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arXiv:cond-mat/0512465 · Physical Review B

Voltage probe model of spin decay in a chaotic quantum dot with applications to spin-flip noise and entanglement production

B. Michaelis, C. W. J. Beenakker

Published Mar 23, 2006• 10 citations
arXivHTMLPDF

Abstract

The voltage probe model is a model of incoherent scattering in quantum transport. Here we use this model to study the effect of spin-flip scattering on electrical conduction through a quantum dot with chaotic dynamics. The spin decay rate $\ensuremath{\gamma}$ is quantified by the correlation of spin-up and spin-down current fluctuations (spin-flip noise). The resulting decoherence reduces the ability of the quantum dot to produce spin-entangled electron-hole pairs. For $\ensuremath{\gamma}$ greater than a critical value ${\ensuremath{\gamma}}_{c}$, the entanglement production rate vanishes identically. The statistical distribution $P({\ensuremath{\gamma}}_{c})$ of the critical decay rate in an ensemble of chaotic quantum dots is calculated using the methods of random-matrix theory. For small ${\ensuremath{\gamma}}_{c}$ this distribution is $\ensuremath{\propto}{\ensuremath{\gamma}}_{c}^{\ensuremath{-}1+\ensuremath{\beta}∕2}$, depending on the presence $(\ensuremath{\beta}=1)$ or absence $(\ensuremath{\beta}=2)$ of time-reversal symmetry. To make contact with experimental observables, we derive a one-to-one relationship between the entanglement production rate and the spin-resolved shot noise, under the assumption that the density matrix is isotropic in the spin degrees of freedom. Unlike the Bell inequality, this relationship holds for both pure and mixed states. In the tunneling regime, the electron-hole pairs are entangled if and only if the correlator of parallel spin currents is at least twice larger than the correlator of antiparallel spin currents.

DOI
10.1103/physrevb.73.115329

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The voltage probe model is a model of incoherent scattering in quantum transport. Here we use this model to study the effect of spin-flip scattering on electrical conduction through a quantum dot with chaotic dynamics. The spin decay rate \ensuremathγ\ensuremath{\gamma}\ensuremathγ is quantified by the correlation of spin-up and spin-down current fluctuations (spin-flip noise). The resulting decoherence reduces the ability of the quantum dot to produce spin-entangled electron-hole pairs. For \ensuremathγ\ensuremath{\gamma}\ensuremathγ greater than a critical value \ensuremathγc{\ensuremath{\gamma}}_{c}\ensuremathγc​, the entanglement production rate vanishes identically. The statistical distribution P(\ensuremathγc)P({\ensuremath{\gamma}}_{c})P(\ensuremathγc​) of the critical decay rate in an ensemble of chaotic quantum dots is calculated using the methods of random-matrix theory. For small \ensuremathγc{\ensuremath{\gamma}}_{c}\ensuremathγc​ this distribution is \ensuremath∝\ensuremathγc\ensuremath−1+\ensuremathβ∕2\ensuremath{\propto}{\ensuremath{\gamma}}_{c}^{\ensuremath{-}1+\ensuremath{\beta}∕2}\ensuremath∝\ensuremathγc\ensuremath−1+\ensuremathβ∕2​, depending on the presence (\ensuremathβ=1)(\ensuremath{\beta}=1)(\ensuremathβ=1) or absence (\ensuremathβ=2)(\ensuremath{\beta}=2)(\ensuremathβ=2) of time-reversal symmetry. To make contact with experimental observables, we derive a one-to-one relationship between the entanglement production rate and the spin-resolved shot noise, under the assumption that the density matrix is isotropic in the spin degrees of freedom. Unlike the Bell inequality, this relationship holds for both pure and mixed states. In the tunneling regime, the electron-hole pairs are entangled if and only if the correlator of parallel spin currents is at least twice larger than the correlator of antiparallel spin currents.
Electron-hole pairs produced by tunneling in a degenerate electron gas lose their spin entanglement by spin-orbit coupling, which transforms the fully entangled Bell state into a partially entangled mixed density matrix of the electron and hole spins. We calculate the dependence of the entanglement (quantified by the concurrence) on the spin-orbit coupling time ${\ensuremath{\tau}}_{\mathrm{so}}$ and on the diffusion time (or dwell time) ${\ensuremath{\tau}}_{\text{dwell}}$ of electrons and holes in the conductors (with conductances ⪢e2∕h⪢{e}^{2}∕h⪢e2∕h) at the two sides of the tunnel barrier (with conductance ⪡e2∕h⪡{e}^{2}∕h⪡e2∕h). The entanglement disappears when the ratio ${\ensuremath{\tau}}_{\text{dwell}}∕{\ensuremath{\tau}}_{\mathrm{so}}$ exceeds a critical value of order unity. The results depend on the type of conductor (disordered wire or chaotic quantum dot), but they are independent of other microscopic parameters (number of channels, level spacing). Our analytical treatment relies on an ``isotropy approximation'' (no preferential basis in spin space), which allows us to express the concurrence entirely in terms of spin correlators. We test this approximation for the case of chaotic dynamics with a computer simulation (using the spin-kicked rotator) and find good agreement.
It has recently been shown theoretically that elastic scattering in the Fermi sea produces quantum mechanically entangled states. The mechanism is similar to entanglement by a beam splitter in optics, but a key distinction is that the electronic mechanism works even if the source is in local thermal equilibrium. An experimental realization was proposed using tunneling between two edge channels in a strong magnetic field. Here we investigate a low-magnetic field alternative, using multiple scattering in a quantum dot. Two pairs of single-channel point contacts define a pair of qubits. If the scattering is chaotic, a universal statistical description of the entanglement production (quantified by the concurrence) is possible. The mean concurrence turns out to be almost independent on whether time-reversal symmetry is broken or not. We show how the concurrence can be extracted from a Bell inequality using low-frequency noise measurements, without requiring the tunneling assumption of earlier work.
The production of orbitally entangled electrons in quantum-chaotic dots is investigated from a statistical point of view. The degree of entanglement is quantified through the concurrence and the entanglement of formation. We calculate the complete statistical distributions of the entanglement measures by using random matrix theory. Simple analytical expressions are provided for the concurrence distributions. We identify clear signatures of time-reversal invariance in the production of entanglement at the level of the entanglement-measure distributions, such as the ability of producing maximally entangled (Bell) states, which passed unnoticed in previous works, where only the first two moments of the distributions were studied.
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, ${\ensuremath{\lambda}}_{F}∕L⪡1$. 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 $\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}∕{\ensuremath{\tau}}_{\ensuremath{\phi}}]$, with the dephasing rate ${\ensuremath{\tau}}_{\ensuremath{\phi}}^{\ensuremath{-}1}$. The parameter $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}$ depends strongly on the source of dephasing. For a voltage probe, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}$ is of order the Ehrenfest time $\ensuremath{\propto}\mathrm{ln}[L∕{\ensuremath{\lambda}}_{F}]$. In contrast, for a chaotic environment or a classical source of noise, it has the correlation length $\ensuremath{\xi}$ of the coupling or noise potential replacing the Fermi wavelength ${\ensuremath{\lambda}}_{F}$. 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.
Charge and spin current correlations are analyzed in a source of spin-entangled electrons built from a superconductor and two quantum dots in parallel. In addition to the ideal (crossed Andreev) channel, parasitic channels (direct Andreev and cotunneling) and spin flip processes are fully described in a density matrix framework. The way they reduce both the efficiency and the fidelity of the entangler is quantitatively described by analyzing the zero-frequency noise correlations of charge current as well as spin current in the two output branches. Spin current noise is characterized by a spin Fano factor, equal to 0 (total current noise) and -1 (crossed correlations) for an ideal entangler. The violation of the Bell inequalities, as a test of non-locality (entanglement) of split pairs, is formulated in terms of the correlations of electron charge and spin numbers counted in a specific time window τ\tauτ. The efficiency of the test is analyzed, comparing τ\tauτ to the various time scales in the entangler operation.
Charge and spin current correlations are analyzed in a source of spin-entangled electrons built from a superconductor and two quantum dots in parallel. In addition to the ideal (crossed Andreev) channel, parasitic channels (direct Andreev and cotunneling) and spin flip processes are fully described in a density matrix framework. The way they reduce both the efficiency and the fidelity of the entangler is quantitatively described by analyzing the zero-frequency noise correlations of charge current as well as spin current in the two output branches. Spin current noise is characterized by a spin Fano factor, equal to 0 (total current noise) and $\ensuremath{-}1$ (crossed correlations) for an ideal entangler. The violation of the Bell inequalities, as a test of nonlocality (entanglement) of split pairs, is formulated in terms of the correlations of electron charge and spin numbers counted in a specific time window $\ensuremath{\tau}$. The efficiency of the test is analyzed, comparing $\ensuremath{\tau}$ to the various time scales in the entangler operation.