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arXiv:1103.2710 · Physical Review B

Feedback stabilization of pure states in quantum transport

Christina Pöltl, Clive Emary, Tobias Brandes

Published Aug 19, 2011• 28 citations
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Abstract

We propose a feedback control scheme for generating and stabilizing pure states of transport devices, such as charge qubits, under nonequilibrium conditions. The purification of the device state is conditioned on single electron jumps and leaves a clear signal in the full counting statistics which can be used to optimize control parameters. As an example of our control scheme, we are presenting the stabilization of pure transport states in a double-quantum-dot setup with the inclusion of phonon dephasing.

DOI
10.1103/physrevb.84.085302

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We propose a feedback control scheme for generating and stabilizing pure states of transport devices, such as charge qubits, under nonequilibrium conditions. The purification of the device state is conditioned on single electron jumps and leaves a clear signal in the full counting statistics which can be used to optimize control parameters. As an example of our control scheme, we are presenting the stabilization of pure transport states in a double-quantum-dot setup with the inclusion of phonon dephasing.
A Markovian open quantum system which relaxes to a unique steady state ${\ensuremath{\rho}}_{\mathrm{ss}}$ of finite rank can be decomposed into a finite physically realizable ensemble (PRE) of pure states. That is, as shown by R. I. Karasik and H. M. Wiseman [Phys. Rev. Lett. 106, 020406 (2011)], in principle there is a way to monitor the environment so that in the long-time limit the conditional state jumps between a finite number of possible pure states. In this paper we show how to apply this idea to the dynamics of a double quantum dot arising from the feedback control of quantum transport, as previously considered by C. P\"oltl, C. Emary, and T. Brandes [Phys. Rev. B 84, 085302 (2011)]. Specifically, we consider the limit where the system can be described as a qubit, and show that while the control scheme can always realize a two-state PRE, in the incoherent-tunneling regime there are infinitely many PREs compatible with the dynamics that cannot be so realized. For the two-state PREs that are realized, we calculate the counting statistics and see a clear distinction between the coherent and incoherent regimes.
We propose an all-electronic technique to manipulate and control interacting quantum systems by unitary single-jump feedback conditioned on the outcome of a capacitively coupled electrometer and, in particular, a single-electron transistor. We provide a general scheme for stabilizing pure states in the quantum system and use an effective Hamiltonian method for the quantum master equation to elaborate on the nature of stabilizable states and the conditions under which state purification can be achieved. The state engineering within the quantum feedback scheme is shown to be linked with the solution of an inverse eigenvalue problem. Two applications of the feedback scheme are presented in detail: (i) stabilization of delocalized pure states in a single charge qubit and (ii) entanglement stabilization in two coupled charge qubits. In the latter example, we demonstrate the stabilization of a maximally entangled Bell state for certain detector positions and local feedback operations.
The current through nanostructures like quantum dots can be stabilized by a classical feedback loop that continuously adjusts system parameters as a function of the number of tunnelled particles n. At large times, the feedback loop freezes the fluctuations of n, which leads to highly accurate, continuous single particle transfers. For the simplest case of feedback acting simultaneously on all system parameters, we show how to reconstruct the original full counting statistics from the frozen distribution.
We propose the manipulation of an isolated qubit by a simple instantaneous closed-loop feedback scheme in which a time-dependent electronic detector current is directly back-coupled into qubit parameters. As a specific detector model, we employ a capacitively coupled single-electron transistor. We demonstrate the stabilization of pure delocalized qubit states above a critical detector-qubit coupling. This electronic purification is independent of the initial qubit state and is accomplished after few electron jumps through the detector. Our simple scheme can be used for the efficient and robust initialization of solid-state qubits in quantum computational algorithms at arbitrary temperatures.
Feedback control in quantum transport has been predicted to give rise to several interesting effects, among them quantum state stabilization and the realization of a mesoscopic Maxwell's daemon. These results were derived under the assumption that control operations on the system are affected instantaneously after the measurement of electronic jumps through it. In this contribution, I describe how to include a delay between detection and control operation in the master equation theory of feedback-controlled quantum transport. I investigate the consequences of delay for the state stabilization and Maxwell's daemon schemes. Furthermore, I describe how delay can be used as a tool to probe coherent oscillations of electrons within a transport system and how this formalism can be used to model finite detector bandwidth.
A controllable low noise current lies at the heart of high-precision measurements in quantum transport and metrology. While the previous research dealt with the suppression of noise in transport through a tunneling junction or a single quantum dot (QD) device, the present work investigates noise inhibition of a double quantum dot (DQD) transport system based on closed-loop feedback control. The unique advantage of a DQD device is that bidirectional transport at low bias can be measured by a nearby quantum point contact. However, the continuous monitoring of the DQD states inevitably leads to a measurement-induced dephasing. To appropriately characterize its transport properties in the presence of feedback action, we here develop a numerical method dubbed auxiliary density matrix approach, motivated by the hierarchical expansion of the moment-generating function in the hierarchy equations of motion [J. Cerrillo et al., Phys. Rev. B 94, 214308 (2016)]. This generic method has no restriction on the system structure and parameters and is able to evaluate the feedback current cumulants to an arbitrary order. It is revealed that the feedback control of the tunnel coupling between the two dots is the most effective to suppress the noise under various tunnel coupling configurations. The influence of interdot Coulomb interaction, measurement-induced dephasing, and finite time delay on feedback is also analyzed in detail.