[Paper Review] Decoherence in Josephson qubits

G. Falci, Elisabetta Paladino+1 more|arXiv (Cornell University)|Dec 20, 2003

Quantum and electron transport phenomena2 references7 citations

TL;DR

This paper investigates decoherence in charge Josephson qubits due to internal and external noise sources, modeling $1/f$ noise from bistable fluctuators and external electromagnetic environments via the spin-boson model. It derives exact expressions for dephasing under non-Markovian conditions, showing that memory effects are critical and that only one additional microscopic parameter beyond the power spectrum is needed to characterize decoherence in many cases.

ABSTRACT

A high degree of quantum coherence is a crucial requirement for the implementation of quantum logic devices. Solid state nanodevices seem particularly promising from the point of view of integrability and flexibility in the design. However decoherence is a serious limitation, due to the presence of many types low energy excitations in the ``internal'' environment and of ``external'' sources due to the control circuitery. Here we study both kind of dephasing in a special implementation, the charge Josephson qubit, however many of our results are applicable to a large class of solid state qubits. This is the case of 1/f noise for which we introduce and study a model of an environment of bistable fluctuatiors. External sources of noise are analized in terms of a suitable harmonic oscillator environment and the explicit mapping on the spin boson model is presented. We perform a detailed investigation of various computation procedures (single shot measurements, repeated measurements) and discuss the problem of the information needed to characterize the effect of the environment. For a fluctuator environment with 1/f spectrum memory effects turn out to be important. Although in general information beyond the power spectrum is needed, in many situations this results in the knowledge of only one more microscopic parameter of the environment. This allows to determine which degrees of freedom of the environment are effective sources of decoherence in each different physical situation considered.

Motivation & Objective

• To understand and model decoherence in charge Josephson qubits arising from both internal low-energy excitations and external control circuitry.
• To investigate the role of $1/f$ noise from bistable fluctuators in dephasing, particularly in non-Markovian regimes with memory effects.
• To determine the minimal information required to characterize environmental effects on qubit coherence, beyond the power spectrum.
• To analyze the impact of repeated and single-shot measurements on decoherence dynamics in solid-state qubits.
• To map external electromagnetic noise sources onto the spin-boson model for quantitative analysis of dephasing rates.

Proposed method

• Models $1/f$ noise using an environment of bistable fluctuators with transition rates $\gamma_{+}$ and $\gamma_{-}$, leading to non-Markovian dephasing.
• Derives exact probability densities $\pi_{\alpha,\beta}(t; t_m - t_1 | 0)$ for fluctuator state transitions over time, accounting for multiple switches.
• Uses path integral-like summation over all possible switching sequences and transition times to compute the decoherence factor $Z(t)$.
• Applies second cumulant expansion to approximate the exact result as a Gaussian process with the same power spectrum $S(\omega)$, recovering Eq. (8).
• Introduces the full time-dependent decoherence factor $Z(t) = A e^{-\frac{\gamma}{2}(1-\alpha)t} + (1-A)e^{\frac{\gamma}{2}(1+\alpha)t}$, valid in the relaxation regime.
• Maps external electromagnetic noise to a harmonic oscillator bath, enabling explicit connection to the spin-boson model for dephasing.

Experimental results

Research questions

• RQ1 How does $1/f$ noise from bistable fluctuators affect qubit dephasing in charge Josephson qubits, especially when memory effects are non-negligible?
• RQ2 What is the minimal set of environmental parameters needed to fully characterize decoherence beyond the power spectrum?
• RQ3 How do repeated versus single-shot measurements influence the effective decoherence rate in solid-state qubits?
• RQ4 In what regime does the non-Markovian dynamics of fluctuators lead to significant deviations from Markovian approximations?
• RQ5 To what extent can the spin-boson model accurately describe both internal and external sources of decoherence in Josephson qubits?

Key findings

• The exact decoherence factor $Z(t)$ is derived as $Z(t) = A e^{-\frac{\gamma}{2}(1-\alpha)t} + (1-A)e^{\frac{\gamma}{2}(1+\alpha)t}$, valid for fluctuators undergoing relaxation dynamics.
• Memory effects are significant in $1/f$ noise environments, and the full non-Markovian dynamics must be considered for accurate modeling.
• Beyond the power spectrum, only one additional microscopic parameter (e.g., $\overline{\delta p}$) is needed to characterize the effective decoherence in most physical scenarios.
• The second cumulant expansion of the exact result reproduces the standard Markovian dephasing formula (Eq. 8), validating its use under certain conditions.
• The model shows that both internal fluctuators and external circuitry contribute significantly to decoherence, with the latter mapped to a harmonic oscillator bath.
• The analysis reveals that the effective decoherence rate depends on the interplay between fluctuator transition rates and the qubit's energy splitting, with non-trivial time dependence in the non-Markovian regime.