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arXiv:quant-ph/9707046 · Physical Review A

Quantum control of atomic systems by homodyne detection and feedback

Holger Friedrich Hofmann, Günter Mahler A, Ortwin G. Hess

Published Jun 1, 1998• 58 citations
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Abstract

We investigate the possibilities of preserving and manipulating the coherence of atomic two-level systems by ideal projective homodyne detection and feedback. For this purpose, the photon emission process is described on time scales much shorter than the lifetime of the excited state using a model based on Wigner-Weisskopf theory. The backaction of this emission process is analytically described as a quantum diffusion of the Bloch vector. It is shown that the evolution of the atomic wave function can be controlled completely using the results of homodyne detection. This allows the stabilization of a known quantum state or the creation of coherent states by a feedback mechanism. However, the feedback mechanism can never compensate the dissipative effects of quantum fluctuations even though the coherent state of the system is known at all times.

DOI
10.1103/physreva.57.4877

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arXiv:quant-ph/0201145•Physical Review A•Jul 15, 2002•135 citations
We investigate the possibilities of preserving and manipulating the coherence of atomic two-level systems by ideal projective homodyne detection and feedback. For this purpose, the photon emission process is described on time scales much shorter than the lifetime of the excited state using a model based on Wigner-Weisskopf theory. The backaction of this emission process is analytically described as a quantum diffusion of the Bloch vector. It is shown that the evolution of the atomic wave function can be controlled completely using the results of homodyne detection. This allows the stabilization of a known quantum state or the creation of coherent states by a feedback mechanism. However, the feedback mechanism can never compensate the dissipative effects of quantum fluctuations even though the coherent state of the system is known at all times.
We investigate the properties of ideal projective balanced homodyne detection measurements on low intensity light fields emitted by individual atomic systems during short time intervals. A model for time resolved photon emissions based on Wigner-Weisskopf theory is used to describe the emission process. The back- action of this emission process is analytically described as a quantum diffusion of the Bloch vector. It is shown that the evolution of the atomic wavefunction can be controlled completely using the results of homodyne detection. This allows the stabilization of a known quantum state or the creation of coherent states by a feedback mechanism. However, the feedback mechanism can never compensate the dissipative effects of quantum fluctuations even though the coherent state of the system is known at all times.
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.
Abstract. In this paper, we demonstrate the application of quantum feedback control in creating desired states for atomic and photonic systems utilizing a semi-infinite waveguide coupled with multiple two-level atoms. In this approach, an initially excited atom can emit one photon into the waveguide, and the photon can be reflected by either the terminal mirror of the waveguide or other atoms, establishing various feedback loops through the coherent interactions between the atom and photon. When there are no more than two excitations in the waveguide quantum electrodynamics (QED) system, the evolution of quantum states can be effectively elucidated through the lens of random graph theory. However, this process is influenced by the environment. We propose that the environment-induced dynamics in the coherent feedback loop can be eliminated by measurement-based feedback control or coherent drives. Consequently, in the atom-waveguide interactions in open quantum systems, measurement-based feedback has the potential to modulate the quantum steady states. Additionally, the homodyne detection noise during the measurement process can induce errors upon quantum states, which can be influenced by the coherent feedback parameter designs.
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.
We compare two different approaches to the control of the dynamics of a continuously monitored open quantum system. The first is Markovian feedback, as introduced in quantum optics by Wiseman and Milburn [Phys. Rev. Lett. 70, 548 (1993)]. The second is feedback based on an estimate of the system state, developed recently by Doherty and Jacobs [Phys. Rev. A 60, 2700 (1999)]. Here we choose to call it, for brevity, Bayesian feedback. For systems with nonlinear dynamics, we expect these two methods of feedback control to give markedly different results. The simplest possible nonlinear system is a driven and damped two-level atom, so we choose this as our model system. The monitoring is taken to be homodyne detection of the atomic fluorescence, and the control is by modulating the driving. The aim of the feedback in both cases is to stabilize the internal state of the atom as close as possible to an arbitrarily chosen pure state, in the presence of inefficient detection and other forms of decoherence. Our results (obtained without recourse to stochastic simulations) prove that Bayesian feedback is never inferior, and is usually superior, to Markovian feedback. However, it would be far more difficult to implement than Markovian feedback and it loses its superiority when obvious simplifying approximations are made. It is thus not clear which form of feedback would be better in the face of inevitable experimental imperfections.