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arXiv:1102.3283 · Physical Review A

Synthetic magnetic field effects on neutral bosonic condensates in quasi-three-dimensional anisotropic layered structures

T. A. Zaleski, T. P. Polak

Published Feb 10, 2011• 15 citations
arXivHTMLPDF

Abstract

We discuss a system of dilute Bose gas confined in a layered structure of stacked square lattices (slab geometry). A derived phase diagram reveals a nonmonotonic dependence of the ratio of tunneling to on-site repulsion on the artificial magnetic field applied to the system. The effect is reduced when more layers are added, which mimics a two- to quasi-three-dimensional geometry crossover. Furthermore, we establish a correspondence between anisotropic infinite (quasi-three-dimensional) and isotropic finite (slab geometry) systems that share exactly the same critical values, which can be an important clue for choosing experimental setups that are less demanding, but still leading to the identical results. Finally, we show that the properties of the ideal Bose gas in a three-dimensional optical lattice can be closely mimicked by finite (slab) systems when the number of two-dimensional layers is larger than 10 for isotropic interactions, or even less when the layers are weakly coupled.

DOI
10.1103/physreva.83.023607

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Since the experimental realization of Bose–Einstein condensation of ultra cold atoms, extensive studies have been conducted on boson systems. One of recent epochmaking technologies is the synthesis of a fictitious gauge field acting on neutral atoms, which opened another wide field of studies on bosons and/or fermions in a synthetic magnetic field. Charged bosons were studied more than half a century ago by Schafroth. Charged bosons in a magnetic field were then studied in some respects and a rigorous proof was given by Angelescu and Corviovei, who showed absence of Bose– Einstein condensation of free charged bosons in a uniform magnetic field. This is due to the divergence of the density of states at the lowest band energy, i.e., the lowest Landau level. However the situation changes in the presence of a periodic potential. Namely some external periodic potentials are able to restore the Bose–Einstein condensation of the free charge bosons in a magnetic field in threedimensions (3D), although there has been no actual calculation of the condensation temperature of bosons in a typical lattice structure under a magnetic field. According to these previous studies, it is expected that free cold bosonic atoms under a synthetic magnetic field, if realized experimentally, would not condensate in a continuum space but would have a finite Bose–Einstein condensation temperature in the presence of an optical lattice potential. The aim of this short note is to demonstrate the restoration of the Bose–Einstein condensation under a magnetic field by the introduction of a periodic potential, taking typical lattice structures as examples and calculating the condensation temperature analytically. In what follows, with focus on bosons in a 3D lattice under a uniform magnetic field, we calculate the condensation temperature for different lattices and different magnitude of the magnetic field. In particular we consider a layered structure of two dimensional lattices such as square, triangular and honeycomb ones, with a magnetic field applied in the direction normal to the two-dimensional (2D) lattices. The energy spectrum of a particle on a two dimensional lattice, which we assume to be a square lattice for a moment, under a uniform magnetic field shows a characteristic pattern, so-called Hofstadter’s diagram. The one-particle energy of this system is described by the following Hamiltonian in the tight-binding approximation.

Journal of the Physical Society of Japan•Sep 12, 2013•0 citations
We discuss a system of dilute Bose gas confined in a layered structure of stacked square lattices (slab geometry). A derived phase diagram reveals a nonmonotonic dependence of the ratio of tunneling to on-site repulsion on the artificial magnetic field applied to the system. The effect is reduced when more layers are added, which mimics a two- to quasi-three-dimensional geometry crossover. Furthermore, we establish a correspondence between anisotropic infinite (quasi-three-dimensional) and isotropic finite (slab geometry) systems that share exactly the same critical values, which can be an important clue for choosing experimental setups that are less demanding, but still leading to the identical results. Finally, we show that the properties of the ideal Bose gas in a three-dimensional optical lattice can be closely mimicked by finite (slab) systems when the number of two-dimensional layers is larger than 10 for isotropic interactions, or even less when the layers are weakly coupled.
We show that the properties of the ideal Bose gas in three-dimensional optical lattice can be closely mimicked by nite two dimensional systems with only ten of layers.The match between critical properties strongly depends on the anisotropy of the hopping amplitudes in and between layers which we fully control.The theory we provided can be directly used in the experiments and results in less challenging requirements of the setups.We also present the phase diagram with its non-monotonic dependence of the ratio of tunneling to on-site repulsion when articial magnetic eld is applied to the system.
We investigate systems of interacting bosonic particles confined within slablike boxes of size ${L}^{2}\ifmmode\times\else\texttimes\fi{}Z$ with $Z\ensuremath{\ll}L$, at their three-dimensional (3D) Bose-Einstein-condensation (BEC) transition temperature Tc{T}_{c}Tc​, and below Tc{T}_{c}Tc​ where they experience a quasi-two-dimensional (quasi-2D) Berezinskii-Kosterlitz-Thouless (BKT) transition (at TBKT<Tc{T}_{\mathrm{BKT}}<{T}_{c}TBKT​<Tc​ depending on the thickness ZZZ). The low-temperature phase below TBKT{T}_{\mathrm{BKT}}TBKT​ shows quasi-long-range order: the planar correlations decay algebraically as predicted by the 2D spin-wave theory. This dimensional crossover, from a 3D behavior for $T\ensuremath{\gtrsim}{T}_{c}$ to a quasi-2D critical behavior for $T\ensuremath{\lesssim}{T}_{\mathrm{BKT}}$, can be described by a transverse finite-size scaling limit in a slab geometry. Numerical evidence of the $3\mathrm{D}\ensuremath{\rightarrow}2\mathrm{D}$ dimensional crossover is presented for the Bose-Hubbard model defined in anisotropic ${L}^{2}\ifmmode\times\else\texttimes\fi{}Z$ lattices with $Z\ensuremath{\ll}L$. We extend this scaling analysis to the case the slab geometry of the gas is effectively realized by a transverse (inhomogeneous) harmonic trap. Finally, we discuss off-equilibrium behaviors arising from slow time variations of the temperature across the BEC transition of gases confined within a slab geometry. We argue that the system develops an off-equilibrium transverse finite-size scaling under these time-dependent protocols.
Abstract The Bose–Einstein condensation (BEC) in three-dimensional (3D) anisotropic lattices is studied. We present theoretical results for the critical temperature for BEC, chemical potential, condensate fraction and relevant thermodynamic quantities like: internal energy, entropy, specific heat and compressibility as a function of anisotropy parameter being the ratio of the nearest-neighbor in-plane ( t∥t_\parallelt∥​ t ‖ ) and out-of-plane ( t⊥t_\perpt⊥​ t ⊥ ) hopping amplitudes. In particular, considered scenarios include weakly coupled two-dimensional (2D) planes ( t⊥/t∥≪1t_\perp /t_\parallel \ll 1t⊥​/t∥​≪1 t ⊥ / t ‖ ≪ 1 , relevant for layered structures) as well as a rod-like geometry of interacting one-dimensional (1D) chains ( t∥/t⊥≪1t_\parallel /t_\perp \ll 1t∥​/t⊥​≪1 t ‖ / t ⊥ ≪ 1 ). The impact of the dimensional crossover as the system is tuned away from a set of disconnected 2D layers, or traverses from a set of separate 1D chains to a regime where a fully isotropic 3D structure emerges is elucidated. Both numerical and analytic approaches are employed, (the latter in a form of series expansions involving t∥,t⊥t_\parallel ,t_\perpt∥​,t⊥​ t ‖ , t ⊥ amplitudes) for internal energy, entropy, specific heat and isothermal compressibility. The theoretical outcome of the present study may be of interest to a number of scenarios in solid-state physics, where the relevant quasi-particles are bosonic-like, as well as might be applicable to the physics of cold bosons loaded in artificially engineered 3D optical lattices.
We study dynamical conductivity of strongly correlated bosons loaded in an optical lattice with restricted geometry in which gauge fields are present. We show that dynamics influenced by the uniform synthetic magnetic field combined with layered lattice structures changes into rich insulator-metallic behavior in the strongly correlated regime. Especially, the amplitude of optical conductivity for a given frequency is a nonmonotonous function of the number of layers LLL. In particular, conductivity for frequency corresponding to on-site interaction energy can abruptly vanish for a special number of applied layers. Moreover, such an insulating behavior is stable in the whole range of parameters in the Mott phase. This robustness arises from the complex gaplike behavior or from Dirac-like physics reflected in the quasiparticle energy spectra. Furthermore we show that a large interlayer tunneling anisotropy destabilizes the absence of conducting state. We also investigate the critical conductivity on the Mott-insulator--superfluid phase boundary and show the correspondence between the number of Hofstadter subbands and the number of layers. The obtained results also reveal that the value of critical conductivity gradually goes to zero when a three-dimensional system is approached. The experimental setup for generation of layered optical lattices is also proposed.
We study ultracold bosonic atoms with synthetic three-dimensional spin-orbit (SO) coupling in a cubic optical lattice. In the superfluidity phase, the lowest energy band exhibits one, two, or four pairs of degenerate single-particle ground states depending on the SO-coupling strengths, which can give rise to condensate states with spin stripes for weak atomic interactions. In the deep Mott-insulator regime, the effective spin Hamiltonian of the system combines three-dimensional Heisenberg exchange interactions, anisotropy interactions, and Dzyaloshinskii-Moriya interactions. Based on Monte Carlo simulations, we numerically demonstrate that the resulting Hamiltonian with an additional Zeeman field has a rich phase diagram with spiral, stripe, vortex crystal, and especially Skyrmion crystal spin textures in each xyxyxy-plane layer. The obtained Skyrmion crystals can be tunable with square and hexagonal symmetries in a columnar manner along the zzz axis, and moreover are stable against the interlayer spin-spin interactions in a large parameter region.
Since the experimental realization of Bose–Einstein condensation of ultra cold atoms, extensive studies have been conducted on boson systems. One of recent epochmaking technologies is the synthesis of a fictitious gauge field acting on neutral atoms, which opened another wide field of studies on bosons and/or fermions in a synthetic magnetic field. Charged bosons were studied more than half a century ago by Schafroth. Charged bosons in a magnetic field were then studied in some respects and a rigorous proof was given by Angelescu and Corviovei, who showed absence of Bose– Einstein condensation of free charged bosons in a uniform magnetic field. This is due to the divergence of the density of states at the lowest band energy, i.e., the lowest Landau level. However the situation changes in the presence of a periodic potential. Namely some external periodic potentials are able to restore the Bose–Einstein condensation of the free charge bosons in a magnetic field in threedimensions (3D), although there has been no actual calculation of the condensation temperature of bosons in a typical lattice structure under a magnetic field. According to these previous studies, it is expected that free cold bosonic atoms under a synthetic magnetic field, if realized experimentally, would not condensate in a continuum space but would have a finite Bose–Einstein condensation temperature in the presence of an optical lattice potential. The aim of this short note is to demonstrate the restoration of the Bose–Einstein condensation under a magnetic field by the introduction of a periodic potential, taking typical lattice structures as examples and calculating the condensation temperature analytically. In what follows, with focus on bosons in a 3D lattice under a uniform magnetic field, we calculate the condensation temperature for different lattices and different magnitude of the magnetic field. In particular we consider a layered structure of two dimensional lattices such as square, triangular and honeycomb ones, with a magnetic field applied in the direction normal to the two-dimensional (2D) lattices. The energy spectrum of a particle on a two dimensional lattice, which we assume to be a square lattice for a moment, under a uniform magnetic field shows a characteristic pattern, so-called Hofstadter’s diagram. The one-particle energy of this system is described by the following Hamiltonian in the tight-binding approximation.