Interference between fluctuating condensates Anatoli Polkovnikov, Boston University Collaboration: Ehud Altman-Weizmann Eugene Demler - Harvard Vladimir Gritsev - Harvard

What do we observe interfering ideal condensates? a) Correlated phases (  = 0)   I (x)   N cos(Qx) Measure (interference part): d xTOF Andrews et. al b) Uncorrelated, but well defined phases   I(x)   0 Hanbury Brown- Twiss Effect

c) Initial number state. No phases? Work with original bosonic fields: The same answer as in the case b) with random but well defined phases! First theoretical explanation: I. Casten and J. Dalibard (1997): showed that the measurement induces random phases in a thought experiment. Experimental observation of interference between ~ 30 condensates in a strong 1D optical lattice: Hadzibabic et.al. (2004). Easy to check that at large N: The interference amplitude does not fluctuate!

Polar plots of the fringe amplitudes and phases for 200 images obtained for the interference of about 30 condensates. (a) Phase-uncorrelated condensates. (b) Phase correlated condensates. Insets: Axial density profiles averaged over the 200 images. Z. Hadzibabic et. al., Phys. Rev. Lett. 93, (2004).

1.Access to correlation functions. a)Scaling of  A Q 2  with L and : power-law exponents. Luttinger liquid physics in 1D, Kosterlitz-Thouless phase transition in 2D. b)Probability distribution W(A Q 2 ): all order correlation functions. c)Fermions: cusp singularities in  A Q 2 ( )  corresponding to k f. 2.Direct simulator (solver) for interacting problems. Quantum impurity in a 1D system of interacting fermions (an example). 3.Potential applications to many other systems. This talk: Imaging beam L What if the condensates are fluctuating?

What are the advantages compared to the conventional TOF imaging? 1.TOF relies on free atom expansion. Often not true in strongly correlated regimes. Interference method does not have this problem. 2.It is often preferable to have a direct access to the spatial correlations. TOF images give access either to the momentum distribution or the momentum correlation functions. 3.Free expansion in low dimensional systems occurs predominantly in the transverse directions. This renders bad signal to noise. In the interference method this is advantage: longitudinal correlations remain intact.

1.Algebraic correlations at zero temperature (Luttinger liquids). Exponential decay of correlations at finite temperature. 2.Fermionization of bosons, bosonization of fermions. (There is not much distinction between fermions and bosons in 1D). 3.1D systems are well understood. So they can be a good laboratory for testing various ideas. One dimensional systems.

Scaling with L: two limiting cases Ideal condensates: L x z Interference contrast does not depend on L. L x z Dephased condensates: Contrast scales as L -1/2.

The phase distribution of an elongated 2D Bose gas. (courtesy of Zoran Hadzibabic) Matter wave interferometry very low temperature: straight fringes which reveal a uniform phase in each plane “atom lasers” from time to time: dislocation which reveals the presence of a free vortex higher temperature: bended fringes S. StockS. Stock, Z. Hadzibabic, B. Battelier, M. Cheneau, and J. Dalibard: Phys. Rev. Lett. 95, (2005)Z. HadzibabicB. BattelierM. CheneauJ. Dalibard 0 

Formal derivation. L z Independent condensates: for identical homogeneous systems Long range order: short range correlations:

Intermediate case (quasi long-range order). L z 1D condensates (Luttinger liquids): Repulsive bosons with short range interactions: Finite temperature:

Angular Dependence. q is equivalent to the relative momentum of the two condensates (always present e.g. if there are dipolar oscillations). z x(z1)x(z1) x(z2)x(z2) (for the imaging beam orthogonal to the page, is the angle of the integration axis with respect to z.)

Angular (momentum) Dependence. has a cusp singularity for K<1, relevant for fermions.

Higher moments (need exactly two condensates). Relative width of the distribution (no dependence on L at large L): Wide Poissonian distribution in the fermionized regime (and at finite temperatures). Narrow distribution in the weakly interacting regime. Absence of amplitude fluctuations for true condensates.

Evolution of the distribution function.

Connection to the impurity in a Luttinger liquid problem. Boundary Sine-Gordon theory: Same integrals as in the expressions for P. Fendley, F. Lesage, H.~Saleur (1995).

Sounds complicated? Not really. 1.Do a series of experiments and determine the distribution function. T. Schumm, et. al., Nature Phys. 1, 57 (2005). Distribution of interference amplitudes (and phases) from two 1D condensates. 2.Evaluate the integral. 3.Read the result. Direct experimental simulation of a quantum impurity problem.

Spinless Fermions. Note that K+K -1  2, so and the distribution function is always Poissonian. However for K+K -1  3 there is a universal cusp at nonzero momentum as well as at 2k f : There is a similar cusp at There is a similar cusp at 2k f

Two dimensional condensates at finite temperature. Similar setup to 1D. S. Stock et.al : Phys. Rev. Lett. 95, (2005) LyLy LxLx Can also study size or angular (momentum) dependence.

Observing the Kosterlitz-Thouless transition Above KT transition LyLy LxLx Below KT transition Universal jump in  at T KT Expect a similar jump in the distribution function.

Conclusions. 1.Analysis of interference between independent condensates reveals a wealth of information about their internal structure. a)Scaling of interference amplitudes with the system size or the probing beam angle gives the correlation functions exponents. b)Probability distribution ( = full counting statistics in TOF) of amplitudes for two condensates contains information about higher order correlation functions. c)Interference of two Luttinger liquids directly realizes the statistical partition function of a one-dimensional quantum impurity problem. 2.Vast potential applications to many other systems, e.g.: a)Spin-charge separation in spin ½ 1D fermionic systems. b)Rotating condensates (instantaneous measurement of the correlation functions in the rotating frame). c)Correlation functions near continuous phase transitions. d)Systems away from equilibrium.