We review phase space techniques based on the Wigner representation that provide an approximate description of dilute ultra-cold Bose gases. In this approach the quantum field evolution can be represented using equations of motion of a similar form to the Gross-Pitaevskii equation but with stochastic modifications that include quantum effects in a controlled degree of approximation. These techniques provide a practical quantitative description of both equilibrium and dynamical properties of Bose gas systems. We develop versions of the formalism appropriate at zero temperature, where quantum fluctuations can be important, and at finite temperature where thermal fluctuations dominate. The numerical techniques necessary for implementing the formalism are discussed in detail, together with methods for extracting observables of interest. Numerous applications to a wide range of phenomena are presented.

Phase transitions are ubiquitous in nature, ranging from protein folding and denaturisation, to the superconductor-insulator quantum phase transition, to the decoupling of forces in the early universe. Remarkably, phase transitions can be arranged into universality classes, where systems having unrelated microscopic physics exhibit identical scaling behaviour near the critical point. Here we present an experimental and theoretical study of the Bose-Einstein condensation phase transition of an atomic gas, focusing on one prominent universal element of phase transition dynamics: the spontaneous formation of topological defects during a quench through the transition [1, 2, 3]. While the microscopic dynamics of defect formation in phase transitions are generally difficult to investigate, particularly for superfluid phase transitions [4, 5, 6, 7], Bose-Einstein condensates (BECs) offer unique experimental and theoretical opportunities for probing such details. Although spontaneously formed vortices in the condensation transition have been previously predicted to occur [8, 9], our results encompass the first experimental observations and statistical characterisation of spontaneous vortex formation in the condensation transition. Using microscopic theories [10, 11, 12, 13, 14, 15, 16, 17] that incorporate atomic interactions and quantum and thermal fluctuations of a finite-temperature Bose gas, we simulate condensation and observe vortex formation in close quantitative agreement with our experimental results. Our studies provide further understanding of the development of coherence in superfluids, and may allow for direct investigation of universal phase-transition dynamics.

We report experimental observations and numerical simulations of the formation, dynamics, and lifetimes of single and multiply charged quantized vortex dipoles in highly oblate dilute-gas Bose-Einstein condensates (BECs). We nucleate pairs of vortices of opposite charge (vortex dipoles) by forcing superfluid flow around a repulsive gaussian obstacle within the BEC. By controlling the flow velocity we determine the critical velocity for the nucleation of a single vortex dipole, with excellent agreement between experimental and numerical results. We present measurements of vortex dipole dynamics, finding that the vortex cores of opposite charge can exist for many seconds and that annihilation is inhibited in our highly oblate trap geometry. For sufficiently rapid flow velocities we find that clusters of like-charge vortices aggregate into long-lived dipolar flow structures.

Lead-free, potassium sodium niobate piezoelectric ceramics substituted with lithium (K0.5−x∕2,Na0.5−x∕2,Lix)NbO3 or lithium and tantalum (K0.5−x∕2,Na0.5−x∕2,Lix)(Nb1−y,Tay)O3 have been synthesized by traditional solid state sintering. The compositions chosen are among those recently reported to show high piezoelectric properties [Y. Saito, H. Takao, T. Tani, T. Nonoyama, K. Takatori, T. Homma, T. Nagaya, and M. Nakamura, Nature (London) 42, 84 (2004); Y. Guo, K. Kakimoto, and H. Ohsato, Appl. Phys. Lett. 85, 4121 (2004); Mater. Lett. 59, 241 (2005)]. We show that high densities and piezoelectric properties can be obtained for all compositions by pressureless sintering in air, without cold isostatic pressing, and without any sintering aid or special powder treatment. Resonance and converse piezoelectric (strain-field) measurements show a thickness coupling coefficient kt of 53% and converse piezoelectric coefficient d33 around 200pm∕V for the Li-substituted ceramics, and a kt of 52% and d33 over 300pm∕V for the Li- and Ta-modified samples. The unipolar strain-field hysteresis is small and comparable to that measured under similar conditions in hard Pb(Zr,Ti)O3. A peak of piezoelectric properties can be noted close to the morphotropic phase boundary. These ceramics look very promising as possible, practicable, lead-free replacements for lead zirconate titanate.

We provide a derivation of a more accurate version of the stochastic Gross-Pitaevskii equation, as introduced by Gardiner et al. [1]. The derivation does not rely on the concept of local energy and momentum conservation, and is based on a quasi-classical Wigner function representation of a "high temperature" master equation for a Bose gas, which includes only modes below an energy cutoff E R that are sufficiently highly occupied (the condensate band). The modes above this cutoff (the non-condensate band) are treated as being essentially thermalized. The interaction between these two bands, known as growth and scattering processes, provide noise and damping terms in the equation of motion for the condensate band, which we call the stochastic Gross-Pitaevskii equation. This approach is distinguished by the control of the approximations made in its derivation, and by the feasibility of its numerical implementation.

A parameter whose coupling to a quantum probe of n constituents includes all two-body interactions between the constituents can be measured with an uncertainty that scales as 1=n 3=2 , even when the constituents are initially unentangled. We devise a protocol that achieves the 1=n 3=2 scaling without generating any entanglement among the constituents, and we suggest that the protocol might be implemented in a two-component Bose-Einstein condensate. DOI: 10.1103/PhysRevLett.101.040403 PACS numbers: 03.65.Ta, 03.75.Mn, 03.75.Nt Quantum mechanics determines the fundamental limits on measurement precision. In the prototypal quantum metrology scheme, the value of a parameter is imprinted on a quantum probe through an interaction in which the parameter appears as a coupling constant [1]. The number n of constituents in the probe is often considered to be the most important resource for such schemes. We denote the parameter to be estimated by , and we write the interaction Hamiltonian as H @ H, where H is a dimensionless coupling Hamiltonian. The measurement precision is quantified by the units-corrected root-mean-square deviation of the estimate est from its true value [2]. The scaling of with n depends on the probe dynamics as expressed in H [3][4][5][6][7][8][9][10]. For an interaction that acts independently on the probe constituents, the optimal measurement precision scales as 1=n, a scaling often called the ''Heisenberg limit.'' In contrast, a nonlinear Hamiltonian that includes all possible k-body couplings gives an optimal sensitivity that scales as 1=n k . To achieve this requires that the initial probe state be entangled. If practical considerations preclude initializing the probe in an entangled state, sensitivity that scales as 1=n kÿ1=2 is possible using a probe that is initially in a product state [3,4,6,8,10]. Both of these scalings can be achieved with separable measurements.Practical interest in using nonlinear interactions for quantum metrology comes from the fact that, even with two-body couplings and initial product states, it is possible to obtain a 1=n 3=2 scaling for [3,4,6,[8][9][10]. In all such schemes proposed until now, (particle) entanglement is generated during the protocol that leads to better than 1=n scaling. We formulate here a protocol that generates no entanglement among the probe constituents yet still achieves the 1=n 3=2 scaling; in this protocol, it is clearly the dynamics alone that leads to improvement over the 1=n scaling. The dynamics has the same quantum-mechanical character as linear (k 1) metrology schemes but is n times faster. Even though this Letter is mainly about improving on the Heisenberg scaling, any experimental demonstration of a scaling better than 1=n 1=2 would be of considerable interest to the metrology community.A typical k 2 choice for a probe made of qubits is H J 2 z , where J z 1 2 P n j 1 Z j is the z component of the ''total angular momentum,'' with Z j being the Pauli Z operator for the jth qubit. We denote the eigenvectors of Z by j0i and j1i. An...

We develop a stochastic Gross-Pitaveskii theory suitable for the study of Bose-Einstein condensation in a rotating dilute Bose gas. The theory is used to model the dynamical and equilibrium properties of a rapidly rotating Bose gas quenched through the critical point for condensation, as in the experiment of Haljan et al. ͓Phys. Rev. Lett. 87, 210403 ͑2001͔͒. In contrast to stirring a vortex-free condensate, where topological constraints require that vortices enter from the edge of the condensate, we find that phase defects in the initial noncondensed cloud are trapped en masse in the emerging condensate. Bose-stimulated condensate growth proceeds into a disordered vortex configuration. At sufficiently low temperature the vortices then order into a regular Abrikosov lattice in thermal equilibrium with the rotating cloud. We calculate the effect of thermal fluctuations on vortex ordering in the final gas at different temperatures, and find that the BEC transition is accompanied by lattice melting associated with diminishing long-range correlations between vortices across the system.

We study the relaxation dynamics of an isolated zero temperature quasi-two-dimensional superfluid Bose-Einstein condensate (BEC) that is imprinted with a spatially random distribution of quantum vortices. Following a period of vortex annihilation, we find that the remaining vortices self-organise into two macroscopic coherent 'Onsager vortex' clusters that are stable indefinitely. We demonstrate that this occurs due to a novel physical mechanism -the evaporative heating of the vortices -that results in a negative temperature phase transition in the vortex degrees of freedom. At the end of our simulations the system is trapped in a non-thermal state. Our computational results provide a pathway to observing Onsager vortex states in a superfluid Bose gas.The question of how thermodynamics arises from unitary quantum evolution [1] has been debated since the early days of quantum mechanics. The closest laboratory realisation of an isolated quantum system of many particles is perhaps the ultracold quantum gas, and recent progress in the control and manipulation of these systems mean that the question is no longer simply an academic one [2]. A particular focus has been one-dimensional Bose gases as described by the Lieb-Liniger model [3], as the integrability of the model suggests that it may be prevented from attaining thermal equilibrium following a quench [4]. Indeed, two groundbreaking experiments on the dynamics of one-dimensional Bose gases [5,6] sparked a rush of further activity due to their seemingly contradictory results on whether the experiments returned to standard thermal equilibrium. These experiments have generated significant recent theoretical interest in the nonequilibrium dynamics and relaxation of idealised isolated quantum systems following a disturbance [2,[7][8][9].The quantum relaxation of higher dimensional isolated quantum systems is difficult to address computationally due to their exponential complexity. Recent experiments have quenched the interatomic interaction strength of 3D BECs in oblate and spherical harmonic traps and followed the subsequent dynamics [10,11], including apparent saturation of the momentum distribution [10]. However the relevance of quantum dynamics in their relaxation is not clear. In some situations the classical field approximation [12,13] can provide insight to quench dynamics. Indeed, recent theoretical work has considered the classical equilibration dynamics of 2D superfluids following a quench from the perspectives of turbulence and non-thermal fixed points [14][15][16][17][18][19].The thermalisation of isolated classical systems is generally understood in terms of ergodicity and chaotic dynamics [20]. Even though the equations of motion of a physical system are entirely reversible, if a system with a sufficiently large number of degrees of freedom begins in an 'atypical' state -such as a gas with all particles in one half of the container -it will quickly relax to a more 'typical' state consistent with thermal equilibrium as predicted by statistical mechanics...