Interference between quantum gases

Transcription

1 Anderson s question, and its answer Interference between quantum gases P.W. Anderson: do two superfluids which have never "seen" one another possess a relative phase? MIT Jean Dalibard, Laboratoire Kastler Brossel*, ENS Paris How to eplain it? How to use it? * Research unit of CNRS, ENS, and UPMC Outline of the talk The relative phase of two ideal Bose-Einstein condensates. The relative phase of two BECs Coherent states or number states? Javanainen and Yoo Cirac, Gardiner, Naraschewski, Zoller Wong, Collett, Walls Moelmer Castin and Dalibard 2. Interference between multiple sources 3. Using interference patterns to probe the properties of a many-body system For each BEC, the atom number N j ( j=,2 ) and the phase ϕ j are complementary variables in the quantum sense. Does the presence of fringes with high contrast prove that the BECs were in a coherent state? What happens if a measurement of the atom numbers N and N 2 is performed before the interference eperiment, i.e. if one works with number states:?

2 An appetier: the Hanburry-Brown & Twiss effect The relative phase of two condensates Two-boson problem (mass m) initially after time of flight of duration t L beam splitter 50% 50% R Probability to detect one particle in and the other one in Counter + Counter - Interference between two processes The relative phase is a random variable taking a different value at each shot of the eperiment. Eample: The counting probability for Fock states () The counting probability for Fock states (2) Naive argument: detections are independent events L + - R First detection of an atom in the (+) channel where and annihilate a particle in the left and right BEC. Second detection in (+) or (-)? proportional to Different result for coherent and Fock states??? 0-30 for k = 00 proportional to The squares of the norms are in the ratio 3: Once a first atom has been detected in (+) (probability ½), the probability to detect the second atom in (+) is 3/4 Is it correct that one can discriminate between the two points of view? If not, where is the mistake? For k detections identical to coherent states

3 Summary of the relative phase problem The continuous case Following Polkovnikov 07 Two equivalent points of view on the system: Modulation with period The phases and are pre-eisting before the measurement, and the measurements just reveal the difference The condensates are in number states (i.e. Fock states) before the measurements and the relative phase is built by the measurement sequence. Eact identity for any given N : Atomic field : Local density : Measured signal: Fourier component at momentum Observable for the real part : Initial state : : no contrast after average over many shots A single shot gives a result of order Each single shot shows a high contrast interference! Outline of the talk Preparation of a set of equidistant BECs. The relative phase of two BECs λ=532 nm P=200 mw w=00 μm V() Laser 2. Interference between multiple sources Laser d Atoms in a lattice (superfluid to Mott insulator transition) A fast rotating Bose gas: vortices as a result of interference of matter waves Distance d between adjacent planes: 3μm V 0 3. Using interference patterns to probe the properties of a many-body system N=30 planes are populated, with N at =0 4 atoms per plane Negligeable tunneling rate Z. Hadibabic, S. Stock, B. Battelier, V. Bretin, and J. Dalibard, PRL 2004

4 Interference of 30 independent BECs The interference signal between N independent BECs Z. Hadibabic et al, PRL 93, (2004) random phase for each BEC free flight propagator gaussian envelope Assume that all amplitudes α n are equal : d F() is a periodic function of period The measured value agrees well with q-th harmonic of amplitude A q : sum of interferences between BECs n and n±q The phases B q are random and the average amplitudes are given by 0 t time Remarkable features of the interference of N BECs The superfluid Mott insulator transition first harmonic: N=30 is not a large number : lattice with step d Fisher et al. 989 Jaksch et al. 998 Clear st order modulation epected for most images V 0 = 0 Er Greiner et al V 0 = 6 Er Each point represents one image The average A is in good agreement with the prediction above The phase B are indeed random The underlying order of the lattice can still be detected by performing an intensity correlation analysis ehibits peaks for Fölling et al, /28

5 Why is the noise structured? Situation similar to the Hanbury-Brown & Twiss effect An another eample of multiple interferences: Vortices in a rotating ideal gas Initially : d A reminder: Abrikosov lattice of vortices in an interacting, rotating fluid Picture of a rotating BEC Finally : 2 If are randomly chosen, If is a multiple of, constructive interference! ENS, MIT, Boulder, Oford Each vorte is a place of ero density, around which the phase of the wave function rotates by a multiple of Boulder What happens for an ideal rotating gas, at finite temperature? Landau levels for a 2D rotating gas The lowest Landau level Isotropic harmonic trapping in the y plane with frequency ω Hamiltonian in the rotating frame: Ω ω If the temperature is much smaller than, the physics is restricted to the lowest Landau level Ω=0 Ω < ω Ω ω LLL m=-2 m=- m=0 m=2 m=0 m= ω When Ω ω, one approaches a situation with a macroscopically degenerate ground state for the one-body hamiltonian Same physics as for a charged particle in a magnetic field General one-particle state in the LLL: Polynomial

6 The ideal gas in the LLL Vorte position in the ideal gas case State of the gas: superposition of independent BEC s in the various orbitals with random comple amplitudes C m A realiation of the eperiment corresponds to a particular choice of the C m s. m=0 2 3 Equal weight for up to m=40 vorte average density r/a v-v correlation function r/a Vortices (i.e. eroes of a random polynomial) repel each other (like the eigenvalues of a random matri). Vortices are still present and clearly visible: eroes of this polynomial Each vorte correspond to a destructive interference between the BEC s 2/28 Atom interactions provide long range order for the Abrikosov lattice Y. Castin, Z. Hadibabic, S. Stock, J. Dalibard, and S. Stringari, Phys. Rev. Lett. 96, (2006) Outline of the talk Investigation of long range order:. The relative phase of two BECs 2. Interference between multiple sources Two similar D (along ) or 2D (y plane) systems Time-of-flight + Imaging along the y direction 0 Interference signal: 3. Using interference patterns to probe the properties of a many-body system Measurement of Access to higher correlation functions Theory: Altman, Demler, Polkovnikov, Eperiments D: Heidelberg-Vienna Eperiments 2D: ENS Paris

7 Statistical distribution of the interference contrast What can be epected for the distribution of contrasts? Time of y flight imaging laser The local contrast contains information on the phase fluctuations of the two gases along the line of sight y Gritsev, Altman, Demler, Polkovnikov, Imambekov ( ) True BEC Pattern resulting from many uncorrelated elements along the line of sight and are two independent gaussian variables. Work with the distribution of the normalied contrast Close relation between this distribution and subtle theoretical problems. For a D system : quantum impurity in a D electron liquid some models for 2D quantum gravity Eponential distribution for: Eperiment in Heidelberg-Vienna for D systems Eperiment in ENS-Paris for 2D systems Distributions of the local contrast for various For each image, useful columns, each providing a mean phase space density D along the line of sight and a contrast C 300 images all at the same temperature We sort all 9000 couples (D,C) into groups of increasing D Low D Medium D high D Conclusions Two independent BECs can interfere, even if the initial state is a number state Symmetry breaking is a useful tool, not a requirement This interference process can be generalied to a large number of BECs Superfluid-Mott insulator transition LLL physics with an ideal gas Apparition of the phase order as the phase space density along the line of sight increases ENS group: P. Krüger, Z. Hadibabic, M. Cheneau, P. Rath, J.D. Interference of (quasi-)becs is a powerful method to investigate the properties of a many-body quantum system is obtained readily Higher correlation functions can be obtained via the statistical distribution of contrasts after integration along some line of sight.