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Non-Markovian homodyne-mediated feedback on a two-level atom: a quantum trajectory treatment

Jin Wang, Howard M. Wiseman, Gerard J Milburn

Published Nov 17, 2000• 14 citations
OpenAlex

Abstract

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 $\\tau$ 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 ($P=2{\\rm Tr}[\\rho^{2}]-1$) of the average state is given by $P=1-4\\gamma\\tau$ (where $\\gamma$ is the spontaneous emission rate) is found to agree very well with the numerical results.

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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.

Quantum Electronics and Laser Science Conference•Nov 13, 2002•1 citation
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.
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.
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.
Abstract The no-knowledge quantum feedback, introduced in Phys. Rev. Lett., 113, 020407 (2014), is a measurement-based feedback protocol for decoherence suppression in a quantum system coupled to noisy environment. By continuously measuring the environmental noise, without directly gathering any information about the system, the decoherence effect can be suppressed by feeding back quantum controls proportional to the measured signal. In the original work, the feedback control was assumed instantaneous, leading to perfect cancellation of noise backaction on the quantum system. However, the instantaneous feedback is difficult to achieve in practice, and close-loop feedback protocols are always associated with finite delayed time. Therefore, in this work, we consider the effects of the delay between the time at which the measurement signal is acquired and the time that such signal is fed back to the system. We investigate the problem with an example of a two-level system (qubit) coupled to a Markovian reservoir, via a Hermitian coupling operator, where a homodyne detection is used to measure the environmental noise. We numerically simulate quantum stochastic trajectories of the qubit and analyse their averaged dynamics. We find that the feedback control with time delay can either enhance or reduce the decoherence effects, depending on whether the delayed time is in-phase or out-of-phase with the unitary dynamics of the qubit system.
The non-Markovian correlation spectra of spontaneous emission of an excited two-level atom are derived including the effect of finite size of the atom and all the possible contribution of allowed multipole radiations. The emission process is then analyzed by the quantum stochastic trajectory approach. The non-Markovian effect is counted in by expanding the original system to an enlarged system with Markovian reservoirs. In the case of a hydrogenlike atom with large atomic number Z, the deviation from the Weisskopf-Wigner result is quite evident.
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.
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.