Bloch oscillations in the absence of a lattice

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Science  02 Jun 2017:
Vol. 356, Issue 6341, pp. 945-948
DOI: 10.1126/science.aah6616

Detecting unusual oscillations

Under the influence of a constant force, an electron in the periodic potential of a crystal lattice undergoes so-called Bloch oscillations. The same phenomenon has been seen with ultracold atoms in optical lattices, but it is not expected to occur in a uniform system. Meinert et al. observed Bloch oscillations of an impurity atom in one-dimensional tubes of strongly interacting cesium atoms—a system without built-in periodicity. Owing to the strong interactions, the bosonic atoms stayed away from one another, forming an effective lattice. The researchers observed reflections of the impurity atoms of this effective lattice in momentum space, with the lattice constant corresponding to the interatomic distance of the host gas.

Science, this issue p. 945


The interplay of strong quantum correlations and far-from-equilibrium conditions can give rise to striking dynamical phenomena. We experimentally investigated the quantum motion of an impurity atom immersed in a strongly interacting one-dimensional Bose liquid and subject to an external force. We found that the momentum distribution of the impurity exhibits characteristic Bragg reflections at the edge of an emergent Brillouin zone. Although Bragg reflections are typically associated with lattice structures, in our strongly correlated quantum liquid they result from the interplay of short-range crystalline order and kinematic constraints on the many-body scattering processes in the one-dimensional system. As a consequence, the impurity exhibits periodic dynamics, reminiscent of Bloch oscillations, although the quantum liquid is translationally invariant. Our observations are supported by large-scale numerical simulations.

A quantum particle accelerated in a periodic crystal potential does not move on average; rather, it undergoes a periodic motion known as Bloch oscillations (1, 2). Such an oscillatory motion is a direct consequence of the periodic momentum dependence of the eigenstates in a lattice potential and arises from the breaking of continuous translational symmetry. Bloch oscillations have been observed for electrons in solid-state systems (3) and have been investigated in detail with ultracold atoms in optical lattices (48). One might expect that such striking dynamics would not occur in a quantum liquid, which is fully translationally invariant, in contrast to lattice systems, for which translational symmetry is broken down to discrete lattice displacements. However, recent theoretical studies (9, 10) suggest that Bloch oscillations can also emerge in the presence of a continuous translational symmetry. In particular, for impurity atoms immersed in one-dimensional (1D) quantum liquids, such dynamics are expected to arise owing to strong quantum correlations, which lead to effective crystal-like properties. However, there is debate about the conditions in which this phenomenon can occur (11). Ultracold quantum gases provide an ideal setting to experimentally study the dynamics of impurity particles coupled to host environments (1216) because of excellent parameter control, precise initial-state preparation, and decoupling from the environment.

Here, we used a degenerate gas of cesium atoms to study the quantum motion of impurities that are immersed in a strongly correlated 1D Bose gas and are accelerated by a constant force. Although the system is a translationally invariant quantum liquid, we observed Bloch oscillations in the impurity dynamics with characteristic Bragg reflections at a wave vector corresponding to the bosonic interparticle distance, which defines an emergent Brillouin zone. We demonstrate that the phenomenon arises for a wide range of system parameters and is robust.

We consider 1D gases of short-range repulsively interacting bosons with mass m, prepared in an array of tubes formed by interfering laser beams (Fig. 1A). Each 1D system is characterized by the dimensionless parameter γ (17). For γ <<1 the gas is weakly interacting; γ >> 1 signifies the strongly correlated Tonks-Girardeau (TG) regime (1820). The impurity immersed in the quantum liquid is of identical mass m and interacts with the host particles with strength γi. When subjected to a force F, the dynamics of the impurity are strongly affected by the spectrum of collective excitations of the coupled system. The lower edge of the spectrum is cosine-shaped and periodic, resembling the conventional dispersion of a lattice system (Fig. 1B). The existence of the lower edge is a unique feature for one dimension and results from kinematic constraints (21, 22). Furthermore, the excitation spectrum of interacting 1D bosons can be recast in terms of fermions, and a Fermi momentum kF can be introduced to characterize the periodicity of the spectral edge. As a consequence of Luttinger’s theorem (23), the periodicity is twice the Fermi momentum 2kF = 2πn1D, where n1D is the density of host atoms, independent of the interactions. This defines an effective Brillouin zone, which is restricted to momenta between –kF and kF.

Fig. 1 Concept of the experiment.

(A) We realize an ensemble of 1D Bose gases in tubes formed by two pairs of counterpropagating and interfering laser beams. In each tube, a single strongly interacting impurity (green sphere) is immersed in the correlated host gas (black spheres) and is accelerated by gravity (green arrow). Inset: Scattering length as for collisions between the atoms in the host gas (dashed line) and between the impurity and the host atoms (solid line) as a function of the magnetic field B. (B) The excitation spectrum of the impurity coupled to the 1D Bose liquid is a 2kF periodic function of the system’s total momentum p, bounded from below by a spectral edge (solid line). For comparison, the dashed line indicates the lower bound of excitations in the background gas without impurity. A force acting on the impurity gradually increases p and induces a population of the continuous many-body spectrum above the spectral edge as a result of nonadiabatic scattering processes (green shading). When the impurity approaches the edge of the correlation-induced Brillouin zone (k = kF), the background gas can absorb excitations with momentum 2kF without energy cost, which manifests itself in the impurity’s momentum distribution by Bragg reflections. Inset: Numerical simulations of a Bragg reflection for infinitely strong background gas interactions γ = ∞, strong but finite impurity-host interactions γi = 12, and a weak force ℱ = 1.

The force F, characterized by the dimensionless parameter Embedded Image, that acts on the impurity increases the total momentum p of the many-body system linearly in time. Yet the impurity exhibits Bragg reflections at the edge of the emergent Brillouin zone that change its momentum by –2kF (Fig. 1B, inset). The excess momentum is transferred into low-energy particle-hole excitations of the host gas. The resulting oscillatory impurity dynamics can be interpreted as Bloch oscillations. However, this picture is substantially challenged by the gapless continuum of excitations above the spectral edge. These continuum states get excited in the course of the quantum evolution even for weak external forces, which can be understood by a breakdown of adiabaticity in gapless quantum systems. In our experiments, we study to what extent the oscillatory impurity dynamics can prevail.

Starting from a 3D cesium Bose-Einstein condensate (24), we use two pairs of counterpropagating and interfering laser beams to confine the atoms to an array of approximately 3500 vertically oriented and highly elongated 1D systems (17). The combined trapping potentials cause an inhomogeneous atom distribution across the array of tubes with a peak occupancy of about 60 particles. Our confined atoms are initially prepared in their lowest magnetic hyperfine state |F, mF〉 = |3, 3〉 and are levitated against gravity by a vertical magnetic field gradient ∇B ≈ 31.1 G/cm. We set γ by adiabatically raising the scattering length as via an offset magnetic field B using a Feshbach resonance (Fig. 1A, inset). The impurity is encoded in a different Zeeman substate |3, 2〉. Applying a short (50 μs) resonant radio-frequency pulse, we create about 3500 impurities in total (i.e., on average one per tube). The value of B set prior to the radio-frequency transfer also determines the scattering length Embedded Image for impurity collisions with the host gas atoms (Fig. 1A, inset) and thereby the interaction parameter γi. Owing to the smaller magnetic moment, the impurity particles are accelerated by one-third of gravity g. We let the system evolve for a variable hold time t up to 3 ms before we determine the impurity momentum distribution n(k) in a time-of-flight (TOF) measurement. To this end, the magnetic field is rapidly ramped to B ≈ 21 G within 50 μs. Here, Embedded Image is sufficiently close to zero that the subsequent motion of the impurities in the tubes is not affected by the host gas atoms. This allows us to reconstruct n(k) from images of the spatially separated spin states after TOF (17).

Results of such measurements are shown in Fig. 2, A to C, for increasing γ and γi, which we determine using the calculated peak density in each 1D system and averaging over the ensemble of tubes (17). Momentum and time are expressed in units of Fermi momentum kF and Fermi time Embedded Image, likewise evaluated via an ensemble average. For all data sets, we observe initial dynamics consistent with free fall caused by the residual gravitational force F = mg/3 (dashed line). As the impurity momentum approaches kF, a considerable fraction of the impurity is Bragg-reflected by –2kF to smaller momenta, resulting in clearly bimodal distributions [see (17) for cuts through n(k) at discrete time steps]. These atoms again accelerate and a similar second scattering process of weaker contrast is apparent at later times. For stronger interactions, the Bragg-reflected fraction of the distribution increases (Fig. 2B), and deep in the TG regime (Fig. 2C) almost the entire ensemble is scattered multiple times to lower momenta.

Fig. 2 Momentum distribution n(k) of the accelerated impurity as a function of time.

(A to F) Experimental data for n(k) (top row) and numerical simulations (bottom row) are shown for [γ, γi] = [7.8 ± 0.3, 3.4 ± 0.1] [(A) and (D)], [γ, γi] = [15.2 ± 0.5, 7.9 ± 0.3] [(B) and (E)], and [γ, γi] = [38 ± 1, 19.4 ± 0.7] [(C) and (F)]. In the experiment, the calculated values for [kF, tF] averaged over the sample are [4.8 ± 0.2 μm–1, 0.18 ± 0.01 ms] (A), [4.5 ± 0.2 μm–1, 0.21 ± 0.01 ms] (B), and [4.2 ± 0.1 μm–1, 0.24 ± 0.02 ms] (C). The dashed lines in (A) to (C) indicate the impurity momentum corresponding to a free fall in the residual gravitational field of magnitude g/3. Arrows in (D) to (F) depict Bragg reflection of the impurity with momentum transfer –2kF. Standard errors given for the experimental parameters reflect a ±10% uncertainty in the total atom number.

Our experimental findings are supported by numerical simulations based on matrix product states (25). We study a single 1D system described by the many-body HamiltonianEmbedded Image(1)where Embedded Image denotes the Lieb-Liniger Hamiltonian accounting for the host gas of N bosons with coordinates zn, interacting via repulsive contact interactions of strength g1D = ħ2n1Dγ/m (17). The remaining terms describe the accelerated impurity with coordinate z interacting with all host particles (gi = ħ2n1Dγi/m). The impurity is initially injected into a gas of N = 60 particles in its equilibrated lowest-momentum state, before we apply the force F. The ensuing time evolution of the impurity momentum distribution (Fig. 2, D to F) agrees well with the experiment, including the characteristic Bragg reflections. The fact that the measurements can be described by a single 1D system demonstrates the robustness and universality of the observed phenomenon. Moreover, we have numerically verified that for our parameters the dynamics are insensitive to the presence of the harmonic trapping potential (17). We also identify slight quantitative differences: (i) The fraction of n(k) that falls freely through the host gas is systematically larger in the experiment than in the theoretical data. We attribute this to contributions from outer tubes filled with only a few particles, or to impurities that are initially excited near the lower edge of the ensemble. Thereby these realizations are less affected by the host atoms. (ii) The idealized situation with the impurity initiated in its ground state is not fully realized experimentally, as the dynamics start immediately after its creation. (iii) The initial spread of the experimentally measured n(k) is larger than the theoretical predictions, which can be mostly attributed to the finite TOF that sets the measurement resolution (17).

As a consequence of the observed Bragg reflections in n(k), we find that the mean impurity momentum 〈k〉 exhibits clear damped oscillations before the impurity approaches a finite drift momentum 〈kd (Fig. 3). We compare the time evolution of the measured impurity momentum with our simulations, and find good agreement for different values of γ and γi. To obtain the experimental data points for the mean impurity momentum, we evaluate n(k) in the interval |k| ≤ 2kF spanning two effective Brillouin zones, which removes the contribution of residual free-falling impurities caused by the experimental sample inhomogeneity (17). The chosen momentum range essentially comprises the full simulated momentum distribution.

Fig. 3 Mean impurity momentum 〈k〉 as a function of time.

The interaction parameters are (A) [γ, γi] = [7.8 ± 0.3, 3.4 ± 0.1], (B) [γ, γi] = [15.2 ± 0.5, 7.9 ± 0.3], (C) [γ, γi] = [24.7 ± 0.9, 13.2 ± 0.5], and (D) [γ, γi] = [38 ± 1, 19.4 ± 0.7]. Error bars indicate the standard deviation extracted from typically five realizations per data point. Solid lines show the result of numerical simulations; dashed lines show the momentum of a free-falling impurity.

Our simulations indicate that during the time evolution, excitations are generated in the scattering continuum above the spectral edge even at weak external force. This manifests in energy that is continuously dissipated in the system at a rate that increases with increasing ℱ. As a consequence, n(k) broadens as time evolves (Fig. 2) and the Bloch oscillating impurity experiences damping toward finite drift velocities (Fig. 3). Remarkably, the drift is characterized by a constant rate of dissipated energy, reminiscent of Joule heating (17).

Additional insight can be obtained from investigating the Bloch oscillation frequency fB and drift momentum 〈kd. We obtain fB by Fourier analysis of the measured mean impurity momentum 〈k〉(t) in Fig. 3. The results (Fig. 4A) are compared with the prediction fBtF = FtF/2ħkF = ℱ/π3, which only takes into account the 2kF periodicity of the spectral edge. This simple model holds for the experimentally probed forces, as we also verified numerically. The long-time drift momentum 〈kd, extracted by taking the average of 〈k〉(t) in the range 8 ≤ t/tF ≤ 13 to remove the residual oscillatory behavior, is shown in Fig. 4B as a function of γi. The reduction of 〈kd with increasing coupling strength is a result of stronger Bragg reflections of the impurity with increasing interactions.

Fig. 4 Bloch oscillation frequency and drift momentum.

(A) Measured Bloch oscillation frequency fB multiplied by the Fermi time tF as a function of the dimensionless force ℱ. The solid line shows the linear dependence ℱ/π3 predicted from a model that considers the 2kF periodicity of the spectral edge. (B) Drift momentum 〈kd of the impurity as a function of interactions γi, γ. Values of the interaction parameter γ are 4.7 ± 0.2, 6.2 ± 0.2, 7.8 ± 0.3, 15.2 ± 0.5, 24.7 ± 0.9, and 38 ± 1 for the data points from left to right. The solid line shows the prediction extracted from numerical simulations (17); the shaded region indicates an estimated error mainly resulting from residual oscillations. Horizontal error bars in (A) reflect a ±10% uncertainty in the total atom number. Vertical error bars reflect the uncertainty for extracting fB in (A) and errors mainly resulting from residual oscillations in (B).

Our observations demonstrate a striking dynamical phenomenon arising from the interplay of strong quantum correlations and far-from-equilibrium conditions. The controlled realization of transient many-body states marked by a high degree of coherence counters the simplest perspective of fast relaxation predicted by hydrodynamics. Moreover, our experiments emphasize the importance of coherence on transport in correlated quantum matter (26) and support the notion that coherence can play an important role in far-from-equilibrium dynamics of many-body systems, as recently discussed in the context of physical, chemical, and biological systems (25, 2729). Apart from that, our results reveal how a mobile impurity can probe many-body effects, which here are crystalline correlations resulting from the fermionized nature of the interacting 1D Bose gas. Controlling impurity dynamics may play a role in exploring exotic properties of quantum matter, including fractional quantum Hall states and associated topological invariants (30) or Anderson’s orthogonality catastrophe (16, 31).

Supplementary Materials

Supplementary Text

Figs. S1 to S7

References (3240)

References and Notes

  1. See supplementary materials.
Acknowledgments: We are indebted to R. Grimm for generous support by allocating financial resources in the form of a Ph.D. position to the project. We thank O. Gamayun, O. Lychkovskiy, C. Mathy, and M. Schecter for fruitful discussions; P. S. Julienne for providing scattering length data; and E. Haller for contributions in the early stage of the experiment. Supported by CNRS grant PICS06738 (M.B.Z.); the Harvard-MIT Center for Ultracold Atoms, NSF grant DMR-1308435, the Air Force Office of Scientific Research Quantum Simulation Multidisciplinary University Research Initiative, the Humboldt Foundation, and the Max Planck Institute for Quantum Optics (E.D.); European Research Council project 278417; Austrian Science Foundation (FWF) project I1789-N20; and the Technical University of Munich–Institute for Advanced Study, funded by the German Excellence Initiative and the European Union FP7 under grant agreement 291763.
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