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arXiv:cond-mat/0202303 · Physical review. B, Condensed matter

Spectrum of qubit oscillations from generalized Bloch equations

Rusko Ruskov, Alexander N. Korotkov

Published Feb 5, 2003• 37 citations
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Abstract

We have developed a formalism suitable for calculation of the output spectrum of a detector continuously measuring quantum coherent oscillations in a solid-state qubit, starting from microscopic generalized Bloch equations. The results coincide with those obtained using Bayesian and master equation approaches. The previous results are generalized to the cases of arbitrary detector response and finite detector temperature.

DOI
10.1103/physrevb.67.075303

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We have developed a formalism suitable for calculation of the output spectrum of a detector continuously measuring quantum coherent oscillations in a solid-state qubit, starting from microscopic generalized Bloch equations. The results coincide with those obtained using Bayesian and master equation approaches. The previous results are generalized to the cases of arbitrary detector response and finite detector temperature.
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.
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 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.
We employ a quantum Langevin-equation approach to establish Bloch-type dynamical equations, on a fully microscopic basis, to investigate the measurement of the state of a coupled quantum dot qubit by a nearby quantum point contact. The ensuing Bloch equations allow us to analyze qubit relaxation and decoherence induced by measurement, and also to determine the noise spectrum of the meter output current with the help of a quantum regression theorem, at arbitrary bias voltage and temperature. Our analyses attempt to provide a clear resolution of a recent debate concerning the occurrence of a quantum oscillation peak in the noise spectrum.
The quantum evolution of an individual solid-state qubit during the process of its continuous measurement can be described by the recently developed Bayesian formalism. In contrast to the conventional ensemble-averaged formalism, it takes into account the measurement record (in a way similar to the standard Bayesian analysis) and therefore is able to consider individual realizations of the measurement process. The formalism provides testable experimental predictions and can be used for the analysis of a quantum feedback control of solid-state qubits. The Bayesian formalism can be also applied to the continuous measurement of entangled qubits; in particular, it shows how to create a fully entangled pair of qubits without their direct interaction, just by measuring them with an equally coupled detector.