Sensitivity of quantum gate fidelity to laser phase and intensity noise

The fidelity of gate operations on neutral atom qubits is often limited by fluctuations of the laser drive. Here, we quantify the sensitivity of quantum gate fidelities to laser phase and intensity noise. We first develop models to identify features observed in laser self-heterodyne noise spectra, focusing on the effects of white noise and servo bumps. In the weak-noise regime, characteristic of well-stabilized lasers, we show that an analytical theory based on a perturbative solution of a master equation agrees very well with numerical simulations that incorporate phase noise. We compute quantum gate fidelities for one- and two-photon Rabi oscillations and show that they can be enhanced by an appropriate choice of Rabi frequency relative to spectral noise peaks. We also analyze the influence of intensity noise with spectral support smaller than the Rabi frequency. Our results establish requirements on laser noise levels needed to achieve desired gate fidelities.

Figure 1

Self-heterodyne setup. The laser signal is split equally between two paths. One path passes through a single-mode fiber (SMF), where it is delayed by time $t_d$. It then passes through an acousto-optic modulator (AOM), where its frequency is shifted by ${\nu }_s$. The interfering signals are combined and measured by a photodiode (PD), and finally processed through a spectrum analyzer (SA).

Figure 2

White-noise power spectral densities, $S_{\varphi }\left(f\right)$ (blue), $2S_E\left(f\right)/E_0^2$ (gold), and $S_i\left(f\right)/4$ (red), defined in Eqs. (32), (35), and (38), respectively, for noise strength $h_0=100{\mathrm{Hz}}^2/\mathrm{Hz}$. Here we have omitted the $\delta$-function peak in $S_i\left(f\right)$. The inset shows the same quantities plotted on a logarithmic frequency scale. An approximate form for $S_i/4$ (red, dotted) is obtained from Eq. (29). The cutoff $f_x$ (vertical black), defined in Eq. (15), indicates the frequency where Eq. (14) begins to fail.

Figure 3

Self-heterodyne data from a 1040-nm solid-state Ti:Sa laser (MSquared SolsTiS) with fits to white and Gaussian-bump noise models. The laser was locked to a reference cavity with linewidth of approximately 5 kHz using the Pound-Drever-Hall method [26]. (a) Self-heterodyne power spectral density, obtained in a 50-kHz frequency window with $\text{RBW}=100$ Hz (open blue circles) and in a 600-kHz window with $\text{RBW}=300$ Hz (closed red markers). An optical delay fiber of 11 km was used for both data sets, corresponding to a delay time of $t_d=5.445\times {10}^{-5}$ s. The data were shifted horizontally to center their peaks at zero frequency, and the peak data were then fit to Eq. (49) (black line), obtaining $\alpha =5/2$ and $\sigma =240$ Hz. The data were finally renormalized (i.e., shifted vertically) to ensure the correct total power, as described in the main text. (b) Same red data set as (a), plotted over a wider frequency window. The data were fit to Eq. (48), including white noise and two Gaussian servo bumps, obtaining $h_0=13{\mathrm{Hz}}^2/\mathrm{Hz}, h_{g1}=25{\mathrm{Hz}}^2/\mathrm{Hz}, {\sigma }_{g1}=18\mathrm{kHz}, f_{g1}=130\mathrm{kHz}, h_{g2}=2.0\times {10}^3{\mathrm{Hz}}^2/\mathrm{Hz}, {\sigma }_{g12}=1.5\mathrm{kHz}$, and $f_{g2}=234\mathrm{kHz}$. (c) Phase and frequency power spectral densities, $S_{\varphi }\left(f\right)$ and $S_{\delta \nu }\left(f\right)$ (inset), resulting from the fitting.

Figure 4

Coupling scheme for two-photon Rabi oscillations in a ladder configuration, for states $|g\rangle , |e\rangle$, and $|r\rangle$. Two lasers are employed, with photon angular frequencies ${\omega }_{1,2}$ and associated Rabi angular frequencies ${\mathrm{\Omega }}_{1,2}$. The lasers are detuned away from ladder excitations by angular frequencies ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$.

Figure 5

Coupling scheme for two-photon Rabi oscillations in a $\mathrm{\Lambda }$ configuration, for states $|g\rangle , |e\rangle$, and $|p\rangle$. In the setup considered here, the two driving frequencies are assumed to differ by the qubit frequency: $\hslash ({\omega }_1-{\omega }_2)=E_e-E_g$. The figure also shows the associated Rabi angular frequencies ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$, and the detuning $\mathrm{\Delta }$.

Figure 6

Rabi errors for one-photon Rabi oscillations due to white phase noise, plotted as a function of noise amplitude $h_0$. Results are obtained following the procedure described in Sec. 5a for (a) $\pi$ rotations and (b) $2\pi$ rotations. Red markers represent averages from numerical simulations, while pink shading shows the corresponding $1\sigma$ error bars. Blue curves show theoretical results for $\mathcal{E}$ from Eq. (79).

Figure 7

Rabi errors for one-photon Rabi oscillations due to servo-bump phase noise. Results are plotted as a function of the center frequency of the servo bump $f_g$, scaled by the Rabi frequency ${\mathrm{\Omega }}_0/2\pi =1$ MHz, with additional noise parameters described in Sec. 5a. Similar to Fig. 6, (a) shows results for $\pi$ rotations and (b) shows results for $2\pi$ rotations. Theory results are obtained from Eq. (81).

Figure 8

Rabi errors for one-photon Rabi oscillations due to servo-bump phase noise: (a) $\pi$ rotations and (b) $2\pi$ rotations. Calculations are very similar to Fig. 7, except that here, the central peak frequency of the servo bump is held fixed at $1.2\text{MHz}$, while the total power $s_g$ is varied, where $s_g$ is defined in Eq. (41).

Figure 9

Rabi errors for two-photon Rabi oscillations due to white phase noise: (a) $\pi$ rotations and (b) $2\pi$ rotations. All simulations and plots are analogous to Fig. 6, while theory results are given by Eq. (96).

Figure 10

Rabi errors for two-photon Rabi oscillations due to servo-bump phase noise: (a) $\pi$ rotations and (b) $2\pi$ rotations. All simulations and plots are analogous to Fig. 7, while theory results are given by Eq. (98).

Figure 11

One-photon Rabi errors for white, band-limited frequency noise with fixed amplitude $h_0=3.18{\mathrm{kHz}}^2/\mathrm{kHz}$, as a function of the noise bandwidth, $f_c$. (a) $\pi$ rotations. (b) $2\pi$ rotations. Here, red data correspond to numerical simulations, while the blue curves correspond to master-equation theory, Eq. (111), and the green curves correspond to the quasistatic theory, Eq. (121). The master-equation result fails only for the special case of $2\pi$ rotations, for bandwidths below the (vertical-dashed) crossover frequency, $f_c\approx 1.43h_0$. The quasistatic theory works well for both $\pi$ and $2\pi$ rotations, for the case of narrow bandwidths.

Figure 12

Measured RIN spectrum of the Ti:Sa laser used in Fig. 3. The orange curve shows the integrated variance at frequency $f$, and indicates that nearly all intensity noise occurs below 10 Hz, so the quasistatic approximation should be excellent for this laser. The variance integrated from 0.1 to ${10}^5$ Hz is ${\sigma }_{{\alpha }_I}^2=3.7\times {10}^{-6}$.

Figure 13

Tilted rotation axis ($\widehat{\mathrm{\Omega }}$) for two-photon Rabi gates, due to intensity fluctuations.

Figure 14

Rabi errors due to RIN for one-photon gate operations. (a, c) $\pi$ rotations. (b, d) $2\pi$ rotations. (a, b) Results for narrow-bandwidth noise, $f_c=0.1\times {\mathrm{\Omega }}_0/2\pi$. (c, d) We sweep the cutoff frequency $f_c=10\times {\mathrm{\Omega }}_0/2\pi$ holding ${\sigma }_{{\alpha }_I}=0.05$ constant. In all cases we choose ${\mathrm{\Omega }}_0=2\pi \times 1\mathrm{MHz}$.

Figure 15

Rabi errors due to RIN for two-photon gate operations. All panels and simulations parameters are the same as in Fig. 14, but with ${\mathrm{\Omega }}_0$ replaced by ${\tilde{\mathrm{\Omega }}}_0$.