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

We develop and implement a model for decoherence in time-dependent transport. Inspired in a dynamical formulation of the Landauer-Büttiker 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 $|0\rangle$ that undergoes decoherence. Its numerical resolution shows the equivalence with the decoherent steady-state transport in presence of a Büttiker'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 $|0\rangle \equiv |\uparrow \downarrow \rangle$ and $|1\rangle \equiv |\downarrow \uparrow \rangle$. 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 $|0\rangle \equiv |\uparrow \downarrow \rangle$ and $|1\rangle \equiv |\downarrow \uparrow \rangle$ are quite robust against the local decoherence. In contrast, the LE, and hence coherence, decays faster when the system is in a superposition state $\left(\right|\uparrow \downarrow \rangle \pm |\downarrow \uparrow \rangle )/\sqrt{2}$, 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.

Figure 1

Decoherent transmittances obtained for two values of the decoherence strength. For ${\Gamma }_{\varphi }/V=0.01$, the Büttiker transmittance is plotted with the short-dot line and the transmittance of the QD model is plotted with the red triangles for a number $N_S=5$ of realizations in the average. For ${\Gamma }_{\varphi }/V=0.3$, the Büttiker transmittance is showed by the solid line and the QD result, with the blue circles for a number of $N_S=50$ realizations. The other parameters are: $V=1,E_0=0,V_L=V_R=0.15,E_r=0$.

Figure 2

(a) Theoretical rates of decay predicted by the Eqs. (17) and (20) . Red points indicate the rates of the GLBE solution for (b)–(d). The QDPT occurs as a bifurcation in the rates for the critical value ${\Gamma }_{\varphi }/V=2$. The dotted line represents the asymptotic value $2{\Gamma }_{\varphi }/\hslash$. In (b)–(d), we show the survival probabilities ${\tilde{P}}_{00}\left(t\right)$ (thick blue line), the GLBE solutions (thin black line), and individual realizations (thin gray line) in the underdamped regime, in (b) with ${\Gamma }_{\varphi }/V=0.1$ and (c) with ${\Gamma }_{\varphi }/V=1$, and in the overdamped regime, (d) with ${\Gamma }_{\varphi }/V=2.1$. Individual realizations tend to preserve the oscillations. These do not have jumps in the densities but in the slopes. The ensemble average is taken over $N_S=100$ realizations. For ${\tilde{P}}_{00}\left(t\right)\simeq 1/2$ are clear the typical fluctuations of the order of $1/\sqrt{N_S}$. The average ${\tilde{P}}_{00}\left(t\right)$ tends to the GLBE solution by increasing $N_S$.

Figure 3

Density as function of time at each spin for (a) the coherent evolution, (b) an average evolution with a finite coherence time $T_{\mathrm{coh}}=3\hslash /J$, (c) a single realization of the QD model, and (d) a single realization of the QJ model. When the spin wave reaches the edge of the chain, it is reflected constituting the mesoscopic echo. Note that the curves for QD are smooth in density whereas those for QJ present a sudden change when the jump is produced.

Figure 4

The standard error $\sigma$ times ${\sqrt{N}}_s$ as function of the evolution time, for $N_s=10,100$, and 500 . Note that $\sigma N_s^{1/2}$ does not depend on $N_s$ but in the QD or QJ method. Note that, for the same $N_s,{\sigma }^{\mathrm{QJ}}/{\sigma }^{\mathrm{QD}}\approx 2$.

Figure 5

Survival probability (thick blue line) and the Loschmidt echo (thin black line) as function of the time of action of the environment for: (a) $\eta /V=0.1$ and (b) $\eta /V=2.1$, averaged over $N_S=10000$ realizations. (c): the same as in (b) but in log scale. The rate of decay of the LE, $1/{\tau }_{\mathrm{SE}}=1.12V/\hslash$, is smaller than the minimum decay rate of the ${\tilde{P}}_{00},1/{\tau }_2=1.46V/\hslash$.