Dephasing in quantum chaotic transport: A semiclassical approach

We investigate the effect of dephasing (decoherence) on quantum transport through open chaotic ballistic conductors in the semiclassical limit of small Fermi wavelength to system size ratio, ${\lambda }_F∕L⪡1$. We use the trajectory-based semiclassical theory to study a two-terminal chaotic dot with decoherence originating from (i) an external closed quantum chaotic environment, (ii) a classical source of noise, and (iii) a voltage probe, i.e., an additional current-conserving terminal. We focus on the pure dephasing regime, where the coupling to the external source of dephasing is so weak that it does not induce energy relaxation. In addition to the universal algebraic suppression of weak localization, we find an exponential suppression of weak localization $\propto \exp [-\tilde{\tau }∕{\tau }_{\varphi }]$, with the dephasing rate ${\tau }_{\varphi }^{-1}$. The parameter $\tilde{\tau }$ depends strongly on the source of dephasing. For a voltage probe, $\tilde{\tau }$ is of order the Ehrenfest time $\propto \ln [L∕{\lambda }_F]$. In contrast, for a chaotic environment or a classical source of noise, it has the correlation length $\xi$ of the coupling or noise potential replacing the Fermi wavelength ${\lambda }_F$. We explicitly show that the Fano factor for shot noise is unaffected by decoherence. We connect these results to earlier works on dephasing due to electron-electron interactions and numerically confirm our findings.

Figure 1

(Color online) Schematic of the system-environment model. The system is an open quantum dot that is coupled to an environment in the shape of a second, closed dot.

Figure 2

(Color online) A semiclassical contribution to weak localization for the system-environment model. The paths are paired everywhere except at the encounter, where one path crosses itself at angle $ϵ$, while the other one does not (going the opposite way around the loop). Here, we show $\xi >ϵL$, so the dephasing (dotted path segment) starts in the loop $(T_{\xi }>0)$.

Figure 3

(Color online) A semiclassical contribution to coherent backscattering for the system-environment model. It involves paths which return to close, but antiparallel to themselves at lead $L$. The two solid paths are paired (within $W$ of each other) in the cross-hatched region. Here, we show $\xi >ϵL$, so the dephasing (dotted path segment) starts in the loop $(T_{\xi }>0)$. In the basis parallel and perpendicular to $\gamma$ at injection, the initial position and momentum of path $\gamma$ at exit are $r_{0\perp }=(y_0-y)\cos {\theta }_0$, $r_{0\parallel }=(y_0-y)\sin {\theta }_0$, and $p_{0\perp }=p_F(\theta -{\theta }_0)$.

Figure 4

(Color online) Dephasing of weak localization when $\xi ⪡Lϵ\sim {\left(L{\lambda }_F\right)}^{1∕2}$, and hence $T_{\xi }\left(ϵ\right)$ is negative, see Eq. (19). The dephasing starts and ends in the “legs” rather than the loop. In the language of disordered systems, this means that the dephasing affects the diffusons as well as the cooperon.

Figure 5

(Color online) Sketch of typical trajectories which contribute to shot noise in the presence of an environment. In both (a) and (b), the system (environment) paths are sketched above (below) the time axis. In (a), we show a contribution which will survive system averaging because the system paths are paired almost everywhere with an encounter at time $t_1^{\prime }$. There is no constraint on the environment paths, as yet. For simplicity, we show only system paths $\gamma 1$ and $\gamma 3$, with paths $\gamma 2$ and $\gamma 4$ being the same as $\gamma 1$ and $\gamma 3$ except that they cross at the encounter. Depending on the choice of $t_1^{\prime }$ with respect to the other time scales, $t_1,t_2,t_3$, this pair of system paths could represent any of the contributions in Fig. 1 of Ref. 50. In (b), additional constraints are imposed on the ${\Gamma }_i$’s, after one has integrated over all initial and final positions of the environment. This integration removes all contributions from environment propagation when the system paths are paired.

Figure 6

(Color online) Sketch of a typical chaotic quantum dot in a two-dimensional electron gas (2DEG). The dot is defined by metallic gates which are biased to deplete the 2DEG everywhere except in the regions defining the leads and the dot. These gates are a distance $D$ above the 2DEG (with $D⪢{\lambda }_F^{\text{gate}}$). At finite temperature, the electrons at the surface of these metallic gates will fluctuate, leading to noise which will be felt by the electrons in the chaotic quantum dot.

Figure 7

Schematic of the dephasing-lead model. The system is an open quantum dot with an extra lead (lead 3), whose voltage is chosen to make a zero current flow (on average) in that lead. This extra lead thereby causes dephasing without loss of particles.

Figure 8

(a) Magnetoconductance curves $\Delta g\left(x_0\right)=g\left(x_0\right)-g\left(0\right)$ for the double kicked rotator model (see text) with $K_{\mathrm{sys}}=K_{\mathrm{env}}=34.08$ $({\lambda }_{\mathrm{eff}}\approx 2)$, ${\tau }_D∕{\tau }_0=8$, $\xi ∕L=1$, and Hilbert space sizes $M_{\mathrm{sys}}=256$, $M_{\mathrm{env}}=16$. Different symbols correspond to different dephasing times ${\tau }_{\varphi }∕{\tau }_D=\infty$ (${\hslash }_{\mathrm{eff}}^{-1}\epsilon =0$, circles), ${\tau }_{\varphi }∕{\tau }_D=4.8$ (${\hslash }_{\mathrm{eff}}^{-1}\epsilon \simeq 0.25$, squares), ${\tau }_{\varphi }∕{\tau }_D=1.2$ (${\hslash }_{\mathrm{eff}}^{-1}\epsilon \simeq 0.5$, diamonds), ${\tau }_{\varphi }∕{\tau }_D=0.3$ (${\hslash }_{\mathrm{eff}}^{-1}\epsilon \simeq 1$, downward triangles), and ${\tau }_{\varphi }∕{\tau }_D=0.07$ (${\hslash }_{\mathrm{eff}}^{-1}\epsilon \simeq 2$, upward triangles). Data are averaged over 25 different lead positions, each with 25 different quasienergies and 10 different initial environment states. (b) Magnetoconductance curves for the open kicked rotator with transparent dephasing lead (see text) with $K=34.08$, ${\tau }_D∕{\tau }_0=8$, and Hilbert space size $M=256$. Different symbols correspond to different dephasing times ${\tau }_{\varphi }∕{\tau }_D=\infty$ (circles), ${\tau }_{\varphi }∕{\tau }_D=5$ (squares), ${\tau }_{\varphi }∕{\tau }_D=1.25$ (diamonds), ${\tau }_{\varphi }∕{\tau }_D=0.5$ (downward triangles), and ${\tau }_{\varphi }∕{\tau }_D=0.25$ (upward triangles). Data are averaged over 225 different lead positions, each with 50 different quasienergies.

Figure 9

(Color online) Amplitude $\delta g(m_0∕m_c=2.4)-\delta g(m_0∕m_c=0)$ of the weak-localization correction to the conductance as a function of ${\tau }_D∕{\tau }_{\varphi }$ for the double kicked rotator model (circles) and the open kicked rotator with transparent dephasing lead (squares). The black line gives the universal algebraic behavior ${(1+{\tau }_D∕{\tau }_{\varphi })}^{-1}∕4$, and the red line is a guide to the eyes, including an exponential decay with ${\tau }_E^{\mathrm{cl}}=2.78$, on top of the universal decay.

Figure 10

Variance of the conductance vs ${\tau }_D∕{\tau }_{\varphi }$ for the open kicked rotator with $K=14$ and ${\tau }_D∕{\tau }_0=5$ (empty symbols), transparently coupled to a dephasing lead. Different symbols correspond to different Hilbert space sizes (and hence different ${\tau }_E^{\mathrm{cl}}$) $M=128$ (squares, ${\tau }_E^{\mathrm{cl}}∕{\tau }_D=0.6$), $M=512$ (diamonds, ${\tau }_E^{\mathrm{cl}}∕{\tau }_D=0.75$), $M=2048$ (upward triangles, ${\tau }_E^{\mathrm{cl}}∕{\tau }_D=0.9$), and $M=8192$ (downward triangles, ${\tau }_E^{\mathrm{cl}}∕{\tau }_D=1.1$). Additional data for $K=144$, ${\tau }_D=25$, and $M=2048$ are also shown (full circles, ${\tau }_E^{\mathrm{cl}}∕{\tau }_D=0.08$). The dashed line shows the universal behavior of Eq. (2). Unlike for weak localization (see Fig. 8) and for the dephasing-lead model with partial transparency (Ref. 48), the behavior of $\delta g^2$ remains universal and shows no noticeable dependence on ${\tau }_E^{\mathrm{cl}}∕{\tau }_D$. Data are averaged over 50 different quasienergies and from 50 (for $N=8192$) to 500 (for $N=128$ and 512) different lead positions.