[Paper Review] Reply to Comment on "Continuous quantum measurement: inelastic tunneling and lack of current oscillations"

Thomas M. Stace, S. D. Barrett|arXiv (Cornell University)|Jun 30, 2004

Quantum and electron transport phenomena4 citations

TL;DR

This paper responds to criticisms of its earlier work on continuous quantum measurement in a charge qubit coupled to a point contact detector, demonstrating that coherent oscillations in the detector current are suppressed due to inelastic tunneling, especially in the low-bias regime ($eV < \phi$). The suppression arises naturally from a microscopic model using a Markovian master equation with three Lindblad jump operators, not from arbitrary assumptions, and the analysis is valid for experimentally accessible parameters.

ABSTRACT

We reply to a comment by Averin and Korotkov http://uk.arxiv.org/abs/cond-mat/0404549 on Stace and Barrett, PRL 92, 136802 (2004) http://link.aps.org/abstract/PRL/v92/e136802, showing that their specific criticisms are unfounded, and clarifying some of our results.

Motivation & Objective

• To resolve a dispute regarding the presence or absence of coherent oscillations in the current of a point contact detector coupled to a charge qubit.
• To clarify that the three-jump-operator structure in the conditional master equation is not an assumption but a consequence of the microscopic model and the rotating wave approximation.
• To validate the theoretical framework for continuous measurement in the regime $\Gamma_d \ll \phi$, where measurement-induced dephasing is weak compared to qubit energy splitting.
• To assess the validity of the Markovian approximation at high frequencies, particularly in the high-bias regime ($eV > \phi$), where fast dynamics may be neglected.
• To reaffirm the accuracy of the model in describing experimental observations of partial localization in double-well systems.

Proposed method

• Derivation of an unconditional master equation (UME) using the Born-Markov approximation and factorized initial conditions.
• Application of the rotating wave approximation (RWA) to the UME, valid in the limit $\Gamma_d / \phi \ll 1$, yielding a Markovian UME with three Lindblad super-operators.
• Unraveling the UME into a conditional master equation (CME) with three distinct jump operators, consistent with stochastic measurement trajectories.
• Derivation of the jump operators from a microscopic model of electron tunneling in the point contact, ensuring physical consistency with energy conservation.
• Use of explicit current measurement models to validate the coarse-graining in time scales $\sim \phi^{-1}$, justifying the Markovian approximation.
• Comparison of theoretical predictions with prior work (e.g., Shnirman et al., 2002) and experimental results (e.g., Elzerman et al., 2003) to assess consistency.

Experimental results

Research questions

• RQ1 Does inelastic tunneling in the point contact detector suppress coherent oscillations in the measured current, as claimed in the original work?
• RQ2 Are the three jump operators in the conditional master equation an arbitrary assumption or a necessary outcome of the microscopic model and approximations?
• RQ3 Is the Markovian description valid for high-frequency dynamics near the qubit splitting $\phi$, particularly in the high-bias regime ($eV > \phi$)?
• RQ4 How does the low-bias regime ($eV < \phi$) affect the qubit's relaxation and the absence of oscillatory signals in the detector current?
• RQ5 Can the theoretical framework accurately describe experimental observations of partial localization in double-well systems?

Key findings

• The three-jump-operator structure in the conditional master equation is not an assumption but a necessary consequence of the rotating wave approximation applied to the derived master equation.
• In the low-bias regime ($eV < \phi$), the qubit relaxes to the pure ground state $|g\rangle$, which is stationary, leading to no oscillatory signal in the detector current.
• The absence of peaks in the low-frequency power spectrum $S_{\mathrm{lb}}(\omega)$ at $\omega = \phi$ is consistent with the suppression of coherent oscillations.
• In the high-bias regime ($eV > \phi$), the Markovian approximation may not be valid for frequencies comparable to $\phi$, limiting the reliability of predictions at high frequencies.
• The model remains valid for timescales longer than $\phi^{-1}$, and the framework accurately describes continuous measurement in experimentally accessible parameter regimes.
• The original claim of suppressed oscillations is justified in the low-bias regime but remains uncertain in the high-bias regime due to the limitations of the RWA at fast timescales.