# Parametric instability of a vortex ring oscillations in a -periodic Bose-Einstein condensate and the recurrence to starting state

###### Abstract

The dynamics of deformations of a quantum vortex ring in a Bose-Einstein condensate with periodic equilibrium density has been considered within the local induction approximation. Analysis of small deviations has revealed parametric instabilities of the normal modes with azimuthal numbers at the energy integral near values , where is the resonance order. Numerical simulations have shown that already at a rapid growth of unstable modes with , to magnitudes of order of unity is typical, which is then followed, after a few large oscillations, by fast return to a weakly excited state. Such a behavior corresponds to an integrable Hamiltonian of the form for two complex envelopes .

###### pacs:

03.75.Kk, 67.85.DeIntroduction. The dynamics of quantum vortices in a trapped atomic Bose-Einstein condensate with spatially inhomogeneous equilibrium density differs significantly from their dynamics in a uniform system, and the differences are not only quantiative but also qualitative (see review F2009 and references therein). Development of experimental methods in this field makes actual new and diverse profiles . Therefore, vortices in nonuniform systems continue to attract interest in experiment as well as in the theory SF2000 ; FS2001 ; R2001 ; AR2001 ; GP2001 ; A2002 ; RBD2002 ; AD2003 ; AD2004 ; SR2004 ; D2005 ; Kelvin_vaves ; ring_istability ; reconn-2017 ; top-2017 . In the general case the problem is quite complicated, because vortices interact with potential excitations and with non-condensate atoms. But if the condensate at zero temperature is in the Thomas-Fermi regime (the vortex core width is much smaller than a typical scale of the inhomogeneity and the vortex size ), then one can neglect the potential degrees of freedom and use the “anelastic” hydrodynamic approximation SF2000 ; FS2001 ; R2001 ; A2002 ; SR2004 ; R2017-1 ; R2017-2 . If, besides that, a vortex line configuration is far from self-intersections, then a simple mathematical model is applicable, the local induction equation SF2000 ; FS2001 ; R2001

(1) |

where is the geometric shape of the filament depending on arbitrary longitudinal parameter and time , the parameter is the velocity circulation quantum, const is a large logarithm, is a local curvature of the vortex line, is the unit binormal vector, and is the unit tangent vector. To make formulas clean, below we use dimensionless quantities, so that , . It is a well known fact that in the case const, the local induction equation is reduced by the Hasimoto transform Hasimoto to the one-dimensional (1D) focusing nonlinear Schrödinger equation, so the vortex line dynamics against a uniform background is nearly integrable. For nonuniform density profiles investigation of this model is still in the very beginning Kelvin_vaves ; ring_istability ; R2016-1 ; R2016-2 . Even the simplest 1D-periodic density profile

(2) |

was not considered so far in the framework of Eq.(1), although, by the way it is easily realized in optical traps. The purpose of this work is to fill this gap in the theory by considering the propagation of a deformed quantum vortex ring through Bose-Einstein condensate with density (2). By theoretical analysis and numerical simulations we shall identify here such interesting phenomena as parametric resonance and a quasi-recurrence to a weakly excited starting state. To the best author’s knowledge, an idea about possibility of these effects in the system under consideration was not put forward previously by anyone.

Parametric instability. For our purposes it will be convenient to take the angle in the cylindrical coordinate system as the longitudinal parameter, while the two other coordinates will be considered as unknown functions and (apparently, both -periodic on ) which determine geometric shape of the vortex ring at an arbitrary time moment. In order to write equations of motion for and , we take advantage of the variational formulation of Eq.(1). With 1D dependence , the Lagrangian has the following form

(3) |

where primes denote the partial derivatives on . The corresponding variational equations of motion are equivalent to the vector equation (1):

(4) | |||||

(5) | |||||

Unperturbed propagation of a perfectly circular ring along axis is described by solutions of the form and , which satisfy a simple system of ordinary differential equations

(6) |

Obviously, this system has integral of motion const. Let us consider now the dynamics of small deviations from the perfect shape, by writing

(7) | |||||

(8) |

where and are small complex Fourier coefficients. A linearized system for them follows from Eqs.(4)-(5). With taking into account the relation and the presence of integral of motion , we easily obtain

(9) | |||||

(10) |

It is convenient to introduce here instead of a new independent variable in accordance with . Then the linearized system looks very simple:

(11) | |||||

(12) |

where function has been introduced. In our case this dependence is -periodic, so after reduction of (11)-(12) to a single differential equation of the second order we obtain a Hill equation,

(13) |

which is widely known as the main mathematical model describing parametric resonance in linear systems. From here conditions of parametric resonance of order immediately follow:

(14) |

In this work we mainly concentrate on the case , . Let us note that at small values of the density modulation depth we have approximately , i.e. the Hill equation takes form of the Mathieu equation. At the same time, the modulation depth of the nonuniform coefficient in Eq.(13) is equal to . The spatial increment of the instability at exact resonance is given, as can be easily shown, by formula . It corresponds to the growth of the elliptic mode of the ring by a factor of per one period of the density modulation. Even with relatively small we thus have a very rapid growth of deviations.

Numerical simulations. In order to investigate a nonlinear stage of the parametric instability development, solutions of the evolutionary system (4)-(5) with different initial conditions were found numerically by a pseudo-spectral method using a Runge-Kutta 4-th order procedure for the time stepping. Since it follows from the linear analysis of perturbations that at small near the parametric resonance the dependencies and have an oscillating character with period near , while their linear combinations and are mainly proportional to (when higher harmonics are neglected), then for better understanding of the system dynamics it is useful to study behaviour of “slow” complex-valued functions

(15) | |||||

(16) |

Let us note that and are complex envelopes for the amplitudes of standing modes and respectively, while correspond to decomposition of elliptic perturbations of the vortex ring on the propagating modes .

Two typical numerical examples of the ring perturbation dynamics are presented in Fig.1. The main features there which catch our eye are the periodic synchronous returns of the system to a weakly excited state, alternating with strongly deformed ring configurations, the last ones having different angular orientation on the plane. Only with increase of the parameter to values , the regular behaviour is destructed (not shown in the figures).

Such a recurrent dynamics is typical of integrable systems with a few degrees of freedom. Therefore it makes sense to derive a simplified model which could reproduce at least semi-quantitatively the dependencies observed in the numerical experiment.

Explanation of the recurrence phenomenon. In order to explain theoretically the recurrent dynamics of ring deformations, we introduce new canonically conjugate variables

(17) | |||||

(18) |

and expand on small disturbances the corresponding Hamiltonian of the ring,

(19) |

with [at small we have ]. At that we obtain , where ,

(20) | |||||

and . Let us separate in the terms of zeroth order on and in a standard way construct on them the normal complex variables

(21) |

where the frequency in spatially uniform system is

(22) |

The variable is slightly renormalized at that, but we keep the same notation.

It is very important that at our system is completely integrable. Therefore there exist such renormalized normal variables , that three-wave interactions are excluded, while fourth-order terms have the form .

Let us consider mode excitations for resonance number at and introduce slow envelopes for the corresponding normal variables:

(23) |

After that we average on the density oscillations with an accuracy up to the first order on . Nontrivial averaging is required for the term proportional to , and also for . As the result of substitution (23) and subsequent averaging, an effective Lagrangian takes the following form:

(24) | |||||

It is important that we have here an integrable Hamiltonian system with three degrees of freedom. Apparent integrals of motion, besides the Hamiltonian itself, are

(25) | |||||

(26) |

Formula (25) shows that a mean size of the ring is decreased as its deformation is increased. Of course, it is consistent with conservation of the total energy. The conservation law (26) is actually for -component of the angular momentum. Excluding , we obtain an effective Hamiltonian of perturbations in the form

(27) |

In terms of canonically conjugate variables and it is reduced to

(28) |

The recurrence phenomenon then corresponds to quasi-closed phase trajectories in the complex plane of variable , as shown in Fig.2. Of course, the real phase trajectory only “in average” is described by the simplified model.

It is necessary also to say that applicability of the Hamiltonian (27) is limited by close-to-resonance values of the parameter . Taking and making expansion up to 4-th order on , we obtain a quite simplified model Hamiltonian as written in Abstract.

Conclusions. Thus, in this work for the first time we predicted the parametric instability of oscillations of quantum vortex ring in spatially periodic Bose-Einstein condensate, and numerically simulated its nonlinear stage. We applied the local induction approximation which works quite well in the situation under consideration, since the density of condensate does not vanish anywhere. The found here phenomenon of quasi-recurrence was theoretically explained. Such kind nontrivial behaviour of vortex ring definitely deserves further studies within more accurate models, in particular, for more realistic, finite in space condensates with 1D density modulations. Besides that, it is very desirable to reproduce the parametric instability at moderate immediately in numerical solution of the 3D Gross-Pitaevskii equation with periodic external potential. After that, organization of some real-world experiments could become actual.

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