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arXiv:1501.06610 · Physical Review A

Decoherent time-dependent transport beyond the Landauer-Büttiker formulation: A quantum-drift alternative to quantum jumps

Lucas J. Fernández-Alcázar, Horacio M. Pastawski

Published Feb 19, 2015• 16 citations
arXivHTMLPDF

Abstract

We develop and implement a model for decoherence in time-dependent transport. Inspired in a dynamical formulation of the Landauer-B\"uttiker equations, it boils down into a form of wave function that undergoes a smooth stochastic drift of the phase in a local basis, the quantum-drift (QD) model. This drift is nothing else but a local energy fluctuation. Unlike quantum-jumps (QJ) models, no jumps are present in the density as the evolution is unitary. As a first application, we address the transport through a resonant state $|0\ensuremath{\rangle}$ that undergoes decoherence. Its numerical resolution shows the equivalence with the decoherent steady-state transport in presence of a B\"uttiker's voltage probe. In order to test the dynamics we consider two many-spin systems, which are cases of experimental interest, where a local energy fluctuation is a natural phenomenon. A two-spin system is reduced to a two-level system (TLS) that oscillates among $|0\ensuremath{\rangle}\phantom{\rule{0.16em}{0ex}}\ensuremath{\equiv}\phantom{\rule{0.16em}{0ex}}|\ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\rangle}$ and $|1\ensuremath{\rangle}\ensuremath{\equiv}\phantom{\rule{0.16em}{0ex}}|\ensuremath{\downarrow}\ensuremath{\uparrow}\ensuremath{\rangle}$. We show that the QD model recovers not only the exponential damping of the oscillations in the low perturbation regime, but also the nontrivial bifurcation of the damping rates at a critical point, i.e., the quantum dynamical phase transition. We also address the spin-wave-like dynamics of local polarization in a spin chain. By averaging over ${N}_{\mathbf{s}}$ realizations, the QD solution has about half the dispersion respect to the mean dynamics than QJ. By evaluating the Loschmidt echo (LE), we find that the pure states $|0\ensuremath{\rangle}\phantom{\rule{0.16em}{0ex}}\ensuremath{\equiv}\phantom{\rule{0.16em}{0ex}}|\ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\rangle}$ and $|1\ensuremath{\rangle}\ensuremath{\equiv}\phantom{\rule{0.16em}{0ex}}|\ensuremath{\downarrow}\ensuremath{\uparrow}\ensuremath{\rangle}$ are quite robust against the local decoherence. In contrast, the LE, and hence coherence, decays faster when the system is in a superposition state $(|\ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|\ensuremath{\downarrow}\ensuremath{\uparrow}\ensuremath{\rangle})/\sqrt{2}$, which is consistent with the general trend recently observed in spin systems through NMR. Because of its simple implementation, the method is well suited to assess decoherent transport problems as well as to include decoherence in both one-body and many-body dynamics.

DOI
10.1103/physreva.91.022117

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We develop and implement a model for decoherence in time-dependent transport. Inspired in a dynamical formulation of the Landauer-B"uttiker equations, it boils down into a form of wave function that undergoes a smooth stochastic drift of the phase in a local basis, the quantum-drift (QD) model. This drift is nothing else but a local energy fluctuation. Unlike quantum-jumps (QJ) models, no jumps are present in the density as the evolution is unitary. As a first application, we address the transport through a resonant state ∣0\ensuremath⟩|0\ensuremath{\rangle}∣0\ensuremath⟩ that undergoes decoherence. Its numerical resolution shows the equivalence with the decoherent steady-state transport in presence of a B"uttiker's voltage probe. In order to test the dynamics we consider two many-spin systems, which are cases of experimental interest, where a local energy fluctuation is a natural phenomenon. A two-spin system is reduced to a two-level system (TLS) that oscillates among ∣0\ensuremath⟩\ensuremath≡∣\ensuremath↑\ensuremath↓\ensuremath⟩|0\ensuremath{\rangle}\phantom{\rule{0.16em}{0ex}}\ensuremath{\equiv}\phantom{\rule{0.16em}{0ex}}|\ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\rangle}∣0\ensuremath⟩\ensuremath≡∣\ensuremath↑\ensuremath↓\ensuremath⟩ and ∣1\ensuremath⟩\ensuremath≡∣\ensuremath↓\ensuremath↑\ensuremath⟩|1\ensuremath{\rangle}\ensuremath{\equiv}\phantom{\rule{0.16em}{0ex}}|\ensuremath{\downarrow}\ensuremath{\uparrow}\ensuremath{\rangle}∣1\ensuremath⟩\ensuremath≡∣\ensuremath↓\ensuremath↑\ensuremath⟩. We show that the QD model recovers not only the exponential damping of the oscillations in the low perturbation regime, but also the nontrivial bifurcation of the damping rates at a critical point, i.e., the quantum dynamical phase transition. We also address the spin-wave-like dynamics of local polarization in a spin chain. By averaging over Ns{N}_{\mathbf{s}}Ns​ realizations, the QD solution has about half the dispersion respect to the mean dynamics than QJ. By evaluating the Loschmidt echo (LE), we find that the pure states ∣0\ensuremath⟩\ensuremath≡∣\ensuremath↑\ensuremath↓\ensuremath⟩|0\ensuremath{\rangle}\phantom{\rule{0.16em}{0ex}}\ensuremath{\equiv}\phantom{\rule{0.16em}{0ex}}|\ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\rangle}∣0\ensuremath⟩\ensuremath≡∣\ensuremath↑\ensuremath↓\ensuremath⟩ and ∣1\ensuremath⟩\ensuremath≡∣\ensuremath↓\ensuremath↑\ensuremath⟩|1\ensuremath{\rangle}\ensuremath{\equiv}\phantom{\rule{0.16em}{0ex}}|\ensuremath{\downarrow}\ensuremath{\uparrow}\ensuremath{\rangle}∣1\ensuremath⟩\ensuremath≡∣\ensuremath↓\ensuremath↑\ensuremath⟩ are quite robust against the local decoherence. In contrast, the LE, and hence coherence, decays faster when the system is in a superposition state (∣\ensuremath↑\ensuremath↓\ensuremath⟩\ifmmode±\else\textpm\fi∣\ensuremath↓\ensuremath↑\ensuremath⟩)/2(|\ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|\ensuremath{\downarrow}\ensuremath{\uparrow}\ensuremath{\rangle})/\sqrt{2}(∣\ensuremath↑\ensuremath↓\ensuremath⟩\ifmmode±\else\textpm\fi∣\ensuremath↓\ensuremath↑\ensuremath⟩)/2​, which is consistent with the general trend recently observed in spin systems through NMR. Because of its simple implementation, the method is well suited to assess decoherent transport problems as well as to include decoherence in both one-body and many-body dynamics.
The decoherence of quantum states defines the transition between the quantum world and classical physics. Decoherence or, analogously, quantum mechanical collapse events pose fundamental questions regarding the interpretation of quantum mechanics and are technologically relevant because they limit the coherent information processing performed by quantum computers. We have discovered that the transition regime enables a novel type of matter transport. Applying this discovery, we present nanoscale devices in which decoherence, modeled by random quantum jumps, produces fundamentally novel phenomena by interrupting the unitary dynamics of electron wave packets. Noncentrosymmetric conductors with mesoscopic length scales act as two-terminal rectifiers with unique properties. In these devices, the inelastic interaction of itinerant electrons with impurities acting as electron trapping centers leads to a novel steady state characterized by partial charge separation between the two leads, or, in closed circuits to the generation of persistent currents. The interface between the quantum and the classical worlds therefore provides a novel transport regime of value for the realization of a new category of mesoscopic electronic devices.
I will focus on the Loschmidt Echo [LE], a concept developed and pursed at Cordoba. It is the recovered fraction of a localized excitation after a spreading period followed by an imperfect time reversal procedure [1]. In Solid State NMR, the LE has allowed us to quantify the decoherence and irreversibility induced by an uncontrolled environment. Notably complex many-body dynamics makes the system particularly sensitive to environmental disturbances presenting a decoherence rate that becomes perturbation independent beyond some small threshold. These experiments and the theoretical analysis based on the Feynman's path integral, summarized at a tutorial level, fueled the field of dynamical quantum chaos. The quest for a perturbation independent decoherence as an emergent phenomenon in thermodynamic limit, lead us to discuss other dynamical observables that depend non-analytically on the environment strength, i.e. that undergo a quantum dynamical phase transition QDPT [2]. GPU based high performance computing boosts the evaluation of the LE [3], allowing us to assess thermalization and how the Metal-Insulator transition (also a QDPT) emerges in interacting many-body systems. [1] Loschmidt Echo A. Gousev, R.A. Jalabert, HMP and D.A. Wisniacki. Scholarpedia 7, 11687 (2012) http://www.scholarpedia.org/article/Loschmidt_echo [2] Environmentally induced quantum dynamical phase transition in the spin swap operation, G.A.Alvarez, E.P.Danieli, P.R. Levstein, and HMP, J. Chem.Phys. 124, 1 (2006) [3] Interaction-disorder competition in a spin system evaluated through the Loschmidt echo P.R. Zangara, A.D. Dente, A. Iucci, P.R. Levstein, and HMP, Phys. Rev. B 88, 195106 (2013)
Classical Brownian motion has well been investigated since the pioneering work of Einstein, which inspired mathematicians to lay the theoretical foundation of stochastic processes. A stochastic formulation for quantum dynamics of dissipative systems described by the system-plus-bath model has been developed and found many applications in chemical dynamics, spectroscopy, quantum transport, and other fields. This article provides a tutorial review of the stochastic formulation for quantum dissipative dynamics. The key idea is to decouple the interaction between the system and the bath by virtue of the Hubbard-Stratonovich transformation or Itô calculus so that the system and the bath are not directly entangled during evolution, rather they are correlated due to the complex white noises introduced. The influence of the bath on the system is thereby defined by an induced stochastic field, which leads to the stochastic Liouville equation for the system. The exact reduced density matrix can be calculated as the stochastic average in the presence of bath-induced fields. In general, the plain implementation of the stochastic formulation is only useful for short-time dynamics, but not efficient for long-time dynamics as the statistical errors go very fast. For linear and other specific systems, the stochastic Liouville equation is a good starting point to derive the master equation. For general systems with decomposable bath-induced processes, the hierarchical approach in the form of a set of deterministic equations of motion is derived based on the stochastic formulation and provides an effective means for simulating the dissipative dynamics. A combination of the stochastic simulation and the hierarchical approach is suggested to solve the zero-temperature dynamics of the spin-boson model. This scheme correctly describes the coherent-incoherent transition (Toulouse limit) at moderate dissipation and predicts a rate dynamics in the overdamped regime. Challenging problems such as the dynamical description of quantum phase transition (local- ization) and the numerical stability of the trace-conserving, nonlinear stochastic Liouville equation are outlined.
Building upon our prior work [1], we present a unified stochastic drift model (SdM) for superconducting charge qubits based on memory multi-fractional Brownian motion (mmFBM). The classical sector employs a time-dependent Hurst exponent H(t) and adaptive memory kernel K(t,s), capturing non-stationary 1/f^beta noise and long-range temporal correlations inaccessible to conventional models. The quantum extension is formulated via a time-dependent Caldeira--Leggett environment with spectral density J(omega;t) = eta(t) omega_c^{1-s(t)} omega^{s(t)} exp(-omega/omega_c), where s(t) = 2H(t)-1, consistently reproducing beta(t) = 2H(t)-1. Four central results emerge: (1) relaxation and noise amplitudes act independently on energy decay; (2) time-varying H(t) matches experimental 1/f spectra more accurately than any constant exponent; (3) adaptive kernel dynamics preserve correlations without artificial damping; and (4) simulations predict coherence times (T1 ~ 5.00 x 10^6 ns, T2 ~ 4.18 x 10^5 ns) consistent with theory when charge noise dominates. The qubit exhibits stretched-exponential Ramsey and echo decay, non-Markovian dephasing, and a temperature-driven quantum-to-classical crossover. We derive the effective time-local Lindblad master equation, establish the classical mmFBM limit at high temperatures, and provide experimentally testable scaling relations. The non-exponential decay patterns reveal fundamental limitations of Markovian approaches, and the framework guides the design of noise-resilient qubit architectures.
In the long quest to identify and compensate the sources of decoherence in many-body systems far from the ground state, the varied family of Loschmidt echoes (LEs) became an invaluable tool in several experimental techniques. A LE involves a time-reversal procedure to assess the effect of perturbations in a quantum excitation dynamics. However, when addressing macroscopic systems one is repeatedly confronted with limitations that seem insurmountable. This led to the formulation of the central hypothesis of irreversibility stating that the timescale of decoherence, T3{T}_{3}T3​, is proportional to the timescale of the many-body interactions we reversed, T2{T}_{2}T2​. We test this by implementing two experimental schemes based on Floquet Hamiltonians where the effective strength of the dipolar spin-spin coupling, i.e., 1/T21/{T}_{2}1/T2​, is reduced by a variable scale factor kkk. This extends the perturbation timescale, ${T}_{\mathrm{\ensuremath{\Sigma}}}$, in relation to T2{T}_{2}T2​. Strikingly, we observe the superposition of the normalized Loschmidt echoes for the bigger values of kkk. This manifests the dominance of the intrinsic dynamics over the perturbation factors, even when the Loschmidt echo is devised to reverse that intrinsic dynamics. Thus, in the limit where the reversible interactions dominate over perturbations, the LE decays within a timescale, ${T}_{3}\ensuremath{\approx}{T}_{2}/R$ with $R=(0.15\ifmmode\pm\else\textpm\fi{}0.01)$, confirming the emergence of a perturbation independent regime. These results support the central hypothesis of irreversibility.
The investigation of the phenomenon of dephasing assisted quantum transport, which happens when the presence of dephasing benefits the efficiency of this process, has been mainly focused on Markovian scenarios associated with constant and positive dephasing rates in their respective Lindblad master equations. What happens if we consider a more general framework, where time-dependent dephasing rates are allowed, thereby, permitting the possibility of non-Markovian scenarios? Does dephasing-assisted transport still manifest for non-Markovian dephasing? Here, we address these open questions in a setup of coupled two-level systems. Our results show that the manifestation of non-Markovian dephasing-assisted transport depends on the way in which the incoherent energy sources are locally coupled to the chain. This is illustrated with two different configurations, namely non-symmetric and symmetric. Specifically, we verify that non-Markovian dephasing-assisted transport manifested only in the non-symmetric configuration. This allows us to draw a parallel with the conditions in which time-independent Markovian dephasing-assisted transport manifests. Finally, we find similar results by considering a controllable and experimentally implementable system, which highlights the significance of our findings for quantum technologies.