We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg--de Vries, equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg--de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.Comment: 25 pages, 24 figures, slightly revised and corrected versio

Atomic Bose-Einstein condensates confined to a dual-ring trap support Josephson vortices as topologically stable defects in the relative phase. We propose a test of the scaling laws for defect formation by quenching a Bose gas to degeneracy in this geometry. Stochastic Gross-Pitaevskii simulations reveal a -1/4 power-law scaling of defect number with quench time for fast quenches, consistent with the Kibble-Zurek mechanism. Slow quenches show stronger quench-time dependence that is explained by the stability properties of Josephson vortices, revealing the boundary of the Kibble-Zurek regime. Interference of the two atomic fields enables clear long-time measurement of stable defects and a direct test of the Kibble-Zurek mechanism in Bose-Einstein condensation.

We investigate the dynamics of turbulent flow in a two-dimensional trapped Bose-Einstein condensate by solving the Gross-Pitaevskii equation numerically. The development of the quantum turbulence is activated by the disruption of an initially embedded vortex quadrupole. By calculating the incompressible kinetic-energy spectrum of the superflow, we conclude that this quantum turbulent state is characterized by the Kolmogorov-Saffman scaling law in the wave-number space. Our study predicts the coexistence of two subinertial ranges responsible for the energy cascade and enstrophy cascade in this prototype of two-dimensional quantum turbulence.

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We propose a scheme for generating mesoscopic superpositions of two distinguishable pair coherent states of motion in a two-dimensional ion trap. In our scheme, the trapped ion is excited bichromatically by five laser beams along different directions in the X-Y plane of the ion trap. Four of these have the same frequency and can be derived from the same source, reducing the demands on the experimentalist. We show that if the initial vibrational state is given by a two-mode Fock state, as demonstrated in recent experiments, these highly correlated two-mode ''Schrödinger cat'' states are realized when the system reaches its steady state, which is indicated by the extinction of the fluorescence emitted by the ion. ͓S1050-2947͑96͒02611-X͔ PACS number͑s͒: 42.50. Vk, 42.50.Dv, 32.80.Pj Ever since Schrödinger suggested his famous cat experiment in 1935 ͓1͔, superpositions of macroscopically distinguishable quantum states ͑which are also known as Schrö-dinger cat states͒ have become a longstanding exemplar of the peculiarities which occur in the interpretation of quantum reality. To explore this subtlety and to gain insight into the fundamental issues of quantum theory, a number of schemes have been proposed for the realization of such states. In quantum optics, the Schrödinger cat states are usually described as superpositions of different coherent states ͓2͔, as coherent states are the closest quantum states to a classical description of the field of definite complex amplitude. Specifically, the archetype of a Schrödinger cat state is given by the superposition ͉⌿͘ cat ϭN͓͉␣͘ϩexp(i)͉Ϫ␣͘], where ͉␣͘ is a coherent state of the single-mode quantized field and N is a normalization coefficient. In particular, these states are referred to the even, odd, and Yurke-Stoler ͓3͔ coherent states when ϭ0,, and /2, respectively. They have been extensively studied and shown to exhibit nonclassical properties such as squeezing and sub-Poissonian statistics ͓2͔.In a recent paper Gerry and Grobe ͓4͔ have proposed a two-mode generalization of Schrödinger cat states which are defined as superpositions of two different pair coherent states ͑PCS͒. For two independent boson annihilation operators â and b , a pair coherent state ͉,q͘ is defined as an eigenstate of both the pair annihilation operator â b and the number difference operator Q ϭâwhere is a complex number and q is the ''charge'' which is a fixed integer. Without loss of generality, we may set qу0 and the PCS can be explicitly expanded as a superposition of the two-mode Fock states, i.e.,where N q ϭ͓͉͉ Ϫq I q (2͉͉)͔ Ϫ1/2 is the normalization constant (I q is the modified Bessel function of the first kind of order q). Pair coherent states are regarded as an important type of correlated two-mode state with prominent nonclassical properties such as sub-Poissonian statistics, strong intermode correlation in the number fluctuations, squeezing of quadrature variances, and violations of Cauchy-Schwarz inequalities ͓5͔.The correlated two-mode Schrödinger cat states ͉,q,͘ are defined as...

We propose a scheme for generating vibrational pair coherent states of the motion of an ion in a twodimensional trap. In our scheme, the trapped ion is excited bichromatically by three laser beams along different directions in the X-Y plane of the ion trap. We show that if the initial vibrational state is given by a two-mode Fock state, the final steady state, indicated by the extinction of the fluorescence emitted by the ion, is a pure state. The motional state of the ion in the equilibrium realizes that of the highly correlated pair coherent state. ͓S1050-2947͑96͒50408-7͔PACS number͑s͒: 42.50. Vk, 42.50.Dv, 32.80.Pj A variety of generalized coherent states has been constructed to describe different physical phenomena ͓1͔. Mathematically, the constructions of these sets of generalized coherent states are associated with particular Lie groups. The Glauber coherent states, defined as the right eigenstates of a single-mode boson annihilation operator, are associated with the Heisenberg group. Beyond these canonical coherent states, particular generalized coherent states associated with the noncompact SU͑1,1͒ group have been extensively studied ͓1-6͔. According to Barut and Girardello ͓3͔, these associated coherent states are defined as the eigenstates of the SU͑1,1͒ lowering operator, whereas in the definition given by Perelomov ͓4͔, they are generated by the SU͑1,1͒ analogue of the displacement operator. These two sets of coherent states are different, though they are closely related to the SU͑1,1͒ group.Regarding the two-mode boson realization of the SU͑1,1͒ group, a special set of coherent states of the Barut-Girardello type known as pair coherent states ͑PCS͒ can be formulated ͓1͔. If â (â † ) and b (b † ) denote two independent boson annihilation ͑creation͒ operators, then â b (â † b † ) stands for the pair annihilation ͑creation͒ operator for the two modes. The pair coherent states ͉,q͘ PCS are defined as eigenstates of both the pair annihilation operator â b and the number difference operator Q ϭâwhere is a complex number and q is the ''charge'' parameter, which is a fixed integer. Furthermore, the PCS can be expanded as a superposition of the two-mode Fock states,where N q ϭ͓͉͉ Ϫq I q (2͉͉)͔ Ϫ1/2 is the normalization constant (I q is the modified Bessel function of the first kind of order q). Pair coherent states were introduced by Horn and Silver ͓5͔ to describe the production of pions and applied to other problems in quantum field theory ͓1͔. In quantum optics, PCS are regarded as an important type of correlated twomode state, which possess prominent nonclassical properties such as sub-Poissonian statistics, correlation in the number fluctuations, squeezing, and violations of Cauchy-Schwarz inequalities ͓2͔.The experimental realization of such nonclassical states is of practical importance. Agarwal ͓2͔ suggested that the optical PCS can be generated via the competition of four-wave mixing and two-photon absorption in a nonlinear medium. This is the only scheme proposed to generate PCS known to us. In ...

The formation of an equilibrium quantum state from an uncorrelated thermal one through the dynamical crossing of a phase transition is a central question of non-equilibrium many-body physics. During such crossing, the system breaks its symmetry by establishing numerous uncorrelated regions separated by spontaneously-generated defects, whose emergence obeys a universal scaling law with the quench duration. Much less is known about the ensuing re-equilibrating or "coarse-graining" stage, which is governed by the evolution and interactions of such defects under system-specific and external constraints. In this work we perform a detailed numerical characterization of the entire non-equilibrium process, addressing subtle issues in condensate growth dynamics and demonstrating the quench-induced decoupling of number and coherence growth during the re-equilibration process. Our unique visualizations not only reproduce experimental measurements in the relevant regimes, but also provide valuable information in currently experimentally-inaccessible regimes.arXiv:1712.08074v1 [cond-mat.quant-gas]

A scheme for preparing vibrational SU͑2͒ states of motion in a two-dimensional ion trap is described. These anticorrelated two-mode states are formally equivalent to the output photon states of a lossless SU͑2͒ interferometer with number-state inputs. Nontrivial statistics such as the binomial distribution and the discrete ''arcsine'' distribution can be generated in the vibrational states of trapped ions, and detected by measuring the population inversion of the ion driven by a laser field along a specific direction. ͓S1050-2947͑96͒03308-2͔ PACS number͑s͒: 42.50. Dv, 42.50.Vk, 32.80.Pj Recent developments in the cooling and trapping ions ͓1-4͔ have opened a new research realm for both atomic physics and quantum optics. Theoretically, an ion confined in an electromagnetic trap is equivalent to a particle moving in a harmonic potential in which the center-of-mass ͑c.m.͒ motion of the ion is quantized as a harmonic oscillator. When the internal atomic states of the trapped ion are excited or deexcited by the classical laser driving field, the vibrational states of the c.m. motion are changed, as the atomic stimulated absorption or emission processes are always accompanied by momentum exchange of the driving field. For the most general case, the Schrödinger equation of this model is described by a set of linear differential equations that couple the probability amplitudes for the different vibrational states ͓5͔. However, if the vibrating amplitude of the ion is much smaller than the laser wavelength, i.e., the Lamb-Dicke limit is satisfied, and the driving field is tuned to one of the vibrational sidebands of the atomic transition, then this model can be simplified to a form similar to the Jaynes-Cummings model ͑JCM͒ ͓5-7͔ except that the quantized radiation field is replaced by the quantized c.m. motion of the ion. As the vibrational mode in the ion trap does not couple to the external optical modes, the dissipative effects inevitable from cavity damping in the optical regime can now be significantly suppressed. This prominent feature thus leads to the possibility of realizing some cavity QED experiments without using an optical cavity. There have been several schemes proposed recently following this approach to produce nonclassical vibrational states inside an ion trap. For example, Fock states can be prepared by methods involving quantum jumps ͓8͔, adiabatic passage ͓9͔, trapping states ͓10͔, or a sequence of Rabi pulses driving the ion ͓11͔. Coherent states of motion can be produced from the vacuum by a spatially uniform classical driving field or by a ''moving standing wave'' ͓11͔. Using bichromatic Raman excitation of the ion, one is able to produce squeezed states of motion inside the trap ͓11-13͔. In particular, quantum superpositions of two microscopically distinguishable states ͑the Schrödinger cat states͒ of the trapped ions can also be prepared ͓14-16͔.In this paper, we describe how to prepare and observe the anticorrelated two-mode SU͑2͒ vibrational states by using trapped ions. Consider a two-leve...