Effect of spin-orbit coupling on tunnelling escape of Bose-Einstein condensate

TL;DR

This paper theoretically explores how spin-orbit coupling influences the quantum tunnelling escape of Bose-Einstein condensates, revealing phase-dependent behaviors and potential control mechanisms for atom lasers.

Contribution

It introduces a detailed analysis of tunnelling dynamics in SO-coupled BECs across different phases, highlighting the suppressing effect of SO coupling on tunnelling.

Findings

Tunnelling behavior varies with phase: continuous in single minimum and separated, pulsed in stripe.

SO coupling suppresses tunnelling, nearly eliminating escape near phase transition.

System tuning can control tunnelling, useful for atom laser applications.

Abstract

We theoretically investigate quantum tunnelling escape of a spin-orbit (SO) coupled Bose-Einstein condensate (BEC) from a trapping well. The condensate is initially prepared in a quasi-one-dimensional harmonic trap. Depending on the system parameters, the ground state can fall in different phases --- single minimum, separated or stripe. Then, suddenly the trapping well is opened at one side. The subsequent dynamics of the condensate is studied by solving nonlinear Schr\"{o}dinger equations. We found that the diverse phases will greatly change the tunneling escape behavior of SO coupled BECs. In single minimum and separated phases, the condensate escapes the trapping well continuously, while in stripe phase it escapes the well as an array of pulses. We also found that SO coupling has a suppressing effect on the tunnelling escape of atoms. Especially, for BECs without inter-atom interaction, the tunnelling escape can be almost completely eliminated when the system is tuned near the transition point between single minimum and stripe phase. Our work thus suggests that SO coupling may be a useful tool to control the tunnelling dynamic of BECs, and potentially be applied in realization of atom lasers and matter wave switches.

Equations

$$i\hslash \frac{\partial {\psi }_{\downarrow }}{\partial t}$$

$$i\hslash \frac{\partial {\psi }_{\uparrow }}{\partial t}$$

$$V_0\left(x\right)=\frac{1}{2}m{\omega }_0^2{x}^2.$$

$$\mu {\phi }_{\downarrow ,\uparrow }=\left(\frac{p_x^2}{2m}+V_0\left(x\right)\right){\phi }_{\downarrow ,\uparrow }+\hslash \Omega {\phi }_{\uparrow ,\downarrow },$$

$${\Phi }_0^G=\frac{1}{\sqrt{2\sqrt{\pi }{x}_0}}{\left(1,-1\right)}^T\exp \left[-\frac{x^2}{2x_0^2}\right].$$

$$V\left(x\right)=\left\{\begin{array}[]{c@{\quad}r}m\omega_{0}^{2}x^{2}/2,&x\leq x_{b},\\ V_{0}\exp\left[-\left(\frac{x-x_{c}}{\delta}\right)^{2}\right],&x>x_{b}.\end{array}\right.$$

$$N_{\downarrow ,\uparrow }^{out}={\int }_{x_f}^{\infty }{\left|{\psi }_{\downarrow ,\uparrow }\left(x\right)\right|}^{2}dx,$$

$$\Psi \left(x,t\right)=\psi \left(x,t\right){\left(1,-1\right)}^T{e}^{i\Omega t}.$$

$$i\hslash \frac{\partial }{\partial t}\psi \left(x,t\right)$$

$$\Psi \left(x,t\right)$$

$$\Psi \left(x,t\right)=\varphi \left(x,t\right){\left(1,0\right)}^T{e}^{-i{p}_{c}x/\hslash },$$

$$i\hslash \frac{\partial }{\partial t}\varphi \left(x,t\right)$$

Full text

††: J. Phys. B: At. Mol. Phys.

,

• December 2018

Keywords: Tunnelling, Bose-Einstein condensate, Spin-orbit coupling

1 Introduction

Quantum tunnelling, the phenomenon where a particle penetrates and in most cases passes through a potential barrier which it classically cannot surmount, is one of the most surprising effects in quantum mechanics. In the early days, it is mainly studied in microscopic level [1, 2]. In pace with the discovery and development of various macroscopic quantum systems, macroscopic quantum tunnelling (for example, Josephson tunnelling [3]) also begins to attract great research interest [2, 4]. Specially, Bose-Einstein condensate (BEC) which is macroscopic matter wave with very good coherence and manipulability is very suitable for studying quantum tunnelling (and many other macroscopic quantum phenomena). Atomic Josephson junctions have been realized a long time ago [5, 6]. Furthermore, atomic analogy of superconducting quantum interference devices (SQUID) has also been experimentally demonstrated and proposed to be a good compact rotation sensor [7]. Tunnelling escape of BECs from trapping well has also been extensively studied [8, 9, 10, 11, 12, 13, 14] and recently has been experimentally observed as well [15, 16].

Interference plays a key role in quantum physics. It can have significant influence on quantum tunnelling. These influences have been reported in various fields. Fundamentally, it can lead to breakdown of the exponential decay law [2, 17, 18, 19], and this phenomenon has already been observed in BEC systems [20]. It can also give rise to a modification of tunneling time [21]. In nanomechanical system, it can cause a suppression of the tunneling between opposite magnetizations [22]. In atomic and molecular physics, recently it is reported to have significant effect on the process of photoassociation [23] (which is also a quantum tunnelling related phenomenon).

Spin-orbital (SO) coupling will greatly enrich the quantum properties of BECs. There exist diverse ground state phases in which the matter wave properties are quite different [24, 25, 26, 27]. In zero momentum phase, the condensate stays in a state with zero momentum, while in separated phase it possess momentum of finite value either $+p_{0}$ or $-p_{0}$. And specially in stripe phase, its state is an superposition of $+p_{0}$ and $-p_{0}$ states. The simultaneous existence of these two states will cause interference between them. Subsequently, the quantum tunneling phenomena of SO coupled BECs show new features. In recent years, Josephson tunnelling of SO coupled BEC attracts much research interests [28, 29, 30, 31, 32]. A new kind of Josephson effect — momentum space Josephson effect has been predicted [33]. It is also reported that SO coupling will lead to rich dynamical phenomena of BEC Josephson vortices [34, 35].

However, tunnelling escape of a SO coupled BEC from a trapping well has not been carefully examined yet. In this work, we will study this phenomenon, and try to find out what effect will SO coupling have on the system. Tunnelling escape dynamics of SO coupled BEC in different phases will be studied by solving nonlinear Schrödinger equations, and the results will be compared. The dependence of tunnelling escaped atoms number on SO coupling strength will examined. Possible applications of the system in the field of atom optics will also be discussed.

The following contents of this article is organized as follows. In section 2, we introduce the model for studying tunnelling escape of SO coupled BEC. Then in section 3, tunnelling escape dynamic of the system is studied both analytically and numerically, and the results are discussed. At last, the work is summarized in section 4.

2 Model

We consider a system of SO coupled Bose-Einstein condensate realized by Raman coupling scheme [36], as schematically shown in figure 1. The condensate is assumed to be confined in a parabolic trap with frequencies $\omega_{x}\ll\omega_{\bot}$, so that the system can be reduced to one-dimension [37], and the dynamic can be described by the following two coupled nonlinear Schrödinger equations

[TABLE]

[TABLE]

where $\psi_{\downarrow}$ and $\psi_{\uparrow}$ are wave functions of the two spin components obeying normalization condition ${\displaystyle\int\left(\left|\psi_{\downarrow}\right|^{2}+\left|\psi_{\uparrow}\right|^{2}\right)dx}=1$, $\hbar$ is reduced Planck’s constant , $m$ is the mass of atom, $p_{x}=-i\hbar\frac{\partial}{\partial x}$ is the momentum operator in $x$-direction, $p_{c}$ is the SO coupling strength determined by momentum transfer of Raman lasers, $\Omega$ is the Rabi coupling strength accounting for the transition between the two spin states, $g_{0},g_{1}$ are effective one-dimensional contact interaction strength (their values are $g_{i}=2N_{0}a_{i}\hbar\omega_{\bot}$, where $a_{0}$ and $a_{1}$ are s-wave scattering lengths for collision of two atoms with same or different spins, $N_{0}$ is number of atoms in the condensate), and $U\left(x\right)$ is the external trap potential. Experimentally, such a system is highly controllable, SO coupling can be tuned by manipulating the two Raman lasers [38, 39], and contact interaction can be tuned using the well-known Feshbach resonance technique [40].

Initially, the condensate is confined in harmonic trap

[TABLE]

Thus, the initial state is assumed to be ground state $\Psi_{0}$=$\left(\psi_{\downarrow,0},\psi_{\uparrow,0}\right)^{T}$ of this trapping potential. For vanishing of both SO coupling and inter-atom interactions ($p_{c}=g_{0}=g_{1}=0$), equations (1) and (2) are simplified to

[TABLE]

with $\psi_{\downarrow,\uparrow}\left(x,t\right)=\varphi_{\downarrow,\uparrow}\left(x\right)e^{-i\mu t/\hbar}$. Solving this time-independent Schrödinger equation, the ground state can be calculated precisely to be a Gaussian wave packet

[TABLE]

Otherwise, for no vanishing of SO coupling and inter-atom interaction, the initial state will be obtained by numerically evolving a trial wave function in imaginary time dimension.

The ground state properties of SO coupled BEC have been carefully studied in reference [26]. Here we give a brief review. Omitting the nonlinear and trapping terms, at the same time assuming a plane wave solution $\psi_{\downarrow,\uparrow}\left(x,t\right)=\psi_{\downarrow,\uparrow}\exp\left[i\left(p_{x}x-Et\right)/\hbar\right]$, the dispersion relation of SO coupled matter wave can be obtained by diagonalizing equations (1) and (2), which reads $E_{\pm}\left(p_{x}\right)=\left(p_{x}^{2}+p_{c}^{2}\right)/\left(2m\right)\pm\sqrt{\hbar^{2}\Omega^{2}+p_{c}^{2}p_{x}^{2}/m^{2}}.$ This dispersion relation splits into two branches — $E_{+}\left(p_{x}\right)$ and $E_{-}\left(p_{x}\right)$. The higher energy branch $E_{+}\left(p_{x}\right)$ is quite normal. It always has a single minimum at $p_{x}=0$, and will have little influence on ground state property of the system. Interestingly, the lower energy branch $E_{-}\left(p_{x}\right)$ behaves very differently for different strength of SO and Rabi coupling. When $\hbar\Omega>p_{c}^{2}/m$, it also has only one minimum at $p_{x}=0$. Therefore, the lowest energy state will carry zero momentum, and the corresponding phase is called “zero momentum phase”. While in the case of $\hbar\Omega<p_{c}^{2}/m$, the lower energy branch dispersion curve has two equal minima at $p_{x}=-p_{0},+p_{0}$ (with $p_{0}=\sqrt{p_{c}^{2}-\left(m\hbar\Omega/p_{c}\right)^{2}}$). As a result, the lowest energy state may carry separated momentum $-p_{0}$ or $+p_{0}$ (separated phase), or a superposition of them (due to the interference between these two states, there exist interference stripes, thus the corresponding phase is called “stripe phase”), and these states are degenerate. This degeneracy may be lifted by the external potential or interaction between atoms. When there is only the external harmonic trap potential, the system prefers to stay in stripe phase. And when interaction is considered, as the SO coupling strength getting strong, the system can firstly approach separated phase, then reaches the stripe phase. In the following content, these conclusions will be used directly without more explanation.

At time $t=0$ trapping potential (3) is suddenly opened at the right side. Mathematically speaking, the trapping potential is suddenly changed to the following form (see figure 1)

[TABLE]

Here parameters $V_{0},x_{c},\delta$ are determined by matching the potential function and its first derivative at point $x=x_{b}$. They represent height, location and width of the barrier respectively, and their values are set to $V_{0}=1.57E_{0}$, $x_{b}=1.5x_{0}$, $x_{c}=2x_{0}$, $\delta=0.87x_{0}$, with $E_{0}=\hbar\omega_{0}$ and $x_{0}=\sqrt{\hbar/\left(m\omega_{0}\right)}$ being the harmonic oscillator (with frequency $\omega_{0}$) energy and length units. Consistently, frequency, time and momentum will be measured in units $\omega_{0}$, $t_{0}=1/\omega_{0}$ and $\hbar k_{0}=\sqrt{\hbar m\omega_{0}}$.

After the trapping potential well is opened, i.e., changed to the form of equation (6), the atoms begin to escape from the trapping well due to quantum tunnelling. The dynamic is studied by solving the time dependent nonlinear Schrödinger equations (1) and (2) with a Crank-Nicolson scheme.

To characterize the tunnelling escape, we define the number of escaped spin-down/up atoms as

[TABLE]

where $x_{f}=3x_{0}$ is the demarcation point between the potential area and free space. And the total escaped atoms number is a sum of the two spin components $N^{out}=N_{\downarrow}^{out}+N_{\uparrow}^{out}$. It should be noted, the total wave-function is normalized to $1$ (as previously mentioned), therefore the escaped atoms number defined here is indeed a normalized atoms number or in other words — fraction to the total atoms number.

3 Results

Firstly, for compare purpose and most basic understanding of the system, we examine the no SO coupling case. Without any SO coupling ($p_{c}=0$), the symmetry between spin-down and spin-up components suggests the following form of wave function

[TABLE]

Substituting it into nonlinear Schrödinger equations (1) and (2), dynamic of the system can be simplified to a single governing equation

[TABLE]

which has no dependence on the Rabi frequency $\Omega$. This is to say that without presence of SO coupling, Rabi coupling will not affect dynamic of the system solely. This fact is also confirmed by our numerical results. In the first row (a1-d1) of figure 2, we set SO coupling parameter $p_{c}=0$, and for each spin components show the initial wave packet, evolution of atomic density, wave packet at time $t=20t_{0}$ and escaped atoms number during the evolution. We see that for all these quantities the spin-up and spin-down components have the same values.

As SO coupling is added to the system, symmetry between spin-down and spin-up components is broken, see figure 2 (a2-d4). For single minimum (figures a2-d2) and stripe phases (figures a3-d3), although the two spin components have the same initial wave packets, they gain an evident difference during the evolution. And for separated phase (figures a4-d4), both the initial wave packets and their subsequent evolutions for the two spin components are very different. Interaction is essential in achieving the separated phase [26], and the repulsive interaction can considerably reinforce the escape of the condensate [11, 15], resulting a much larger value of escaped atoms number and atomic density compared to the non-interacting cases.

In figure 2, we also noticed that an initial single minimum or separated phase wave packet escapes from the trapping well continuously; while because of interference between $+p_{0}$ and $-p_{0}$ momentum components, an initial stripe phase wave packet escapes from the trapping well as a series of pulses (for a better sight, see figure 3 in which (b3.1), (b3.2), (c3) of figure 2 are enlarged to shown details of the pulsive property). This indicates tunneling escape of a SO coupled BEC from trapping well may be useful in realizing an atom laser which can be operated in both continuous and pulsed modes.

From figure 2, one can also somewhat see that SO coupling will affects the escaping speed of the condensate. This fact can be more clearly demonstrated by figure 4 where we plot the escaped atoms number $N^{out}$ as a function of time for different SO coupling strength in the case of no inter-atom interaction. From the figure, we see the line for $p_{c}=0$ (no SO coupling) locates above all the other ones, i.e., without SO coupling there are the most escaped atoms, that is to say SO coupling has a suppressing effect on tunnelling escape of atoms. In the figure, we also noticed that in the single minimum phase ($p_{c}=0.0,1.0,1.5,3.0\hbar k_{0}$), a stronger SO coupling tends to more reduce the tunnelling escaped atoms number; while in the stripe phase ($p_{c}=3.5,4.0,4.5,8.0\hbar k_{0}$), thing is quite the opposite — as the coupling gets stronger, it tends to less reduce the tunneling escaped atoms number. Around the phase transition point $p_{c}=3\hbar k_{0}$, the escaped atoms number reaches a minimum with value almost zero. Using this feature, an efficient matter wave switch may be realized. In the case with inter-atom interaction, a similar suppression phenomenon can be observed as well (see figure 5). In single minimum phase ($p_{c}=0.0,1.0,2.0,2.5\hbar k_{0}$), the escaped atoms number also decreases with the increasing of coupling strength. As the SO coupling getting stronger ($p_{c}=3.0,3.5,4.5,5.0\hbar k_{0}$), the system goes into separated phase. In separated phase, the escaped atoms number increases with the increasing of coupling strength. However, due to the escape reinforcement caused by repulsive interaction, even near the phase transition point a sufficient large amount of atoms will escape the trapping well.

In strong SO coupling limit, according to the dynamical equations (1) and (2), spin-down/up component tend to carry a large gauge momentum in the negative/positive direction. Thus, we can assume the wave function approximately having the following form

[TABLE]

for stripe phase, and

[TABLE]

for separated phase, with $\phi\left(x,t\right)$ being a spatially slow varying wave function. Inserting them into nonlinear coupled Schrödinger equations (1) and (2), we found that for the stripe phase wave packet, $\phi\left(x,t\right)$ obeys the same equation as non-coupling equation (9); while for the separated phase wave packet, $\phi\left(x,t\right)$ obey a slightly different equation

[TABLE]

When inter and intra-components interaction strength are close to each other $g_{0}\approx g_{1}$, equation (12) is approximately the same as non-coupling equation (9). So, for strong SO coupling, the tunnelling escape dynamic will go back to the non-coupling case. This conclusion is also demonstrated by the fact that in figure 4 lines for $p_{c}=0,8\hbar k_{0}$ and in figure 5 lines for $p_{c}=0,5\hbar k_{0}$ are nearly overlapping with each other.

For inter/intra-component interaction parameters $g_{0}>g_{1}$, as repulsive interaction will reinforce the tunnelling escape, equations (9) and (12) suggest that more atoms will escape the trapping well for a wave packet in separated phase than in stripe phase. This can be seen from figure 6, where we compared the escaped atoms number during the evolution for initial separated (violet solid line) and stripe (green dashed line) phase wave packets under parameters $p_{c}=5\hbar k_{0}$, $\Omega=8\omega_{0}$, $g_{0}=10E_{0}x_{0}$, $g_{1}=9E_{0}x_{0}$. Here, it should be pointed out that under these parameters, the ground state really fall in the separated phase with energy $E=-12.03E_{0}$. But since the system is nearly degenerate, a stripe phase wave packet can also been obtained with a slightly higher energy $E=-11.91E_{0}$.

4 Summary

In summary, we have analyzed the tunnelling escape of a SO coupled BEC wave packet from a trapping well. Our results show that the dynamics of the system are quite different in different phases: a single minimum or separated phase wave packet escape the trapping well continuously, while a stripe phase one escape in a pulsed manner. This feature may be used in realizing atom laser which can be operated both in continuous and pulsed modes. We also found that SO coupling can suppress the tunnelling escape of atoms. Especially, in non-interacting case, by tuning SO coupling strength to the transition point between single minimum and stripe phase, the tunnelling escape can be almost totally suppressed, thus a matter wave switch may potentially be realized. In the strong SO coupling limit, the dynamics go back to the non-coupling case except for a phase factor.

Acknowledgment

The author thanks Prof. Dong Guangjiong from East China Normal University for very helpful conversations. This work is supported by National Natural Science Foundation of China (Grant No. 11847059 and 11874127).