Interference between Bose-Einstein condensates:

classical phase, quantum angle(quantum non-local effects)

Symposium to celebrate the 60th birthdays of Jean-Paul Blaizot and Larry McLerran

Quantum field theory in extreme environmentsIPhT, CEA, Saclay

Franck Laloë, LKB/ENS, 23/04/20091

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 ».

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. 2

1. Spontaneous symmetry breaking (Anderson)

• When a system of bosons undergoes the superfluid transition (BEC), spontaneous symmetry breaking takes place; the order parameter <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) phase?

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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. 4

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Take for instance spin condensates:

AliceBob

Carole

Experiment: interferences between two condensates

M.R. Andrews, C.G. Townsend, H.J. Miesner, D.S. Durfee, D.M. Kurn andW. 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

A.J. Leggett and F. Sols, « on the concept of spontaneous broken gauge symmetry in condensed matter physics », Foundations of Physics 21, 353 (1991)

E. Siggia and A. Rückenstein, « Bose-Einstein condensation in atomic hydrogen », Phys. Rev. Lett. 44, 1423 (1980)

« 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, 161-164 (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 iis, if M<<N = N++N-:

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An additional (or « hidden ») variable 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).

The phase distribution becomes narrower and narrower

W.J. Mullin, R. Krotkov and F. Laloë, cond-mat/0604371 11

Two large condensates

A

12

?

Enormous amplification effect, discussed by Leggett and Sols for a Josephson junction between two superconductors

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 »

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A.J. Leggett and F. Sols, « On the concept of spontaneously broken gauge symmetry in condensed matter physics », Found. Physics, vol. 21, 353 (1991).

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.

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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 remote 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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Alice Bob

Three condensates (transitivity)

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.)

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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.

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 :

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one now obtains:

where 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.

Alice and Bob make measurements with various combinations of angles

Alice Bob

Measurements of the average of the product of transverse spin components in two different directions

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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

a b

Violations of BCHSH inequalities

F. Laloë and W.J. Mullin, Phys. Rev. Lett. 99, 150401 (2007).

Q

The BCHSH inequalities are violated even for arbitrarily large BE condensates

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The EPR argument with interferometers

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Alice

Bob

.1.2

.3

.4

N

N

D

Population oscillations

• The interference measurement in D makes the two condensates to choose a relative phase

• 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

J.A. Dunningham, K. Burnett, R. Roth and W. PhillipsNew journal of physics, vol. 8, 182 (2006)

Conclusion

• 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!

BE condensates provide new light on fundamental quantum mechanics, and the old debate of the « measurement problem ».

Conclusion of the conclusion

Happy birthday Jean-Paul and Larry!!