Semiclassical theory of quantum chaotic transport: Phase-space splitting, coherent backscattering, and weak localization

We investigate transport properties of quantized chaotic systems in the short-wavelength limit. We focus on noncoherent quantities such as the Drude conductance, its sample-to-sample fluctuations, shot noise, and the transmission spectrum, as well as coherent effects such as weak localization. We show how these properties are influenced by the emergence of the Ehrenfest time scale ${\tau }_{\mathrm{E}}$. Expressed in an optimal phase-space basis, the scattering matrix acquires a block-diagonal form as ${\tau }_{\mathrm{E}}$ increases, reflecting the splitting of the system into two cavities in parallel, a classical deterministic cavity (with all transmission eigenvalues either 0 or 1) and a quantum mechanical stochastic cavity. This results in the suppression of the Fano factor for shot noise and the deviation of sample-to-sample conductance fluctuations from their universal value. We further present a semiclassical theory for weak localization which captures nonergodic phase-space structures and preserves the unitarity of the theory. Contrarily to our previous claim [Phys. Rev. Lett. 94, 116801 (2005)], we find that the leading off-diagonal contribution to the conductance leads to the exponential suppression of the coherent backscattering peak and of weak localization at finite ${\tau }_{\mathrm{E}}$. This latter finding is substantiated by numerical magnetoconductance calculations.

Figure 1

(Color online) Classical phase-space (color) plot of the transmission probability from the phase-space projection of the injection lead (see Fig. 2). The phase space has been coarse grained by a rectangular grid with $9\times {10}^4$ cells, and the transmission probability in each cell has been calculated by time-evolving ${10}^4$ classical trajectories per cell. Dark areas are fully transmitting or fully reflecting (red and blue, respectively), while light areas (yellow/green) indicate a mixture of transmission and reflection . The dynamical system used is the open kicked rotator as defined in Sec. 6.

Figure 2

(Color online) Sketch of a two-terminal open chaotic cavity (top left) and its phase space represented as a Poincaré surface of section on the boundary of the cavity (bottom right). All momenta on the energy surface $(E=p_{\mathrm{F}}^2∕2m)$ are parametrized by the tangential momentum $p$ running from $-p_{\mathrm{F}}$ to $p_{\mathrm{F}}$. All possible positions on the boundary are parametrized by $q$ running from zero to $L$, where $L$ is the circumference of the cavity. The phase space is made dimensionless by normalizing momenta and real-space coordinates with $p_{\mathrm{F}}$ and $L$, respectively.

Figure 3

(Color online) Quantum phase-space plots of the transmission probability from the injection lead. The system is the quantum equivalent of the classical system of Fig. 1. The phase space has been coarse grained by a rectangular grid with $9\times {10}^4$ points. Starting from each point of the grid, an isotropic Gaussian wave packet has been time evolved and its transmission probability calculated. From top to bottom, the three panels correspond to decreasing effective Planck’s constant ${\hslash }_{\mathrm{eff}}=2\pi ∕M$ with $M=512$ (with a conductance $g=22.4$ and a Fano factor $F=0.193$), 8192 ($g=375.9$ and $F=0.121$), and $131072$ ($g=5990.8$ and $F\approx 0.08$), respectively. More and more fine-structured details of the classical phase space are resolved as ${\hslash }_{\mathrm{eff}}\to 0$. Dark areas are fully transmitting or fully reflecting (red and blue, respectively), while light areas (yellow/green) indicate a mixture of transmission and reflection.

Figure 4

(Color online) Schematic of the different transport regimes through a ballistic chaotic cavity with perimeter $L$ coupled to leads of width $W$. Above the separatrix, the system splits into two effective cavities, one purely classical and the other quantum mechanical. Only the latter contains quantum effects such as nondeterministic transmission and quantum interferences. Below the separatrix $W=({\lambda }_{\mathrm{F}}L{)}^{1∕2}$ (solid curve), all modes are quantum mechanical (gray area). The (red) dashed arrows indicate the semiclassical limit of ${\hslash }_{\mathrm{eff}}\to 0$ at fixed classical parameters. The slope of the arrows is given by the inverse dwell time ${\tau }_{\mathrm{D}}$.

Figure 5

(Color online) Time evolution of an L to R transmission band (parallelograms) on the boundary of the cavity of Fig. 2. The leads are indicated by the shaded rectangles.

Figure 6

(Color online) Main plot: Real-space wave-function components $\mathbf{\mid }⟨Q\mid \mathrm{ps};0,0⟩{\mathbf{\mid }}^2$ of a PS basis state (black line), and $\mathbf{\mid }⟨Q\mid \mathrm{cs};0,0⟩{\mathbf{\mid }}^2$ of a coherent state [gray (red) line]. Both wave functions are symmetric in the dimensionless phase space under $Q\leftrightarrow P$, up to a scaling factor. Inset: Logarithmic plot of $\mathbf{\mid }⟨Q\mid \mathrm{ps};0,0⟩{\mathbf{\mid }}^2$, showing that the PS state decays exponentially in position (dashed line). The oscillations in $\mathbf{\mid }⟨Q\mid \mathrm{ps};0,0⟩{\mathbf{\mid }}^2$ ensure it is orthogonal to PS states centered at finite $Q$, while its broader central peak ensures that it is orthogonal to PS states centered at finite $P$.

Figure 7

(Color online) Sketch of two parallelogram bands [dark (green) areas] with the PS states superimposed on them (ellipses). The upper band has a phase-space area which is a bit more than 16 times $2\pi \hslash$, while the lower band has a phase-space area a bit more than $2\pi \hslash$. In both cases the basis is optimized; the lattice of PS states is stretched and rotated such that the maximum number of PS states can be fitted into each band (solid-edged ellipses) with the minimum number partially in the band (dashed-edged ellipses). Note that these optimally chosen PS states have the same aspect ratio as the classical band in which they sit, thus their longitudinal and transversal extensions $x$ and $y$ are given by Eq. (11).

Figure 8

(Color online) Sketch of a trapezoidal and a folded scattering band with a lattice of PS states superimposed on each of them. Both bands cover a phase-space area larger than $2\pi \hslash$. In both cases the basis is optimized; the lattice of PS states is locally stretched and rotated such that the maximum number of PS states can be fitted into each band. Some of them are indicated by solid-edged ellipses.

Figure 9

(Color online) Sketch of the leading-order quantum correction to the conductance. Trajectory $\gamma 1$ (solid line) is injected at ${\mathbf{Y}}_0=(y_0,{\theta }_0)$, crosses itself, and escapes at $\mathbf{Y}=(y,\theta )$. Its first visit to the crossing (the open dot) occurs at ${\mathbf{R}}_1=({\mathbf{r}}_1,{\varphi }_1)$, where ${\mathbf{r}}_1$ is the position in the cavity, and ${\varphi }_1$ is the angle of the momentum to a reference axis (not shown). Trajectory $\gamma 2$ (dashed line) starts and ends at the same positions as $\gamma 1$; however, it avoids the crossing. We divide $\gamma 1$ into three segments: leg1, loop, and leg2.

Figure 10

(Color online) Sketches of the trajectory pairings which give the leading off-diagonal contributions to (a) transmission (conductance) and (b) reflection. All contributions involve a trajectory $\gamma 1$ crossing itself at an angle $ϵ$, and a trajectory $\gamma 2$ that avoids the crossing. The action difference between the two trajectories is thus small and does not fluctuate under averaging. For transmission, $y_0$ is on L lead and $y$ is on R lead; for reflection both $y_0$ and $y$ are on L lead. There are two reflection contributions. On the left is weak localization, and on the right is coherent backscattering (details of the latter are in Fig. 12).

Figure 11

(Color online) Sketch of a trajectory (solid line) which crosses itself at point ${\mathbf{r}}_1=(x,y)$, visiting this point first at time $t_1$ and second at time $t_2$. Superimposed is an infinitesimally different trajectory (dashed line) which also visits the point $(x,y)$ at $t_1$, but is at point ${\mathbf{r}}_2=(x+\delta x,y+\delta y)$ at time $t_2$. This trajectory also intersects itself; however, it visits the self-intersection [which is no longer $(x,y)$] at times $t_1+\delta t_1$ and $t_2+\delta t_2$.

Figure 12

(Color online) Trajectories for the backscattering contributions to reflection. Trajectory $\gamma 1$ (solid black line) start on the cross section of the L lead at position $y_0$ with momentum angle ${\theta }_0$ and ends at $y$ with momentum angle $\theta$. In the basis parallel and perpendicular to $\gamma 1$ at injection, the initial position and momentum of path $\gamma 1$ at exit are $r_{0\perp }=(y_0-y)\cos {\theta }_0$, $r_{0\Vert }=(y_0-y)\sin {\theta }_0$, and $p_{0\perp }\simeq -p_{\mathrm{F}}(\theta -{\theta }_0)$.

Figure 13

Magnetoconductance curves $\mathrm{\Delta }g\left(m_0\right)=g\left(m_0\right)-g\left(0\right)$ for the open kicked rotator model (defined in the text) at fixed classical configuration $K=14$ $(\lambda \approx 1.95)$, ${\tau }_{\mathrm{D}}∕{\tau }_{\mathrm{f}}=5$, and different Hilbert space sizes $M=256$ (squares, ${\tau }_{\mathrm{E}}^{\mathrm{cl}}∕{\tau }_{\mathrm{D}}\simeq 0.57$), 512 (diamonds, ${\tau }_{\mathrm{E}}^{\mathrm{cl}}∕{\tau }_{\mathrm{D}}\approx 0.64$), 1024 (upward triangles, ${\tau }_{\mathrm{E}}^{\mathrm{cl}}∕{\tau }_{\mathrm{D}}\approx 0.71$), 2048 (downward triangles, ${\tau }_{\mathrm{E}}^{\mathrm{cl}}∕{\tau }_{\mathrm{D}}\approx 0.78$), and 4096 (circles, ${\tau }_{\mathrm{E}}^{\mathrm{cl}}∕{\tau }_{\mathrm{D}}\approx 0.85$). The solid line gives the best Lorentzian fit for the $M=512$ curve, $\mathrm{\Delta }g\left(m_0\right)=0.15-0.15∕[1+1.03(m_0∕m_c{)}^2]$. Data have been obtained after averaging over $100000$ (1000 classically different samples, each with 100 different quasienergies for $M=256$) to $25000$ (500 classically different samples, each with 50 different quasienergies for $M=4096$) different samples. Inset: rescaled magnetoconductance data $\mathrm{\Delta }g\left(m_0\right)∕\mathrm{\Delta }g\left(2.2m_c\right)$. The data collapse onto a single curve, confirming our theory that $m_c$ does not depend on ${\tau }_{\mathrm{E}}^{\mathrm{cl}}$.

Figure 14

Amplitude of the weak-localization correction to the conductance as a function of the size of Hilbert space. Data have been extracted from the curves shown on Fig. 13 via either a fitting of the magnetoconductance curve with the Lorentzian $\mathrm{\Delta }g\left(m_0\right)=\gamma -\gamma ∕[1+{\gamma }^{\prime }(m_0∕m_c{)}^2]$ (squares) or taking $\gamma =\mathrm{\Delta }g\left(2.2m_c\right)$ (circles). Both methods confirm the exponential suppression $\gamma \propto \exp (-{\tau }_{\mathrm{E}}^{\mathrm{cl}}∕{\tau }_{\mathrm{D}})$ (dashed and dot-dashed lines) with an effectively smaller Lyapunov exponent $\lambda \approx 1.26$ [compared to $\ln (K∕2)\approx 1.95$] (Ref. 44).

Figure 15

(Color online) A sketch of the area enclosed by the correlated part of the weak-localization loop. The area ${\tilde{\mathcal{A}}}_n$ is defined by the segment of the loop-forming trajectory between the $(n-1)$th and $n$th collisions and the segment between the $(N-n+1)$th and the $(N-n+2)$th collisions (dashed region).