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arXiv:cond-mat/0003225 · Physical review. B, Condensed matter

Output spectrum of a detector measuring quantum oscillations

Alexander N. Korotkov

Published Feb 5, 2001• 97 citations
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Abstract

We consider a two-level quantum system (qubit) which is continuously measured by a detector, and calculate the spectral density of the detector output. In the weakly coupled case the spectrum exhibits a moderate peak at the frequency of quantum (Rabi) oscillations and a Lorentzian-shape increase of the detector noise at low frequency. As the coupling increases, the spectrum transforms into a single Lorentzian corresponding to random jumps between two states. We prove that the Bayesian formalism for the selective evolution of the density matrix gives the same spectrum as the conventional master equation approach, despite the significant difference in interpretation. The effects of the detector nonideality and the finite-temperature environment are also discussed.

DOI
10.1103/physrevb.63.085312

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We consider a two-level quantum system (qubit) which is continuously measured by a detector, and calculate the spectral density of the detector output. In the weakly coupled case the spectrum exhibits a moderate peak at the frequency of quantum (Rabi) oscillations and a Lorentzian-shape increase of the detector noise at low frequency. As the coupling increases, the spectrum transforms into a single Lorentzian corresponding to random jumps between two states. We prove that the Bayesian formalism for the selective evolution of the density matrix gives the same spectrum as the conventional master equation approach, despite the significant difference in interpretation. The effects of the detector nonideality and the finite-temperature environment are also discussed.
We calculate the noise spectrum of the output signal of a quantum detector during continuous measurement of a two-level system (qubit). We generalize the previous results obtained for the regime of high voltages (when eVeVeV is much larger than the qubit's energy level splitting ΔΔΔ) to the case of arbitrary voltages and temperatures. When V∼ΔV \sim ΔV∼Δ the output spectrum is essentially asymmetric in frequency, i.e., the output signal is no longer classical. In the emission (negative frequency) part of the spectrum the peak due to the qubit's coherent oscillations can be 8 times higher that the background pedestal. For V<ΔV < ΔV<Δ and T=0 the coherent peaks do not appear at all.
The Bayesian formalism for a continuous measurement of solid-state qubits is extended to a model which takes into account several factors of the detector nonideality. In particular, we consider additional classical output and backaction noises (with finite correlation), together with quantum-limited output and backaction noises, and take into account possible asymmetry of the detector coupling. The formalism is first derived for a single qubit and then generalized to the measurement of entangled qubits.
We have studied quantum coherent oscillations of two qubits under continuous measurement by a symmetrically coupled mesoscopic detector. The analysis is based on a Bayesian formalism that is applicable to individual quantum systems. Measurement continuously collapses the two-qubit system to one of the subspaces of the Bell basis. For a detector with linear response this corresponds to measurement of the total spin of the qubits. In the other extreme of purely quadratic response the operator ${\ensuremath{\sigma}}_{y}^{1}{\ensuremath{\sigma}}_{y}^{2}+{\ensuremath{\sigma}}_{z}^{1}{\ensuremath{\sigma}}_{z}^{2}$ is measured. In both cases, collapse naturally leads to spontaneous entanglement which can be identified by measurement of the power spectrum and/or the average current of the detector. Asymmetry between the two qubits results in evolution between the different measurement subspaces. However, when the qubits are even weakly coupled to the detector, a kind of quantum Zeno effect cancels the gradual evolution and replaces it with rare, abrupt switching events. We obtain the asymptotic switching rates for these events and confirm them with numerical simulations. We show how such switching affects the observable power spectrum on different time scales.
We consider the continuous measurement of a double quantum dot by a weakly coupled detector (tunnel point contact nearby). While the conventional approach describes the gradual system decoherence due to the measurement, we study the situation when the detector output is explicitly recorded that leads to the opposite effect: gradual purification of the double-dot density matrix. The nonlinear Langevin equation is derived for the random evolution of the density matrix which is reflected and caused by the stochastic detector output. Gradual collapse, gradual purification, and the quantum Zeno effect are naturally described by the equation. We also discuss the possible experiments to confirm the theory.
We consider a continuous measurement of a two-level system (double-dot) by weakly coupled detector (tunnel point contact nearby). While usual treatment leads to the gradual system decoherence due to the measurement, we show that the knowledge of the measurement result can restore the pure wavefunction at any time (this can be experimentally verified). The formalism allows to write a simple Langevin equation for the random evolution of the system density matrix which is reflected and caused by the stochastic detector output. Gradual wavefunction ``collapse'' and quantum Zeno effect are naturally described by the equation.