Control Design for Inhomogeneous-Broadening Compensation in Single-Photon Transducers

A transducer of single photons between microwave and optical frequencies can be used to realize quantum communication over optical fiber links between distant superconducting quantum computers. A promising scalable approach to constructing such a transducer is to use ensembles of quantum emitters interacting simultaneously with electromagnetic fields at optical and microwave frequencies. However, inhomogeneous broadening in the transition frequencies of the emitters can be detrimental to this collective action. In this paper, we utilize a gradient-based optimization strategy to design the temporal shape of the laser field driving the transduction system to mitigate the effects of inhomogeneous broadening. We study the improvement of transduction efficiencies as a function of inhomogeneous broadening in different single-emitter cooperativity regimes and correlate it with a restoration of super-radiance effects in the emitter ensembles. Furthermore, to assess the optimality of our pulse designs, we provide certifiable bounds on the design problem and compare them to the achieved performance.

Figure 1

(a) Schematic of a three-level system ensemble-based transducer device. (b) Scaling of transduction efficiency with increasing number $(N)$ of three-level systems in a homogeneous ensemble for different cooperativities $C$ (we keep $\gamma$ fixed and change $\mathrm{\Gamma }$ to vary cooperativity). (c) Decrease in the transduction efficiency through randomly inhomogeneously broadened ensembles of $N=10$ emitters with increasing inhomogeneous broadening $\mathrm{\Delta }$ for different cooperativities $C$. For each value of the inhomogeneous broadening $\mathrm{\Delta }$, $100$ randomly broadened ensembles are created by sampling the emitter detunings ${\delta }_{\mu }^{\left(i\right)},{\delta }_{\mathrm{opt}}^{\left(i\right)}$ from a Gaussian distribution with standard deviation equal to $\mathrm{\Delta }$. Each plot point corresponds to the mean over the 100 ensembles with inhomogeneous broadening equal to the corresponding value of $\mathrm{\Delta }$ and the shaded regions represent the standard deviation. (d) Transduction spectra of ensembles ($N=10,C=0.1$) with varying inhomogeneous broadening $\mathrm{\Delta }$.

Figure 2

(a) Fourier transform of the input microwave field (Gaussian waveform). Dashed lines are representative of the individual emitter frequencies in a random ensemble ($N=10,\mathrm{\Delta }=200\gamma$). (b) Amplitudes of the harmonic components of the optimized $\mathrm{\Omega }\left(t\right)$ designed for the same ensemble. (c) Comparison of the transduction spectrum of the same ensemble with and without optimized drives applied—the transduction spectrum with the optimized drive is computed using a Floquet scattering theory approach [99].

Figure 3

Optimized drives countering inhomogeneous broadening. (a) Transduction efficiency and (b) improvement in the transduction efficiency through randomly inhomogeneously broadened ensembles of $N=10$ emitters with increasing inhomogeneous broadening for different cooperativities $C$ when the optimized drives are applied. For each $\mathrm{\Delta }$, optimized drives are designed for each of the same 100 randomly generated ensembles with inhomogeneous broadening equal to $\mathrm{\Delta }$ as used in Fig. 1. Before running the optimizations, for each ensemble, the input photon is frequency-shifted to match the highest peak of the unoptimized transduction spectrum. Also, the initial condition for the optimization is $\mathrm{\Omega }\left(t\right)=\left(N\gamma +\mathrm{\Gamma }\right)/2$, which is a constant drive that maximizes the transduction efficiency through a homogeneous ensemble with the same decay rates (see Appendix pp3). Improvement is defined as the ratio of the efficiencies with and without the optimized drive applied. Each plot point corresponds to the mean over the 100 ensembles with inhomogeneous broadening equal to the corresponding value of $\mathrm{\Delta }$ and the shaded regions represent the standard deviation.

Figure 4

(a) Comparison of the super-radiance metric for ensembles with inhomogeneous broadening $\mathrm{\Delta }=200\gamma$ with and without optimized drives applied (data for optimized and unoptimized cases are dodged in the plot for visual clarity). After generating the optimized drives used in Fig. 3, we compute the metric for all eigenstates of each of the 100 random ensembles with inhomogeneous broadening $\mathrm{\Delta }=200\gamma$ by numerically diagonalizing the propagator over one time period of the effective Hamiltonian. Each plot point and associated error bars correspond to the mean and standard deviation (over the collection of ensembles with $\mathrm{\Delta }=200\gamma$) of the maximum value of the super-radiance measure $f[|\varphi ⟩]$ over all Floquet eigenstates $|\varphi ⟩$. The dashed line denotes the same for a homogeneous ensemble. As we increase $\mathrm{\Gamma }$ to decrease the cooperativity, the metric is larger on average in the unoptimized case. We attribute this to the simultaneous increase in the unoptimized drive $\mathrm{\Omega }\left(t\right)=\left(N\gamma +\mathrm{\Gamma }\right)/2$ overshadowing the constant inhomogeneous broadening $\mathrm{\Delta }=200\gamma$ (see Appendix pp4). (b),(c),(d) Density plots (obtained by kernel density estimation using Gaussian kernels [104]) of the super-radiance measure for eigenstates of the 100 ensembles with inhomogeneous broadening $\mathrm{\Delta }=200\gamma$, (b) $C=0.01$, (c) $C=0.1$, (d) $C=1$.

Figure 5

Transduction efficiency improvement with uncustomized optimization. (a) Amplitudes of the frequency components comprising the uncustomized drive. (b) Density plots of the transduction efficiency through 100 ensembles (test set) with $\mathrm{\Delta }=200\gamma ,C=0.1$ for three cases—(green) no optimized drive is applied and the input photon is fixed at the resonance of a homogeneous ensemble, (orange) no optimized drive is applied but the input photon is frequency shifted to match the highest peak of the unoptimized transduction spectrum for each inhomogeneous ensemble, and (blue) the uncustomized optimized drive is applied and the input photon is fixed at the resonance of a homogeneous ensemble.

Figure 6

Heuristic and certifiable upper bounds and unoptimized and optimized transduction efficiencies calculated for ensembles with $N=3$ emitters and cooperativities (a) $C=0.01$, (b) $C=0.1$, (c) $C=1$. For each $\mathrm{\Delta }$, $100$ random ensembles are generated with inhomogeneous broadening equal to $\mathrm{\Delta }$. For each such ensemble, optimized drives are designed to improve transduction efficiency by using a local optimizer to solve problem (2). Then, using the state obtained by solving the input-output equation with the aforementioned optimized drive as the reference state, heuristic and certifiable bounds are calculated. Each plot point corresponds to the mean over the 100 ensembles with inhomogeneous broadening equal to the corresponding value of $\mathrm{\Delta }$ and the shaded regions represent the standard deviation.

Figure 7

Temporal mode overlap-based design of drives. The amplitude ${\left|a_{\mathrm{opt}}\left(t\right)\right|}^2$ of the output photon’s temporal wavepacket after transduction by an ensemble of emitters with $N=10$, $C=0.1$, $\mathrm{\Delta }\approx 61.61\gamma$, and under the application of drives obtained by locally solving problem (G1) for (a) $n=0$, $a_{\mu }\left(t\right)={\phi }_0\left(t\right)$ and (b) $n=1$, $a_{\mu }\left(t\right)={\phi }_1\left(t\right)$. The curves labeled “Target” represent the (amplitude of the) Hermite-Gaussian modes ${\phi }_0\left(t\right),{\phi }_1\left(t\right)$, respectively. The improvement in the transduced output power in the desired target modes are (a) $\approx 61.05\times$ and (b) $\approx 62.43\times$. The ratio of the output power in the target mode to the total output power are (a) approximately equal to $99\mathrm{\%}$ and (b) approximately equal to $98.6\mathrm{\%}$. The input photon’s central frequency is fixed at the resonance of a homogeneous ensemble for both cases.

Figure 8

Effect of noise on optimized drives. (a) Density plots (obtained by kernel density estimation using Gaussian kernels [104]) of the transduction efficiency through $100$ ensembles with inhomogeneous broadening $\mathrm{\Delta }=200\gamma$, $C=0.1$ for three cases—(green) no optimized drive is applied, (blue) optimized drives (custom designed for each of the $100$ random ensembles) are applied, (red) noise ($e=50$) is added to the customized optimized drives. For all cases, the input photon is frequency shifted to match the highest peak of the unoptimized transmission spectrum. (b) Statistics of the transduction efficiency with increasing noise percentage $\left(e\right)$ in the optimized drive compared with the noiseless optimized and unoptimized cases. Lines denote the mean of the transduction efficiency over $100$ ensembles and the shaded areas and error bars denote the standard deviation.