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Non-Markovian homodyne-mediated feedback on a two-level atom: a quantum trajectory treatment
Quantum Physics · Submitted 17 Nov 2000
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The solution of the non-Markovian quantum trajectory will not remain fixed, so that the time-averaged state will be mixed, not pure.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Quantum feedback can stabilize a two-level atom against decoherence (spontaneous emission), putting it into an arbitrary (specified) pure state. | background 1.00 | 0.13 | 0.10 | asserted 0.89 |
| s1 | This requires perfect homodyne detection of the atomic emission, and instantaneous feedback. | background 0.96 | 0.09 | 0.03 | asserted 1.00 |
| s2 | Inefficient detection was considered previously by two of us. | background 0.97 | 0.30 | 0.02 | asserted 0.99 |
| s3 | Here we allow for a non-zero delay time $\tau$ in the feedback circuit. | contribution 0.93 | 0.74 | 0.03 | asserted 0.92 |
| s4 | Because a two-level atom is a nonlinear optical system, an analytical solution is not possible. | limitation 0.73 | 0.23 | 0.02 | asserted 1.00 |
| s5 | However, quantum trajectories allow a simple numerical simulation of the resulting non-Markovian process. | method 0.97 | 0.49 | 0.05 | asserted 0.73 |
| s6 | We find the effect of the time delay to be qualitatively similar to that of inefficient detection. | result 0.89 | 0.95 | 0.36 | asserted 1.00 |
| s7 | The solution of the non-Markovian quantum trajectory will not remain fixed, so that the time-averaged state will be mixed, not pure. | result 0.96 | 0.88 | 0.44 | asserted 1.00 |
| s8 | In the case where one tries to stabilize the atom in the excited state, an approximate analytical solution to the quantum trajectory is possible. | contribution 0.41 | 0.84 | 0.06 | hedged possibility 0.72 |
| s9 | 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. | result 1.00 | 0.94 | 0.42 | asserted 1.00 |
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Original 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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