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International Journal of Modern Physics B

Characteristics of information entropy in spin–orbit coupled spin-1 Bose–Einstein condensates

Qiang Zhao, Bin‐Zhou Mi, Yushan Li

Published May 10, 2019• 10 citations
Publisher

Abstract

In this paper, we study the properties of information entropy in spin–orbit coupled spin-1 Bose–Einstein condensates. Our discussion is divided into two parts, stationary state and dynamics. For the former, we find that the coordinate component Sr and total entropy S increase with the strength of the spin–orbit coupling [Formula: see text]. The momentum component Sk shows a discontinuity behavior, it increases with [Formula: see text] when [Formula: see text] is lesser than a critical value [Formula: see text], while it decreases with [Formula: see text] when [Formula: see text] is greater than [Formula: see text]. In order to characterize the disorder-to-order transition, we introduce an order parameter [Formula: see text]. Numerical results display that [Formula: see text] gradually increases with increasing [Formula: see text] and reaches its saturation, which implies that the system becomes more and more ordered. For the latter, the Sr increases and Sk decreases with increasing [Formula: see text] at the same time. The entropic uncertainty relation is well obeyed at fixed [Formula: see text]. Moreover, [Formula: see text] decreases as a function of t. The strength effect of spin–orbit coupling on order-to-disorder transition becomes weak.

DOI
10.1142/s0217979219501194

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arXiv:2112.13417•Dec 26, 2021•0 citations
In this paper, we study the properties of information entropy in spin–orbit coupled spin-1 Bose–Einstein condensates. Our discussion is divided into two parts, stationary state and dynamics. For the former, we find that the coordinate component Sr and total entropy S increase with the strength of the spin–orbit coupling [Formula: see text]. The momentum component Sk shows a discontinuity behavior, it increases with [Formula: see text] when [Formula: see text] is lesser than a critical value [Formula: see text], while it decreases with [Formula: see text] when [Formula: see text] is greater than [Formula: see text]. In order to characterize the disorder-to-order transition, we introduce an order parameter [Formula: see text]. Numerical results display that [Formula: see text] gradually increases with increasing [Formula: see text] and reaches its saturation, which implies that the system becomes more and more ordered. For the latter, the Sr increases and Sk decreases with increasing [Formula: see text] at the same time. The entropic uncertainty relation is well obeyed at fixed [Formula: see text]. Moreover, [Formula: see text] decreases as a function of t. The strength effect of spin–orbit coupling on order-to-disorder transition becomes weak.
In this paper, we study the formation of Shannon information entropy in spin–orbit coupled (SOC) spin-1 antiferromagnetic dipolar Bose–Einstein condensates with external magnetic field. Our results show that, in the absence of magnetic field and with an increase in dipole–dipole interaction (DDI), information entropy in position space [Formula: see text] and momentum space [Formula: see text] remains almost unchanged and increases, respectively. Meanwhile, the order parameter [Formula: see text] decreases, which implies that the system develops toward a disorder state. With the increase of SOC strength, [Formula: see text], [Formula: see text] and [Formula: see text] show similar dynamics behavior. Whereas, in the presence of magnetic field, [Formula: see text] and [Formula: see text] are localized in the small scope by increasing the dipole and SOC strength. In addition, the value of [Formula: see text] is nearly the same in this process. These results embody that the introduction of external magnetic field suppresses the role of SOC and DDI, and impedes the condensates towards disorder state. At last, the influence of geometric structure and atom number on information entropy is investigated. It is seen that a narrower trap and fewer atom number make [Formula: see text] decrease and [Formula: see text] increase.
In this study, we investigate the dynamics properties of spin-1 ferromagnetic Bose–Einstein condensates (BECs) with dipole–dipole interaction (DDI). Numerical results are obtained by solving the mean-field nonlocal Gross–Pitaevskii equation. We find that the population of the information entropy in momentum space [Formula: see text] is greatly affected by geometric structure and dipole strength. The prolate condensate is favorable to obtain bigger [Formula: see text] than oblate condensate. The density distribution verifies that the formation of magnetic domain is associated with the variation of [Formula: see text]. In addition, the kinetic energy displays that dramatical oscillation appears for strong dipolar interaction and it is larger in prolate condensate. Also, we use the order parameter to describe the order–disorder crossover. It is shown that BECs develop toward the disordered state with prolate condensate under strong DDI.
In this paper, we investigate the dynamic properties of spin–orbit coupling spin-1 ferromagnetic Bose–Einstein condensates with different trap geometry. Our results are obtained in terms of the three-component Gross–Pitaevskii equation of mean-field theory. Two kinds of trap potential are discussed: isotropic and anisotropic. It is shown that the spin-exchange dynamics are greatly influenced by trap geometry. For the latter with weak spin–orbit coupling strength, we find that the three-component oscillation accelerates and some small difference emerges between component m=1 and m=-1. With the increase in spin–orbit coupling strength, the three components reach almost the same population. In addition, the kinetic energy of the system changes within a small scope for strong spin–orbit coupling, as opposed to a constant value in an isotropic trap. The density distributions display that the stripe phase appears with the increase in spin–orbit coupling strength. The method of generating stripe structure is different from the ground state of ferromagnetic condensate. For isotropic trap, the spatial separation of top and bottom spin–orbit condensates in component m=1 and m=-1 occurs at weak spin–orbit coupling, and square lattice appears at strong spin–orbit coupling.
Abstract We consider a spin-orbit coupled (SOC) one dimensional Bose–Einstein condensate (BEC) with attractive atom-atom interaction and make use of Shannon entropy ( S ), and Fisher information ( I ) to study the response of atomic density profiles to changes in κ L , the strength of the spin–orbit coupling for fixed values of Rabi frequency (Ω). Density profiles of our interest are characterized by (i) κ L 2 < Ω (region 1) and (ii) κ L 2 > Ω (region 2). We also pay attention to a spatially modulated density profile representing stripe phase of the condensate. Our numbers for S and I show that in region 1 the profile becomes localized as κ L increases while in region 2 we observe delocalization. Values of I in the stripe phase provide an evidence for the supersolid properties of the SOC BEC and indicate that supersolidity is a purely quantum phenomenon.
In this article, we study the vortex formation in spin-1 spin–orbit coupling rotating Bose–Einstein condensates. Numerical results are obtained by solving the spinor Gross–Pitaevskii equation. We mainly focus on the influences of external magnetic fields on vortex structures and dynamics properties. With the increase in magnetic field strength, the populations of magnetic components j=±1 reach the identical value. For the density profile, the three components present identical density structures, and the size of condensates is nearly the same. In addition, some related physical quantities, such as the time taken for the arrival of a steady population and root-mean-square size, kinetic energy, and total angular momentum, are calculated. The results show that these quantities decrease as the magnetic field strength increases. Moreover, we also investigate the time evolution of angular momentum. It is seen that the dynamic behavior of the magnetic components j=±1 is exactly consistent, and the total angular momentum reduces in the presence of the strong magnetic field. This reflects the fact that the introduction of the strong magnetic field makes it difficult to rotate the condensate, and thus, it is disadvantageous for generating more vortices.
We make use of Shannon entropy (SSS) and Fisher information (III) to study the response of atomic density profiles of a spin-orbit coupled Bose-Einstein condensate to changes in the wave number (κLκ_LκL​) of the Raman laser that couples two hyperfine states of atoms in the condensate. The choice for values of κLκ_LκL​, the so-called spin-orbit parameter, and Rabi frequency (ΩΩΩ) leads to two distinct regions in the system's energy spectrum with different order parameters and/or probability densities. In addition, we can have a spatially modulated density profile, reminiscent of the so called stripe phase. Our numbers for SSS and III demonstrate that for κL2Ωκ_L^2ΩκL2​Ω (region 2) for increasing values of κLκ_LκL​. In the stripe phase the nature of SSS and III to changes in κLκ_LκL​ is similar to that found for the condensate in region 2. The results for information theoretic quantities in the stripe phase are, in general, augmented compared to those of region 2. In particular, the highly enhanced values of position-space Fisher information imply an extremely concentrated atomic density distribution to provide an evidence for supersolid properties of Bose-Einstein condensates in the presence of spin-orbit coupling.