We present an approach to induce localization of a Bose-Einstein condensate in a one-dimensional lattice under the influence of unitary quantum walk evolution using disordered quantum coin operation. We introduce a discrete-time quantum walk model in which the interference effect is modified to diffuse or strongly localize the probability distribution of the particle by assigning a different set of coin parameters picked randomly for each step of the walk, respectively. Spatial localization of the particle/state is explained by comparing the variance of the probability distribution of the quantum walk in position space using disordered coin operation to that of the walk using an identical coin operation for each step. Due to the high degree of control over quantum coin operation and most of the system parameters, ultracold atoms in an optical lattice offer opportunities to implement a disordered quantum walk that is unitary and induces localization. Here we present a scheme to use a Bose-Einstein condensate that can be evolved to the superposition of its internal states in an optical lattice and control the dynamics of atoms to observe localization. This approach can be adopted to any other physical system in which controlled disordered quantum walk can be implemented.
We analytically investigate the recently proposed and implemented discrete-time quantum walk based on a kicked Bose-Einstein condensate. We extend previous work on the effective dynamics by taking into account spontaneous emission due to the kicking light. Spontaneous emission affects both the internal and external degrees of freedom, arising from the entanglement between them during the walk dynamics. The result is a measurable degrading of the experimental walk signal that we characterise.
We simulate ultra-cold interacting Bosons in quasi-one-dimensional, incommensurate optical lattices. In the tight-binding limit, these lattices have pseudo-random on-site energies and thus can potentially lead to Anderson localization. We explore the parameter regimes that lead to Anderson localization and investigate the role of repulsive interactions, harmonic confinement and finite temperature. We find that interactions can obscure the exponential localization characteristic of Anderson localization, thus impeding the direct observation of this phenomenon when interactions are present.
We study the dynamical evolution of a two-dimensional Bose gas after a disorder potential quench. Depending on the initial conditions, the system evolves either to a thermal or a superfluid state. Using extensive quasi-exact numerical simulations, we show that the two phases are separated by a Kosterlitz-Thouless transition. The thermalization time is shown to be longer in the superfluid phase, but no critical slowing down is observed at the transition. The long-time phase diagram is well reproduced by a simple theoretical model. The spontaneous emergence of Kosterlitz-Thouless transitions fol…
We analytically investigate the recently proposed and implemented discrete-time quantum walk based on kicked ultra-cold atoms. We show how the internal level structure of the kicked atoms leads to the emergence of a relative light-shift phase immediately relevant for the experimental realization. Analytical solutions are provided for the momentum distribution for both the case of quantum resonance and the near-resonant quasimomenta.
We study the zero temperature phase diagram of the disordered spin-1 Bose-Hubbard model in a 2-dimensional square lattice. To this aim, we use a mean field Gutzwiller ansatz and a probabilistic mean field perturbation theory. The spin interaction induces two different regimes corresponding to a ferromagnetic and antiferromagnetic order. In the ferromagnetic case, the introduction of disorder reproduces analogous features of the disordered scalar Bose-Hubbard model, consisting in the formation of a Bose glass phase between Mott insulator lobes. In the antiferromagnetic regime the phase diagram …
Using the exact Bose-Fermi mapping, we study universal properties of ground-state density distributions and finite-temperature quantum critical behavior of one-dimensional hard-core bosons in trapped incommensurate optical lattices. Through the analysis of universal scaling relations in the quantum critical regime, we demonstrate that the superfluid to Bose glass transition and the general phase diagram of disordered hard-core bosons can be uniquely determined from finite-temperature density distributions of the trapped disordered system.
We realize a pair of simultaneous ten-step one-dimensional quantum walks with two walkers sharing coins, which we prove is analogous to the ten-step two-dimensional quantum walk with a single walker holding a four-dimensional coin. Our experiment demonstrates a ten-step quantum walk over an 11x11 two-dimensional lattice with a line defect, thereby realizing a localized walker state.
We propose a realistic scheme to implement discrete-time quantum walks in the Brillouin zone (i.e., in quasimomentum space) with a spinor Bose-Einstein condensate. Relying on a static optical lattice to suppress tunneling in real space, the condensate is displaced in quasimomentum space in discrete steps conditioned upon the internal state of the atoms, while short pulses periodically couple the internal states. We show that tunable twisted boundary conditions can be implemented in a fully natural way by exploiting the periodicity of the Brillouin zone. The proposed setup does not suffer from …
The present letter gives a rigorous way from quantum to classical random walks by introducing an independent random fluctuation and then taking expectations based on a path integral approach.
We present a novel scheme for universal quantum computation based on spinless interacting bosonic quantum walkers on a piecewise-constant graph, described by the two-dimensional Bose-Hubbard model. Arbitrary X and Z rotations are constructed, as well as an entangling two-qubit CPHASE gate and a SWAP gate. Quantum information is encoded in the positions of the walkers on the graph, as in previous quantum walk-based proposals for universal quantum computation, though in contrast to prior schemes this proposal requires a number of vertices only linear in the number of encoded qubits. It allows si…
Symmetrically evolving discrete quantum walk results in dynamic localization with zero mean displacement when the standard evolution operations are replaced by a temporal disorder evolution operation. In this work we show that the quantum ratchet action, that is, a directed transport in standard or disordered discrete-time quantum walk can be realized by introducing a pawl like effect realized by using a fixed coin operation at marked positions that is, different from the ones used for evolution at other positions. We also show that the combination of standard and disordered evolution operatio…
Different regimes of entanglement growth under measurement have been demonstrated for quantum many-body systems, with an entangling phase for low measurement rates and a disentangling phase for high rates (quantum Zeno effect). Here we study entanglement growth on a disordered Bose-Fermi mixture with the bosons playing the role of the effective self-induced measurement for the fermions. Due to the interplay between the disorder and a non-Abelian symmetry, the model features an entanglement growth resonance when the boson-fermion interaction strength is varied. With the addition of a magnetic f…
The understanding of disordered quantum systems is still far from being complete, despite many decades of research on a variety of physical systems. In this review we discuss how Bose-Einstein condensates of ultracold atoms in disordered potentials have opened a new window for studying fundamental phenomena related to disorder. In particular, we point our attention to recent experimental studies on Anderson localization and on the interplay of disorder and weak interactions. These realize a very promising starting point for a deeper understanding of the complex behaviour of interacting, disord…
We study a 2-D disordered time-discrete quantum walk based on 1-D `generalized elephant quantum walk' where an entangling coin operator is assumed and which paves the way to a new set of properties. We show that considering a given disorder in one direction, it is possible to control the degree of spreading and entanglement in the other direction. This observation helps assert that the random quantum walks of this ilk serve as a controllable decoherence channel with the degree of randomness being the tunable parameter and highlight the role of dimensionality in quantum systems regarding inform…
We present a discrete-time, one-dimensional quantum walk based on the entanglement between the momentum of ultracold rubidium atoms (the walk space) and two internal atomic states (the "coin" degree of freedom). Our scheme is highly flexible and can provide a platform for a wide range of applications such as quantum search algorithms, the observation of topological phases, and the realization of walks with higher dimensionality. Along with the investigation of the quantum-to-classical transition, we demonstrate the distinctive features of a quantum walk and contrast them to those of its classi…
We theoretically study the out-of-equilibrium dynamics in momentum space of a weakly interacting disordered Bose gas launched with a finite velocity. In the absence of interactions, coherent multiple scattering gives rise to a background of diffusive particles, on top of which a coherent backscattering interference emerges. We revisit this scenario in the presence of interactions, using a diagrammatic quantum transport theory. We find that the dynamics is governed by coupled kinetic equations describing the thermalization of the diffusive and coherent components of the gas. This phenomenon lea…
We investigate the properties of the superfluid phase in the three-dimensional disordered Bose-Hubbard model using Quantum Monte-Carlo simulations. The phase diagram is generated using Gaussian disorder on the on-site potential. Comparisons with box and speckle disorder show qualitative similarities leading to the re-entrant behavior of the superfluid. Quantitative differences that arise are controlled by the specific shape of the disorder. Statistics pertaining to disorder distributions are studied for a range of interaction strengths and system sizes, where strong finite-size effects are obs…
We study the quantum ground state phases of the one-dimensional disordered Bose--Hubbard model with attractive interactions, realized by a chain of superconducting transmon qubits or cold atoms. We map the phase diagram using perturbation theory and exact diagonalization. Compared to the repulsive Bose--Hubbard model, the quantum ground state behavior is dramatically different. At strong disorder of the on-site energies, all the bosons localize into the vicinity of a single site, contrary to the Bose glass behavior of the repulsive model. At weak disorder, depending on hopping, the ground stat…
We design a quantum probing protocol using Quantum Walks to investigate the Quantum Information spreading pattern. We employ Quantum Fisher Information, as a figure of merit, to quantify extractable information about an unknown parameter encoded within the Quantum Walk evolution. Although the approach is universal, we focus on the coherent static and dynamic disorder to investigate anomalous and classical transport as well as Anderson localization. Our results show that a Quantum Walk can be considered as a readout device of information about defects and perturbations occurring in complex netw…
We report on the theoretical analysis of bosonic and fermionic non-interacting systems in a discrete two-particle quantum walk affected by different kinds of disorder. We considered up to 100-step QWs with a spatial, temporal and space-temporal disorder observing how the randomness and the wavefunction symmetry non-trivially affect the final spatial probability distribution, the transport properties and the Shannon entropy of the walkers.
Transport phenomena play a crucial role in modern physics and applied sciences. Examples include the dissipation of energy across a large system, the distribution of quantum information in optical networks, and the timely modeling of spreading diseases. In this work, we experimentally prove the feasibility of disordered quantum walks to realize a quantum simulator that is able to model general subdiffusive phenomena, exhibiting a sublinear spreading in space over time. Our experiment simulates such phenomena by means of a finely controlled insertion of various levels of disorder during the evo…
We study the effects of long-range hopping and long-range inter-particle interactions on quantum walk of hard-core bosons in ideal and disordered one-dimensional lattices. We find that the range of hopping has a much more significant effect on the particle correlation dynamics than the range of interactions. We illustrate that long-range hopping makes the correlation diagrams asymmetric with respect to the sign of the interaction and examine the relative role of repulsive and attractive interactions on the dynamics of scattering by isolated impurities and Anderson localization in dis- ordered …
Each step in a quantum random walk is typically understood to have two basic components; a `coin-toss' which produces a random superposition of two states, and a displacement which moves each component of the superposition by different amounts. Here we suggest the realization of a walk in momentum space with a spinor Bose-Einstein condensate subject to a quantum ratchet realized with a pulsed, off-resonant optical lattice. By an appropriate choice of the lattice detuning, we show how the atomic momentum can be entangled with the internal spin states of the atoms. For the coin-toss, we propose …
Many disordered systems show a superdiffusive dynamics, intermediate between the diffusive one, typical of a classical stochastic process, and the so called ballistic behaviour, which is generally expected for the spreading in a quantum process. We have experimentally investigated the superdiffusive behaviour of a quantum walk (QW), whose dynamics can be related to energy transport phenomena, with a resolution which is high enough to clearly distinguish between different disorder regimes. By our experimental setup, the region between ballistic and diffusive spreading can be effectively scanned…
We investigate the emerging properties of quantum walks with temporal disorder engineered from a binary Markov chain with tailored correlation, $C$, and disorder strength, $r$. We show that when the disorder is weak -- $r \ll 1$ -- the introduction of negative correlation leads to a counter-intuitive higher production of spin-lattice entanglement entropy, $S_e$, than the setting with positive correlation, that is $S_e(-|C|)>S_e(|C|)$. These results show that negatively correlated disorder plays a more important role in quantum entanglement than it has been assumed in the literature.
We study Lévy walks in quenched disordered one-dimensional media, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling relations for the random-walk probability and for the resistivity in the equivalent electric problem, we obtain the asymptotic behavior of the mean square displacement as a function of the exponent characterizing the scatterers distribution. We demonstrate that in quenched media different average procedures can display different asymptotic behavior. In particular, we estimate the moments of the displacement averaged over processes starting f…
We show that the entanglement between the internal (spin) and external (position) degrees of freedom of a qubit in a random (dynamically disordered) one-dimensional discrete time quantum random walk (QRW) achieves its maximal possible value asymptotically in the number of steps, outperforming the entanglement attained by using ordered QRW. The disorder is modeled by introducing an extra random aspect to QRW, a classical coin that randomly dictates which quantum coin drives the system's time evolution. We also show that maximal entanglement is achieved independently of the initial state of the …
We observe large density modulations in time-of-flight images of elongated Bose-Einstein condensates, initially confined in a harmonic trap and in the presence of weak disorder. The development of these modulations during the time-of-flight and their dependence with the disorder are investigated. We render an account of this effect using numerical and analytical calculations. We conclude that the observed large density modulations originate from the weak initial density modulations induced by the disorder, and not from initial phase fluctuations (thermal or quantum).
One-dimensional discrete-time quantum walk has played an important role in development of quantum algorithms and protocols for different quantum simulations. The speedup observed in quantum walk algorithms is attributed to quantum interference and coherence of the wave packet in position space. Similarly, localization in quantum walk due to disorder is also attributed to quantum interference effect. Therefore, it is intriguing to have a closer look and understand the way quantum interference manifests in different forms of quantum walk dynamics. Quantum coherence in the system is responsible f…
We study the equilibrium properties of the one-dimensional disordered Bose-Hubbard model by means of a gauge-adaptive tree tensor network variational method suitable for systems with periodic boundary conditions. We compute the superfluid stiffness and superfluid correlations close to the superfluid to glass transition line, obtaining accurate locations of the critical points. By studying the statistics of the exponent of the power-law decay of the correlation, we determine the boundary between the superfluid region and the Bose glass phase in the regime of strong disorder and in the weakly in…
We propose a scheme to implement the simplest and best-studied version of quantum random walk, the discrete Hadamard walk, in one dimension using coherent macroscopic sample of ultracold atoms, Bose-Einstein condensate (BEC). Implementation of quantum walk using BEC gives access to the familiar quantum phenomena on a macroscopic scale. This paper uses rf pulse to implement Hadamard operation (rotation) and stimulated Raman transition technique as unitary shift operator. The scheme suggests implementation of Hadamard operation and unitary shift operator while the BEC is trapped in long Rayleigh…
We report our research on disordered complex systems using cold gases and trapped ions, and address the possibility of using complex systems for quantum information processing. Two simple paradigmatic models of disordered complex systems are revisited here. The first one corresponds to a short range disordered Ising Hamiltonian (spin glasses), which can be implemented with a Bose-Fermi (Bose-Bose) mixture in a disordered optical lattice. The second model we address here is a long range disordered Hamiltonian, characteristic of neural networks (Hopfield model), which can be implemented in a cha…
Systems with purely off-diagonal disorder have peculiar features such as the localization-delocalization transition and long-range correlations in their wavefunctions. To motivate possible experimental studies of the physics of off-diagonal disorder, we study in detail disordered discrete-time quantum walk in a finite chain, where the diagonal disorder can be set to zero by construction. Starting from a transfer matrix approach, we show, both theoretically and computationally, that the dynamics of the quantum walk with disorder manifests all the main features of off-diagonal disorder. We also …
Quantum walks constitute a versatile platform for simulating transport phenomena on discrete graphs including topological material properties while providing a high control over the relevant parameters at the same time. To experimentally access and directly measure the topological invariants of quantum walks we implement the scattering scheme proposed by Tarasinski et al.[Phys. Rev. A 89, 042327 (2014)] in a photonic time multiplexed quantum walk experiment. The tunable coin operation provides opportunity to reach distinct topological phases, and accordingly to observe the corresponding topolo…
The continuous-time quantum walk on the underlying graphs of association schemes have been studied, via the algebraic combinatorics structures of association schemes, namely semi-simple modules of their Bose-Mesner and (reference state dependent) Terwilliger algebras. By choosing the (walk) starting site as a reference state, the Terwilliger algebra connected with this choice turns the graph into the metric space, hence stratifies the graph into a (d+1) disjoint union of strata, where the amplitudes of observing the continuous-time quantum walk on all sites belonging to a given stratum are the…
Recent experiments with ultracold atoms in an optical lattice have realized cavity-mediated long-range interaction and observed the emergence of a supersolid phase and a density wave phase in addition to Mott insulator and superfluid phases. Here we consider theoretically the effect of uncorrelated disorder on the phase diagram of this system and study the two-dimensional Bose-Hubbard model with cavity-mediated long-range interactions and uncorrelated diagonal disorder. We also study the phase diagram of the extended Bose-Hubbard model with nearest-neighbor interactions in the presence of unco…
We investigate numerically and theoretically the effect of spatial disorder on two-dimensional split-step discrete-time quantum walks with two internal "coin" states. Spatial disorder can lead to Anderson localization, inhibiting the spread of quantum walks, putting them at a disadvantage against their diffusively spreading classical counterparts. We find that spatial disorder of the most general type, i.e., position-dependent Haar random coin operators, does not lead to Anderson localization but to a diffusive spread instead. This is a delocalization, which happens because disorder places the…
We present a detailed numerical study of the dynamics of a disordered one-dimensional Bose-Einstein condensates in position and momentum space. We particularly focus on the region where non-linearity and disorder simultaneously effect the time propagation of the condensate as well as the possible interference between various parts of the matter wave. We report oscillation between spatially extended and localized behavior for the propagating condensate which dies down with increasing non-linearity. We also report intriguing behavior of the phase fluctuation and the coherence properties of the m…
We theoretically investigate the localization of an expanding Bose-Einstein condensate with repulsive atom-atom interactions in a disordered potential. We focus on the regime where the initial inter-atomic interactions dominate over the kinetic energy and the disorder. At equilibrium in a trapping potential and for small disorder, the condensate shows a Thomas-Fermi shape modified by the disorder. When the condensate is released from the trap, a strong suppression of the expansion is obtained in contrast to the situation in a periodic potential with similar characteristics. This effect crucial…
We study a spin-1/2-particle moving on a one dimensional lattice subject to disorder induced by a random, space-dependent quantum coin. The discrete time evolution is given by a family of random unitary quantum walk operators, where the shift operation is assumed to be deterministic. Each coin is an independent identically distributed random variable with values in the group of two dimensional unitary matrices. We derive sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization. Put differently, the tunneling probability between two…
We investigate how the introduction of different types of disorder affects the generation of entanglement between the internal (spin) and external (position) degrees of freedom in one-dimensional quantum random walks (QRW). Disorder is modeled by adding another random feature to QRW, i.e., the quantum coin that drives the system's evolution is randomly chosen at each position and/or at each time step, giving rise to either dynamic, fluctuating, or static disorder. The first one is position-independent, with every lattice site having the same coin at a given time, the second has time and positi…
