Protecting collective qubits from non-Markovian dephasing

TL;DR

This work tackles non-Markovian inhomogeneous dephasing in collectively-encoded qubits by mapping the dephasing spectrum $P( obreak\omega)$ to a time-frequency phase-space displacement and constructing a sequential Dicke-state ladder. By truncating this ladder and treating higher-lying states as a continuum, the authors formulate a resource-efficient GKSL-based description that captures non-Markovian dynamics and enables analytic insights. They predict a driving-protection regime where strong driving suppresses dephasing, with scaling laws $gamma_{ m G,Omega}\,\propto e^{-Omega^2/8}$ for Gaussian spectra and analogous but slower suppression for light-shift spectra, confirmed by an experiment using a Rydberg superatom in a cavity. The experimental results show coherence times extended up to ~14x the undriven dephasing time, matching numerical simulations and illustrating a practical, scalable protection mechanism for quantum technologies relying on collective qubits.

Abstract

Collectively-encoded qubits, involving ensembles of atomic or solid-state emitters, present many practical advantages for quantum technologies. However, they suffer from uncontrolled inhomogeneous dephasing which couples them to a quasi-continuum of dark states. In most cases, this process cannot be encompassed in a standard master equation with time-independent coefficients, making its description either tedious or inaccurate. We show that it can be understood as a displacement in time-frequency phase space and accurately included in resource-efficient numerical simulations of the qubit's dynamics. This description unveils a regime where the qubit becomes protected from dephasing through a combination of strong driving and non-Markovianity. We experimentally investigate this regime using a Rydberg superatom and extend its coherent dynamics beyond the inhomogeneous-dephasing characteristic time by an order of magnitude.

Figures (7)

• Figure 1: Modeling inhomogeneous dephasing and driving protection. The effect of an inhomogeneous dephasing spectrum $P(\omega)$ (a) is mapped to a displacement in a phase space where frequency and time play the roles of the usual position and momentum quadratures $x$ and $p$ (b). This allows one to derive the corresponding ladder of $n+1$ dephasing-coupled Dicke states $\left| E_{k\leq n} \right\rangle$ and their coupling coefficients (c), included in the Hamiltonian to describe the non-Markovian features, and to determine the irreversible decay of $\left| E_n \right\rangle$ into the "continuum" $C=\{\left| E_{k>n} \right\rangle\}$. In the Markovian case, $\left| E_0 \right\rangle$ decays directly in the continuum (d). When driving the qubit transition with a Rabi frequency $\Omega$ (e), the dressed eigenstates $\left| \psi_\pm \right\rangle=(\left| G \right\rangle\pm\left| E_0 \right\rangle)/\sqrt{2}$ also decay in the continuum, owing to its broad spectral extent. In a non-Markovian case, the decay out of the qubit's subspace decreases drastically between the free dephasing from $\left| E_0 \right\rangle$ (f) and the driven case (g), due to the energy difference between $\left| \psi_\pm \right\rangle$ and the $\left| E_{k\geq 1} \right\rangle$ asymmetric Dicke states.
• Figure 2: Dicke basis truncation. (a) and (b) Energy levels $H_k$ of truncated basis states with $n=3$ for the Gaussian and lighshift-broadened cases, respectively. Arrow linewidths are proportional to coupling coefficients $|V_k|$ or decay rates $\gamma_3$ towards the continuum $\left| C \right\rangle$. The dashed line for $E_3$ represent the energy $H_3$ before the addition of the Lamb-like energy shift $\delta_3$. (c) and (d) Related evolutions of the population $|s(t)|^2$. The black curves are the analytic solutions given by Eq. \ref{['EqDecaySvsX']}. The blue dotted and dashed lines are computed in a truncated basis with $n=1$, without and with the continuum state, respectively. The green curves use the basis with $n=3$, again without or with the continuum.
• Figure 3: Driving protection. Evolution of the population $|s_\Omega(t)|^2$ (a) for the Gaussian spectrum $\rho_\mathrm{G}$ and (b) for the lightshift-broadened spectrum $\rho_\mathrm{L}$, with and without driving. The black curves are the analytic solutions of Eq. \ref{['EqDecaySvsX']} for an undriven qubit ($\Omega=0$). The blue lines are computed in a Dicke basis with a Rabi frequency $\Omega=10$. The dotted red curves are the corresponding approximations given by Eqs. \ref{['eq:SGOmegaAppr']} and \ref{['eq:SLOmegaAppr']}. In (b), the dotted and dashed green curves are the populations of $\left| \psi_+ \right\rangle$ and $\left| \psi_- \right\rangle$ states, respectively.
• Figure 4: Explanation of the driving protection mechanism. The eigenstates $\left| \psi_\pm \right\rangle$ of the driven qubit, shifted by $\pm\Omega/2$, remain resonantly coupled to a "continuum" $\{\left| x \right\rangle\}$ for a Lorentzian spectrum (blue, left). This coupling becomes off-resonant for a Gaussian spectrum (red, center), which decreases the decay rate of the Rabi oscillations. For a light-shift spectrum, $\left| \psi_+ \right\rangle$ decays faster than $\left| \psi_- \right\rangle$.
• Figure 5: Experimental protocol. (a) Schematic of the experimental setup featuring a Rydberg superatom in an 4-mirror optical cavity, with the laser pulses and the transition scheme used in the experiment, described in Section \ref{['sec:Expmeth']}. (b) Measured pulse shapes (dots) and best-fit functions (lines) used in the simulations. The lower panel is a zoom on the times between the dotted vertical lines in the upper panel. The blue and dark red curves represent the blue and red drive pulses, respectively. In the lower panel, we represent the short red drive pulse used to prepare the cloud in the singly-excited symmetric Dicke state $\left| E_0 \right\rangle$ with a fast $\pi$ pulse. The cyan curve represents the blue read beam mapping $\left| E_0 \right\rangle$ to a photon, while the red line indicates the photon's temporal intensity profile seen by the single-photon detector. The gray dashed lines represent the average times for the $\pi$ pulse and the detected photon.