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arXiv:2411.03151 · Journal of Physics B Atomic Molecular and Optical Physics

Modulation of spin–orbit coupled Bose–Einstein condensates: analytical characterization of acceleration-induced transitions between energy bands

Jose M. Gomez Llorente, Jesús Plata

Published Oct 23, 2024• 1 citation
arXivHTMLPDF

Abstract

Abstract The effects of modulating spin-orbit coupled Bose–Einstein condensates are analytically studied. A sinusoidal driving of the coupling amplitude is shown to induce significant changes in the energy bands and in the associated spin-momentum locking. Moreover, in agreement with recent experimental results, gravitational acceleration of the modulated system is found to generate transitions between the modified energy bands. The applicability of the Landau–Zener (LZ) model to the understanding of the experimental findings is rigorously traced. Through a sequence of unitary transformations and the reduction to the spin space, the modulated Hamiltonian, with the gravitational potential incorporated, is shown to correspond to an extended version of the LZ scenario. The generalization of the basic LZ model takes place along two lines. First, the dimensionality is enlarged to combine the description of the external dynamics with the internal-state characterization. Second, the model is extended to incorporate two avoided crossings emerging from the changes induced in the energy bands by the modulation. Our approach allows a first-principle derivation of the effective model-system parameters. The obtained analytical results provide elements to control the transitions.

DOI
10.1088/1361-6455/ad8ab1

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Abstract The effects of modulating spin-orbit coupled Bose–Einstein condensates are analytically studied. A sinusoidal driving of the coupling amplitude is shown to induce significant changes in the energy bands and in the associated spin-momentum locking. Moreover, in agreement with recent experimental results, gravitational acceleration of the modulated system is found to generate transitions between the modified energy bands. The applicability of the Landau–Zener (LZ) model to the understanding of the experimental findings is rigorously traced. Through a sequence of unitary transformations and the reduction to the spin space, the modulated Hamiltonian, with the gravitational potential incorporated, is shown to correspond to an extended version of the LZ scenario. The generalization of the basic LZ model takes place along two lines. First, the dimensionality is enlarged to combine the description of the external dynamics with the internal-state characterization. Second, the model is extended to incorporate two avoided crossings emerging from the changes induced in the energy bands by the modulation. Our approach allows a first-principle derivation of the effective model-system parameters. The obtained analytical results provide elements to control the transitions.
We analytically study the effect of gravitational and harmonic forces on ultracold atoms with synthetic spin-orbit coupling (SOC). In particular, we focus on the recently observed transitions between internal states induced by acceleration of the external modes. Our description corresponds to a generalized version of the Landau-Zener (LZ) model: the dimensionality is enlarged to combine the quantum treatment of the external variables with the internal-state characterization; additionally, atomic-interaction effects are considered. The emergence of the basic model is analytically traced. Namely, by using a sequence of unitary transformations and a subsequent reduction to the spin space, the SOC Hamiltonian, with the gravitational potential incorporated, is exactly converted into the primary LZ scenario. Moreover, the transitions induced by harmonic acceleration are approximately cast into the framework of the basic LZ model through a complete analytical procedure. We evaluate how the validity of our picture depends on the system preparation and on the magnitude of atomic-interaction effects. The identification of the regime of applicability and the rigorous characterization of the parameters of the effective model provide elements to control the transitions.
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The Landau-Zener (LZ) transition is one of the most fundamental phenomena in quantum dynamics. It describes nonadiabatic transitions between quantum states near an avoided crossing that can occur in diverse physical systems. Here we report experimental measurements and tuning of LZ transitions between the dressed eigenlevels of a Bose-Einstein condensate (BEC) that is synthetically spin-orbit (SO) coupled. We measure the transition probability as the BEC is accelerated through the SO avoided crossing and study its dependence on the coupling between the diabatic (bare) states, eigenlevel slope, and eigenstate velocity---the three parameters of the LZ model that are independently controlled in our experiments. Furthermore, we performed time-resolved measurements to demonstrate the breaking down of the spin-momentum locking of the spin-orbit-coupled BEC in the nonadiabatic regime, and we determined the diabatic switching time of the LZ transitions. Our observations show quantitative agreement with the LZ model and numerical simulations of the quantum dynamics in the quasimomentum space. The tunable LZ transition may be exploited to enable a spin-dependent atomtronic transistor.
Dynamical characteristics of internal and external states of a Bose-Einstein condensate are generally different and independent, thus requiring different experimental manipulation techniques. The spin-orbit coupling recently achieved in Bose-Einstein condensates essentially connects spin and motion degree of freedom, endowing spin states with the ability to respond to orbital manipulation, and vice versa. In this work, a dynamical response effect, induced by simultaneously manipulating the internal and external states of a spin-orbit-coupled Bose-Einstein condensate, is predicted. Here, the “simultaneously manipulating the internal and external states” means that the driving field combines the Zeeman field applied to the internal state of the atom and the orbital potential affecting the external states of the atom. Specifically, the Bose-Einstein condensate is assumed to be activated by an abruptly applied Zeeman field and a sudden shake of the trapping potential. After some reasonable simplification and approximation of the model (i.e. neglecting the inter-atomic interactions and modelling the shake of the trapping potential by a short time-dependent pulse), an analytical relationship connecting spin frequency spectrum and the parameters of the driving fields is derived. The numerical calculations based on directly integrating the Gross-Pitaevskii equation are in good agreement with the results from the analytical relationship. The physical origin of the predicted spin dynamical response can be traced back to the quantum interference among different spin-orbit states. Due to the fact that a series of characteristic parameters of the condensate can be manifested in the spin frequency spectrum, the dynamical response effect predicted here provides a candidate method for determining and calibrating various system parameters by measuring the spin frequency spectrum.
We investigate the Landau-Zener (LZ) process in spin-orbit coupled systems of single or multiple two-level (spin-12\frac{1}{2}21​) particles. The coupling between internal spin states and external vibrational states, a simple spin-orbit coupling (SOC), is induced by applying a spin-dependent harmonic trap. Because of the SOC, the single-particle energy-level structures are modified by the Franck-Condon (FC) effects, in which some avoided energy-level-crossings (ELCs) are almost closed and some ELCs are opened. The close of avoided ELCs and the open of ELCs result in the FC blockade and the vibrational transitions, respectively. For a given low sweeping rate, the sequential LZ transitions of ladder-like population transition can be induced by strong SOC. We derive an analytical formula for the final population which is well consistent with the numerical results. For a given strong SOC, the sequential LZ transitions are submerged in the non-adiabatic transitions if the sweeping rate is sufficiently high. Further, we study LZ transitions of multiple interacting two-level Bose particles in a spin-dependent harmonic trap. The interplay between the SOC effects and the interaction effects is explored.
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