Simulating Dirac Hamiltonian in curved space-time by split-step quantum walk

Abstract

Dirac particle represents a fundamental constituent of our nature. Simulation of Dirac particle dynamics by a controllable quantum system using quantum walks will allow us to investigate the nonclassical nature of dynamics in its discrete form. In this work, starting from a modified version of onespatial dimensional general inhomogeneous split-step discrete quantum walk we derive an effective Hamiltonian which mimics a single massive Dirac particle dynamics in curved (1+1) space-time dimension coupled to U(1) gauge potential-which is a forward step towards the simulation of the unification of electromagnetic and gravitational forces in lower dimension and at the single particle level. Implementation of this simulation scheme in simple qubit-system has been demonstrated. We show that the same Hamiltonian can represent (2+1) space-time dimensional Dirac particle dynamics when one of the spatial momenta remains fixed. We also discuss how we can include U(N) gauge potential in our scheme, in order to capture other fundamental force effects on the Dirac particle. The emergence of curvature in the two-particle split-step quantum walk has also been investigated while the particles are interacting through their entangled coin operation.operations-mimics the Dirac evolution under influence of gravitational waves in (2+1) dimension-was also recently reported in [24]. In [25], it is shown that the SS-DQW, where the coin parameters are space and timestep independent, can capture properties of the discretized Dirac particle dynamics in flat (1+1) dimension, while conventional DQW is unable to capture all properties of it. This motivates us to generalize the SS-DQW operation and study the consequences of it.In this paper starting from a slightly modified version of the single-step split-step DQW (SS-DQW) [26] whose coin operators are time and position-step dependent (inhomogeneous both in time and space), we derive a SS-DQW version of the (1+1) dimensional massive Dirac particle Hamiltonian under the influence of the U(1) gauge potential in curved space-time. This scheme is realizable in various physical table-top system as the SS-DQW has been proposed and successfully implemented in various systems like cold atoms [27], superconducting qubits [28,29], photonic systems [30,31]. Our scheme can also describe the (2+1) dimensional Dirac Hamiltonian in curved space-time when one component of momentum of the particle remains fixed. We provide realization of our simulation scheme using qubit systems. This scheme doesn't require any prior encoding or decoding, nor it demands extra conditions on the coin parameters in order to satisfy the boundedness (well-defined eigenvalues) of the generator which is the effective Hamiltonian in our case, i.e.the unitary operator should start evolution from identity while the parameter of the corresponding lie group evolves from zero to a nonzero real value. Our coin operations are general U(2) group elements in coinspace. After considering all the terms up to first order in time-step s...

Pseudospectral computational methods for the time-dependent Dirac equation in static curved spaces

“…where the Cauchy-Schwarz inequality was used to obtain (55). Expanding (55) ψ n k 2 . This concludes the proof for C 1.…”

“…One of the goals of this article is to extend this numerical scheme to the Dirac equation in curved space.Recently, the Dirac equation in curved space-time has gained important interest in some applications such as condensed matter physics for describing the dynamics of charge carriers in deformed 2D Dirac materials [44,45], as well as astrophysics for fermion tunneling in black holes [46,47,48,49]. In its discrete version, it has been considered from the lattice Boltzmann technique point of view [50,51,52] and as continuous limits of quantum walks [53,54,55]. However, the literature on numerical methods for this equation is scarse.…”

Unruh effect for interacting particles with ultracold atoms

“…This strategy allows for a systematic quantum simulation approach. As shown in [54] (see also [86][87][88]) one can engineer the motion of Dirac fermions in optical metrics [89] by making the Fermi velocity position dependent, that is by engineering position-dependent tunneling rates on the lattice, which can be equivalently translated in random walk processes [90][91][92][93]. One can apply such procedure to any lattice formulation of Dirac Hamiltonian [94], as for instance artificial graphene obtained from ultracold fermions in a brick-wall lattice [95] (see [96,97] and [98] for realizations of hexagonal optical lattices).…”

Quantum walks and Dirac cellular automata on a programmable trapped-ion quantum computer

“…The correspondence between DQWs and the dynamics of Dirac particles suggests that the QWs formalism is as a viable approach to reproduce a variety of phenomena underpinned by Dirac particle dynamics in both the high-and low-energy regime 22,39,43 . Quantum simulations of free quantum field theory 43 , Yang-Mills gauge-field on fermionic matter 55 , as well as the effect of mass and space-time curvature on entanglement between accelerated particles 20,56,57 have been reported and probing quantum field theory from the algorithmic perspective in an active field of research. However, the circuit complexity for position-dependent coin operations needed for simulating these effects will increase with the complexity of the evolution, which means further improvements in quantum hardware will be necessary for their realization.…”

“…uantum walks (QWs) are the quantum analog of classical random walks, in which the walker steps forward or backward along a line based on a coin flip. In a QW, the walker proceeds in a quantum superposition of paths, and the resulting interference forms the basis of a wide variety of quantum algorithms, such as quantum search [1][2][3][4][5] , graph isomorphism problems [6][7][8] , ranking nodes in a network [9][10][11][12] , and quantum simulations, which mimic different quantum systems at the low and high energy scale [13][14][15][16][17][18][19][20][21][22] . In the discrete-time QW (DQW) 23,24 , a quantum coin operation is introduced to prescribe the direction in which the particle moves in position space at each discrete step.…”