Observation and uses of position-space Bloch oscillations in an ultracold gas
Zachary A. Geiger, Kurt M. Fujiwara, Kevin Singh, Ruwan Senaratne, Shankari V. Rajagopal, Mikhail Lipatov, Toshihiko Shimasaki, Rodislav Driben, Vladimir V. Kon… 展开作者
Zachary A. Geiger, Kurt M. Fujiwara, Kevin Singh, Ruwan Senaratne, Shankari V. Rajagopal, Mikhail Lipatov, Toshihiko Shimasaki, Rodislav Driben, Vladimir V. Konotop, Torsten Meier, David M. Weld
We report the direct observation and characterization of position-space Bloch oscillations using an ultracold gas in a tilted optical lattice. While Bloch oscillations in momentum space are a common feature of optical lattice experiments, the real-space center-of-mass dynamics are typically too small to resolve. Tuning into the regime of rapid tunneling and weak force, we observe real-space Bloch oscillation amplitudes of hundreds of lattice sites, in both ground and excited bands. We demonstrate two unique capabilities enabled by tracking of Bloch dynamics in position space: measurement of the full position-momentum phase-space evolution during a Bloch cycle, and direct imaging of the lattice band structure. These techniques, along with the ability to exert long-distance coherent control of quantum gases without modulation, may open up new possibilities for quantum control and metrology.
We simulate and analyze an experimental method of loading interacting ultracold atoms onto nontrivial quantum states such as nonlinear Bloch wave and soliton solutions in a 1-dimensional bichromatic lattice. Of standard bands, inverted bands, and bands with Dirac-like points permitted by a bichromatic lattice, we consider the case of an inverted band and examine the loading process in terms of nonlinear Bloch waves formed by an aggregate of ultracold atoms described by the mean-field model. Specifically, we solved the Gross-Pitaevskii equation numerically and found an appropriate standing wave…
The standard Bloch oscillation normally refers to the oscillatory tunneling dynamics of quantum particles in a periodic lattice plus a linear gradient. In this work we theoretically investigate the generalized form of the Bloch oscillation in the presence of additional higher order gradients, and demonstrate that the higher order gradients can significantly modify the tunneling dynamics, particularly in the spectrum of the density oscillation. The spectrum of the standard Bloch oscillation is composed of a single prime frequency and its higher harmonics, while the higher-order gradients in the…
The dispersion relation of ultracold atoms in variably shaped optical lattices can be tuned to resemble that of a relativistic particle, i.e. be linear instead of the usual nonrelativistic quadratic dispersion relation of a free atom. Cold atoms in such a lattice can be used to carry out quantum simulations of relativistic wave equation predictions. We begin this article by describing a Raman technique that allows to selectively load atoms into a desired Bloch band of the lattice near a band crossing. Subsequently, we review two recent experiments with quasirelativistic rubidium atoms in a bic…
We describe a scheme for probing a gas of ultracold atoms trapped in an optical lattice and moving in the presence of an external potential. The probe is non-destructive and uses the existing lattice fields as the measurement device. Two counter-propagating cavity fields simultaneously set up a conservative lattice potential and a weak quantum probe of the atomic motion. Balanced heterodyne detection of the probe field at the cavity output along with integration in time and across the atomic cloud yield information about the atomic dynamics in a single run. The scheme is applied to a measureme…
Bloch oscillations are a hallmark of coherent wave dynamics in periodic potentials. They occur as the response of quantum mechanical particles in a lattice if a weak force is applied. In optical lattices with their perfect periodic structure they can be readily observed and employed as a quantum mechanical force sensor, for example, for precise measurements of the gravitational acceleration. However, the destructive character of the measurement process in previous experimental implementations poses serious limitations for the precision of such measurements. In this article we show that the use…
Bloch-sphere rotations in driven double-well for ultracold atoms
University of the Basque Country (EHU)（巴斯克大学）；IKERBASQUE, Basque Foundation for Science（伊卡巴斯基奎科学基金会）；EHU Quantum Center, University of the Basque Country (EHU)（巴斯克大学量子中心）；Università di Firenze（佛罗伦萨大学）；Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche (CNR-INO)（国家光学研究所，意大利国家研究委员会）；European Laboratory for Nonlinear Spectroscopy (LENS)（欧洲非线性光谱学实验室）；INFN Sezione di Firenze（佛罗伦萨国家核物理研究所）
We provide a one-to-one map between the spin correlations and certain three-dimensional shapes, analogous to the map between single spins and Bloch vectors, and demonstrate its utility. Much as one can reason geometrically about dynamics using a Bloch vector -- e.g. a magnetic field causes it to precess and dissipation causes it to shrink -- one can reason similarly about the shapes we use to visualize correlations. This visualization demonstrates its usefulness by unveiling the hidden structure in the correlations. For example, seemingly complex correlation dynamics can be described as simple…
We numerically investigate the damping of Bloch oscillations in a one-dimensional lattice potential whose translational symmetry is broken in a systematic manner, either by making the potential bichromatic or by introducing scatterers at distinct lattice sites. We find that the damping strongly depends on the ratio of lattice constants in the bichromatic potential, and that even a small concentration of scatterers can lead to strong damping. Moreover, mean-field interactions are able to counteract aperiodicity-induced damping of Bloch oscillations.
We study super Bloch oscillations of ultracold atoms in a shaken lattice potential, subjected to a harmonically modulated mean-field interaction. Usually, any interaction leads to the decay of the wave packet and its super Bloch oscillation. Here, we use the phases of interaction and shaking with respect to the free Bloch oscillation as control parameters. We find two types of long-living cases: (i) suppression of the immediate broadening of the wave packet, and (ii) dynamical stability of all degrees of freedom. The latter relies on the rather robust symmetry argument of cyclic time [Gaul et …
Berry's phase is investigated for ultracold atoms in a frequency modulated optical lattice. It is shown that Berry's phase appears due to Bloch oscillation and the periodic motion of the optical lattice. Particularly, Berry's phase for ultracold atoms under the gravitational force in an oscillating tight-binding optical lattice is calculated analytically. It is found that the Berry's phase depends linearly on the amplitude of the oscillation of the optical lattice.
We report the observation of quantum dynamical oscillations of ultracold atomic gases in the F and D bands of a single-well optical lattice. We are able to control the Bragg reflections at the Brillouin zone edge up to the third order. As a result, we can switch the quantum dynamics from oscillations across both the F and D bands to oscillations only within the F-band. Our capability to observe these remarkable oscillations comes from the innovative non-adiabatic technique which allows us to load ultracold atoms efficiently to the G-band of an optical lattice.
Particles in a perfect lattice potential perform Bloch oscillations when subject to a constant force, leading to localization and preventing conductivity. For a weakly-interacting Bose-Einstein condensate (BEC) of Cs atoms, we observe giant center-of-mass oscillations in position space with a displacement across hundreds of lattice sites when we add a periodic modulation to the force near the Bloch frequency. We study the dependence of these "super" Bloch oscillations on lattice depth, modulation amplitude, and modulation frequency and show that they provide a means to induce linear transport …
We present a new technique for stabilizing and monitoring Bloch oscillations of ultracold atoms in an optical lattice under the action of a constant external force. In the proposed scheme, the atoms also interact with a unidirectionally pumped optical ring cavity whose one arm is collinear with the optical lattice. For weak collective coupling, Bloch oscillations dominate over the collective atomic recoil lasing instability and develop a synchronized regime in which the atoms periodically exchange momentum with the cavity field.
We study atomic Bloch oscillations in an ensemble of one-dimensional tilted superfluids in the Bose-Hubbard regime. For large values of the tilt, we observe interaction-induced coherent decay and matter-wave quantum phase revivals of the Bloch oscillating ensemble. We analyze the revival period dependence on interactions by means of a Feshbach resonance. When reducing the value of the tilt, we observe the disappearance of the quasi-periodic phase revival signature towards an irreversible decay of Bloch oscillations, indicating the transition from regular to quantum chaotic dynamics.
We consider Bloch oscillations of Bose-Einstein condensates in presence of a time-modulated s-wave scattering length. Generically, interaction leads to dephasing and decay of the wave packet. Based on a cyclic-time argument, we find---additionally to the linear Bloch oscillation and a rigid soliton solution---an infinite family of modulations that lead to a periodic time evolution of the wave packet. In order to quantitatively describe the dynamics of Bloch oscillations in presence of time-modulated interactions, we employ two complementary methods: collective-coordinates and the linear stabil…
We report on experiments investigating quantum transport and band interferometry of an atomic Bose-Einstein condensate in an optical lattice with a two-band miniband structure, realized with a Fourier-synthesized optical lattice potential. Bloch-Zener oscillations, the coherent superposition of Bloch oscillations and Landau-Zener tunneling between the two bands are observed. When the relative phase between paths in different bands is varied, an interference signal is observed, demonstrating the coherence of the dynamics in the miniband system. Measured fringe patterns of this Stückelberg inter…
Investigating Friedel oscillations in ultracold gases would complement the studies performed on solid state samples with scanning-tunneling microscopes. In atomic quantum gases interactions and external potentials can be tuned freely and the inherently slower dynamics allow to access non-equilibrium dynamics following a potential or interaction quench. Here, we examine how Friedel oscillations can be observed in current ultracold gas experiments under realistic conditions. To this aim we numerically calculate the amplitude of the Friedel oscillations which a potential barrier provokes in a 1D …
We study Rabi oscillations between the bands of an arbitrary biased superlattice in a tight-binding model. We reduce the problem to an equation of Whittaker--Hill type and, in absence of any known solutions in closed form, discuss different approximations to describe the oscillations between the Bloch bands. We identify regimes of weak and strong inter-band coupling and compare predictions for these Rabi oscillations to numerical results.
We analyze the dynamics of an initially trapped cloud of interacting quantum particles on a lattice under a linear (Stark) potential. We reveal a dichotomy: initially trapped interacting systems possess features typical of both many-body-localized and self-thermalizing systems. We consider both fermions ($t$-$V$ model) and bosons (Bose-Hubbard model). For the zero and infinite interaction limits, both systems are integrable: we provide analytic solutions in terms of the moments of the initial cloud shape, and clarify how the recurrent dynamics (many-body Bloch oscillations) depends on the init…
We study the magnetic Bloch oscillations performed by a quantum particle moving in a two-dimensional lattice in the presence of a strong (synthetic) magnetic field and a uniform force. An elementary derivation of the Berry curvature effect on the semiclassical trajectory is given as well as an explicit connection to the classical Hall effect in the continuum limit. Perspectives to observe these effects in optical systems using synthetic gauge fields for photons are discussed.
We investigate Bloch-Zener oscillations and mean-field Bloch bands of a Bose-Einstein condensate (BEC) in a Lieb optical lattice. We find that the atomic interaction will break the point group symmetry of the system, leading to the destruction of the Dirac cone structure, while the flat band is preserved on the highly symmetric lines. Due to the nonlinear effect, a tubular band structure with a flat band will appear in the system. Furthermore, comparing with that the tight-binding (TB) model fails to describe the interacting bosonic systems in the honeycomb lattice, we show that the TB model i…
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 modelling of this system, identify experimental challenges and explore a new 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: effective three-body interaction…
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-t…
We study the Bloch oscillation dynamics of a spin-orbit-coupled cold atomic gas trapped inside a one-dimensioanl optical lattice. The eigenspectra of the system is identified as two interpenetrating Wannier-Stark ladder. Based on that, we carefully analyzed the Bloch oscillation dynamics and found out that intraladder coupling between neighboring rungs of Wannier-Stark ladder give rise to ordinary Bloch oscillation while interladder coupling lead to small amplitude high frequency oscillation superimposed on it. Specifically spin-orbit interaction breaks Galilean invariance, which can be reflec…
We analyze theoretically the transport properties of a weakly-interacting ultracold Bose gas enclosed in two reservoirs connected by a constriction. We assume that the transport of the superfluid part is hydrodynamic, and we describe the ballistic transport of the normal part using the Landauer-Buttiker formalism. Modeling the coupled evolution of the phase, atom number, and temperature mismatches between the reservoirs, we predict that Helmholtz (plasma) oscillations, induced by an initial imbalance in atom numbers, can be observed at non-zero temperatures below Tc. We show that, because of i…
Several recent studies have investigated the dynamics of cold atoms in optical lattices subject to AC forcing; the theoretically predicted renormalization of the tunneling amplitudes has been verified experimentally. Recent observations include global motion of the atom cloud, such as giant "Super-Bloch Oscillations" (SBOs). We show that, in order to understand unexplained features of SBOs, in addition to the renormalization of the tunneling, a new and important phase correction must be included. For Fermionic systems with strong attractive interactions, one may engineer different types of col…
We identify a new type of periodic evolution that appears in driven quantum systems. Provided that the instantaneous (adiabatic) energies are equidistant we show how such systems can be mapped to (time-dependent) tilted single-band lattice models. Having established this mapping, the dynamics can be understood in terms of Bloch oscillations in the instantaneous energy basis. In our lattice model the site-localized states are the adiabatic ones, and the Bloch oscillations manifest as a periodic repopulation among these states, or equivalently a periodic change in the system's instantaneous ener…
We study dynamics of a mobile impurity moving in a one-dimensional quantum liquid. Such an impurity induces a strong non-linear depletion of the liquid around it. The dispersion relation of the combined object, called depleton, is a periodic function of its momentum with the period 2πn, where n is the mean density of the liquid. In the adiabatic approximation a constant external force acting on the impurity leads to the Bloch oscillations of the impurity around a fixed position. Dynamically such oscillations are accompanied by the radiation of energy in the form of phonons. The ensuing energy …
We propose to use Bloch oscillation of ultra-cold atoms in a spin-dependent optical lattice to prepare schrodinger cat states. Depending on its internal state, an atom feels different periodic potentials and thus has different energy band structures for its center-of-mass motion. Consequently, under the same gravity force, the wave packets associated with different internal states perform Bloch oscillation of different amplitudes in space and in particular they can be macroscopically displaced with respect to each other. In this way, a cat state can be prepared.
Bloch oscillations are a powerful tool to investigate spectra with Dirac points. By varying band parameters, Dirac points can be manipulated and merged at a topological transition towards a gapped phase. Under a constant force, a Fermi sea initially in the lower band performs Bloch oscillations and may Zener tunnel to the upper band mostly at the location of the Dirac points. The tunneling probability is computed from the low energy universal Hamiltonian describing the vicinity of the merging. The agreement with a recent experiment on cold atoms in an optical lattice is very good.
We discuss the method for the measurement of the gravity acceleration g by means of Bloch oscillations of an accelerated BEC in an optical lattice. This method has a theoretical critical point due to the fact that the period of the Bloch oscillations depends, in principle, on the initial shape of the BEC wavepacket. Here, by making use of the nearest-neighbor model for the numerical analysis of the BEC wavefunction, we show that in real experiments the period of the Bloch oscillations does not really depend on the shape of the initial wavepacket and that the relative uncertainty, due to the fa…
We review the derivation of the effective Dirac equation for ultracold atoms in one-dimensional bichromatic optical lattices, following the proposal by Witthaut et al. Phys. Rev. A 84, 033601 (2011). We discuss how such a derivation - based on a suitable rotation of the Bloch basis and on a coarse graining approximation - is affected by the choice of the Wannier functions entering the coarsening procedure. We show that in general the Wannier functions obtained by rotating the maximally localized Wannier functions for the original Bloch bands can be sufficiently localized for justifying the coa…
We show that the motion of an accelerated atomic impurity immersed in a spin-orbit coupled Bose-Einstein condensate in the supersolid stripe phase undergoes oscillations, similar to the well-known phenomenon of Bloch oscillations in solids. While the back-action of the oscillatory movement onto the condensate excites phonon modes inside the supersolid, it does not affect the position of the roton minimum and therefore not the periodicity of the matter wave lattice. The ultimate decay of the oscillations is mostly due to the dispersion of the wavepacket and we show that this can be counteracted…
Recent experimental progress in the creation of synthetic electric and magnetic fields, acting on cold atoms in a two-dimensional lattice, has attracted renewed interest to the problem of a quantum particle in the Hall configuration. The present work contains a detailed analysis of the eigenstates of this system, called Landau-Stark states, and of the associated dynamical phenomenon of cyclotron-Bloch oscillations. It is shown that Landau-Stark states and cyclotron-Bloch oscillations crucially depend on two factors. The first is the orientation of the electric field relative to the primary axe…
We investigate the interaction-induced superfluid-to-Mott insulator transition in the one-dimensional Bose-Hubbard model (BHM) for fillings $n=1$, $n=2$, and $n=3$ by studying the single-particle gap, the fidelity susceptibility, and the amplitude of Bloch oscillations via density-matrix renormalization-group methods. We apply a generic scaling procedure for the gap, which allows us to determine the critical points with very high accuracy. We also study how the fidelity susceptibility behaves across the phase transition. Furthermore, we show that in the BHM, and in a system of spinless fermion…
Cold atoms in an optical lattice execute Bloch-Zener oscillations when they are accelerated. We have performed a theoretical investigation into the case when the optical lattice is the intra-cavity field of a driven Fabry-Perot resonator. When the atoms oscillate inside the resonator, we find that their back-action modulates the phase and intensity of the light transmitted through the cavity. We solve the coupled atom-light equations self-consistently and show that, remarkably, the Bloch period is unaffected by this back-action. The transmitted light provides a way to observe the oscillation c…
We consider the angular momentum of a harmonically trapped, noninteracting Fermi gas subject to either rotation or to an artificial gauge field. The angular momentum of the gas is shown to display oscillations as a function of the particle number or chemical potential. This phenomenon is analogous to the de Haas - van Alphen oscillations of the magnetization in the solid-state context. However, key differences exist between the solid-state and ultracold atomic gases that we point out and analyze. We explore the dependence of the visibility of these oscillations on the physical parameters and p…
We point out that in higher dimensions, in contrast to the one dimensional case considered usually, Bloch oscillation driven by a static force can induce transport of the wave packet. The wave packet oscillates constantly, but on a larger time scale it drifts at a constant velocity permanently. As a noteworthy feature, the net transport in the long run is always normal to the external force and thus controlled by it. We verify this prediction numerically and discuss its experimental realization both with cold atoms in optical lattices and with two dimensional photonic lattices.
Variational solutions of the Boltzmann equation usually rely on the concept of linear response. We extend the variational approach for tight-binding models at high entropies to a regime far beyond linear response. We analyze both weakly interacting fermions and incoherent bosons on a lattice. We consider a case where the particles are driven by a constant force, leading to the well-known Bloch oscillations, and we consider interactions that are weak enough not to overdamp these oscillations. This regime is computationally demanding and relevant for ultracold atoms in optical lattices. We deriv…
Adiabatic quantum pumping in one-dimensional lattices is extended by adding a tilted potential to probe better topologically nontrivial bands. This extension leads to almost perfectly quantized pumping for an arbitrary initial state selected in a band of interest, including Bloch states. In this approach, the time variable offers not only a synthetic dimension as in the case of the Thouless pumping, but it assists also in the uniform sampling of all momenta due to the Bloch oscillations induced by the tilt. The quantized drift of Bloch oscillations is determined by a one-dimensional time integ…
We propose to measure band topology via quantized drift of Bloch oscillations in a two-dimensional Harper-Hofstadter lattice subjected to tilted fields in both directions. When the difference between the two tilted fields is large, Bloch oscillations uniformly sample all momenta, and hence the displacement in each direction tends to be quantized at multiples of the overall period, regardless of any momentum of initial state. The quantized displacement is related to a reduced Chern number defined as a line integral of Berry curvature in each direction, providing an almost perfect measurement of…
