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Ultra-Cold Atoms on Optical Lattices

Parag Ghosh

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

In the second half of the dissertation, we investigate the problem of vortex trapping in cyclically coupled Bose-Josephson junctions. Starting with N independent BECs we couple the condensates through Josephson links and allow the system to reach a stable circulation by adding a dissipative term in our semiclassical equations of motion. The central question we address is what is the probability to trap a vortex with winding number m . Our numerical simulations reveal that the final distribution of winding numbers is narrower than the initial distribution of total phases, indicating an increased probability for no-vortex configurations. Specifically, the final width of the distribution of winding numbers for N sites scales as lambdaNalpha, where alpha = 0.47 +/- 0.01 and lambda < 0.67 (value predicted for the initial distribution). The actual value of lambda is found to depend on the strength of dissipation. The nonlinearity of the problem also manifests itself in the result that it is possible to obtain a non-zero circulation starting with zero total phase around the loop.

Related papers

Vortex trapping in suddenly connected Josephson junctions of Bose-Einstein condensates

Parag Ghosh, Fernando Sols

We investigate the problem of vortex trapping in cyclically coupled Bose-Josephson junctions. Starting with $N$ independent Bose-Einstein condensates we couple the condensates through Josephson links and allow the system to reach a stable circulation by adding a dissipative term in our semiclassical equations of motion. The central question we address concerns the probability of trapping a vortex with winding number $m$. Our numerical simulations reveal that the final distribution of winding numbers is narrower than the initial distribution of total phases, indicating an increased probability for no-vortex configurations. Specifically, the final width of the distribution of winding numbers for $N$ sites scales as $\ensuremath{\lambda}{N}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}=0.47\ifmmode\pm\else\textpm\fi{}0.01$ and $\ensuremath{\lambda}<0.67$ (the value predicted for the initial distribution). The actual value of $\ensuremath{\lambda}$ is found to depend on the strength of dissipation. The nonlinearity of the problem also manifests itself in the result that it is possible to obtain a nonzero circulation starting with zero total phase around the loop.

arXiv:0704.2431•Physical Review A•Mar 10, 2008•10 citations

Topics in Ultra-cold Bose Gases: the Bose-Hubbard Model; Analogue Models for an Expanding Universe and for an Acoustic Black Hole

Jain, Piyush

In this thesis we consider the application of phase-space methods to Bose-Einstein condensates; the work comprises of three main parts: Part I: A phase-space method for the Bose-Hubbard model; Part II: An analogue model of an expanding universe in Bose-Einstein condensates. and Part III: An analogue model of an acoustic Black Hole in Bose-Einstein condensates. In part I we present a phase-space method for the Bose-Hubbard model based on the Qfunction representation. In particular, we consider two model Hamiltonians in the mean-field approximation; the first is the standard “one site” model where quantum tunneling is approximated entirely using mean-field terms; the second “two site” model explicitly includes tunneling between two adjacent sites while treating tunneling with other neighbouring sites using the meanfield approximation. The ground state is determined by minimising the classical energy functional subject to quantum mechanical constraints, which take the form of uncertainty relations. For each model Hamiltonian we compare the ground state results from the Q-function method with the exact numerical solution. The results from the Q-function method, which are easy to compute, give a good qualitative description of the main features of the Bose-Hubbard model including the superfluid to Mott insulator. We find the quantum mechanical constraints dominate the problem and show there are some limitations of the method particularly in the weak lattice regime. Analogue models of gravity have been motivated by the possibility of investigating phenomena not readily accessible in their cosmological counterparts. In particular, the prediction of quasiparticle creation in ultra-cold Bose gases in specific configurations can be viewed as an analogue to either cosmological particle creation or the Hawking effect. In part II of this thesis we investigate the analogue of cosmological particle creation in an expanding universe by numerically simulating a Bose-Einstein condensate with a time-dependent scattering length. In particular, we simulate a 2D homogeneous condensate using the classical field method via the truncated Wigner approximation. We show that for several different expansion scenarios the calculated particle production is consistent with the underlying theory. For inflationary models we find the particle production for long wavelength modes coincides with the analytic theory within the acoustic approximation, whereas the particle production is suppressed for short wavelength (ie. free-particle like) modes. Moreover, particle production is enhanced for faster expansions, approaching the analytic result for the sudden expansion in the limit of a very fast expansion. For the case of a cyclic expansion, particle production peaks for a mode frequency that is approximately half of the driving frequency as expected for parametric resonance. In part III of this thesis we investigate an acoustic black hole in a Bose-Einstein condensate, formed by two de Laval nozzles in a ring configuration — a system we refer to as the quantum de Laval nozzle. Our model is formulated in one dimension with a sinusoidal potential. For nonzero superfluid flow, this system can exhibit stable transonic flow with both black and white hole sonic horizons. Stationary states are found by solving the time-independent Gross-Pitaevskii equation subject to a phase quantisation constraint. By solving the projected Bogoliubov-de Gennes equations for the system, we also find the discrete spectrum and quasiparticle modes. There are dynamical instabilities for certain values of winding number and potential depth, for which it is possible to construct pairs of normalisable modes. We further investigate the dynamics of the system using a classical field method based on the truncated Wigner approximation. For a low winding number and unstable configuration, we find exponential growth for the pair of unstable modes, whereas there is no growth in these modes for a stable configuration. This can be interpreted as non-degenerate parametric amplification, valid for short times. In contrast, for a large winding number, there is significant growth in modes for both stable and unstable configurations. This is indicative of higher order processes neglected in the quasiparticle picture, which is further reinforced by that fact that large winding number solutions require large nonlinearities. Finally, we consider the connection of our results with the usual semi-classical prediction of the Hawking effect. For an unstable configuration, the normalised unstable modes couple equal and opposite real frequencies, so that the growth in these modes represents the closest analogy with the Hawking effect for our quantum system.

•Jan 1, 2007•0 citations

Statistics of vortex trapping in cyclically coupled Bose-Josephson junctions

Parag Ghosh, Fernando Sols, Tony Leggett

APS•Mar 1, 2007•0 citations

Generation and pinning of high winding-number vortices in Bose-Einstein condensates

E. Carlo Samson, Kali Wilson, Zachary L. Newman +2 more

Bulletin of the American Physical Society•Jun 15, 2011•0 citations

Bénard-von Kármán vortex street in dipolar Bose-Einstein condensate trapped by square-like potential

Xi Zhong-Hong, Xue-Ying Yang, Tang Na +3 more

Bénard-von Kármán vortex street in dipolar Bose-Einstein Condensate (BEC) trapped by a square-like potential is investigated numerically. In the frame of mean-field theory, the nonlinear dynamic of the dipolar BEC can be described by the so-called two-dimensional Gross-Pitaevskii (GP) equation with long-range interaction. In this paper, we only consider the case that all the dipoles are polarized along the z-axis, which is perpendicular to the plane of disc-shaped BEC. Firstly, the stationary state of the BEC is obtained by the imaginary-time propagation approach. Secondly, the nonlinear dynamic of the BEC, when a moving Gaussian potential exists in such a system, is numerically investigated by the time-splitting Fourier spectral method, in which the stationary state obtained before is set to be the initial state. The results show that when the velocity of the cylindrical obstacle potential reaches a critical value, which depends on interaction strength and the shape of the potential, the vortex-antivortex pairs will be generated alternately in the super-flow behind the obstacle potential. However, in general, such a vortex-antivortex pair structure is dynamically unstable. When the velocity of the obstacle potential increases to a certain value and for a suitable potential width, a stable vortex structure called Bénard-von Kármán vortex street will be formed. While this phenomenon emerges, the vortices in pairs created by the obstacle potential have the same circulation. The pairs with opposite circulations are alternately released from the moving obstacle potential. For larger potential width and velocity, the shedding pattern becomes irregular. We also numerically investigate the effects of the dipole interaction strength, the width and the velocity of the obstacle potential on the vortex structures arising in the wake flow. As a result, the phase graph is presented by lots of numerical calculations for a group of given physical parameters. Thirdly, the drag force on the obstacle potential is also calculated and the mechanical mechanism of vortex pair is analyzed. Finally, we discuss how to find the phenomenon of Bénard-von Kármán vortex street in dipolar BEC experimentally.

Acta Physica Sinica•Jan 1, 2018•2 citations

Vortex lattice of a Bose-Einstein condensate in a rotating anisotropic trap (12 pages)

Mehmet Özgür Oktel

Physical Review A•Jan 1, 2004•0 citations
In the second half of the dissertation, we investigate the problem of vortex trapping in cyclically coupled Bose-Josephson junctions. Starting with N independent BECs we couple the condensates through Josephson links and allow the system to reach a stable circulation by adding a dissipative term in our semiclassical equations of motion. The central question we address is what is the probability to trap a vortex with winding number m . Our numerical simulations reveal that the final distribution of winding numbers is narrower than the initial distribution of total phases, indicating an increased probability for no-vortex configurations. Specifically, the final width of the distribution of winding numbers for N sites scales as lambdaNalpha, where alpha = 0.47 +/- 0.01 and lambda < 0.67 (value predicted for the initial distribution). The actual value of lambda is found to depend on the strength of dissipation. The nonlinearity of the problem also manifests itself in the result that it is possible to obtain a non-zero circulation starting with zero total phase around the loop.
We investigate the problem of vortex trapping in cyclically coupled Bose-Josephson junctions. Starting with NNN independent Bose-Einstein condensates we couple the condensates through Josephson links and allow the system to reach a stable circulation by adding a dissipative term in our semiclassical equations of motion. The central question we address concerns the probability of trapping a vortex with winding number mmm. Our numerical simulations reveal that the final distribution of winding numbers is narrower than the initial distribution of total phases, indicating an increased probability for no-vortex configurations. Specifically, the final width of the distribution of winding numbers for NNN sites scales as $\ensuremath{\lambda}{N}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}=0.47\ifmmode\pm\else\textpm\fi{}0.01$ and $\ensuremath{\lambda}<0.67$ (the value predicted for the initial distribution). The actual value of $\ensuremath{\lambda}$ is found to depend on the strength of dissipation. The nonlinearity of the problem also manifests itself in the result that it is possible to obtain a nonzero circulation starting with zero total phase around the loop.
In this thesis we consider the application of phase-space methods to Bose-Einstein condensates; the work comprises of three main parts: Part I: A phase-space method for the Bose-Hubbard model; Part II: An analogue model of an expanding universe in Bose-Einstein condensates. and Part III: An analogue model of an acoustic Black Hole in Bose-Einstein condensates. In part I we present a phase-space method for the Bose-Hubbard model based on the Qfunction representation. In particular, we consider two model Hamiltonians in the mean-field approximation; the first is the standard “one site” model where quantum tunneling is approximated entirely using mean-field terms; the second “two site” model explicitly includes tunneling between two adjacent sites while treating tunneling with other neighbouring sites using the meanfield approximation. The ground state is determined by minimising the classical energy functional subject to quantum mechanical constraints, which take the form of uncertainty relations. For each model Hamiltonian we compare the ground state results from the Q-function method with the exact numerical solution. The results from the Q-function method, which are easy to compute, give a good qualitative description of the main features of the Bose-Hubbard model including the superfluid to Mott insulator. We find the quantum mechanical constraints dominate the problem and show there are some limitations of the method particularly in the weak lattice regime. Analogue models of gravity have been motivated by the possibility of investigating phenomena not readily accessible in their cosmological counterparts. In particular, the prediction of quasiparticle creation in ultra-cold Bose gases in specific configurations can be viewed as an analogue to either cosmological particle creation or the Hawking effect. In part II of this thesis we investigate the analogue of cosmological particle creation in an expanding universe by numerically simulating a Bose-Einstein condensate with a time-dependent scattering length. In particular, we simulate a 2D homogeneous condensate using the classical field method via the truncated Wigner approximation. We show that for several different expansion scenarios the calculated particle production is consistent with the underlying theory. For inflationary models we find the particle production for long wavelength modes coincides with the analytic theory within the acoustic approximation, whereas the particle production is suppressed for short wavelength (ie. free-particle like) modes. Moreover, particle production is enhanced for faster expansions, approaching the analytic result for the sudden expansion in the limit of a very fast expansion. For the case of a cyclic expansion, particle production peaks for a mode frequency that is approximately half of the driving frequency as expected for parametric resonance. In part III of this thesis we investigate an acoustic black hole in a Bose-Einstein condensate, formed by two de Laval nozzles in a ring configuration — a system we refer to as the quantum de Laval nozzle. Our model is formulated in one dimension with a sinusoidal potential. For nonzero superfluid flow, this system can exhibit stable transonic flow with both black and white hole sonic horizons. Stationary states are found by solving the time-independent Gross-Pitaevskii equation subject to a phase quantisation constraint. By solving the projected Bogoliubov-de Gennes equations for the system, we also find the discrete spectrum and quasiparticle modes. There are dynamical instabilities for certain values of winding number and potential depth, for which it is possible to construct pairs of normalisable modes. We further investigate the dynamics of the system using a classical field method based on the truncated Wigner approximation. For a low winding number and unstable configuration, we find exponential growth for the pair of unstable modes, whereas there is no growth in these modes for a stable configuration. This can be interpreted as non-degenerate parametric amplification, valid for short times. In contrast, for a large winding number, there is significant growth in modes for both stable and unstable configurations. This is indicative of higher order processes neglected in the quasiparticle picture, which is further reinforced by that fact that large winding number solutions require large nonlinearities. Finally, we consider the connection of our results with the usual semi-classical prediction of the Hawking effect. For an unstable configuration, the normalised unstable modes couple equal and opposite real frequencies, so that the growth in these modes represents the closest analogy with the Hawking effect for our quantum system.
Bénard-von Kármán vortex street in dipolar Bose-Einstein Condensate (BEC) trapped by a square-like potential is investigated numerically. In the frame of mean-field theory, the nonlinear dynamic of the dipolar BEC can be described by the so-called two-dimensional Gross-Pitaevskii (GP) equation with long-range interaction. In this paper, we only consider the case that all the dipoles are polarized along the z-axis, which is perpendicular to the plane of disc-shaped BEC. Firstly, the stationary state of the BEC is obtained by the imaginary-time propagation approach. Secondly, the nonlinear dynamic of the BEC, when a moving Gaussian potential exists in such a system, is numerically investigated by the time-splitting Fourier spectral method, in which the stationary state obtained before is set to be the initial state. The results show that when the velocity of the cylindrical obstacle potential reaches a critical value, which depends on interaction strength and the shape of the potential, the vortex-antivortex pairs will be generated alternately in the super-flow behind the obstacle potential. However, in general, such a vortex-antivortex pair structure is dynamically unstable. When the velocity of the obstacle potential increases to a certain value and for a suitable potential width, a stable vortex structure called Bénard-von Kármán vortex street will be formed. While this phenomenon emerges, the vortices in pairs created by the obstacle potential have the same circulation. The pairs with opposite circulations are alternately released from the moving obstacle potential. For larger potential width and velocity, the shedding pattern becomes irregular. We also numerically investigate the effects of the dipole interaction strength, the width and the velocity of the obstacle potential on the vortex structures arising in the wake flow. As a result, the phase graph is presented by lots of numerical calculations for a group of given physical parameters. Thirdly, the drag force on the obstacle potential is also calculated and the mechanical mechanism of vortex pair is analyzed. Finally, we discuss how to find the phenomenon of Bénard-von Kármán vortex street in dipolar BEC experimentally.