,
the state vector is a Fock state (number state) to
a very good approximation.
• We assume that the particles are repulsive
(stability); in 3D, the system is superfluid.
• Anderson’s question applies to Fock states.
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Take for instance spin condensates:
Bob
Alice
Carole
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Experiment: interferences between two
condensates
M.R. Andrews, C.G. Townsend, H.J. Miesner, D.S. Durfee, D.M. Kurn and
W. Ketterle, Science 275, 637 (1997).
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Symmetry breaking is wonderful, but..
• What is the physical mechanism?
• For the ferromagnetic transition, the mechanism is clear.
But, in a BE condensate, how is it possible to create a
coherent superposition of different population numbers?
• Does symmetry breaking occur each time one reaches a
Fock state, without involving any phase transition?
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2. Interference beween condensates with, or
without, spontaneous symmetry breaking
E. Siggia and A. Rückenstein, « Bose-Einstein condensation in atomic
hydrogen », Phys. Rev. Lett. 44, 1423 (1980)
A.J. Leggett and F. Sols, « on the concept of spontaneous broken gauge
symmetry in condensed matter physics », Foundations of Physics 21, 353
(1991)
« We argue that the study of this question (how seriously shoud we take the
idea of spontaneous symmetry breaking in the Josephson effect) pushes us
toward the frontiers of what we understand about the quantum measurement
process, and underline the need for a new theoretical framework that keeps
with modern technological capabilities ».
A.J. Leggett, « Broken gauge symmetry in a Bose condensate » in « Bose
Einstein condensation »; ed. by A. Griffin, D.W. Snoke and S. Stringari,
Cambridge University Press (1995)
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Interference beween condensates without
spontaneous symmetry breaking
- J. Javanainen and Sun Mi Ho, "Quantum phase of a Bose-Einstein
condensate with an arbitrary number of atoms", Phys. Rev. Lett. 76, 161164 (1996).
- T. Wong, M.J. Collett and D.F. Walls, "Interference of two Bose-Einstein
condensates with collisions", Phys. Rev. A 54, R3718-3721 (1996)
- J.I. Cirac, C.W. Gardiner, M. Naraschewski and P. Zoller, "Continuous
observation of interference fringes from Bose condensates", Phys. Rev. A
54, R3714-3717 (1996).
- Y. Castin and J. Dalibard, "Relative phase of two Bose-Einstein
condensates", Phys. Rev. A 55, 4330-4337 (1997)
- K. Mølmer, "Optical coherence: a convenient fiction", Phys. Rev. A 55,
3195-3203 (1997).
- K. Mølmer, "Quantum entanglement and classical behaviour", J. Mod.
Opt. 44, 1937-1956 (1997)
- C. Cohen-Tannoudji, Collège de France 1999-2000 lectures, chap. V et
VI "Emergence d'une phase relative sous l'effet des processus de
détection" http://www.phys.ens.fr/cours/college-de-france/.
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How Bose-Einstein condensates acquire a phase under the
effect of successive quantum measurements
Initial state before measurement:
No phase at all ! One measures the spin a point r1 along the transverse
direction 1 , the spin a point r2 along the transverse direction 2 , etc.
The combined probability for M measurements at points ri,, with angle i, and
with results hi is, if M<<N = N++N-:
An additional (or « hidden ») variable l appears very naturally in the calculation,
within perfectly orthodox quantum mechanics. Ironically, it appears
mathematically as a consequence of the number conservation rule!
F. Laloë, “The hidden phase of Fock states; quantum non-local effects”,
European Physical Journal 33, 87-97 (2005).
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The phase distribution becomes narrower and
narrower
W.J. Mullin, R. Krotkov and F. Laloë, cond-mat/0604371
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Two large condensates
?
A
Enormous amplification effect, discussed by Leggett and Sols for
a Josephson junction between two superconductors
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Leggett and Sols:
« We argue that the study of this question pushes us toward the frontiers of
what we understand about the quantum measurement process, and underline
the need for a new theoretical framework that keeps with modern technological
capabilities ».
« Can it really be that, by placing a minuscule compass needle (measurement
apparatus) next to the system, we can force the large system to realize a
definite macroscopic value of the current (angular momentum)? Common sense
rebels against this conclusion, and we believe that common sense is right.
This idea (a small system should force, by state vector reduction, the
macrocopic state of another large system) sounds bizarre in the extreme »
A.J. Leggett and F. Sols, « On the concept of spontaneously broken
gauge symmetry in condensed matter physics », Found. Physics, vol.
21, 353 (1991).
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3. The EPR argument with spin
condensates
Alice
Bob
Orthodox quantum mechanics tells us that it is the measurement performed by
Alice that creates the transverse orientation observed by Bob.
N.B: It is just the relative phase of the mathematical wave functions that is
determined by measurements; the physical states themselves remain unchanged;
it is not a matter of propagating of something physical along the condensates, for
instance phonons etc.
EPR argument: the « elements of reality » contained in Bob’s region of space can not
change under the effect of a measurement performed in Alice’s arbitrarily remote14
region. They necessarily pre-exited; therefore quantum mechanics is incomplete.
Bohr’s reply to the usual EPR argument
(with two microscopic particles)
The notion of physical reality used by EPR is ambiguous; it
does not apply to the microscopic world; it can only be
defined in the context of a precise experiment involving
macroscopic measurement apparatuses.
But here, the transverse spin orientation may be
macroscopic! Actually, we do not know what Bohr would
have replied to the BEC version of the EPR argument.
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Three condensates (transitivity)

Alice
Bob
The interactions in both regions of space may be of completely different nature,
depending on the matrix elements between the internal states; for instance, the
measurement performed by Alice may involve electric quadrupoles, having nothing
to do with angular momentum. Then, in orthodox quantum mechanics, the angular
momentum observed by Bob emerges really « from nothing ».
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Anderson’s phase; summary
1. If M<<N, phase symmetry breaking provides exactly the same results as
standard quantum mechanics (without symmetry breaking). It is not necessary
from a conceptual point of view, but sometimes be technically convenient !
2. It is natural to think that the phase was there from the beginning, and was not
created by the act of measurement. Anderson’s phase is not a new concept,
associated with quantum phase transitions; it is a special case of
additional/hidden variable theories (de Broglie, Bohm, Goldstein, etc.)
3. If M=N, the results of symmatry breaking disagree with those of standard
quantum mechanics. With a classical phase, no violation of the Bell inequalities
would be obtained, while quantum mechanics predicts that they occur.
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4. Quantum non-local effects with BE
condensates (Bell inequalities)
What happens if the number of measurements M becomes comparable to the
number of particles N? Instead of an expression with a single phase angle l
:
one now obtains:
where L is the quantum angle. Then the « probabilities » can become negative;
quantum effects become possible, for instance violations of the BCHSH local
inequalities. The notion of Anderson’s phase no longer applies.
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Alice and Bob make measurements
with various combinations of angles
a
b
Alice
Bob
Measurements of the average of the product of transverse spin components in
two different directions
Q= <S(a)S(b)> + <S(a’)S(b)> + <S(a)S(b’)> + <S(a’)S(b’)>
The BCHSH inequality states that, if local realism is obeyed:
- 2  Q  +2
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Violations of BCHSH inequalities
Q
2
The BCHSH inequalities are violated even for arbitrarily large BE condensates
F. Laloë and W.J. Mullin, Phys. Rev. Lett. 99, 150401 (2007).
The EPR argument with interferometers
Alice
Bob
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Population oscillations
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D
1
.
.
Nb
.
2
• The interference measurement in D makes the two
condensates to choose a relative phase
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.
• But only the absolute value of this phase is fixed;
one therefore creates a coherent superposition of two
different values of the macroscopic phase
• Then a complementary measurement of the
populations exhibits oscillations
Na
J.A. Dunningham, K. Burnett, R. Roth and W. Phillips
New journal of physics, vol. 8, 182 (2006)
Conclusion
BE condensates provide new light on fundamental quantum mechanics,
and the old debate of the « measurement problem ».
• There is an expected relation between spontaneous
symmetry breaking and additional (hidden) variables in
quantum mechanics (Bohm, etc.)
• The EPR argument can be transposed to a macroscopic
scale, and then becomes even stronger
• Bohr’s reply against additional variables then becomes
less convincing
• We still do not know if quantum mechanics is indeed
incomplete or not. Maybe the postulate of quantum
measurements require some improvements. In any case it
remains a wonderful theory!
Conclusion of the conclusion
Happy birthday Jean-Paul and Larry!!