A theory determining the electric and magnetic properties of vortex states in Bose-Einstein condensates (BECs) is presented. The principal ingredient is the Lagrangian of the system, which we derive correct to the first order in the atomic centre-of-mass velocity. For the first time using centre-of-mass coordinates, a gauge transformation is performed and relevant relativistic corrections are included. The Lagrangian is symmetric in the electric and magnetic aspects of the problem and includes two key interaction terms, namely the Aharanov-Casher and the Röntgen interaction terms. The constitutive relations, which link the electromagnetic fields to the matter fields via their electric polarization and magnetization, follow from the Lagrangian as well as the corresponding Hamiltonian. These relations, together with a generalized Gross-Pitaevskii equation, determine the magnetic (electric) monopole charge distributions accompanying an order n vortex state when the constituent atoms are characterized by an electric dipole (magnetic dipole). Field distributions associated with electric dipole active (magnetic dipole active) BECs in a vortex state are evaluated for an infinite- and a finite-length cylindrical BEC. The predicted monopole charge distributions, both electric and magnetic, automatically satisfy the requirement of global charge neutrality and the derivations highlight the exact symmetry between the electric and magnetic properties. Order of magnitude estimates of the effects are given for an atomic gas BEC, superfluid helium and a spin-polarized hydrogen BEC.
A theory determining the electric and magnetic properties of vortex states in Bose-Einstein condensates (BECs) is presented. The principal ingredient is the Lagrangian of the system, which we derive correct to the first order in the atomic centre-of-mass velocity. For the first time using centre-of-mass coordinates, a gauge transformation is performed and relevant relativistic corrections are included. The Lagrangian is symmetric in the electric and magnetic aspects of the problem and includes two key interaction terms, namely the Aharanov-Casher and the Röntgen interaction terms. The constitutive relations, which link the electromagnetic fields to the matter fields via their electric polarization and magnetization, follow from the Lagrangian as well as the corresponding Hamiltonian. These relations, together with a generalized Gross-Pitaevskii equation, determine the magnetic (electric) monopole charge distributions accompanying an order n vortex state when the constituent atoms are characterized by an electric dipole (magnetic dipole). Field distributions associated with electric dipole active (magnetic dipole active) BECs in a vortex state are evaluated for an infinite- and a finite-length cylindrical BEC. The predicted monopole charge distributions, both electric and magnetic, automatically satisfy the requirement of global charge neutrality and the derivations highlight the exact symmetry between the electric and magnetic properties. Order of magnitude estimates of the effects are given for an atomic gas BEC, superfluid helium and a spin-polarized hydrogen BEC.
A theory determining the electric and magnetic properties of vortex states in Bose–Einstein condensates (BECs) is presented. The principal ingredient is the Lagrangian of the system, which we derive correct to the first order in the atomic centre-of-mass velocity. For the first time using centre-of-mass coordinates, a gauge transformation is performed and relevant relativistic corrections are included. The Lagrangian is symmetric in the electric and magnetic aspects of the problem and includes two key interaction terms, namely the Aharanov–Casher and the Röntgen interaction terms. The constitutive relations, which link the electromagnetic fields to the matter fields via their electric polarization and magnetization, follow from the Lagrangian as well as the corresponding Hamiltonian. These relations, together with a generalized Gross–Pitaevskii equation, determine the magnetic (electric) monopole charge distributions accompanying an order n vortex state when the constituent atoms are characterized by an electric dipole (magnetic dipole). Field distributions associated with electric dipole active (magnetic dipole active) BECs in a vortex state are evaluated for an infinite- and a finite-length cylindrical BEC. The predicted monopole charge distributions, both electric and magnetic, automatically satisfy the requirement of global charge neutrality and the derivations highlight the exact symmetry between the electric and magnetic properties. Order of magnitude estimates of the effects are given for an atomic gas BEC, superfluid helium and a spin-polarized hydrogen BEC.
We show that by virtue of the R\"ontgen interaction, a vortex of order n in a Bose-Einstein condensate (BEC), in which the constituent atoms are characterized by an electric dipole, gives rise to a magnetic monopole distribution, which correctly preserves magnetic charge neutrality. If, on the other hand, the BEC atoms are characterized by a magnetic dipole, we show that the Aharonov-Casher interaction term leads to the prediction of an electric charge distribution associated with the vortex state, which too, exhibits electric charge neutrality. Our theory displays an exact symmetry between the electric and magnetic aspects of the problem and can thus be presented in a unified fashion. Illustrations are given for finite-size atomic gas BECs, superfluid helium, and spin-polarized hydrogen BECs in the n=1 vortex state.
Quantum mechanically, static or applied gauge fields manifest themselves by directly coupling to a system's phase. Artificial gauge field techniques manipulate the phase of a Bose-Einstein condensate's order parameter, yielding effects present in condensed matter systems subject to magnetic fields, electric fields, or spin-orbit coupling. In this chapter, we explore the impact of these artificial gauge fields on Bose- Einstein condensates and explain the consequences. Several analogues to traditional condensed matter systems are enabled by these techniques, including the Hall effect, superfluid vortex nucleation in systems with artificial magnetic fields, and magnetic-like phase transitions in systems with spin-orbit coupling. Introduction Since the first experiments with Bose-Einstein condensates (BECs), researchers have exploited the universality of physical laws to draw analogies between the behavior of ultracold quantum gases and systems ranging from material to astrophysical. These analogous situations are deliberately engineered, giving cold-atom “quantum simulators” [1] that exhibit behavior either observed or predicted in analogous quantum systems. Superfluidity was one of the first many-body quantum phenomena demonstrated in BECs. For example, several experiments showed superfluid behavior by rapidly rotating the condensates and observing the formation of vortices [2, 3, 4, 5] – quantized point-defects in the BEC's order parameter. In the rotating frame of these experiments, this nucleation of vortices could be understood as the consequence of an artificial magnetic field [6] that arises via the formal equivalence of the rotating-frame Coriolis force and the Lorentz force, which we describe in Section 15.4.1. Together with other measurements, these results proved to be in excellent agreement with the predictions of the Gross-Pitaevskii equation (GPE) [7], a decades-old idealization of superfluid helium. Buoyed by the prospect for studying magnetic field–driven strongly correlated systems using these nearly ideal quantum gases, proposals for alternate techniques to simulate gauge fields were developed. These methods primarily employ an internal state–selective atom–light coupling, where the photon momentum provides the forces necessary to mimic a Lorentz force [8, 9, 10, 11, 12, 13, 14, 15, 16]. The first demonstration of this type of technique used a spatially dependent Ramantransition scheme to effect a magnetic field [17], and experiments demonstrating artificial electric fields [18] and spin-orbit coupling [19, 20, 21, 22, 23, 24] soon followed.
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.
The aim of this paper is to derive the equations of motion of quantum vortices in a weakly non-ideal Bose–Einstein condensate (BEC) at finite temperatures and in the presence of macroscopic superfluid and normal velocities. The motivation for this work is that the dynamics of quantum vortices in superfluid helium has been studied at the phenomenological level, and a number of issues require clarification, which can be addressed by studying the corresponding problem in BEC. In BEC, quantum vortices are located at points where the macroscopic wave function (MWF) ψ(r,T), which describes the BEC dynamics, vanishes ψ(r,T)=0. The intersection of the surfaces Reψ(r,T)=0 and Imψ(r,T)=0 forms a one-dimensional line, which is just the quantum vortex line. During the evolution of the MWF, the position of the zero surfaces changes, that is, the vortex lines move. Conversely, the motion of these vortex filaments determines the evolution of the MWF. Thus, there is a mutual correspondence, and the quantity ψ(r,T) can be considered as a functional of the entire vortex configuration, i.e., {s(ξ,t)}. This viewpoint allows us to derive the equations of motion for vortices from the Gross–Pitaevskii equation (GPE), which describes the dynamics of MWF. The key finding is the confirmation that the equation derived from the fundamental quantum description (the GPE) for a BEC has the same mathematical structure as the equation derived from macroscopic, phenomenological reasoning for superfluid helium.
It is shown that the condensate of a degenerated Bose gas consisting of neutral atoms possesses electrical properties which differ from a trivial polarization of the atoms in the electric field.A notion of an isotropic quadrupole moment (IQM) of a neutral atom is introduced.A distribution of IQM reflects a specific spatial ordering in the condensate and produces a distribution of the electric potential.Small vibrations of the Bose gas are considered and a correction to the Bogoliubov spectrum of elementary excitations in the degenerated Bose gas is obtained.An additional term in the Gross-Pitaevskii equation which is responsible for such a correction is found and a new type of the nonlinear Schr ödinger equation (NSE) is constructed.Since the Bose condensate is akin to the superfluid component in He II, a manifestation of its electrical activity could have a relation to the electrical activity of the superfluid liquid observed experimentally.