Quantum walks in weak electric fields and Bloch oscillations

Abstract

Bloch oscillations appear when an electric field is superimposed on a quantum particle that evolves on a lattice with a tight-binding Hamiltonian (TBH), i.e., evolves via what we call an electric TBH; this phenomenon will be referred to as TBH Bloch oscillations. A similar phenomenon is known to show up in so-called electric discrete-time quantum walks (DQWs) [C. Cedzich et al., Phys. Rev. Lett. 111, 160601 (2013);] this phenomenon will be referred to as DQW Bloch oscillations. This similarity is particularly salient when the electric field of the DQW is weak. For a wide, i.e., spatially extended, initial condition, one numerically observes semiclassical oscillations, i.e., oscillations of a localized particle, for both the electric TBH and the electric DQW. More precisely, the numerical simulations strongly suggest that the semiclassical DQW Bloch oscillations correspond to two counterpropagating semiclassical TBH Bloch oscillations. In this work it is shown that, under certain assumptions, the solution of the electric DQW for a weak electric field and a wide initial condition is well approximated by the superposition of two continuous-time expressions, which are counterpropagating solutions of an electric TBH whose hopping amplitude is the cosine of the arbitrary coin-operator mixing angle. In contrast, if one wishes the continuous-time approximation to hold for spatially localized initial conditions, one needs at least the DQW to be lazy, as suggested by numerical simulations and by the fact that this has been proven in the case of a vanishing electric field [F. W. Strauch, Phys. Rev. A 74, 030301(R) ( 2006)].

Bloch-like super-oscillations and unidirectional motion of phase driven quantum walkers

“…1(a). Thanks the presence FE operator, such behavior is usually associated with Bloch Oscillations (BOs) [30,32,36], an emerging phenomenon of solid state physics in which an electron is loaded in a periodic potential subjected to a constant electric field. This driving, which we treat as Bloch-like oscillations, is a quasi-stationary dynamics for a quantum walker that persists over a long-time whenever the phase increment Φ 0 ≪ 1.…”

Clifford algebra from quantum automata and unitary Wilson fermions

“…If h does not depend on the point x, i.e., if h is translationally invariant, then one can check by considering Eq. ( 14) for Expression (21) that each Fourier coefficient satisfies the equation…”

“…That the Hamiltonian H is (ultra)local is in contrast with the case of the well-known effective Hamiltonian of the DQW. Note that both Hamiltonians are related by a proportionality constant in Fourier space [21].…”

Revivals in One-dimensional Quantum Walks with a Time and Spin-dependent Phase Shift

“…Owing to their quantum mechanical resources, i.e., quantum superposition, quantum interference, and entanglement, QWs hold the promise to develop new algorithms for computations on quantum computers [2][3][4][5][6][7][8][9][10]. In Physics QWs provide a versatile platform to simulate various physical phenomena, e.g., topological phases [11][12][13][14][15][16][17][18][19][20][21][22][23][24], Anderson localization [25][26][27][28][29], Bloch Oscillations [30][31][32], molecular binding [33][34][35], and Hofstadter spectrum [36,37], to name just a few. Due to their broad spectrum of applications, QWs have been realized in experiments using different physical systems, e.g., neutral atoms trapped in optical lattices [38,39], trapped ions on a line [40][41][42], photons in free space [43,44], correlated photons on continuously evanescently coupled waveguides [45], and integrated photonics…”