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Quantum Electronics and Laser Science Conference

Decoherence control by homodyne-mediated feedback

J. Wang, Howard M. Wiseman, Gerard J Milburn

Published Nov 13, 2002• 1 citation
Publisher

Abstract

Summary form only given. Decoherence in quantum systems can be loosely defined as loss of purity. Therefore the ultimate success in using feedback to fight decoherence would be to create an arbitrary stable pure state in the presence of dissipation. This goal was realized by Hoffman, Mahler and Hess (1998) for a very simple system: a resonantly driven two-level atom. They showed that by making part of the coherent driving proportional to the homodyne photocurrent can stabilize the state to any point on the bottom half of the sphere. Here we reanalyze their proposal by using the technique of stochastic master equations, allowing their results to be generalized in three ways.

DOI
10.1109/qels.2001.961903

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Summary form only given. Decoherence in quantum systems can be loosely defined as loss of purity. Therefore the ultimate success in using feedback to fight decoherence would be to create an arbitrary stable pure state in the presence of dissipation. This goal was realized by Hoffman, Mahler and Hess (1998) for a very simple system: a resonantly driven two-level atom. They showed that by making part of the coherent driving proportional to the homodyne photocurrent can stabilize the state to any point on the bottom half of the sphere. Here we reanalyze their proposal by using the technique of stochastic master equations, allowing their results to be generalized in three ways.
Unit-efficiency homodyne detection of the resonance fluorescence of a two-level atom collapses the quantum state of the atom to a stochastically moving point on the Bloch sphere. Recently, Hofmann, Mahler, and Hess [Phys. Rev. A 57, 4877 (1998)] showed that by making part of the coherent driving proportional to the homodyne photocurrent one can stabilize the state to any point on the bottom-half of the sphere. Here we reanalyze their proposal using the technique of stochastic master equations, allowing their results to be generalized in two ways. First, we show that any point on the upper- or lower-half, but not the equator, of the sphere may be stabilized. Second, we consider nonunit-efficiency detection, and quantify the effectiveness of the feedback by calculating the maximal purity obtainable in any particular direction in Bloch space.
Quantum feedback can stabilize a two-level atom against decoherence (spontaneous emission), putting it into an arbitrary (specified) pure state. This requires perfect homodyne detection of the atomic emission, and instantaneous feedback. Inefficient detection was considered previously by two of us. Here we allow for a non-zero delay time tau\\tautau in the feedback circuit. Because a two-level atom is a nonlinear optical system, an analytical solution is not possible. However, quantum trajectories allow a simple numerical simulation of the resulting non-Markovian process. We find the effect of the time delay to be qualitatively similar to that of inefficient detection. The solution of the non-Markovian quantum trajectory will not remain fixed, so that the time-averaged state will be mixed, not pure. In the case where one tries to stabilize the atom in the excited state, an approximate analytical solution to the quantum trajectory is possible. The result, that the purity (P=2rmTr[rho2]−1P=2{\\rm Tr}[\\rho^{2}]-1P=2rmTr[rho2]−1) of the average state is given by P=1−4gammatauP=1-4\\gamma\\tauP=1−4gammatau (where gamma\\gammagamma is the spontaneous emission rate) is found to agree very well with the numerical results.
For two two-level atoms coupled to a single Bosonic mode that is driven and heavily damped, the steady state can be entangled by resonantly driving the system [S. Schneider and G. J. Milburn, Phys. Rev. A 65, 042107 (2002)]. We present a scheme to significantly increase the steady-state entanglement by using homodyne-mediated feedback, in which the Bosonic mode is that of an electromagnetic cavity, the output of which is measured and the resulting homodyne photocurrent is used to modulate the field driving the qubits. Such feedback can increase the nonlinear response to both the decoherence process of the two-qubit system and the coherent evolution of individual qubits. We present the properties of the entangled states using the SO(3) QQQ function.
As a specific trait of quantum mechanics, quantum superposition is exploited in many applications such as quantum computation. This paper proposes a feedback scheme to stabilize the desired quantum superposition states. To this aim, a new factorization is derived for the stochastic master equation of a quantum system which is driven by the field in the superposition of coherent states. Then, a feedback scheme is proposed, which consists of a quantum system driven by the field in the superposition of coherent states and a classical feedback channel proportional to the observations of a Homodyne detector. Afterward, the quantum filter of the closed loop system is attained using the SLH framework. Finally, the factorization of the closed loop quantum filter is obtained and the control law is calculated such that the proposed factorization satisfies the stabilization condition. Finally, as an example, the control method is simulated for a two-level atom and the stabilization of the superposition states is achieved.