Strongly correlated quantum walks in optical lattices

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Science  13 Mar 2015:
Vol. 347, Issue 6227, pp. 1229-1233
DOI: 10.1126/science.1260364

Quantum walkers under a microscope

Generations of physics students have been taught to think of one-dimensional random walks in terms of a drunken sailor taking random steps to the right or to the left. But that doesn't compare with the complexity of a quantum walker, who can propagate down multiple paths at the same time. Preiss et al. detected particles in single sites of an optical lattice to study the dynamics of two interacting atoms of 87Rb performing a quantum walk (see the Perspective by Widera). Depending on the initial conditions and the interaction strength between the atoms, the atoms either ignored each other, stuck to each other, or tried to get as far away from each other as possible.

Science, this issue p. 1229; see also p. 1200


Full control over the dynamics of interacting, indistinguishable quantum particles is an important prerequisite for the experimental study of strongly correlated quantum matter and the implementation of high-fidelity quantum information processing. We demonstrate such control over the quantum walk—the quantum mechanical analog of the classical random walk—in the regime where dynamics are dominated by interparticle interactions. Using interacting bosonic atoms in an optical lattice, we directly observed fundamental effects such as the emergence of correlations in two-particle quantum walks, as well as strongly correlated Bloch oscillations in tilted optical lattices. Our approach can be scaled to larger systems, greatly extending the class of problems accessible via quantum walks.

Quantum walks are the quantum mechanical analogs of the classical random walk process, describing the propagation of quantum particles on periodic potentials (1, 2). Unlike classical objects, particles performing a quantum walk can be in a superposition state and take all possible paths through their environment simultaneously, leading to faster propagation and enhanced sensitivity to initial conditions. These properties have generated considerable interest in using quantum walks for the study of position-space quantum dynamics and for quantum information processing (3). Two distinct models of quantum walk with similar physical behavior were devised: (i) the discrete-time quantum walk (1), in which the particle propagates in discrete steps determined by a dynamic internal degree of freedom, and (ii) the continuous-time quantum walk (2), in which the dynamics is described by a time-independent lattice Hamiltonian.

Experimentally, quantum walks have been implemented for photons (4), trapped ions (5, 6), and neutral atoms (79), among other platforms (4). Until recently, most experiments were aimed at observing the quantum walks of a single quantum particle, which are described by classical wave equations.

An enhancement of quantum effects emerges when more than one indistinguishable particle participates in the quantum walk simultaneously. In such cases, quantum correlations can develop as a consequence of Hanbury-Brown-Twiss (HBT) interference and quantum statistics, as was investigated theoretically (10, 11) and experimentally (1217). In the absence of interactions or auxiliary feedforward measurements of the Knill-Laflamme-Milburn type (18), this problem is believed to lack full quantum complexity, although it can still become intractable by classical computing (11).

The inclusion of interaction between indistinguishable quantum walkers (19, 20) may grant access to a much wider class of computationally hard problems, such as many-body localization and the dynamics of interacting quantum disordered systems (21). Similarly, in the presence of interactions, the quantum walk can yield universal and efficient quantum computation (22).

The classical simulation of such correlated quantum dynamics has been achieved with single-particle quantum walks in photonic systems, where effective interactions may be engineered through conditional phase shifts in fiber networks (23) or waveguide arrays (24). Here, we used bosonic atoms in an optical lattice to directly implement continuous two-particle quantum walks with strong, tunable interactions and direct scalability to larger particle numbers. Our system realizes the fundamental building block of interacting many-body systems with atom-resolved access to the strongly correlated dynamics in a quantum gas microscope (25).

In our experiment, ultracold atoms of bosonic 87Rb perform quantum walks in decoupled one-dimensional tubes of an optical lattice with spacing d = 680 nm. The atoms may tunnel in the x direction with amplitude J and experience a repulsive on-site interaction U, realizing the Bose-Hubbard HamiltonianEmbedded Image (1)where Embedded Image and ai are the bosonic creation and annihilation operators, respectively, and Embedded Image gives the atom number on site i. The values of J and U are tunable via the depth Vx of the optical lattice, specified in units of the recoil energy Er = 2π × (h/8md2) ≈ 2π × 1240 Hz, where h is Planck’s constant and m is the atomic mass of 87Rb. The energy shift per lattice site E is set by a magnetic field gradient. We measure time in units of inverse tunneling rates, τ = tJ, and define the dimensionless interaction u = U/J and gradient Δ = E/J.

We set the initial motional state of the atoms through an adaptable single-site addressing scheme, enabling the deterministic preparation of a wide range of few-body states: Using a digital micromirror device (DMD) as an amplitude hologram in a Fourier plane, we generate arbitrary diffraction-limited potentials in the plane of the atoms. Starting from a low-entropy two-dimensional Mott insulator with a fixed number of atoms per site, we project a repulsive Hermite-Gauss profile to isolate atoms in selected rows while a short reduction of the optical lattice depth ejects all other atoms from the system (26). For the quantum walk, we prepare one or two rows of atoms along the y direction of a deep optical lattice with Vx = Vy = 45Er (Fig. 1A). The quantum walk is performed at a reduced lattice depth Vx while the y lattice and the out-of-plane confinement are fixed at Vy = 45Er and ωz = 2π × 7.2 kHz, respectively. The atom positions are recorded with single-site resolution using fluorescence imaging in a deep optical lattice (25). Pairs of atoms residing on the same site are lost during imaging because of light-assisted collisions and cannot be detected directly. Using a magnetic field gradient, we separate pairs of atoms along the direction of the quantum walk prior to imaging (26) and obtain the full two-particle correlator Γi,j = Embedded Image and the density distribution. Only outcomes with the correct number of atoms per row are included in the data analysis (26).

Fig. 1 Coherent single-particle quantum walks.

(A) Left: Starting from a localized initial state (I), individual atoms perform independent quantum walks in an optical lattice (II). Right: The single-particle density distribution expands linearly in time, and atoms coherently delocalize over 20 sites (lower panel shows the averaged density distribution at the end of the quantum walk and a fit to Eq. 2 with the tunneling rate J as a free parameter). Error bars indicate SEM. (B) In the presence of a gradient, a single particle undergoes Bloch oscillations. The atom initially delocalizes (II) but maintains excellent coherence and reconverges to its initial position after one period (III). Densities are averages over ~700 and ~200 realizations for (A) and (B), respectively.

We first consider quantum walks of individual atoms (Fig. 1A). A single particle is initialized at a chosen site in each horizontal tube and propagates in the absence of an external force. For each individual realization, the particle is detected on a single lattice site, while the average over many experiments yields the single-particle probability distribution. In contrast to a classical random walk, for which slow, diffusive expansion of the Gaussian density distribution is expected, coherent interference of all single-particle paths leads to ballistic transport with well-defined wavefronts (4). The measured probability density ρ expands linearly in time (Fig. 1A, right panel), in good agreement with the theoretical expectation Embedded Image (2)(27), where Embedded Image is a Bessel function of the first kind on lattice site i. If a potential gradient is applied to ultracold atoms in an optical lattice, net transport does not occur because of the absence of dissipation and the separation of the spectrum into discrete bands. Instead, the gradient induces a position-dependent phase shift and causes atoms to undergo Bloch oscillations (28). For a fully coherent single-particle quantum walk with gradient Δ, the atom remains localized to a small volume and undergoes a periodic breathing motion in position space (27, 2930) with a maximal half width LB = 4/Δ and temporal period TB = 2π/Δ in units of the inverse tunneling.

Figure 1B shows a single-particle quantum walk with Δ = 0.56, resulting in Bloch oscillations over ~14 lattice sites. We observed a high-quality revival after one Bloch period and detected the particle back at the origin with a probability of up to 0.96 ± 0.03 at τ = TB in individual tubes. The average over six adjacent rows in Fig. 1B displays a revival probability of 0.88 ± 0.02, limited by the temporal resolution of the measurements and inhomogeneous broadening across different rows. The fidelityEmbedded Image (3)for the measured and expected probability distributions px(t) and qx(t), averaged over ~1.5 Bloch oscillations, is 98.1 ± 0.1%, indicating that a high level of coherence is maintained while the particle delocalizes over ~10 μm in the optical lattice.

If two particles perform a quantum walk simultaneously, they undergo HBT interference (12, 32) and their dynamics are sensitive to the underlying particle statistics. All two-particle processes in the system add coherently, leading to quantum correlations between the particles (Fig. 2A): For bosons, the processes bringing both particles into close proximity of each other add constructively, leading to bosonic bunching, as observed in tunnel-coupled optical tweezers (33), expanding atomic clouds (32, 34), and photonic implementations of quantum walks (12, 13). For fermionic particles, on the other hand, particle exchange leads to an additional phase shift of π, causing a reversal of the HBT logic and characteristic antibunching of free fermions (13, 32).

Fig. 2 Hanbury-Brown-Twiss interference and fermionization.

(A) Processes connecting the initial and final two-particle states interfere coherently. Each tunneling step contributes a phase i. For noninteracting bosons, processes of the same length add constructively (I), whereas processes differing in length by two steps interfere destructively (II). (B) Weakly interacting bosons display strong bunching (I). Strong, repulsive on-site interactions cause bosons in one dimension to fermionize and develop long-range anticorrelations (II). (C) Measured correlator Γi,j at time τmax ≈ 2π × 0.5, averaged over ~3200 realizations. The interactions are tuned from weak (u < 1) to strong (u >> 1) by choosing Vx = 1Er, 2.5Er, 4Er, and 6.5Er.

In our experiment, the bunching of free bosonic atoms is apparent in single-shot images of quantum walks with two particles starting from adjacent sites in the state Embedded Image. For weak interactions, the two atoms are very likely to be detected close to each other because of HBT interference, as shown in raw images in Fig. 2B. To characterize the degree of bunching, we used the density-density correlator Γi,j (Fig. 2C), measured at time τmax ≈ 2π × 0.5. Panel I of Fig. 2C shows the two-particle correlator for a quantum walk with weak interactions (u = 0.7). Sharp features are caused by quantum interference and demonstrate the good coherence of the two-particle dynamics. The concentration of probability on and near the diagonal of the correlator Γi,j indicates HBT interference of nearly free bosonic particles.

We used the quantum walk’s sensitivity to quantum statistics to probe the “fermionization” of bosonic particles caused by repulsive interactions in one-dimensional systems. When such interactions are strong, double occupancies are suppressed by the large energy cost U, which takes the role of an effective Pauli exclusion principle for bosonic particles. In the limiting case of infinite, “hard-core” repulsive interactions, one-dimensional bosonic systems “fermionize” and display densities and spatial correlations that are identical to those of noninteracting spinless fermions (35). This behavior has been observed in equilibrium in the pair correlations and momentum distributions of large one-dimensional Bose-Einstein condensates (36, 37). These systems are characterized by the dimensionless ratio of interaction to kinetic energy γ, and the fermionized Tonks-Girardeau regime is entered when γ is large. For Bose-Hubbard systems below unity filling, such as ours, the corresponding parameter is the ratio u = U/J.

We studied the process of fermionization in the fundamental unit of two interacting particles by repeating the quantum walk from initial state Embedded Image at increasing interaction strengths (19). Figure 2C shows Γi,j for several values of u. At intermediate values of the interaction, u = 1.4 and u = 2.4, the correlation distribution is relatively uniform, as repulsive interactions compete with HBT interference. For the strongest interaction strength, u = 5.1, most of the weight is concentrated on the antidiagonal of Γi,j, corresponding to pronounced antibunching. The anticorrelations are strong enough to be visible in raw images of the quantum walk (Fig. 2B, panel II), and Γi,j is almost identical to the expected outcome for noninteracting fermions. Note that although the correlations change markedly with increasing interaction, the densities remain largely unchanged. At all interaction strengths, the observed densities and correlations are in excellent agreement with a numerical integration of the Schrödinger equation with the Bose-Hubbard Hamiltonian in Eq. 1 (26). Interactions in two-particle scattering events, which we observed on a lattice, take on a central role in closely related models that may be solved via the Bethe ansatz (35), such as Heisenberg spin chains (9) and bosonic continuum systems (37): Within integrable models, scattering between arbitrary numbers of particles may be decomposed into two-particle scattering events, and the phase shift acquired in such processes determines the microscopic and thermodynamic properties of the system.

The precise control over the initial state in our system enables the study of strongly interacting bosons in scenarios not described by fermionization, such as the quantum walks of two atoms prepared in the same state. Figure 3 shows the correlations and densities for the initial state Embedded Image. Because both atoms originate from the same site, HBT interference terms are not present. In the weakly interacting regime (u = 0.7), both particles undergo independent free dynamics and the correlator is the direct product of the single-particle densities. As the interaction increases, separation of the individual atoms onto different lattice sites becomes energetically forbidden. The two atoms preferentially propagate through the lattice together, as reflected in increasing weights on the diagonal of the correlation matrix. For the strongest interactions, the particles form a repulsively bound pair with effective single-particle behavior (38). The two-particle dynamics may be described as a quantum walk of the bound pair (19, 20) at a decreased tunneling rate Jpair, which reduces to the second-order tunneling (39) Jpair = (2J2/U) << J for large values of u.

Fig. 3 Formation of repulsively bound pairs.

Two-particle correlations at τmax ≈ 2π × 0.5 for two particles starting on site 0 in state Embedded Image. For weak interactions (u = 0.7), the atoms perform independent single-particle quantum walks. As the interaction strength is increased, repulsively bound pairs form and undergo an effective single-particle quantum walk along the diagonal of the two-particle correlator. Experimental parameters are identical to those in Fig. 2.

The formation of repulsively bound pairs and their coherent dynamics can be observed in two-particle Bloch oscillations. We focused on the dynamics of two particles initially prepared on the same site with a gradient Δ ≈ 0.5 (Fig. 4). In the weakly interacting regime (u = 0.3), both particles undergo symmetric Bloch oscillations as in the single-particle case, and we observed a high-quality revival after one Bloch period. For intermediate interactions (u = 2.4), the density evolution is very complex: In this regime where J, U, and E are similar in magnitude, states both with and without double occupancy are energetically allowed and contribute to the dynamics. The skew to the right against the applied force is caused by resonant long-range tunneling of single particles over several sites (40, 41) and agrees with numerical simulation. When the interactions are sufficiently strong (u = 3.5), the pairs of atoms are tightly bound by the repulsive interaction and behave like a single composite particle. However, the effective gradient has doubled with respect to the single-particle case, and the pairs perform Bloch oscillations at twice the fundamental frequency and reduced spatial amplitude. The frequency doubling of Bloch oscillations was predicted for electron systems (42) and cold atoms (20, 40) and has recently been simulated with photons in a waveguide array (24). Throughout the breathing motion, the repulsively bound pairs themselves undergo coherent dynamics and delocalize without unbinding. The clean revival after half a Bloch period directly demonstrates the entanglement of atom pairs during the oscillation.

Fig. 4 Bloch oscillations of repulsively bound pairs.

In the weakly interacting regime (I), two particles initialized on the same site undergo clean, independent Bloch oscillations. Increasing the interaction strength (II) leads to complex dynamics: Pairs of atoms remain bound near the origin or they separate, breaking left-right symmetry via long-range tunneling. For the largest interactions (III), repulsively bound pairs perform coherent, frequency-doubled Bloch oscillations. Densities are averages over ~220 independent quantum walks.

Quantum walks of ultracold atoms in optical lattices offer an ideal starting point for the “bottom-up” study of many-body quantum dynamics. The present two-particle implementation provides intuitive access to essential features of many-body systems, such as localization caused by interactions or fermionization of bosons. Such microscopic features, when scaled to larger system sizes, manifest in emergent phenomena—for example, quantum phase transitions, quasi-particles, or superfluidity—as observed in other cold atom experiments. The particle-by-particle assembly of interacting systems may give access to the crossover from few- to many-body physics and may reveal the microscopic details of disordered quantum systems (21) and many-body quench dynamics (43).

Supplementary Materials

Materials and Methods

Table S1

References (44, 45)

References and Notes

  1. See supplementary materials on Science Online.
  2. Acknowledgments: We thank S. Aaronson, M. Endres, and M. Knap for helpful discussions. Supported by grants from NSF through the Center for Ultracold Atoms, the Army Research Office with funding from the DARPA OLE program and a MURI program, an Air Force Office of Scientific Research MURI program, the Gordon and Betty Moore Foundation's EPiQS Initiative, the U.S. Department of Defense through the NDSEG program (M.E.T.), a NSF Graduate Research Fellowship (M.R.), and the Pappalardo Fellowship in Physics (Y.L.).
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