Scaling and full counting statistics of interference between

Scaling and full counting statistics of interference between independent fluctuating condensates Anatoli Polkovnikov, Boston University Collaboration: Ehud Altman Eugene Demler Vladimir Gritsev - Weizmann Harvard

Interference between two condensates. TOF d x Free expansion: Andrews et. al. 1997

What do we observe? TOF x a) Correlated phases ( = 0) b) Uncorrelated, but well defined phases int(x) =0 Hanbury Brown-Twiss Effect c) Initial number state. No phases? Work with original bosonic fields:

Interference amplitude squared. Observable! Easy to check that at large N: The interference amplitude does not fluctuate! 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 1 D optical lattice: Hadzibabic et. al. (2004).

Z. Hadzibabic et. al. , Phys. Rev. Lett. 93, 180401 (2004). 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.

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

z 1 z 2 z AQ Identical homogeneous condensates: x Interference amplitude contains information about fluctuations within each condensate.

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

Formal derivation: Ideal condensate: L Thermal gas: L

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

x(z 2) x(z 1) Angular Dependence. z (for the imaging beam orthogonal to the page, is the angle of the integration axis with respect to z. ) q is equivalent to the relative momentum of the two condensates (always present e. g. if there are dipolar oscillations).

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

Two-dimensional condensates at finite temperature CCD z camera z Time of flight y x x (picture by Z. Hadzibabic) imaging laser Elongated condensates: Lx>>Ly.

The phase distribution of an elongated 2 D Bose gas. (courtesy of Zoran Hadzibabic) Matter wave interferometry 0 p very low temperature: straight fringes which reveal a uniform phase in each plane “atom lasers” higher temperature: bended fringes from time to time: dislocation which reveals the presence of a free vortex S. Stock, Z. Hadzibabic, B. Battelier, M. Cheneau, and J. Dalibard: Phys. Rev. Lett. 95, 190403 (2005)

Observing the Kosterlitz-Thouless transition Above KT transition Below KT transition Lx Ly Ly Lx Universal jump of at TKT Always algebraic scaling, easy to detect.

Zoran Hadzibabic, Peter Kruger, Marc Cheneau, Baptiste Battelier, Sabine Stock, and Jean Dalibard (2006). Interference contrast: x z Contrast after integration 0. 4 integration over x axis z 0. 2 integration over x axis X middle T low T high T z 0 0 10 20 30 integration distance X (pixels)

Exponent Z. Hadzibabic et. al. “universal jump in the superfluid density” 0. 5 c. f. Bishop and Reppy 0. 4 0. 3 high T 0 low T 0. 1 0. 2 0. 3 central contrast 1. 0 0 1. 1 Vortex proliferation Fraction of images showing at least one dislocation: 30% 20% 10% 0 low T high T 0 0. 1 0. 2 0. 3 0. 4 central contrast T (K) 1. 2

Higher Moments. is an observable quantum operator Identical condensates. Mean: Similarly higher moments Probe of the higher order correlation functions. Distribution function (= full counting statistics): Non-interacting non-condensed regime (Wick’s theorem): Nontrivial statistics if the Wick’s theorem is not fulfilled!

1 D condensates at zero temperature: Low energy action: Then Similarly Easy to generalize to all orders.

Changing open boundary conditions to periodic find These integrals can be evaluated using Jack polynomials (Fendley, Lesage, Saleur, J. Stat. Phys. 79: 799 (1995)) Explicit expressions are cumbersome (slowly converging series of products).

Two simple limits: Strongly interacting Tonks-Girardeau regime z 1 (also in thermal case) Weakly interacting BEC like regime. z 2 z AQ x

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

Experimental simulation of the quantum impurity problem 1. Do a series of experiments and determine the distribution function. Distribution of interference phases (and amplitudes) from two 1 D condensates. T. Schumm, et. al. , Nature Phys. 1, 57 (2005). 2. Evaluate the integral. 3. Read the result.

can be found using Bethe ansatz methods for half integer K. In principle we can find W: Difficulties: need to do analytic continuation. The problem becomes increasingly harder as K increases. Use a different approach based on spectral determinant: Dorey, Tateo, J. Phys. A. Math. Gen. 32: L 419 (1999); Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102: 567 (2001)

Evolution of the distribution function.

Universal asymmetric distribution at large K ( -1)/

Further extensions: is the Baxter Q-operator, related to the transfer matrix of conformal field theories with negative charge: 2 D quantum gravity, non-intersecting loops on 2 D lattice Yang-Lee singularity

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 2 kf: There is a similar cusp at 2 kf Higher dimensions: nesting of Fermi surfaces, CDW, … Not a low energy probe!

Fermions in optical lattices. Possible efficient probes of superconductivity (in particular, d-wave vs. s-wave). Not yet, but coming! Rapidly rotating two dimensional condensates Time of flight experiments with rotating condensates correspond to density measurements Interference experiments measure single particle correlation functions in the rotating frame

Conclusions. 1. Analysis of interference between independent condensates reveals a wealth of information about their internal structure. a) Scaling of interference amplitudes with L or : correlation function exponents. Working example: detecting KT phase transition. b) Probability distribution of amplitudes (= full counting statistics of atoms): information about higher order correlation functions. c) Interference of two Luttinger liquids: partition function of 1 D quantum impurity problem (also related to variety of other problems like 2 D quantum gravity). 2. Vast potential applications to many other systems, e. g. : a) Fermionic systems: superconductivity, CDW orders, etc. . 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.

Universal adiabatic dynamics across a quantum critical point gap Consider slow tuning of a system through a critical point. t, 0 tuning parameter Gap vanishes at the transition. No true adiabatic limit! How does the number of excitations scale with ? This question is valid for isolated systems with stable excitations: conserved quantities, topological excitations, integrable models.

Use a general many-body perturbation theory. Expand the wave-function in many-body basis. Substitute into Schrödinger equation.

Uniform system: can characterize excitations by momentum: Use scaling relations: Find:

Caveats: 1. Need to check convergence of integrals (no cutoff dependence) Scaling fails in high dimensions. 2. Implicit assumption in derivation: small density of excitations does not change much the matrix element to create other excitations. 3. The probabilities of isolated excitations: should be smaller than one. Otherwise need to solve Landau. Zeener problem. The scaling argument gives that they are of the order of one. Thus the scaling is not affected.

Simple derivation of scaling (similar to Kibble-Zurek mechanism): Breakdown of adiabaticity: From t and we get In a non-uniform system we find in a similar manner:

Example: transverse field Ising model. There is a phase transition at g=1. This problem can be exactly solved using Jordan-Wigner transformation:

Spectrum: Critical exponents: z= =1 d /(z +1)=1/2. Correct result (J. Dziarmaga 2005): Other possible applications: quantum phase transitions in cold atoms, adiabatic quantum computations, etc.