Non-equilibrium transition from dissipative quantum walk to classical random walk

Abstract

We have investigated the time-evolution of a free particle in interaction with a phonon thermal bath, using the tight-binding approach. A dissipative quantum walk can be defined and many important non-equilibrium decoherence properties can be investigated analytically. The non-equilibrium statistics of a pure initial state have been studied. Our theoretical results indicate that the evolving wave-packet shows the suppression of Anderson's boundaries (ballistic peaks) by the presence of dissipation. Many important relaxation properties can be studied quantitatively, such as von Neumann's entropy and quantum purity. In addition, we have studied Wigner's function. The time-dependent behavior of the quantum entanglement between a free particle -in the lattice-and the phonon bath has been characterized analytically. This result strongly suggests the non-trivial time-dependence of the off-diagonal elements of the reduced density matrix of the system. We have established a connection between the quantum decoherence and the dissipative parameter arising from interaction with the phonon bath. The time-dependent behavior of quantum correlations has also been pointed out, showing continuous transition from quantum random walk to classical random walk, when dissipation increases.

Quantum correlations and coherence between two particles interacting with a thermal bath

“…To end this section we comment that (37) agrees with numerical calculations presented in [24] (S (1) is linear for t D << 1). In particular when D = 0 and noting that I n (0) = δ n,0 we get that S (1) = 0, and in general for D = 0 we get S (1) (t) > 0, ∀t > 0.…”

“…This marginal solution corresponds to the one-particle density matrix with IC ρ(t = 0) = |0 0|, and shows when D ≫ Ω/ asymptotically the same behavior than a classical RW [24]. A result that is not entirely surprising because in the Hamiltonian (1) each particle is originally non-interacting between them.…”

“…where ω c is the frequency cut off in the Ohmic approximation. This Hamiltonian is the natural extension of van Kampen's Hamiltonian for two spinless particles in the lattice [9,24]. From the QME (8) we can see that the term −D 2 a † 12 a 12 ρ + ρa 12 a † 12 + a 12 a † 12 ρ + ρa † 12 a 12 ,…”

Bath-induced correlations in an infinite-dimensional Hilbert space

“…is the one-particle Purity (with independent bath [24]). Thus a common bath produces a difference in the total purity…”

“…(8)). Here, Fig.3(b) corresponds to the case when the two particles do not interact with the bath (D = 0), the inset shown the one-axis projection of one tight-binding quantum walk [24]. In Fig.3(d) P s1,s2 (t ′ = t D ) corresponds to the high dissipative regime.…”