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Discrete-Time Quantum Walk of a Bose-Einstein Condensate in Momentum Space

Siamak Dadrasmarani

Published Jan 1, 2018• 1 citation
OpenAlex

Abstract

Randomness is the essence of many processes in nature and human society. It can provide important insights into phenomena as diverse as disease transmission, financial markets, and signal processing. Quantum randomness is intrinsically different from classical stochasticity since it is affected by interference and entanglement. Quantum walks profit from the entanglement that is built up between a walker's position and the "coin" degree of freedom that determines the direction of a walker's motion. This entanglement makes quantum walks promising candidates for the implementation of quantum computational algorithms and as a detector of quantum behavior. While several proof-of-principle quantum walk experiments have recently been implemented, the walk scheme we demonstrate here involves the synthesis of the atom-optics kicked rotor with the dynamics of a state selective ratchet realized with a Bose-Einstein condensate. Here we present a discrete-time quantum walk that uses the momentum of ultra-cold rubidium-87 atoms as the walk space and two internal atomic states as the coin degree of freedom. To implement each step of the walk, i.e. a coin toss followed by a unitary shift operator, we use a microwave pulse, which produces a superposition of two internal states, and a resonant quantum ratchet, which entangles the internal and external degrees of freedom. Our quantum ratchet, whose direction at each step depends on the result of the coin toss, is realized by the atom-optics kicked rotor. We demonstrate the principle features of a standard quantum walk, contrasting them to the behavior of a classical walk. In addition, by manipulating either the walk or coin operator we show how the walk dynamics can be biased or reversed. Furthermore, we show the quantum-to-classical transition in our walk, via randomizing the mixing between the two internal states during each coin toss. Our walk offers distinct advantages arising from the robustness of its dynamics in momentum space, extendability to higher dimensions, and possible dependence on many-body correlations.

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Randomness is the essence of many processes in nature and human society. It can provide important insights into phenomena as diverse as disease transmission, financial markets, and signal processing. Quantum randomness is intrinsically different from classical stochasticity since it is affected by interference and entanglement. Quantum walks profit from the entanglement that is built up between a walker's position and the "coin" degree of freedom that determines the direction of a walker's motion. This entanglement makes quantum walks promising candidates for the implementation of quantum computational algorithms and as a detector of quantum behavior. While several proof-of-principle quantum walk experiments have recently been implemented, the walk scheme we demonstrate here involves the synthesis of the atom-optics kicked rotor with the dynamics of a state selective ratchet realized with a Bose-Einstein condensate. Here we present a discrete-time quantum walk that uses the momentum of ultra-cold rubidium-87 atoms as the walk space and two internal atomic states as the coin degree of freedom. To implement each step of the walk, i.e. a coin toss followed by a unitary shift operator, we use a microwave pulse, which produces a superposition of two internal states, and a resonant quantum ratchet, which entangles the internal and external degrees of freedom. Our quantum ratchet, whose direction at each step depends on the result of the coin toss, is realized by the atom-optics kicked rotor. We demonstrate the principle features of a standard quantum walk, contrasting them to the behavior of a classical walk. In addition, by manipulating either the walk or coin operator we show how the walk dynamics can be biased or reversed. Furthermore, we show the quantum-to-classical transition in our walk, via randomizing the mixing between the two internal states during each coin toss. Our walk offers distinct advantages arising from the robustness of its dynamics in momentum space, extendability to higher dimensions, and possible dependence on many-body correlations.
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 to use a microwave pulse to mix these internal states. We present experimental results showing an optimized quantum ratchet, and through a series of simulations, demonstrate how our proposal gives extraordinary control of the quantum walk. This should allow for the investigation of possible biases, and classical-to-quantum dynamics in the presence of natural and engineered noise.
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 classical counterpart. Also, by manipulating either the walk or coin operator, we show how the walk dynamics can be steered or even reversed.
We report on a discrete-time quantum walk that uses the momentum of ultra-cold rubidium-87 atoms as the walk space and two internal atomic states as the coin degree of freedom. Each step of the walk consists of a coin toss (a microwave pulse) followed by a unitary shift operator (a resonant ratchet pulse). We carry out a comprehensive experimental study on the effects of various parameters, including the strength of the shift operation, coin parameters, noise, and initialization of the system on the behavior of the walk. The walk dynamics can be well controlled in our experiment; potential applications include atom interferometry and engineering asymmetric walks.
We have realized a quantum walk in momentum space with a rubidium spinor Bose-Einstein condensate by applying a periodic kicking potential as a walk operator and a resonant microwave pulse as a coin toss operator. The generated quantum walks appear to be stable for up to ten steps and then quickly transit to classical walks due to spontaneous emissions induced by laser beams of the walk operator. We investigate these quantum to classical walk transitions by introducing well-controlled spontaneous emissions with an external light source during quantum walks. Our findings demonstrate a scheme to control the robustness of the quantum walks and can also be applied to other cold atom experiments involving spontaneous emissions.
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 or 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.
In most widely discussed discrete time quantum walk model, after every unitary shift operator, the particle evolves into the superposition of position space and settles down in one of its basis states, loosing entanglement in the coin space in the new position. The Hadamard operation is applied to let the particle to evolve into the superposition in the coin space and the walk is iterated. We present a model with a additional degree of freedom for the unitary shift operator U′U^{\prime}U′. The unitary operator with additional degree of freedom will evolve the quantum particle into superposition of position space retaining the entanglement in coin space. This eliminates the need for quantum coin toss (Hadamard operation) after every unitary displacement operation as used in most widely studied version of the discrete time quantum walk model. This construction is easily extended to a multiple particle quantum walk and in this article we extend it for a pair of particles in pure state entangled in coin degree of freedom by simultaneously subjecting it to a pair of unitary displacement operators which were constructed for single particle. We point out that unlike for single particle quantum walk, upon measurement of its position after NNN steps, the entangled particles are found together with 1/2 probability and at different positions with 1/2 probability. This can act as an advantage in applications of the quantum walk. A special case is also treated using a complex physical system such as, inter species two-particle entangled Bose-Einstein condensate, as an example.