Multiqubit spectroscopy of Gaussian quantum noise

Gerardo A. Paz-Silva, Leigh M. Norris, and Lorenza Viola

Phys. Rev. A 95, 022121 – Published 23 February 2017

Abstract

We introduce multipulse quantum noise spectroscopy protocols for spectral estimation of the noise affecting multiple qubits coupled to Gaussian dephasing environments including both classical and quantum sources. Our protocols are capable of reconstructing all the noise auto- and cross-correlation spectra entering the multiqubit dynamics, providing access, in particular, to the asymmetric spectra associated with nonclassical environments. Our result relies on (i) an exact analytic solution for the reduced multiqubit dynamics that holds in the presence of an arbitrary Gaussian environment and dephasing-preserving control; (ii) the use of specific timing symmetries, which allow for a frequency comb to be engineered for all filter functions of interest, and for the spectra to be related to experimentally accessible observables. We show that quantum spectra have distinctive dynamical signatures, which we explore in two paradigmatic open-system models describing spin and charge qubits coupled to bosonic environments. Complete noise spectroscopy is demonstrated numerically in a realistic setting consisting of two-exciton qubits coupled to a phonon bath. The estimated spectra allow us to accurately predict the exciton dynamics as well as extract the temperature and spectral density of the quantum environment.

Received 21 September 2016

DOI:https://doi.org/10.1103/PhysRevA.95.022121

Physics Subject Headings (PhySH)

Open quantum systems Quantum control Quantum correlations, foundations & formalism

Quantum Information, Science & Technology

Authors & Affiliations

Gerardo A. Paz-Silva¹, Leigh M. Norris², and Lorenza Viola²

¹Centre for Quantum Dynamics & Centre for Quantum Computation and Communication Technology, Griffith University, Brisbane, Queensland 4111, Australia
²Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, New Hampshire 03755, USA

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Vol. 95, Iss. 2 — February 2017

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Images

Figure 1
Sample pulse patterns of pairs of switching functions $y_{ℓ, ℓ^{'}} (t)$ and $y_{\bar{ℓ}, ℓ^{'}} (t)$ , obeying mirror and displacement (anti)symmetry in an interval $[0, T]$ . For each symmetry (displacement or mirror), $y_{ℓ, ℓ^{'}} (t)$ has been chosen to be symmetric and $y_{\bar{ℓ}, ℓ^{'}} (t)$ antisymmetric, so that their product is antisymmetric.
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Figure 2
Spectral reconstructions and estimates of bath parameters for two exciton qubits coupled to a common 1D phonon bath. (a) Spectral reconstructions and actual spectra for an initially thermal bath at $T_{B} = 5$ K. The reconstructed spectra $S_{11}^{+, R} (ω)$ (black dots), $Re [S_{12}^{+, R} (ω)]$ (green squares), $Im [S_{12}^{+, R} (ω)]$ (blue diamonds), $Re [S_{12}^{-, R} (ω)]$ (red up arrows), and $Im [S_{12}^{-, R} (ω)]$ (purple down arrows) are plotted alongside the actual spectra (numbered solid lines). Inset: Temperature dependence obtained by spectral reconstruction via $S_{12}^{+, R} (ω) / S_{12}^{-, R} (ω)$ (red dots) vs the actual temperature dependence, $coth (β | ω | ℏ / 2)$ , for $T_{B} = 5$ K (red line). The bath temperature estimated via Eq. (68) is 5.02 K. Spectral density $J (ω)$ obtained from the spectral reconstructions, Eq. (69) (blue dots), and actual spectral density (blue line). (b) Contours of the magnitude of the reconstructed cross-correlation spectrum $| S_{12}^{R} (ω) | = | S_{12}^{+, R} (ω) + S_{12}^{-, R} (ω) | / 2$ vs angular frequency and temperature (white dash line corresponding to the estimated bath temperature). Note how the spectral asymmetry about $ω = 0$ , a signature of the quantum bath, becomes more pronounced as the temperature decreases.
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Figure 3
Qubit dynamics under free evolution for the initial product state $| ψ_{1}^{-} {〉 = | + 〉}_{1} \otimes {| - z 〉}_{2}$ . (a) Phase evolution $ϕ (t) = C_{1, 12} (t) - C_{1, 1} (t)$ of qubit 1. Because $ϕ (t)$ depends exclusively on the quantum spectra $S_{12}^{-} (ω)$ and $S_{11}^{-} (ω)$ , it is a signature of the quantum nature of the bath. The actual phase evolution under free evolution (solid line) is plotted along with the phase evolution predicted using (1) all reconstructed spectra, $S$ (asterisks); (2) all reconstructed spectra except the quantum self-spectra (dots), $S_{r}$ ; and (3) only the classical reconstructed spectra, $S_{c}$ (large dashes). Inset: Dephasing of qubit 1. Unlike phase evolution, the decay of coherences depends exclusively on the classical spectra. Actual dephasing (solid line) and dephasing predicted by spectral reconstructions (asterisks) show excellent agreement.
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Figure 4
Two-qubit fidelity decay under free and controlled evolution. The latter consists of repetitions of the product-mirror antisymmetric concatenated sequence ${CDD}_{3} \times {CDD}_{2}$ , with cycle time $T = 2.7$ ps. (a) Estimated average fidelity, $\bar{F} (t)$ , obtained by averaging 1000 Haar-random initial pure states of the two qubits. Under free evolution, the actual $\bar{F} (t)$ (blue solid line) and the $\bar{F} (t)$ predicted using all spectral reconstructions in $S$ (blue asterisks) demonstrate excellent agreement. Predictions produced with $S_{c}$ (small blue dashes) and $S_{r}$ (blue dots) significantly overestimate the fidelity. The classical prediction made with $S_{c}$ and the actual dynamics differ by over $6 %$ . For the controlled evolution, the actual $\bar{F} (t)$ (red solid line), the predicted $\bar{F} (t)$ using $S$ (red diamonds), the $\bar{F} (t)$ predicted using $S_{r}$ (large red dashes) and the $\bar{F} (t)$ predicted using $S_{c}$ (red dot-dashed line) are in closer agreement, as the control partially suppresses the contributions of the quantum spectra. However, the classical prediction and the actual $\bar{F} (t)$ still differ by $1 %$ (inset). (b) Actual and predicted fidelity for the subset of initial states, out of the total sampled 1000 random states, which demonstrate the greatest discrepancies between the classical predictions using $S_{c}$ and the actual dynamics. For free evolution, the actual fidelity and the classical prediction differ by as much as $11 %$ . For the controlled evolution, the maximum difference is about $2 %$ (inset).
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