Quantum walks in artificial electric and gravitational fields

Molfetta, Giuseppe Di; Brachet, Marc; Debbasch, Fabrice

doi:10.1016/j.physa.2013.11.036

Cited by 90 publications

(197 citation statements)

References 51 publications

(66 reference statements)

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“…Let us simply mention two differences. First, unlike the 'U' parallel transporters in LGTs, gauge fields do not have to be added by hand to the DTQW dynamics, as the connection is already part of the basic definition of DTQWs and most DTQWs are by definition locally gauge invariant [29]. Second, the difference operators (discrete derivatives) which arise in conjunction with the local gauge invariance of DTQWs are more complicated than the usual finite difference operators used in lattice gauge theories.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…The formal continuous limit of the DTQWs (1) can be determined by the method used in [28,29,32,[34][35][36]: we first introduce a (dimensionless) spacetime-lattice step l and interpret any (j, p, q)-dependent quantity Q j,p,q as the value taken by a function…”

Section: Electromagnetic Dtqwsmentioning

confidence: 99%

“…Recently, such connexions have been extensively extended to Dirac fermions coupled to gauge fields [25][26][27][28][29][30][31][32]. More precisely, 1D DTQWs have been proposed which reproduce the dynamics of Dirac fermions coupled to arbitrary electric [25][26][27] and/or gravitational [28][29][30][31] fields, and a 2D DTQW simulating the coupling of a Dirac fermion to a constant uniform magnetic field has been proposed in [32].…”

Section: Introductionmentioning

confidence: 99%

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Quantum walks and discrete gauge theories

2016

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A particular example is produced to prove that quantum walks can be used to simulate fullfledged discrete gauge theories. A new family of 2D walks is introduced and its continuous limit is shown to coincide with the dynamics of a Dirac fermion coupled to arbitrary electromagnetic fields. The electromagnetic interpretation is extended beyond the continuous limit by proving that these DTQWs exhibit an exact discrete local U (1) gauge invariance and possess a discrete gauge-invariant conserved current. A discrete gauge-invariant electromagnetic field is also constructed and that field is coupled to the conserved current by a discrete generalization of Maxwell equations. The dynamics of the DTQWs under crossed electric and magnetic fields is finally explored outside the continuous limit by numerical simulations. Bloch oscillations and the so-called E × B drift are recovered in the weak-field limit. Localization is observed for some values of the gauge fields.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Electromagnetic Dtqwsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum walks and discrete gauge theories

2016

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“…Simulation of Dirac particle dynamics in the presence of the external abelian and nonabelian gauge field by DQW has been recently reported [16,17]. Other recent works [18,19] investigated the inhomogeneous DQW that mimics the Dirac particle dynamics under the influence of external gauge-potential and curved space-time as a background. Two-step stroboscopic DQW with space-time dependent U(2) coin operator was used to produce gravitational and gauge potential effect in single Dirac fermion, but their approach was unable to capture mass, gravity and gauge potential in one Hamiltonian [18,19].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2019

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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...

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“…In particular, it is well known that the continuous limit of various QWs formally coincides with the Dirac equation [34][35][36]. It has been shown recently that several QWs can model the dynamics of free Dirac fermions coupled to electromagnetic [37,38] and relativistic gravitational fields [3,[39][40][41].…”

Section: Introductionmentioning

confidence: 99%

Quantum walks as simulators of neutrino oscillations in a vacuum and matter

Molfetta

Perez

2016

New J. Phys.

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We analyze the simulation of Dirac neutrino oscillations using quantum walks, both in a vacuum and in matter. We show that this simulation, in the continuum limit, reproduces a set of coupled Dirac equations that describe neutrino flavor oscillations, and we make use of this to establish a connection with neutrino phenomenology, thus allowing one to fix the parameters of the simulation for a given neutrino experiment. We also analyze how matter effects for neutrino propagation can be simulated in the quantum walk. In this way, important features, such as the MSW effect, can be incorporated. Thus, the simulation of neutrino oscillations with the help of quantum walks might be useful to illustrate these effects in extreme conditions, such as the solar interior or supernovae.

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Quantum walks in artificial electric and gravitational fields

Cited by 90 publications

References 51 publications

Quantum walks and discrete gauge theories

Quantum walks and discrete gauge theories

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

Quantum walks as simulators of neutrino oscillations in a vacuum and matter

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