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arXiv:cond-mat/0101081 · Physical review. B, Condensed matter

Effects of entanglement in controlled dephasing

Alessandro Silva, Shimon Levit

Published May 7, 2001• 21 citations

Abstract

In controlled dephasing a controlled environment causes dephasing of transport through a mesoscopic system. As a result of their interaction the states of the two subsystems are entangled. In this paper we discuss how their entanglement influences dephasing using as an example the ``which path detector,'' where the dephasing of transport through a quantum dot (QD) due to the interaction with a quantum point contact (QPC) is detected by means of an Aharonov-Bohm (AB) interferometer. In particular, we calculate the suppression $\ensuremath{\nu}$ of AB oscillations as a function of the bias $\mathrm{eV}$ applied to the QPC and the coupling $\ensuremath{\Gamma}$ of the QD to the leads. At low temperatures the entanglement produces a smooth crossover from $\ensuremath{\nu}\ensuremath{\propto}(eV/\ensuremath{\Gamma}{)}^{2},$ when $\mathrm{eV}\ensuremath{\ll}\ensuremath{\Gamma}$ to $\ensuremath{\nu}\ensuremath{\propto}eV/\ensuremath{\Gamma},$ for $\mathrm{eV}\ensuremath{\gg}\ensuremath{\Gamma}.$

DOI: 10.1103/physrevb.63.201309

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In controlled dephasing a controlled environment causes dephasing of transport through a mesoscopic system. As a result of their interaction the states of the two subsystems are entangled. In this paper we discuss how their entanglement influences dephasing using as an example the ``which path detector,'' where the dephasing of transport through a quantum dot (QD) due to the interaction with a quantum point contact (QPC) is detected by means of an Aharonov-Bohm (AB) interferometer. In particular, we calculate the suppression

\ensuremath{\nu}

of AB oscillations as a function of the bias

\mathrm{eV}

applied to the QPC and the coupling

\ensuremath{\Gamma}

of the QD to the leads. At low temperatures the entanglement produces a smooth crossover from

\ensuremath{\nu}\ensuremath{\propto}(eV/\ensuremath{\Gamma}{)}^{2},

when

\mathrm{eV}\ensuremath{\ll}\ensuremath{\Gamma}

to

\ensuremath{\nu}\ensuremath{\propto}eV/\ensuremath{\Gamma},

for

\mathrm{eV}\ensuremath{\gg}\ensuremath{\Gamma}.

We study dephasing of electrons induced by a which path detector and thus verify Bohr’s complementarity principle for fermions. We utilize a double path interferometer with two slits, with one slit being replaced by a coherent quantum dot (QD). A short one dimensional channel, in the form of a quantum point contact (QPC), in close proximity to the QD, serves as a which path detector. We find that by varying the properties of the QPC detector we affect the visibility of the interference, inducing thus dephasing. We develop a simple model to explain the dephasing due to the nearby detector and find good agreement with the experiment. Bohr’s complementarity principle excludes the possibility of observation of multiple paths interference simultaneously with which path (WP) determination [1]. Wave-like behavior, or interference, is possible only when the different possible paths a particle can take are indistinguishable, even

Motivated by a recent experiment by Buks et al. ( Nature 391 (1998) 871) we consider electron transport through an Aharonov-Bohm interferometer with a quantum dot in one of its arms. The quantum dot is coupled to a quantum system with a finite number of states acting as a which-path detector. The Aharonov-Bohm interference is calculated using a two-particle scattering approach for the joint transitions in detector and quantum dot. Tracing over the detector yields dephasing and a reduction of the interference amplitude. We show that the interference can be restored by a suitable measurement on the detector and propose a mesoscopic quantum eraser based on this principle.

Recently, the investigations on the coherent tunnelling in a coupled quantum-dot (QD) system have carried out both experimentally (Blick et al., 1998, and Oosterkamp et al., 1998), and theoretically (Tsukada et al., 1997, and Wu et al., 2000). However, all of these studies did not take the influence of the measurement into account. It is well known that measurement itself will certainly induce dephasing. This effect was studied (Aleiner et at., 1997, Levinson, 1997, and E. Buks et al., 1998) in detail. Also, Gurvitz et al. (1996, 1997) derived modified rate equations and studied the dephasing effect induced by measuring the electron state in a coupled QD system via a quantum point contact.

Interference results when a quantum particle is free to choose among a few indistinguishable paths. A canonical example of Bohr's complementarity principle [1] is a two-path interferometer with an external detector coupled to one of the paths. Then, interference between the two paths vanishes (i.e. dephasing) if one is able to detect, even in principle, the path taken by the particle. This type of which path (WP) experiment was already executed with photons, cooled atoms, neutrons, solitons, and more recently with electrons in a mesoscopic system [2]. In the latter experiment path determination was provided by inducing a change in the current flowing through the detector. In the present experiment we perform a more intriguing WP determination: only a phase change is being induced in the detector while the current remains unaffected. We show that such detector-interferometer interaction can be understood both, from the WP information obtained by the detector, or alternatively by understanding the disturbance in the interferometer caused by the detector. Moreover, we address the subtle role of detector coherency and point out difficulties in the intuitive interpretation of the experiment.

We review recent theoretical investigations on the two-particle Aharonov–Bohm effect and its relation to entanglement production and detection. The difficulties of the entanglement detection due to dephasing and finite temperature are discussed regarding a recent experimental realization of a two-particle Aharonov–Bohm interferometer [15]. We also discuss a theoretical proposal for a two-particle Aharonov–Bohm interferometer, which as against the finite bias setup is driven with dynamical single-electron sources allowing for the tunable production of time-bin entanglement.