Bose-Einstein condensates (Fock states): classical or quantum?

Transcription

1 Bose-Einstein condensates (Fock states): classical or quantum? Does the phase of Bose-Einstein condensates exist before measurement? Quantum non-locality, macroscopic superpositions (QIMDS experiments)) Toulouse, 4 avril 2012 Franck Laloë, LKB, ENS Paris William Mullin, U. Mass., Amherst, Massachusetts, USA. 1

2 Related articles by A.J. Leggett «Schrödinger s cat and her laboratory cousins», Contemp. Phys. 25, 583 (1984). A. Leggett and A. Garg, «Quantum mechanics versus macroscopic realism: is the flux there when nobody looks?», PRL 54, 857 (1985). A. Leggett and F. Sols, «On the concept of spontaneous gauge symmetry in condensed matter physics», Found. Phys. 21, 353 (1990). «Broken gauge symmetry in a Bose condensate», BEC conference, Levico-Terme (1995). «Probing quantum mechanics toward the everyday world: where do we stand?», Phys. Scripta, T201, 69 (2002). «The quantum measurement problem», Science, 307, 871 (2002). «Realism and the physical world», Rep. Progr. Phys. 71, (2008). 2

3 Subject of this talk In BE condensates, the wave function is often considered as a classical field. Are the quantum limitations concerning the measurement of a wave function really lifted? Does the relative phase of two such condensates behave as a classical phase, as suggested by the idea of spontaneous phase appearance under the effect of symmetry breaking? (Anderson s phase in superfluids) Can we use this phase to perform experiments creating a quantum superposition of macroscopically distinct states? 3

4 Quantum limitations concerning the measurement of a wave function With a single quantum system, as soon as the wave function is measured, it suddenly changes (state reduction). One cannot perform exclusive measurements on the same system (Bohr s complementarity) One cannot determine the wave function of a single system perfectly well (but one can teleport it without knowing it) One cannot clone the wave function of a single quantum system But, if you have many particles with the same wave function (quantum state), these limitations do not apply anymore. The wave function becomes similar to a classical field. For instance, you could use some particles to make one measurements, the others to make a complementary (exclusive) measurement. 4

5 Many particles in the same wave function S «Complementary» simultaneous measurements 5

6 Introduction Outline of the talk 1. Creating and manipulating Fock states (number states) in dilute gases 2. Single beam splitter: quantum effects, quantum angle 3. Two beam splitters, non-local effects 4. Macroscopic superpositions (QSMDS): population oscillations, NOON states 5. Spin condensates, the macroscopic EPR argument Conclusion 6

7 1. Creating and manipulating Fock states obtained by BE condensation in gases One can control the interactions in dilute gases through Feshbach resonances, and make them attractive of repulsive, with small or large interactions at will. One can make small condensates (for instance many «pancakes» by creating a periodic repulsive potential in an elongated condensate). By adjusting the chemical potential controlled by the interactions, one can create condensates containing a welldefined (small) number of atoms, 20 or 50 for instance. 7

8 Small condensates 8

9 Fock states can also be prepared with photons: measuring the number of photons in a cavity Nature, vol 446, mars

10 Measuring the number of photons in a cavity Nature, vol 448, 23 August

11 Manipulating BEC s in gases One can obtain «beam splitters» by sending BEC s onto gratings created by the interference of two laser beams (Bragg scattering). One can change the phase of a condensate by appliying a potential with a non-resonant laser beam (light shifts). 11

12 Beam splitters for atomic Bose-Einstein condensates 12

13 Optically guided beam splitter 13

14 Counting atoms Physics Today, october 2010, «Putting quantum gases under the microscope» One can count atoms trapped in a periodic lattice potential created by atoms. When the atoms recombine in molecules, the parity of the number of atoms is more easily accessible. Markus Greiner et al. Harvard Univ. Stephan Kuhr, Immanuel Bloch et al., Garching 14

15 2. Spontaneous appearance of a relative phase in one interference region (single beam splitter) Can we consider that the two condensates «who have never seen each other» (Anderson) already have a relative phase? Or do they spontaneously acquire a (classical) phase under the effect of measurement? In both cases, we could make a classical calculation by averaging over a random classical phase. Does this contains all the physics, or is the phase non-classical even after measurement? 15

16 Spontaneous emergence of a relative phase of two condensates under the effect of quantum measurement - J. Javanainen and Sun Mi Ho, "Quantum phase of a Bose-Einstein condensate with an arbitrary number of atoms", Phys. Rev. Lett. 76, (1996). - T. Wong, M.J. Collett and D.F. Walls, "Interference of two Bose-Einstein condensates with collisions", Phys. Rev. A 54, R (1996) - J.I. Cirac, C.W. Gardiner, M. Naraschewski and P. Zoller, "Continuous observation of interference fringes from Bose condensates", Phys. Rev. A 54, R (1996). - Y. Castin and J. Dalibard, "Relative phase of two Bose-Einstein condensates", Phys. Rev. A 55, (1997) - K. Mølmer, "Optical coherence: a convenient fiction", Phys. Rev. A 55, (1997). - K. Mølmer, "Quantum entanglement and classical behaviour", J. Mod. Opt. 44, (1997) -C. Cohen-Tannoudji, Collège de France lectures, chap. V et VI "Emergence d'une phase relative sous l'effet des processus de détection" -etc. 16

17 The Ketterle et al. experiment M.R. Andrews, C.G. Townsend, H.J. Miesner, D.S. Durfee, D.M. Kurn and W. Ketterle, Science 275, 637 (1997). The phase takes completely random values from one realization of the experiment to the next, but for one realization remains consistent with the choice of a single value for a single experiment. The results are fully compatible with the idea of a pre-existing, but completely unknown, relative phase the phase plays the role of a classical variable. 17

18 Beam splitter: classical calculation Equal source intensities Different intensities 18

19 But we know that quantum effects occur with a single beam splitter With two photons only, we already have the Hong-Ou- Mandel effect (HOM). Two input photons, one on each side, always leave in the same direction (never in two different directions). 19

20 Beam splitter: quantum calculation Two angles appear, the classical phase and the quantum angle. The «probabilities» may become negative. If some particles are missed, a [cos appears inside the integral, where N is the total number of particles, and M the number of measured particles. Then to a good approximation and one recovers the classical formula The quantum angle plays a role only when all particles are measured. It contains properties that are beyond the classical phase (Anderson s phase). It is the source of the HOM effect for instance F. Laloë and W.J. Mullin, «Quantum properties of a single beam splitter», Found. Phys. 42, (2012) 20

21 Examples of quantum predictions Populations of the sources: 4, 4 Populations of the sources: 25,25 21

22 Other examples F. Laloë et W. Mullin, Festschrift in the honour of H. Rauch et D. Greenberger, (Vienna, 2009) 22

23 3. Quantum non-local effects Instead of making the two BEC s spontaneously choose a relative phase in one place, we can test how they choose this phase at two different places. If the two places are remote, we can test if this choice is consistent and if locality is obeyed. Anderson: do two superfluids that have never seen each other have a relative phase? Here: do two BEC s chose a consistent phase if they are required to make this choice in remote places at the same time? 23

24 Two interferometers with two phase shifts Alice Bob In 2 remote regions of space, Alice and Bob measure what relative phase the 2 condensates have chosen, and they compare the consistency of their results. 24

25 Quantum calculation; role of the quantum angle Here again, the quantum result looks like the classical result, but with one more angle: the quantum angle. If some particles are missed in the measurement, the quantum angle plays no role. All results can be explained by a random pre-existing phase. If all particles are measured, specific quantum effect take place. F. Laloë and W.J. Mullin, PRL 99, (2007). 25

26 Violating the Bell (CHSH) inequalities Classically: One predicts strong violations of the Bell inequalities, even if the total number of particles is large; but the number of particles in the sources must be sharply defined (Fock states). 26

27 4. Macroscopic superpositions, population oscillations S «Complementary» measurements 27

28 Two «complementrary» measurements at the same time m m N m m W. Mullin and F. Laloë, PRL 104, (2010) N 28

29 If the phase was classical.. m m The measurements of m and m should be completely independent of the relative phase of the two sources. N m m N But what is created by the measurement of m1 and m2 is a quantum superposition of two opposite values of the phase. This 29 creates quantum macroscopic interference.

30 Detecting the quantum macroscopic superposition (QIMDS) The phase measurement process creates fluctuations of the number of particles in each input beam. On sees oscillations in the populations, directly at the output of the particle sources. 30

31 Creating NOON states 3 D 2 D 5 D 1 A 4 D 6 Conditional création of a «NOON state» in arms 5 and 6 W. Mullin and F. Laloë, JLTP 162, (2011). 31

32 Leggett s QSMDS (quantum superpositions of macroscopically different states) 3 5 (D 5 ) D 2 7 D 1 A 4 6 (D 6 ) B 8 Detection in arms 7 and 8 of the NOON state in arms 5 and 6 32

33 5. Spin condensates: Alice Bob F. Laloë, EPJD 33, (2005). Carole measures the transverse directions of individual spins 33

34 The EPR argument with macroscopic variables Alice Bob Orthodox quantum mechanics tells us that it is the measurement performed by Alice that creates the transverse orientation observed by Bob. Spontaneous appearance of angular momentum! It is just the relative phase of the mathematical wave functions that is determined by measurements; the physical states themselves remain unchanged; nothing physical propagates along the condensates, Bogolubov phonons for instance, 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 remote 34 region. They necessarily pre-exited; therefore quantum mechanics is incomplete.

35 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; physical reality can only be defined in the context of a precise experiment involving macroscopic measurement apparatuses. But here, the transverse spin orientation that appears spontaneously under the effect of a remote measurement may be macroscopic! Macroscopic physical quantities should have physical reality, independently of the measurement apparatuses. One can no longer invoke the huge difference of scale between macroscopic and microscopic objects. We do not know what Bohr would have replied to the BEC macroscopic version of the EPR argument. 35

36 Einstein and Anderson agree! Suppose we agree that the transverse spin orientation should exist already before the first measurement. But this is precisely what the spontaneous symmetry breaking argument is saying! the relative phase existed before the measurement, as soon as the condenstates were formed. So, in this case, Anderson s phase appears as a macroscopic version of the «EPR element of reality», applied to the case of relative phases of two condensates. It is an additional variable, a «hidden variable» (Bohm, etc.). 36

37 Spin condensates: 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, within local realism: -2 Q +2 Again quantum mechanics violates this inequality by a large factor 37

38 Conclusion Many quantum effects are possible with Fock states The wave function of highly populated quantum state (BEC s) has classical properties, but also retains strong quantum features if all particles are measured. The measurement of a few particles creates a phase that has classical properties, and keeps them when more and more particles are measured. Nevertheless, when the very last particles are measured, strong quantum properties appear again. 38

39 Neither Anderson, nor HOM.. but both combined Repeating the HOM experiment many times: Populating Fock states: 39