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arXiv:1708.07712

On the hydrodynamics of Bose-condensed fluids subject to density-dependent gauge potentials

Yvan Buggy, Patrik Öhberg

Published Aug 25, 2017• 1 citation
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Abstract

When the energy functional of a Bose-condensed state of matter features an effective gauge potential which depends on the density $ρ$ of the condensate, the kinetic energy density of the matter field becomes nonlinear in $ρ$ and additional flow-dependent terms enter the wave equation for the phase of the condensate wavefunction. To begin with, we consider a certain class of density-dependent `single-component' gauge potentials, and later extend this class to encompass more general `multi-component' potentials. The nonlinear flow terms are cast into the general form of an inner-product between the velocity field of the fluid and the gauge potential. This is achieved by introducing a coupling matrix of dimensionless functions $γ_{ij}\left( ρ\right)$, which characterises the particular functional form of the gauge potential and regulates the strengths of the nonlinear terms accordingly. In the momentum-transport equation of the fluid, two non-trivial terms emerge due to the density-dependent vector potential. A body-force of dilation appears as a product of the gauge potential and the dilation rate of the fluid, while the fluid stress tensor features a flow-dependent pressure contribution given by the inner-product of the gauge potential and the current density of the fluid. This explicit dependence of the fluid pressure on the flow highlights the lack of Galilean invariance of the nonlinear fluid.

DOI
10.48550/arxiv.1708.07712

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When the energy functional of a Bose-condensed state of matter features an effective gauge potential which depends on the density ρρρ of the condensate, the kinetic energy density of the matter field becomes nonlinear in ρρρ and additional flow-dependent terms enter the wave equation for the phase of the condensate wavefunction. To begin with, we consider a certain class of density-dependent single-component' gauge potentials, and later extend this class to encompass more general multi-component' potentials. The nonlinear flow terms are cast into the general form of an inner-product between the velocity field of the fluid and the gauge potential. This is achieved by introducing a coupling matrix of dimensionless functions γij(ρ)γ_{ij}\left( ρ\right)γij​(ρ), which characterises the particular functional form of the gauge potential and regulates the strengths of the nonlinear terms accordingly. In the momentum-transport equation of the fluid, two non-trivial terms emerge due to the density-dependent vector potential. A body-force of dilation appears as a product of the gauge potential and the dilation rate of the fluid, while the fluid stress tensor features a flow-dependent pressure contribution given by the inner-product of the gauge potential and the current density of the fluid. This explicit dependence of the fluid pressure on the flow highlights the lack of Galilean invariance of the nonlinear fluid.
Abstract By constructing a hydrodynamic canonical formalism, we show that the occurrence of an arbitrary density-dependent gauge potential in the meanfield Hamiltonian of a Bose-condensed fluid invariably leads to nonlinear flow-dependent terms in the wave equation for the phase, where such terms arise due to the explicit dependence of the mechanical flow on the fluid density. In addition, we derive a canonical momentum transport equation for this class of nonlinear fluid and obtain an expression for the stress tensor. Further, we study the hydrodynamic equations in a particular nonlinear fluid, where the effective gauge potential results from the introduction of weak contact interactions in an ultracold dilute Bose gas of optically-addressed two-level atoms. In the Cauchy equation of mechanical momentum transport of the superfluid, two non-trivial terms emerge due to the density-dependent vector potential. A body-force of dilation appears as a product of the gauge potential and the dilation rate of the fluid, while the stress tensor features a canonical flow pressure term given by the inner-product of the gauge potential and the canonical current density. By numerical simulation, we illustrate an interesting effect of the nonlinear gauge potential on the groundstate wavefunction of a superfluid in the presence of a foreign impurity. We find that the groundstate adopts a non-trivial local phase, which is antisymmetric under reversal of the gauge potential. The phase profile leads to a canonical-flow or phase-flow dipole about the impurity, resulting in a skirting mechanical flow. As a result, the pressure becomes asymmetric about the object and the condensate undergoes a deformation. Graphical abstract
We investigate certain invariance properties of quantum fluids subject to a nonlinear gauge potential. In particular, we derive the covariant transformation laws for the nonlinear potentials under a space-time Galilean boost and consider U(1)U(1)U(1) gauge transformations. We find that the hydrodynamic canonical field equations are form-invariant in the case of external gauge functions $\ensuremath{\chi}(\mathbf{r},t)$, but not for nonlinear gauge functionals $\ensuremath{\chi}[\ensuremath{\rho}]$. Hence, nonlinear gauge potentials are nontrivial potentials which may not be ``gauged-away.'' Notably, for a superfluid in dimension d=1d=1d=1, attempting to do so generates the gauge pressure of the fluid in the Hamiltonian density. Furthermore, we investigate how the field equations transform under arbitrary Galilean transformations. We find that the immediate lack of Galilean covariance is restored under a suitably chosen transformation rule set for the potentials, which is identical in form to that of a Schr\"odinger particle coupled to external scalar and vector potentials.
The Hamiltonian of a moving atom in electromagnetic fields includes velocity- dependent terms. We show that the leading velocity dependence emerges systematically in the non-relativistic limit from a scheme firmly based on the relativistic invariance of the energy-momentum stress tensor of the coupled matter-fields system. We then extend the Hamiltonian to the many-body situation suitable for describing a Bose-Einstein condensate (BEC). From first principles, we use the equation of motion for the condensate wavefunction to obtain an extended version of the Gross-Pitaevskii (GP) equation and an equation for the internal states of the atoms. It is shown that laser fields modify the GP equation by inclusion of convective terms involving a Rontgen interaction plus a term coupling the centre of mass momentum to the Poynting vector. We also obtain the modified Maxwell equations for the electromagnetic fields coupled to the BEC involving the average velocity of the atoms.
Abstract Nonlinear feedback between the gauge field and the material field can yield novel quantum phenomena. Here, the interplay between a density-dependent artificial gauge field and Bose–Einstein condensates (BECs) trapped in an optical lattice is studied. The energy spectrum and superfluidity represented by energetic and dynamical stabilities of the system are systematically discussed. A density-dependent artificial gauge field with a back-action between the BECs dynamics and the gauge field induces an effective atomic interaction that depends on the quasi-momentum and density of the condensates, resulting in a symmetry-broken energy spectrum and exotic stability phase diagram, that is, the system is only stable in a certain range of atoms density and under a limited lattice strength. The density-dependent artificial gauge field changes the sequence for the emergence of energetic and dynamical instability and the regimes of the energetic and dynamical instabilities are significantly separated, offering an efficient way to examine the energetic and dynamical instabilities of superfluids separately. In particular, the density-dependent artificial gauge field, as a mechanism for transferring momentum to the fluid, results in dynamic instability of the condensates even in free space. Our results provide deep insights into the dynamical response of superfluid systems to gauge fields and have potential applications for the coherent control of exotic superfluid states.
The extended coupled-cluster method (ECCM) of quantum many-body theory, which has been studied and developed in earlier papers in this series, is now applied to condensed Bose systems. The formalism is seen to provide a concise and convenient description of quite general nonstatic states of such systems with arbitrary spatial inhomogeneity. The entire ECCM description is based on the equations of motion for the set of linked-cluster amplitudes which, we have shown, completely characterize an arbitrary quantum system with a Schr\"odinger dynamics. Since all such amplitudes obey the cluster property, and hence may be regarded as a set of quasilocal, many-body, classical order parameters, the formalism is in principle perfectly capable of describing phase transitions and states of topological excitation or deformation and broken symmetry. At the lowest (mean-field) level of truncation, the formalism degenerates to the well-known Gross-Pitaevskii description of the condensate wave function or one-body order parameter. The treatment is developed in a fully gauge-invariant fashion, and is thereby shown to provide a complete hydrodynamical description, valid in the zero-temperature limit. In particular, by studying the off-diagonal one- and two-body density matrices in terms of the basic ECCM amplitudes, we derive balance (or local conservation) equations for such local observables as the number density, momentum density, and energy density. These are shown to be obeyed not only by the exact untruncated formalism but also by most practical truncation schemes in the ECCM configuration space.
The study of systems formed from ultracold atomic gases has emerged to become one of the most active research fields within the condensed matter landscape. These highly controllable macroscopic systems amalgamate ideas from many sub disciplines of physics, including the study of low temperatures, quantum optics and quantum information theory as well as the seemingly disparate field of high energy physics. The central concept of this thesis is gauge theories as applied to systems of bosonic atoms, which at temperatures close to absolute zero form Bose-Einstein condensates. To simulate the mathematical structure of a gauge theory, the geometric (Berry) phase formalism is adopted. This is in turn accomplished by considering the adiabatic following of the eigenstates of the light-matter coupling for an ensemble of atoms forming a Bose-Einstein condensate. These concepts are then applied to show how one can generate a spin-orbit coupling in a one-dimensional condensate, which additionally features a random mass term that allows us to study the physics of Anderson localization in an intriguing “quasi” relativistic regime. One of the features of light induced gauge potentials is that they are static; in the sense that there is no feedback between the light-matter interaction and the matter field. In the second part of this thesis it is demonstrated how such a feedback mechanism can be induced by the appropriate modification of the light-matter interaction. The consequences this has for the condensate are then described at the mean-field level, including the expected experimental signatures of the resulting ‘interacting’ gauge theory, in terms of the expansion of the condensate and also the structure of the solitons of this nonlinear system. Finally, this nonlinear model is applied to a double well system, from which the associated Bose-Hubbard model is derived and analysed; and the nonlinear Josephson problem studied.