Adiabatically tuning quantized supercurrents in an annular Bose-Einstein condensate

TL;DR

This paper demonstrates how to adiabatically generate and tune quantized supercurrents in a ring-shaped Bose-Einstein condensate using spin-orbital-angular-momentum coupling, enabling new atomtronic device functionalities.

Contribution

It introduces a method to prepare and tune quantized supercurrents as ground states via adiabatic processes with SOAM coupling and external potentials, including superfluid hysteresis analysis.

Findings

Quantized supercurrents can be adiabatically tuned in BECs.

Superfluid hysteresis occurs due to nonlinear interactions.

A nonlinear Landau-Zener theory explains the hysteresis.

Abstract

The ability to generate and tune quantized persistent supercurrents is crucial for building superconducting or atomtronic devices with novel functionalities. In ultracold atoms, previous methods for generating quantized supercurrents are generally based on dynamical processes to prepare atoms in metastable excited states. Here we show that arbitrary quantized circulation states can be adiabatically prepared and tuned as the ground state of a ring-shaped Bose-Einstein condensate by utilizing spin-orbital-angular-momentum (SOAM) coupling and an external potential. There exists superfluid hysteresis for tuning supercurrents between different quantization values with nonlinear atomic interactions, which is explained by developing a nonlinear Landau-Zener theory. Our work will provide a powerful platform for studying SOAM coupled ultracold atomic gases and building novel atomtronic circuits.

Figures

Equations

$$H_0^{\mathrm{ring}}=-{\partial }_{\varphi }^2+\left(2il{\partial }_{\varphi }+\delta /2\right){\sigma }_z+\Omega {\sigma }_x,$$

$$E_g=\frac{1}{2\pi }{\int }_0^{2\pi }\left(g_{\uparrow \uparrow }|{\psi }_{\uparrow }{|}^4+g_{\downarrow \downarrow }|{\psi }_{\downarrow }{|}^4+2g_{\uparrow \downarrow }|{\psi }_{\uparrow }{|}^2|{\psi }_{\downarrow }{|}^2\right)d\varphi .$$

$$\mathcal{H}=(l-m+1{)}^2{p}_1+(l-m{)}^2{p}_3+(l+m-1{)}^2{p}_2+$$

$$(l+m{)}^2{p}_4+2V[\sqrt{p_1{p}_3}\cos (q_1-q_3)+\sqrt{p_2{p}_4}\cos q_2]+$$

$$2\Omega [\sqrt{p_1{p}_2}\cos (q_1-q_2)+\sqrt{p_3{p}_4}\cos q_3]+$$

$$(g/\pi )[p_1{p}_3+p_2{p}_4+\sqrt{p_1{p}_2{p}_3{p}_4}\cos (q_1-q_2-q_3)],$$

Full text

Introduction. Quantized supercurrents are one of the most remarkable phenomena of superfluids and superconductors and have been widely studied in solid-state superconductors Tinkham2004 and ultracold atomic gases Ryu2007 ; Corman2014 . Such persistent circulation currents are crucial elements for building many important devices such as SQUID Ryu2013 ; Sato2013 ; Clarke2004 , superfluid gyroscopes Gustavson1997 ; Packard1992 ; Schwab1997 , atomic interferometers Kasevich1991 ; Mathew2015 ; Marti2015 ; Hoskinson2006 ; Simmonds2001 and atomtronic circuits Seaman2007 ; Pepino2009 . These important applications naturally require the experimental ability of coherent generation and manipulation of quantized supercurrents. In this context, ultracold atomic gases possess intrinsic advantages for unprecedented control of experimental parameters and the lack of disorder Bloch2008 .

Great progresses have been made recently for generating quantized circulation supercurrents in a ring-shaped geometry and exploring these properties and device applications Ryu2007 ; Marzlin1997 ; Cooper2010 ; Moulder2012 ; Beattie2013 ; Ramanathan2011 ; Wright2013 ; Jendrzejewski2013 ; Eckel2014 . So far two experimental tools have been applied to prepare quantized circulation currents for an annular Bose-Einstein condensate (BEC): i) a short two-photon Raman pulse with orbital angular momentum (OAM) transfer between two spin states Ryu2007 ; Moulder2012 ; Beattie2013 using Laguerre-Gaussian (LG) laser beams Allen1992 ; ii) periodic rotation of a local repulsive potential barrier along a ring Ramanathan2011 ; Wright2013 ; Jendrzejewski2013 ; Eckel2014 . These methods involve dynamical process to transfer atoms to metastable high OAM states [see Fig. 1 (c)], which could induce complicated excitations, heating, and decays of the BEC. Therefore a nature question is whether quantized supercurrents can be adiabatically prepared and manipulated as the ground state of an annular BEC to circumvent these issues.

To prepare a finite OAM circulation state as the ground state, the OAM need be coupled with other degrees of freedom such as spins, as we see from the spin-orbit coupling Lin2011 ; Zhang2012b ; Qu2013a ; Olson2014 ; Hamner2014 ; Wang2012 ; Cheuk2012 ; Williams2013 . Recently, spin–OAM (SOAM) coupling has been proposed for ultracold atoms using LG Raman lasers with finite OAM to couple two atomic hyperfine spin states Sun2015 ; DeMarco2015 ; Qu2015 . Since OAM is still a good quantum number on a ring, an external nonuniform potential is needed to induce coupling between different OAM states.

In this Letter, we show that the combination of these two ingredients, SOAM coupling and nonuniform potential, allows adiabatic preparation and control of quantized supercurrents with arbitrary OAM as the system ground state in a ring-shaped BEC. Our main results are:

i) A circulation state with arbitrary OAM can be generated on demand with a high precision from a non-rotating state at a time scale $\sim 10\hbar/E_{R}$, where $E_{R}=\hbar^{2}/2MR^{2}$ is the energy unit defined by atomic mass $M$ and ring radius $R$. Large interaction strength or OAM states shorten the time required for the adiabatic process. Such circulation state carries both non-zero particle and spin supercurrents. Because the adiabatic preparation process is free from complicated excitations and the system is on ground state, the generated quantized supercurrents are stable against heating and decaying of the BEC.

ii) The tuning of supercurrents between different circulation states strongly depends on the interatomic interaction and nonuniform potential, which, in a proper parameter region, possesses superfluid hysteresis.

iii) We develop a nonlinear Landau-Zener theory to explain the observed adiabatic preparation and superfluid hysteresis, which exhibits a swallowtail energy structure due to nonlinear interactions.

Model and Hamiltonian. We consider an atomic BEC with two internal spin states \begin{array}[]{cc}{({\psi_{\uparrow}}}&{{\psi_{\downarrow}}{)^{T}}}\end{array}, subject to a pair of co-propagating vertical LG beams with opposite OAM number $\pm l$ [see Fig. 1(a)] and $\Omega_{\pm}(r)=\Omega_{0}\left(\frac{\sqrt{2}r}{w}\right)^{|l|}\exp\left(-\frac{r^{2}}{w^{2}}\pm il\phi-ik_{L}z\right)$, where $(r,\phi,z)$ are cylindrical coordinates, and $\Omega_{0}$, $w$, and $k_{L}$ are the beams’ amplitude, waist, and wavevector in the $\hat{z}$ direction, respectively. The LG beams induce a Raman transition [as in Fig. 1(b)] that generates the SOAM coupling Sun2015 ; DeMarco2015 ; Qu2015 . An additional far-detuned vertical LG laser with the same beam waist and OAM provides the tube potential around the maximum beam intensity $r=\sqrt{l/2}w$ and a horizontal “sheet” beam provides the confinement along the $z$ direction Beattie2013 . Together the SOAM-coupled BEC is confined on an annular geometry of a fixed radius $R$ and can be described by an effective single-particle Hamiltonian Sun2015 in units of $E_{R}$ and $\hbar$ ($=1$), as

[TABLE]

plus interatomic interactions. Here $\delta$ is the Zeeman detuning, $\Omega$ is the Raman coupling strength, and $\{\sigma\}$ are the Pauli matrices.

The Hamiltonian has two energy bands. The eigenstates of the lower band are plane waves in the $\hat{\phi}$ direction $\left|m\right\rangle={(\cos\theta_{m},-\sin\theta_{m}{)^{T}}{e^{im\phi}}}$, where $m$ is an integer due to the periodic boundary condition. Note that $m$ represents a quasi-OAM quantum, while the up and down components have real OAM $\propto(m\mp l)$ Sun2015 , respectively. In other words, the $\left|m\right\rangle$ state physically carries quantized particle supercurrent $J_{c}(m)=m-l\cos 2\theta_{m}$ and spin supercurrent $J_{s}(m)=m\cos 2\theta_{m}-l$. The ground-state supercurrent configurations change with $\Omega$. Given $\delta=0$ and $\Omega=0$, the lower band has two degenerate minima at $m=\pm l$ [as in Fig. 1(c)]. As $\Omega$ increases, the double minima shift toward each other and merge at $m=0$ when $\Omega=4l^{2}-1$, above which the band has a single minimum $m=0$. In the double-minima region, any pair of $\left|\pm m\right\rangle$ states exhibits a $Z_{2}$ symmetry by having opposite spin polarization $\langle\sigma_{z}\rangle=\cos 2\theta_{m}$, opposite $J_{c}$, and same $J_{s}$. Below we focus on the plane-wave state on the $m\geq 0$ side.

Preparation of quantized supercurrent. To create the ground state or manipulate the supercurrent circulation in the ring system, one may properly tune $\Omega$ to load the BEC at a desired quantum $m$, as it naturally pursuits the energy minimum. However, there are two issues preventing this. First, all $m$ states are stationary states and do not couple to each other, so a high-$m$ state can have a very long lifetime. Second, a smooth transition between adjacent $m$ states encounters an energy barrier induced by interatomic interaction, which takes the form of

[TABLE]

In Fig. 1(c), we schematically plot the system energy vs expectation value of $m$ for a superposition of two adjacent $|m\rangle$ states. The ripples reflect the interaction effects: the local minima correspond to pure $|m\rangle$ states, while the local maxima correspond to the equal superposition of two adjacent $|m\rangle$ states, which possess density modulations that cost interaction energy.

To assist the BEC to overcome the barrier and move toward a lower energy state, it is essential to trigger sufficient coupling between adjacent $|m\rangle$ states. We propose the use of a linear external potential $V_{\mathrm{ext}}=V(t)x/R=V(t)\cos\phi$, which couples two states as $\langle m|\cos\phi|m\pm 1\rangle=\frac{1}{2}\langle m|e^{i\phi}+e^{-i\phi}|m\pm 1\rangle\neq 0$. Such potential has already been experimentally applied to annular BECs without SOAM coupling Kumar2016 . In addition, one can use the linear slope around $x=0$ for a Gaussian laser $V\left(x\right)=V_{0}\exp\left(-(x-\xi)^{2}/2\xi^{2}\right)\approx V_{0}\exp\left(-1/2\right)\left(1+x/\xi\right)$ with $\xi\gg R$. We propose a process of tuning both $\Omega$ and $V$ simultaneously, as illustrated in the inset of Fig. 2(a), and simulate the system evolution with the time-dependent Gross-Pitaevskii equation (GPE). Initially, the BEC is prepared in the presence of external potential $V(0)=V_{\mathrm{M}}$. The Raman coupling $\Omega$ is then slowly ramped on to shift the band minimum, while $V$ is gradually turned off to suppress the coupling. Finally, at $\Omega(t)=\Omega_{\mathrm{M}}$ and $V(t)=0$, the BEC stays in the new ground state, the targeted single-$m$ state, which decouples from the others and carries the aforementioned particle and spin supercurrents.

Our simulation of four different cases is presented in Fig. 2. We plot the BEC’s population $P_{m}$ at several associated $m$ states as a function of time. In panel (a) for $l=1$, the BEC starts from a $|1\rangle$ dominant state and ends at an almost single-$m$ state, $|0\rangle$, demonstrating an effective transfer between adjacent states. Although the initial state also couples to $|2\rangle$, the BEC clearly pursuits the lower energy state $|0\rangle$. In general, higher energy states are hardly involved during the transition. The same procedure can be applied to a large $l=10$, as shown in Fig. 2(b), where only $\Omega_{\mathrm{M}}$ is changed for the desired band minimum. The system actually undergoes a smoother transition with a shorter transition time for a large $l$. Using the same strategy, one may possibly reach any $m$ state ($0\leq m\leq l$) by repeating the $\Omega$ and $V$ cycle to lower the $m$ number one by one. However, such a transfer, e.g. from $|10\rangle$ to $|5\rangle$, can be achieved with a single process and hence in a shorter time [see Fig. 2(c)]. The BEC passes through multiple $m$ states with each intermediate $m$ state dominating in a narrow time window, and stops at the final $m$ state. In Fig. 2(d), we quadruple the interatomic interaction and increase $V_{\mathrm{M}}$ accordingly to overcome the interaction energy barrier. The transition goes more smoothly and the transition time is significantly shorter.

Superfluid hysteresis. Hysteresis is a hallmark phenomena of quantized supercurrents Pepino2002 ; Watanabe2011 ; Morsch2006 , and recently a hysteresis loop for supercurrent jumps in an annular BEC has been observed in experiments, where a periodically rotating local laser barrier is used to induce the transition Eckel2014 . In our system, the supercurrent change is driven by the SOAM coupling to change the ground state, which, unlike the local laser barrier, acts on the whole system uniformly. Below we will show the existence of superfluid hysteresis using numerical GPE simulation (Fig. 3), followed by developing a nonlinear Landau-Zener theory Vitanov1999 ; Liu2003 ; Wu2000 to explain the physics, where a swallowtail band structure is ascribed to the origin of the observed hysteretic phenomenon.

The time procedure of $\Omega(t)$ and $V(t)$ in our simulation is illustrated in Fig. 3(a) inset. In the forward evolution, $\Omega(t)$ linearly increases from $\Omega_{\mathrm{m}}$ to $\Omega_{\mathrm{M}}$ with the initial ground state prepared at $\Omega_{\mathrm{m}}$, while $V(t)$ is turned on up to the maximum $V_{\mathrm{M}}$ and then symmetrically turned off. In the backward evolution, $\Omega(t)$ and $V(t)$ take the time-reversal path, with the initial ground state prepared at $\Omega_{\mathrm{M}}$. Since $V(t)$ is the key ingredient that induces the transition, we expect that it plays a crucial role on the hysteresis phenomenon. In Fig. 3(a)–(d), we plot the expectation value $\langle m\rangle$ for transitions between $|2\rangle$ and $|3\rangle$ at various $V_{\mathrm{M}}$. At weak $V_{\mathrm{M}}=0.2$, there is no transition [Fig. 3(a)]. At $V_{\mathrm{M}}=1.6$ [Fig. 3(b)], the system cannot jump from $|3\rangle$ to $|2\rangle$ before the coupling is turned off, leaving the path unclosed. With the increase of the potential to $V_{\mathrm{M}}=1.8$ [Fig. 3(c)], a two-way transition happens with a clear hysteresis loop. At a strong $V_{\mathrm{M}}=3$ [Fig. 3(d)], the coupling completely overwhelms the barrier, and the loop becomes trivial, as the system follows the same path forward and backward.

The above observed phenomena can be intuitively illustrated from a double well energy structure [Fig. 4(a1)] in proper phase space of parameters of the system. Initially, BEC always stays in one of the minimum (ground state). During the evolution, the energy difference $\Delta_{\Omega}$ between the two minima changes with $\Omega(t)$, while the barrier height $\Delta_{V}$ is determined by the interaction and $V(t)$. If $V_{\mathrm{M}}$ is sufficiently weak, the two minima are well separated by the barrier, and the hopping $h_{V}$ is not strong enough to overcome it. As a result, the BEC is trapped in one minimum without any transition. As $V_{\mathrm{M}}$ increases, the minimum on which the BEC originally stays disappears during the process, and the BEC jumps to the other minimum. Such evolution is not symmetric in the time-reversal path, leading to the hysteretic behavior [Fig. 3(c)]. If $V_{\mathrm{M}}$ is strong, the initial minimum continuously evolves to the final minimum, and vice versa, resulting in the trivial loop [Fig. 3(d)].

This intuitive picture is confirmed by developing a nonlinear Landau-Zener theory for our system. As shown in Fig. 2, the transition process between $|m\rangle$ and $|m-1\rangle$ barely involves other higher-energy states, which allows us to build an effective Hamiltonian in a truncated space spanned by two relevant quasi-OAM and two spin states. The system’s wavefunction is determined by four complex amplitudes as \left(\begin{array}[]{c}a_{1}\\ a_{2}\end{array}\right){{e^{i(m-1)\phi}+}}\left(\begin{array}[]{c}a_{3}\\ a_{4}\end{array}\right){{e^{im\phi}}}, subject to a normalization condition $\sum_{j=1}^{4}p_{j}=1$ with $p_{j}=|a_{j}|^{2}$.

Following the general formalism for nonlinear Landau-Zener tunneling Liu2003 ; Wu2000 , we construct a semi-classical Hamiltonian with canonical coordinates ($q_{j}=\arg a_{j}$, $p_{j}$)

[TABLE]

with a global phase chosen for $q_{4}=0$ and $g=g_{\uparrow\uparrow}=g_{\downarrow\downarrow}=g_{\downarrow\uparrow}$.

For each $\Omega\left(t\right)$ and $V\left(t\right)$ during the transition, we can find a global energy minimum (ground-state energy) in the phase space formed by six parameters $\left(p_{i},q_{i}\right)$ ($i=1,2,3$). There also exist other interaction-induced local energy minima, as shown in the energy structure in Fig. 4(a)–(d), with parameters corresponding to Fig. 3(a)–(d), respectively. In particular, a double-well structure with the emergence of two energy minima similar as Fig. 4(a1) is observed in the phase space (see Supplementary Materials), which confirms the above intuitive explanation of the hysteretic behavior. Because of multiple local energy minima for the same parameter, the energy band structure in Fig. 4 shows swallowtail shape in proper parameter region. In Fig. 4(a), two bands simply cross and both minima survive during the procedure. As a result, the condensate remains on its initial minimum, which ends as a metastable state, without undergoing a transition [Fig. 3(a)]. With larger potential [Fig. 4(b)], only the left-hand-side band minimum breaks. Therefore, when starting from $|2\rangle$, the minimum disappears after the cross, forcing BEC to drop to the ground state. This result agrees with the GPE simulation in Fig. 3(b). With a deeper trap, the breakdown of band minimum is symmetric [Fig. 4(c)], therefore abrupt changes in $\langle m\rangle$ happen in both directions, yielding the hysteresis loop observed in Fig. 3(c). After $V_{\mathrm{M}}$ exceeds some critical value, the barrier vanishes and the BEC always stays on the ground state [Fig. 4(d)]. Therefore the system follows exactly the same path during the evolution as observed in Fig. 3(d). Our quantized supercurrent preparation should use this parameter region.

Experimental consideration. In experiments, we can consider a 87Rb gas Ramanathan2011 ; Wright2013 trapped in a ring of radius $R=8~{}\mu$m, and use LG beams $l=10$ Moulder2012 . The energy unit $E_{R}=2\pi\hbar\times 0.924$ Hz. For the transition procedure ($|10\rangle$ to $|5\rangle$) shown in Fig. 2 (d), $\Omega_{M}=2\pi\hbar\times 166$ Hz, $V_{M}=2\pi\hbar\times 14$ Hz and the overall time length $T=3.4$ s. For a small Raman coupling strength $\Omega_{M}$ around $1$ kHz, the heating effect can be neglected within a timescale of few seconds Wei2013 . Considering the same configuration but with 23Na Stenger1998 , the parameters become $E_{R}=2\pi\hbar\times 3.434$ Hz, $\Omega_{M}=2\pi\hbar\times 618$ Hz, $V_{M}=2\pi\hbar\times 52$ Hz, $T=0.9$ s.

Conclusion. We have proposed a method to adiabatically prepare and tune arbitrary quantized circulation states as the ground states of an annular BEC, which carry both quantized atom and spin supercurrents. The whole procedure can be achieved at a time scale $\sim 10\hbar/E_{R}$ with a satisfactory fidelity. Our system provides a powerful platform for studying phenomena and applications of quantized supercurrents, exploring physics of SOAM coupled atomic gases, and building atomtronic devices with novel functionalities.

Acknowledgements: This work is supported by AFOSR (FA9550-16-1-0387), NSF (PHY-1505496), and ARO (W911NF-17-1-0128).

Appendix A Supplementary Materials

A.1 Time scale of the adiabatic process

For the adiabatic process given in Fig. 2 (a), $V_{\rm{M}}$ should be strong enough to overcome the energy barrier between different $|m\rangle$ states induced by the interaction in order to obtain the desired final state. Therefore a larger $V_{\rm{M}}$ is required for stronger interaction with a rough linear relation as shown in the inset in Fig. 5 (a). The red dots show the minimum values of $V_{\rm{M}}$ to achieve a smooth transition between different $|m\rangle$ states, where the energy minima for adjacent $|m\rangle$ states are smoothly connected by the adiabatic path.

The required time length of the adiabatic process is closely related to the choice of the adiabatic path in the parameter space, which is determined by $V_{\rm{M}}$ and $\Omega_{\rm{M}}$. The value of $\Omega_{\rm{M}}$ can be simply determined by examining the single-particle band structure (or phase diagram). As discussed above, for a given interaction strength, a perfect transition can be realized by slowly tuning the parameters along the path with a large $V_{\rm{M}}$. If the overall time length $T$ is too small, excitations may be created during the process, and consequently, imperfections are introduced in the transition. Fig. 5 (a) illustrates the relations between fidelity and overall time length ($T$) of the process in Fig. 2 (a). The fidelity is defined as the modulus square of the overlap between the final state and the target state. As $T$ decreases, excitations and their interference with the ground state lead to the fidelity oscillation near $T=7.5$, and the adiabatic approach breaks down below the time length $T=7$, where a sudden drop of the fidelity occurs.

For nonlinear quantum problems, the adiabaticity is related to not only the energy-level spacings, but also the fundamental frequencies of periodic orbits around the energy minima. These frequencies can be calculated by linearizing the semi-classical Hamiltonian Eq. Adiabatically tuning quantized supercurrents in an annular Bose-Einstein condensate around the energy minima. One should expect to have three pairs of frequencies ($\omega_{1},\omega_{2},\omega_{3}$), since the degrees of freedom $6$ (i.e. the dimension of the phase space is 6). Fig. 5 (b) shows how these fundamental frequencies vary with time during the adiabatic process shown in Fig. 2 (a). The minimum frequency is about 1, indicating that the overall time length should be much longer than 1 to avoid excitations of such periodic orbits. For the processes shown in Fig. 2 (a) and Fig. 3 (d), the energy minimum is far away from any energy maxima in the phase space, therefore the time scales of the adiabatic processes are determined by the fundamental frequencies, as confirmed by the results shown in Fig. 5 (a). For the hysteresis process, the minimum merges with a maximum and both of them disappear after the merging, therefore the periodic orbits may not even exist and these fundamental frequencies lose their physical meanings. In fact, the adiabaticity breaks down in the hysteresis process. Nevertheless, our simulation suggests that a longer time is always preferred to get a smoother hysteresis loop.

A.2 Energy minima in the phase space

The evolution of energy minima in the phase space are crucial for understanding the adiabatic process. As an example, we consider the hysteresis process discussed in Fig. 3 (c). To visualize the energy minima in the phase space with dimension as high as $6$, we first notice that the minima always satisfy $q_{1}=2\pi$, $q_{2}=\pi$ and $q_{3}=\pi$. This gives us the possibility to project the whole phase space to a 3D subspace spanned by $(p_{1},p_{2},p_{3})$. At $t=66$, there is only one minimum that gives the ground state, as shown in Fig. 6 (a). As we change $\Omega$ and $V$, a second local minimum appears [see Fig. 6 (b) at $t=73$]. This process is accompanied by the emergence of a local maximum that forms the barrier between the two minima. Though the second minimum may have a lower energy, the BEC will stay at the first minimum [see Fig. 6 (c) at $t=77$] until it merges with the maximum and disappears, after which the BEC flows towards the second minimum [see Fig. 6 (d) at $t=85$]. Those phase-space portraits shown in Fig. 6 provide us a clear picture of the physics behind the swallowtail band structure and the corresponding hysteresis. They confirm our intuitive explanation of the hysteresis phenomena illustrated in Fig. 4(a).