Engineering protected cavity-QED interactions through pulsed dynamical decoupling

TL;DR

This work introduces a pulsed dynamical decoupling framework to protect cavity-QED systems from low-frequency dephasing while preserving or engineering light–matter couplings. By moving to a toggling frame and applying carefully designed pulse sequences (e.g., XXYY, XY8), the authors recover effective JC, anti-JC, and Rabi interactions with tunable detunings and enhanced robustness to noise. The approach extends to cavity-mediated spin–spin interactions, enabling protected flip-flop, Ising, and squeezing couplings via a Magnus expansion analysis, with numerical benchmarks showing strong agreement between full dynamics and effective models. Practically, the method yields substantial improvements in entangling fidelities and cooperativity, offering a path to realize cavity QED platforms previously hindered by inhomogeneous broadening and slow drifts.

Abstract

We study a generic cavity QED setup under conditions where the coupling between the two-level systems and a single bosonic mode is significantly degraded by low-frequency noise. To overcome this problem, we identify pulsed dynamical decoupling strategies that suppress the effects of noise while still allowing for a coherent exchange of excitations between the individual subsystems. The corresponding pulse sequences can be further designed to realize either Jaynes-Cummings, anti-Jaynes-Cummings, or Rabi couplings, as well as different types of cavity-mediated interactions between the two-level systems. A detailed analysis of the residual imperfections demonstrates that this decoupling strategy can boost the effective cooperativity of the cavity QED system by several orders of magnitude and improve the fidelity of quantum-technologically relevant operations accordingly.

Figures (8)

• Figure 1: (a) Sketch of a cavity QED setup, where a TLS is coupled to a near-resonant bosonic mode. (b) Energy diagram of the relevant states of the JC model. In the presence of noise, the states $|e,n\rangle$ are shifted compared to the states $|g,n\rangle$ by a slowly fluctuating frequency $\xi(t)$, which leads to dephasing of the bare TLS with a rate $1/T_2^*$. (c) Simulation of the vacuum Rabi oscillations for the case where the dephasing rate is comparable to the coupling strength (left panel). Under the same conditions, but interrupting this evolution by an appropriate sequence of $\pi$-pulses, the effect of the noise can be significantly suppressed while preserving the coherent oscillations between the TLS and the cavity mode (right panel). See text for more details.
• Figure 2: (a) Plots of the modulation functions $f_{x,y,z}(t)$ for an example pulse sequence with four $\pi$-rotations (along both $x$ and $y$ axes) applied within a period $T$. (b) Dependence of the absolute values of $\eta_{x,y}$ on the detuning $\Delta$ for the basic sequence $X_{T/2}X_T$.
• Figure 3: (a) Plot of the transfer fidelity as a function of time for the pulse sequences XY8$_{m=2}$ (blue) and XXYY$_{m=0}$ (green) in the absence of noise. The markers indicate the result of exact numerical simulations with an interpulse spacing $\tau=0.1g^{-1}$, a pulse width $\tau_\pi=0.1\tau$ and $t_1=\tau/2$, while solid lines follow an ideal evolution with the corresponding $g_{\rm eff}$. The shaded line represents the transfer fidelity for the bare JC-interaction. (b) For the same sequences the average transfer fidelity at time $T_{\rm t}=\pi/(2g_{\rm eff})$ is plotted as a function of the noise strength $\sigma$ and for a total of $N_\pi=64$ pulses. While the solid lines represent the average values obtained as a result of 1000 independent noise realizations, randomly sampled from the probability distribution $P(\xi)=(2\pi\sigma^2)^{-1/2}\exp{(-\xi^2/2\sigma^2)}$, the shaded area indicates its variation (one standard deviation). The inset shows the corresponding transfer error $\mathcal{E}_{\rm t}$ on a logarithmic scale. (c) Dependence of the average transfer error on the number of pulses $N_\pi$ for the XY8$_{m=2}$ sequence with $\sigma=5g$ (round markers) and $\sigma=20g$ (square markers). The solid lines indicate the analytic prediction from Eq. \ref{['eq:XY8Error']} for the same parameters. (d) Time evolution of the observables $\langle\sigma_y\rangle$ and $N=\langle a^\dagger a\rangle$ for the XX$_{m=0}$ sequence with parameters $\tau=0.1g^{-1}$, $t_1=\tau$, $\tau_\pi=10^{-2}\tau$, and $\Delta=g_{\rm eff}$. The markers indicate the result obtained from exact numerical simulations, while the solid lines follow the prediction of the effective model in Eq. \ref{['eq:QRM']} with $\Delta_{\rm eff}=g_{\rm eff}$, $\upsilon=0$, and $\phi=0$. (e) The same as in (d) but for the sequence YY$_{m=1}$ with a cavity detuning, $\Delta=2\pi /T+g_{\rm eff}$ and pulse area $\pi+\delta \theta$, where $\delta\theta=g_{\rm eff} T/2$. This simulation is compared to the effective model in Eq. \ref{['eq:QRM']} with $\Delta_{\rm eff}=\upsilon=g_{\rm eff}$ and $\phi=-\pi/2$. The small mismatch with the effective model, most visible at the peak values of $N(t)$, is reduced when using a smaller interpulse spacing $\tau$. The insets in panels (d) and (e) illustrate the pulse sequence, where the area enclosed by the dotted lines corresponds to $\pi$. Notably, in panel (e), the pulse area exceeds $\pi$ by a small amount $\delta\theta$.
• Figure 4: (a) Plot of the average entanglement fidelity $\langle \mathcal{F}_{\rm e}\rangle$ as a function of the noise strength $\sigma$ for a cavity QED system with two TLSs. The TLSs are detuned by $\Delta= 30g$ and undergo cavity-mediated flip-flop interactions with strength $J=g^2/\Delta$. The solid lines (shaded areas) represent the average values (standard deviations) obtained from an average over 500 noise realizations for different numbers of $N_\pi$ instantaneous $\pi$-pulses. (b) Dependence of the effective interaction parameters on $\Delta$ for the $X_{T/2}X_T$ pulse sequence. The red, blue and green shaded stripes indicate the regimes leading to flip-flop, Ising and squeezing interactions, respectively. See Sec. \ref{['subsec:EffectiveSS']} and Fig. \ref{['Fig:5']} for more details.
• Figure 5: Numerical benchmarking of the different types of effective flip-flop (left), Ising (middle) and squeezing (right) interactions discussed in Sec. \ref{['subsec:EffectiveSS']}. The upper (lower) panels show the exact time evolution of initial state $|eg\rangle$ ($|gg\rangle$) in terms of the population $P_{eg}$ ($P_{gg}$) in blue and the concurrence $\mathcal{C}$ of the reduced TLSs state in red. The solid lines and shaded areas represent the mean values and standard deviation of these quantities, as obtained from averaging over $500$ realizations of static fluctuations with strength $\sigma=0.3 g$. In all simulations we assume an $X_{T/2}X_{T}$ sequence with a pulse duration of $\tau_\pi=0.01\tau$. The other relevant parameters for the left panel are $m=2$ and $T=0.2g^{-1}$. For the Ising and the squeezing interactions these are $m=0$, $T=0.1g^{-1}$ and $\Delta_{\rm eff}=10g$, and $m=3$, $T=0.2g^{-1}$, and $\Delta_{\rm eff}=-2.16g$, respectively. The relevant detunings $\Delta$ are indicated by the respective colored bars in Fig. \ref{['Fig:4']}(b).