Vortex dynamics in superfluids governed by an interacting gauge theory

We study the dynamics of a vortex in a quasi two-dimensional Bose gas consisting of light matter coupled atoms forming two-component pseudo spins. The gas is subject to a density dependent gauge potential, hence governed by an interacting gauge theory, which stems from a collisionally induced detuning between the incident laser frequency and the atomic energy levels. This provides a back-action between the synthetic gauge potential and the matter field. A Lagrangian approach is used to derive an expression for the force acting on a vortex in such a gas. We discuss the similarities between this force and the one predicted by Iordanskii, Lifshitz and Pitaevskii when scattering between a superfluid vortex and the thermal component is taken into account.


Introduction
Transport of electrons is at the heart of our understanding and everyday usage of electronic devices. A charge neutral matter wave version of the electron dynamics would have to be able to mimic the electronic properties which would include constructing devices such as diodes and switches for atoms. The goal there is to exploit the tunability of atomic Bose-Einstein condensates (BECs) and explore the prospect for new quantum devices for measurements and sensing. In this respect it is often favourable to have a directional dependence built into the system, but this is not always straightforward to achieve with BECs. Recently it has been shown that an optically addressed BEC governed by an interacting gauge theory [1] can have a chiral nature where the strength and even the sign of the nonlinearity depends on the direction of the superfluid flow. For such a superfluid to be useful for atomtronics applications we need to first understand the transport and collective excitations in the quantum gas. One aspect of this is the understanding of the properties of quantised vortices.
The interest in vortex states, and more generally in the rotational properties of fluids, dates back to the early days of hydrodynamics and is historically related to the phenomenon of turbulence in classical fluids. To understand the onset of chaotic dynamics and turbulence has turned out to be a formidable task, and a deep and complete understanding is still far from achieved. The realisation of BEC of 4 He [2,3], and the consequent discovery of superfluidity opened up a new perspective to this aim. As in their classical counterpart, turbulence also shows up in these quantum fluids, with vortices playing a central role in the transition to chaotic motion [4][5][6][7][8].
The advantage of investigating turbulence phenomena in superfluids is due to the constraints that quantum mechanics imposes on the values of the physical quantities that characterise the system, which simplifies to some extent the scenario with respect to its classical counterpart. For example, in order for the condensate wave function to be single valued, the circulation of the velocity field around any closed path, has to be quantised in multiples of ÿ/m, with m the mass of the atomic species composing the condensate itself. This property leads to the concept of quantised vortices, around which the circulation (and the angular momentum as a consequence) is quantised [5,9]. Apart from the discreteness of the values of the angular momentum, a vortex in a superfluid has the remarkable property of being a particle-like stable object that does not easily decay, in contrast to viscous diffusion of vorticity, as in the case of classical fluids. Because of these considerations, superfluids have become

Atoms in artificial gauge fields
We consider a uniform BEC [29] consisting of two-level atoms, which are confined such that their dynamics in the transverse direction is frozen. The system can be considered as a quasi-two-dimensional cloud of atoms, free to move in the 2D plane but the collisions are treated in three dimensions. It has recently been shown in [1] that a collisionally induced detuning between the incident laser and the two atomic levels can give rise to a density dependent synthetic gauge potential. We refer the reader to appendix A for a detailed derivation of the equation of motion. The resulting mean field equation which describes the dynamics of the condensate is given by in which an unconventional nonlinearity appears, proportional to the current The validity of equation (1) relies on the adiabatic approximation where the atoms are assumed to be prepared in one of the dressed states of the light-matter coupled system, and on the assumption that the Rabi frequency is the dominating energy scale. The resulting gauge potential and scalar potential are given by Here, and through the rest of the paper,±refers to the two different dressed states which we can choose to use (see appendix A). The ( )  f = - A 0 2 is the single particle component of the vector potential with f being the phase of the laser. The vector field is the first order nonlinear density dependent contribution where g 11 and g 22 are the corresponding meanfield coupling constants for collisions between atoms in state | ñ 1 and | ñ 2 respectively, and Ω is the Rabi frequency. See figure 1 for a description of the envisaged setup. The vector and scalar potentials in equations (3) and (4) are synthetic and stem from a geometric potential [25,26]. As such there is no real charge associated with the vector potential, and the choice of gauge is explicitly determined by the parameters of the laser which couples the two atomic levels.

Vortex Lagrangian
We consider a cloud which is strongly confined by a potential in the z-direction, such that the dynamics is frozen in this direction and the atoms are nearly free to move in the x-y plane. We assume an incident laser beam which is propagating in the plane of the condensate and with a uniform intensity and phase ( ) · f = r k r. The Rabi frequency is consequently uniform throughout the condensate. This choice of light beam results in zero magnetic field, but the nonlinear part of the gauge potential will, as we show next, influence the dynamics. The zeroth order gauge potential ( ) A 0 can be gauged away by applying the transformation With ( )( ) = -W g g a k 8 1 11 22 . In order to study the dynamics of the vortex in the cloud, it is convenient to consider the cloud having an effective thickness Z in the z-direction. The original condensate wave function can then be rescaled as ( ) y Z r , where ( ) y r is now two-dimensional, and normalised in such a way that , with N the number of atoms in the condensate. We write the wave function in terms of the particle density ρ and the phase S as y r = e S i , so that the Lagrangian density takes the form where we neglected the second order term A 2 /2m and the physical velocity u in the condensate is related to the phase of the wave function as Given equation (7), we seek for an effective Lagrangian which describes the dynamics of a vortex state. We look in particular for the forces which result from the vortex interacting with the synthetic gauge field. In order to properly take into account the vortex velocity field we need to choose the phase S, in such a way that the velocity field characteristic of the vortex state, and | | k k = = h m the quantum of circulation. From an experimental point of view this is equivalent to preparing a vortex in the atomic cloud in absence of the gauge potentials, and then look at its dynamics once the external laser field is switched on. We next consider the vortex moving relative to the bulk condensate, where we indicate by r 0 the position of its core and by = t v r d d 0 its velocity. We assume this velocity is much smaller than the speed of sound in the condensate, so that the density and phase profiles charactering the vortex, adiabatically follow the core during its motion without undergoing any distortion. We make the ansatz ( ) r r = r r 0 0 for the density of the condensate, with ( ) r r 0 the density profile of a vortex state, which is assumed to carry a single quantum of circulation. We write the phase of the condensate as = + S S S 0 v with S 0 the phase of a steady vortex, so that  = S u 0 0 , and S v the shift due to the coreʼs motion. Exploiting the continuity equation we get the equation for S v . To do so we substitute equation (8) into (10), obtaining . The vortex is assumed to be moving in the condensate with constant velocity. We therefore expect that S v gives rise to an uniform field, so that D = S 0 v , and equation (11) reduces to 1 . It is useful now to distinguish between the in-core (in which r  ¹ 0, r » 0) and out-core (in which r  » 0, r ¹ 0) regions of the vortex [30]. With this distinction in mind, equation (12) can be solved, giving The result in equation (13) follows straightforwardly from equation (12). In equation (14) we have chosen the boundary conditions such that the mass current is zero at infinity. In order to take advantage of these results, we need to identify in equation (7) the terms referring to the different regions of the vortex. To do so, we split terms of the type ( ) , with r B the bulk density of the condensate. The first term is different from zero within a distance from the core of the order of the healing length of the condensate, defined as  x r = m g 2 B , and so refers to the in-core region, while the second one is relative to the out-core region. Substituting the expression for S v in the different terms, and noticing that Integrating the expression in equation (15) we get the effective Lagrangian describing the motion of the vortex core, given by where we defined the effective vortex mass M v and the effective vector and scalar potentials A v and U v as The U 0 accounts for the remaining terms that do not give any contribution to the vortex dynamics, since their values do not depend on the position of the core r 0 . The vortex mass M v takes a negative value, and accounts for the missing mass in the condensate due to the presence of the vortex. It diverges logarithmically with the size of the system , and takes the form is the integral in the dimensionless radial length can take values much larger than one, and increases with the size of the system. For large clouds then, the mass of the vortex can attain a value significantly larger than the core mass.

Vortex motion
The Lagrangian in equation (15) describes the core as a point particle of (negative) mass M v and positive unit charge, which feels the action of an effective vector potential A v , and a scalar potential U v . We therefore expect there to be two forces at play: a Lorentz-type force where  = p kis the momentum carried by the laser beam. The expression in equation (22) has the same form as the Iordanskii transverse force acting on a vortex in a superfluid due to the interaction between the velocity field and a phonon excitation with momentum p (see appendix B) and effective particle density ( ) r = n p B 3D , with r r = Z B 3D B the number of particle per unit volume, the perturbative parameter, playing the role of the particle distribution at momentum p, where is the three-dimensional meanfield coupling constant. There is a significant flexibility in order to emulate this type of transverse forces, because the scattering length difference a a 11 22 the Rabi frequency Ω and to some extent the density of the cloud, can be relatively easily changed in an experiment. The magnitude of the wave vector of the laser beam, in analogue to the wave vector of the phonon excitation, is limited by the energy splitting between the two internal states of the atoms constituting the condensate.
With the forces given in equations (21) and (22), the equation of motion for the vortex core takes the form For an initially stationary vortex at = r 0, the coordinates of the vortex core  r 0 andr 0 , parallel and orthogonal to the wave vector k of the laser beam respectively, then becomes (24) and (25) describe a periodic motion for the vortex core, which undergoes a series of curved trajectories of maximum height d and separated by pd 2 , as shown in figure 2.
In terms of the healing length, the characteristic length d of the motion, takes the value For typical values of these parameters in atomic clouds with x m = 0.1 m, -x = L 10 100, | | l p = = k 2 600 nm, and considering a value for the perturbative parameter ~0.01, the ratio between d and the healing length ξ can take values which are significantly larger than one. The characteristic motion of the vortex should therefore be detectable in experiments.
In order to validate the analytical results, we solved numerically the Gross-Pitaevskii equation (A14), with A and W given in equations (5) and (6). We considered a cloud in a square geometry, with periodic boundary conditions in x-direction and confined by a hard-wall potential in the y-direction, giving a homogeneous density which approximates the infinite homogeneous cloud assumed in the analytical description developed above. We determined the initial state of the system by solving equation (A14) in the imaginary time without the current non-linearity, which leads to the situation represented in figure 3, where two vortices with opposite flow circulation appear in order to match the periodic boundary conditions. Starting from this configuration we compared the numerical simulation with the motion predicted by equations (24) and (25).
In , the Rabi frequency W = 60 kHz, the wave length for the incident laser beam to be l = 628 nm, and the density of the cloud´-3 10 cm 14 3 where an effective thickness of the cloud was assumed to be m 0.2 m. Figure 4 shows the numerical simulation for the motion of the vortex core, compared with the analytical solution. The parameters involved in equations (24) and (25), i.e. the bulk density r B of the cloud and the effective vortex mass M v , have been estimated directly from the initial state of the system. The latter in particular takes a value that is in agreement with the one given by equation (17), obtained using the variational ansatz r r = + x x 2 B 2 [31,32] for the density profile of the vortex (with x = x r the dimensionless coordinate), and the ratio x » L 32, where L is the size of the cloud. Accordingly, the value ( ) z x » L 13 has been obtained for the parameter defined in section 3, which defines the effective mass of the vortex in terms of the core mass m core .

Conclusions
In this paper we have calculated the forces acting on a vortex which is subject to a density dependent gauge potential. We identified a standard Magnus force, but also a novel force which stems from the current nonlinearity and gives rise to a transversal force component which is of the same form as the Iordanskii force.
The numerical solution reproduces qualitatively the motion predicted by the variational calculation, showing trajectories for the vortex core similar to the one represented in figure 2. We do not however expect a perfect match between the two approaches for a number of reasons. The main approximation in the analytical model is the assumption that the density and phase profiles around the vortex core remain symmetric, and that   The blue curve corresponds to the second vortex in figure 3, but plotted as a function of -r 0 in order to compare the paths. We see from the numerical curves that the motion of the vortex cores are in opposite direction forr 0 as suggested by equations (24) and (25). The inset shows the path of the vortex core in the limit of w p  t 2 where the vortex mass has been rescaled as  M M 0.3 v v (blue line), in order to capture effects such as an asymmetric vortex core and phonon emission which will affect the overall dynamics as seen in the numerical solution indicated by the red filled circles. its motion is adiabatic in the sense that there are no phonons or density waves induced in the condensate. In reality the vortex core will be distorted by the presence of the current nonlinearity due to an effective local scattering length being different on either side of the vortex core (see figure 5). The stronger the nonlinearity is the more distorted the vortex becomes. This will change the value of the effective vortex mass, which consequently affects both the time scale and the length scale in the dynamics.
The ansatz used for the analytical results is the simplest possible one which is still able to capture the main features of the dynamics, given by the direction of the forces acting on the vortex and the overall trend of the motion which is the cyclic behaviour. A better match between the analytical result and the numerical simulation can in principle be attained in the limit of very weak current nonlinearity. However, in such a limit, the time scale for the dynamics in question becomes longer, and the amplitude of the cyclic motion of the vortex core decreases as seen from equations (24) and (25). Alternatively one can look at the dynamics for short time scales ( ) w p  t 2 and rescale the vortex mass such that the analytical path coincides with the numerical vortex path. This would effectively mean replacing the bare vortex mass from the variational ansatz, with a rescaled mass which captures the effects of phonon creation and of the asymmetry of the vortex core. The inset in figure 4 shows one example of this where the vortex mass is rescaled such that The dynamics presented here indicate that even if the synthetic magnetic field is zero a vortex will still experience a force due to the Galilean invariance not being fulfilled. We chose a particular laser configuration where the laser beam was incident in the 2D plane of the cloud. Other configurations are possible, in particular a symmetric situation where the synthetic gauge potential corresponds to a uniform magnetic field. Such a scenario, with a sufficiently strong synthetic magnetic field, will give rise to a vortex lattice. This lattice will be influenced by the current nonlinearity, and is likely to deviate from the standard triangular Abrikosov lattice seen in standard superfluids. It is still an open question what the resulting vortex lattice will be in the presence of current nonlinearities, and what role it plays if the quantum Hall regime is reached. The first term in equation (A1) is the sum of the non-interacting Hamiltonians, in which the identity operators act on the subspace excluding the particles ℓ ¼ n, , . The coupling between the two internal levels | ñ 1 and | ñ 2 is given by where Ω is the Rabi frequency characterising the strength of the light-matter coupling, ( ) f r is the laser phase at the atomicʼs position r, and we set the laser detuning from the atomic resonance to zero for simplicity. The second term in equation (A1) represents the pairwise interaction between the particles that, in the above assumptions, has the diagonal form [ ] ( ) ℓ ℓ n d = g g g g r r diag , , , n n , 11 12 12 22 , with the coupling constants given by and where a ij are the scattering lengths relative to the three different collision channels.
We consider the limit of weakly interacting atoms, r  a 1 i ij , and we make a variational ansatz by writing the many-body wavefunction ( ) of the system as the symmetrised product of the single particle spinor wave function ( ) f r , satisfying the normalisation condition † We introduce then the Lagrangian of the system Upon substitution of the expression given above for the many-body wave function into equation (A3), we obtain the Lagrangian in terms of the condensate wave function ( ) ( ) where we defined the single particle mean field Hamiltonian H MF as: in which  is the 2×2 identity operator acting in the space of the atomic internal degrees of freedom. In equation (A5) U aa describes the mean field collisional effects, and is given by and where | | r y = i i 2 is the density of atoms in level | ñ i , = i 1, 2. Since we are working in the weakly interacting limit, the coupling energy W between the internal states is typically much larger than the collisional mean field shifts. This allows us to treat the meanfield interaction as a small perturbation to the atom-field coupling. To the order ( ) , its eigenstates are given by are the so called dressed states. The interacting dressed states in equation (A9), represent a basis for the internal Hilbert space of the atoms, so that the condensate wave function | ( ) can be written as | ( ) In order to capture the dynamics of the±component of the condensate we use the adiabatic assumption, according to which ( ) y º  t r, 0(which is valid as long as the detuning induced by the collisional effect is small compared to W), and we consider the projection of the mean field Lagrangian in equation (A4) onto the subspace spanned by the corresponding (| ) c ñ  dressed state. We obtain then the mean field Lagrangian for the relevant condensate component of the form  are respectively the scalar and vector potential arising from the adiabatic projection of the full system onto one of the subspaces spanned by the dressed states.
Substituting equation (A9) in the expression given above for the potentials, together with ( ) n r = +  g g 2 1 11 12 , ( ) n r = +  g g 2 2 22 12 , obtained from equations (A7) and (A8) in the adiabatic assumption ( ) y º  0 , the synthetic potentials are given, to the leading order, by Here ( )  f = - A 0 2 is the single particle component of the vector potential, and the vector field in which a current nonlinearity appears