Efficient high-fidelity flying qubit shaping

Matter qubit to traveling photonic qubit conversion is the cornerstone of numerous quantum technologies such as distributed quantum computing, as well as several quantum internet and networking protocols. We formulate a theory for stimulated Raman emission which is applicable to a wide range of physical systems, including quantum dots, solid-state defects, and trapped ions, as well as various parameter regimes. We find the upper bound for the photonic pulse emission efficiency of arbitrary matter qubit states for imperfect emitters and show a path forward to optimizing the fidelity. Based on these results, we propose a paradigm shift from optimizing the drive to directly optimizing the temporal mode of the flying qubit using a closed-form expression. Protocols for the production of time-bin encoding and spin-photon entanglement are proposed. Furthermore, the mathematical idea to use input-output theory for pulses to absorb the dominant emission process into the coherent dynamics, followed by a non-Hermitian Schrödinger equation approach, has great potential for studying other physical systems.

Figure 1

(a) Illustration of the physical system and (b) energy level diagram of a stimulated Raman emitter. The illustration depicts a quantum system (blue ball) coupled to an electromagnetic cavity mode (pink) as well as an excitation (e.g., laser) field (green). One of the mirrors couples out into a fiber, enabling the emission of a photon pulse (pink glowing droplet). The level diagram in (b) in the rotating frame depicts the excited state $|e\rangle$ (ES) split from the two ground states $|0\rangle$ and $|1\rangle$ by the detuning $\mathrm{\Delta }$ for the relevant states of the cavity ${|0\rangle }_c$ and ${|1\rangle }_c$ with zero and one photon. The $\mathrm{\Lambda }$ system is set up by a controllable time-dependent (excitation field) Rabi amplitude $\mathrm{\Omega }\left(t\right)$ coupling $|1\rangle {|0\rangle }_c$ to the ES, as well as cavity interaction between the ES and $|0\rangle {|1\rangle }_c$ with single-photon coupling strength $g$. The cavity emits the photon wave packet with pulse shape $v\left(t\right)$ via the (right) out-coupling $\kappa$, thus converting the matter qubit (shaded blue) to a flying qubit. Imperfections of the three-level system and cavity can lead to decoherence processes taken into account via the combined rates $\tilde{\gamma }$, ${\mathrm{\Gamma }}_1$, ${\mathrm{\Gamma }}_2$, and $\tilde{\kappa }$.

Figure 2

Emission of a time-bin qubit using (a) an ancillary (e.g., nuclear spin) qubit or (b) an ancillary state. The main focus of this paper is the particular implementation of the emit process via cavity-enhanced stimulated Raman emission. c${}_n$not and swap refer to the two qubit gates between the matter qubit and the ancillary qubit, and $X$ denotes the one-qubit $X$ gate of the matter qubit (or a $\pi /2$ pulse between $|0\rangle$ and $|1\rangle$). Additionally, if one implements the protocol only up to the first emission, matter-photon entanglement is achieved.

Figure 3

Bounds for the (a) average and (b) worst-case fidelity as a function of the pulse duration for a ${sin}^2$ pulse [see Eqs. (12) and (13)]. We compare different bounds, derived from different expressions for $E^2$ (see legend), the pulse form dependent fidelities based on the exact maximum (11) (solid lines), the simplified analytic bound $E^2=1/{\int }_0^{T}d\left(\tau \right)d\tau$ (dotted lines), and the approximation for slowly varying pulses and perfect emitters $E^2=2C/(1+2C)$ [see Eq. (14); dashed line]. The maxima of the fidelity based on the exact maximum are marked by diamonds. The pulse form dependent bounds are shown for different decoherences of the matter qubit $({\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2)/\tilde{\gamma }$; from dark (blue) to light (yellow) the values are (0,0), (0.01,0.005), (0,0.1), (0.1,0), (0.2,0), and (0.1,0.1). We use cavity QED parameters that can describe silicon-vacancy defects in diamond $(g,\kappa ,\gamma )=2\pi \times (6,30,0.1)\mathrm{GHz}$ inside a perfect one-sided cavity $\tilde{\kappa }=0$.

Figure 4

Integrated depletion rate ${\int }_0^{t}d\left(\tau \right)d\tau$ [see Eq. (10)] for a ${sin}^2$ pulse as a function of integration time $t$. We use this integral to calculate the maximum achievable fidelity for a set of parameters. Here, we consider a ${sin}^2$ pulse which corresponds to the ansatz in Eq. (15) with $L=1$. We use the same colors and parameters as in Fig. 3.

Figure 5

Optimal pulse duration $T$ for $L=1$ and worst-case fidelity $F\left(\right|{\alpha }_0|=1)=|\lambda \left(T\right)/{\alpha }_0{|}^2$ as a function of the matter qubit decoherence rates ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ (see legend). The colors correspond to different decoherence processes (see legend). We use the same parameters as in Fig. 3.

Figure 6

Results of optimizing the pulse shape for a maximum worst-case fidelity. (a) shows half of the optimal symmetric pulse shapes $f\left(t\right)$ for different numbers of free parameters $L$ and restrictions on the pulse (see legend). The corresponding worst-case fidelity $F\left(\right|{\alpha }_0|=1)$ and optimal pulse duration $T$ are depicted in (b) and (c); here. the dashed line corresponds to the perfect emitter bound [see Eq. (14)]. (d) Driving Rabi frequency $\mathrm{\Omega }\left(t\right)$ as a function of time for different pulses [different colors; see the legend in (a)] and $E/E_{\max }$ (see legend). Reducing $E$ leads to a larger minimal $\left|\alpha \left(t\right)\right|=e^{-{\mathrm{\Gamma }}_{2}t}r\left(t\right)$ (see inset) and smooths out discontinuities in $\mathrm{\Omega }\left(t\right)$. Parameters are the same as in Fig. 3, ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2=0.1\gamma$, and $\mathrm{\Delta }=0$. See Appendix pp5 for the optimization results.