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arXiv:2306.09270 · Physical Review Research

Chirped Bloch-harmonic oscillations in a parametrically forced optical lattice

Usman Ali, Martin Holthaus, T. Meier

Published Nov 15, 2023• 2 citations
arXivHTMLPDF

Abstract

The acceleration theorem for wave packet propagation in periodic potentials disentangles the $k$-space dynamics and real space dynamics. This is well known and understood for Bloch oscillations and super Bloch oscillations in the presence of position-independent forces. Here, we analyze the dynamics of a model system in which the $k$-space dynamics and the real space dynamics are inextricably intertwined due to a position-dependent force provided by a parabolic trap. We demonstrate that this coupling gives rise to significantly modified and rich dynamics when the lattice is shaken by a modulated parabolic potential. The dynamics range from chirped Bloch-harmonic oscillations to asymmetric spreading oscillations. We analyze these findings by tracing the spatiotemporal dynamics in real space and by visualizing the relative phase in the $k$-space dynamics which leads to an accurate explanation of the obtained phenomena. We also compare our numerical results to a local acceleration model and obtain very good agreement for the case of coherent oscillations, however, deviations for oscillations with spreading dynamics which altogether supports the interpretations of our findings.

DOI
10.1103/physrevresearch.5.043152

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The acceleration theorem for wave packet propagation in periodic potentials disentangles the kkk-space dynamics and real space dynamics. This is well known and understood for Bloch oscillations and super Bloch oscillations in the presence of position-independent forces. Here, we analyze the dynamics of a model system in which the kkk-space dynamics and the real space dynamics are inextricably intertwined due to a position-dependent force provided by a parabolic trap. We demonstrate that this coupling gives rise to significantly modified and rich dynamics when the lattice is shaken by a modulated parabolic potential. The dynamics range from chirped Bloch-harmonic oscillations to asymmetric spreading oscillations. We analyze these findings by tracing the spatiotemporal dynamics in real space and by visualizing the relative phase in the kkk-space dynamics which leads to an accurate explanation of the obtained phenomena. We also compare our numerical results to a local acceleration model and obtain very good agreement for the case of coherent oscillations, however, deviations for oscillations with spreading dynamics which altogether supports the interpretations of our findings.
Super-Bloch oscillations are the outcome of a relative phase between Bloch oscillations and modulations of the periodic lattice. We analyze the dynamics for a model system in which such a relative phase is intrinsically present due to the position-dependent force provided by a parabolic trap and therefore an external detuning is not required. The relative phase is not unique and the realized dynamics depends on the initial phase of the modulated parabolic potential. We provide accurate explanations for the different obtained oscillatory transport and spreading regimes by analyzing the spatio-temporal dynamics in real space and by visualizing the relative phase in the k-space dynamics. We also compare our numerical results to an approximate semiclassical analytical expression for the group velocity for a modulated constant force system and find good agreement for coherent oscillations but deviations for oscillations with spreading dynamics which altogether supports the interpretations of our findings.
We investigate the dynamics of wave packets in a parabolic optical lattice formed by combining an optical lattice with a global parabolic trap. Our study examines the phase space representation of the system's eigenstates by comparing them to the classical phase space of a pendulum, to which the system effectively maps. The analysis reveals that quantum states can exhibit mixed dynamics by straddling the separatrix. A key finding is that the dynamics around the separatrix enables the controlled creation of highly nonclassical states, distinguishing them from the classical oscillatory or rotational dynamics of the pendulum. By considering a finite momentum of the initial wave packet, we demonstrate various dynamical regimes. Furthermore, a slight energy mismatch between nearly degenerate states localized at opposite turning points of the trap potential results in controlled long-range dynamical tunneling. These results can be interpreted as quantum beating between a clockwise rotating and a counterclockwise rotating pendulum.
A quantum particle subjected to a constant force undergoes an accelerated motion following a parabolic path, which differs from the classical motion just because of wave packet spreading (quantum diffusion). However, when a periodic potential is added (such as in a crystal) the particle undergoes Bragg scattering and an oscillatory (rather than accelerated) motion is found, corresponding to the famous Bloch oscillations (BOs). Here, we introduce an exactly-solvable quantum Hamiltonian model, corresponding to a generalized Wannier–Stark Hamiltonian [Formula: see text], in which a quantum particle shows an intermediate dynamical behavior, namely an oscillatory motion superimposed to an accelerated one. Such a novel dynamical behavior is referred to as accelerated BOs. Analytical expressions of the spectrum, improper eigenfunctions and propagator of the generalized Wannier–Stark Hamiltonian [Formula: see text] are derived. Finally, it is shown that acceleration and quantum diffusion in the generalized Wannier–Stark Hamiltonian are prevented for Airy wave packets, which undergo a periodic breathing dynamics that can be referred to as Airy–Bloch oscillations.
A representation is put forward for wave functions of quantum particles in periodic lattice potentials subjected to homogeneous time-periodic forcing, based on an expansion with respect to Bloch-like states which embody both the spatial and the temporal periodicity. It is shown that there exists a generalization of Bloch's famous acceleration theorem which grows out of this representation and captures the effect of a weak probe force applied in addition to a strong dressing force. Taken together, these elements point at a ``dressing and probing'' strategy for coherent wave-packet manipulation, which could be implemented in present experiments with optical lattices.
We study the dynamics of interacting superfluid bosons in a one-dimensional vertical optical lattice after a sudden increase of the lattice potential depth. We show that this system can be exploited to investigate the effects of strong interactions on Bloch oscillations. We perform theoretical modeling of this system, identify experimental challenges, and explore a regime of Bloch oscillations characterized by interaction-induced matter-wave collapse and revivals which modify the Bloch oscillations dynamics. In addition, we study three dephasing mechanisms: finite value of tunneling, effective three-body interactions, and a background harmonic potential. We also find that the center-of-mass motion in the presence of finite tunneling goes through collapse and revivals, giving an example of quantum transport where interaction-induced revivals are important. We quantify the effects of residual harmonic trapping on the momentum distribution dynamics and show the occurrence of an interaction-modified temporal Talbot effect. Finally, we analyze the prospects and challenges of exploiting Bloch oscillations of cold atoms in the strongly interacting regime for precision measurement of the gravitational acceleration ggg.
Bloch oscillation (BO), i.e., the oscillatory motion of a quantum particle in a periodic potential, is one of the most striking effects of coherent quantum transport in matter. In the semiclassical picture, it is well known that BOs can be explained owing to the periodic band structure of the crystal and the so-called acceleration theorem: since in the momentum space the particle wave packet drifts with a constant speed without being distorted, in real space the probability distribution of the particle undergoes a periodic motion following a trajectory which exactly reproduces the shape of the lattice band. In non-Hermitian lattices with a complex (i.e., not real) energy band, extension of the semiclassical model is not intuitive. Here we show that the acceleration theorem holds for non-Hermitian lattices with a complex energy band only on average, and that the periodic wave-packet motion of the particle in real space is described by a trajectory in the complex plane, i.e., it generally corresponds to reshaping and breathing of the wave packet in addition to a transverse oscillatory motion. The concept of BOs involving complex trajectories is exemplified by considering two examples of non-Hermitian lattices with a complex band dispersion relation, including the Hatano-Nelson tight-binding Hamiltonian describing the hopping motion of a quantum particle on a linear lattice with an imaginary vector potential and a tight-binding lattice with imaginary hopping rates.