Photon-recoil and laser-focusing limits to Rydberg gate fidelity

Limits to Rydberg gate fidelity that arise from the entanglement of internal states of neutral atoms with the motional degrees of freedom due to the momentum kick from photon absorption and re-emission is quantified. This occurs when the atom is in a superposition of internal states but only one of these states is manipulated by visible or UV photons. The Schrödinger equation that describes this situation is presented and two cases are explored. In the first case, the entanglement arises because the spatial wave function shifts due to the separation in time between excitation and stimulated emission. For neutral atoms in a harmonic trap, the decoherence can be expressed within a sudden approximation when the duration of the laser pulses are shorter than the harmonic oscillator period. In this limit, the decoherence is given by simple analytic formulas that account for the momentum of the photon, the temperature of the atoms, the harmonic oscillator frequency, and atomic mass. In the second case, there is a reduction in gate fidelity because the photons causing absorption and stimulated emission are in focused beam modes. This leads to a dependence of the optically induced changes in the internal states on the center of mass atomic position. In the limit where the time between pulses is short, the decoherence can be expressed as a simple analytic formula involving the laser waist, temperature of the atoms, the trap frequency, and the atomic mass. These limits on gate fidelity are studied for the standard $\pi \text{−}2\pi \text{−}\pi$ Rydberg gate and a protocol based on a single adiabatic pulse with a Gaussian envelope.

Figure 1

Energy level structure of neutral atom qubits with ground states $|0\rangle ,|1\rangle$. Rydberg states $|R\rangle$ interact with strength $\mathsf{B}$. Rydberg states are excited with Rabi frequency $\mathrm{\Omega }\left(t\right)$ at detuning $\mathrm{\Delta }\left(t\right)$. Level $|d\rangle$ is an uncoupled state that accumulates spontaneous emission from $|R\rangle$ which has lifetime ${\tau }_R=1/{\gamma }_R$.

Figure 2

Rydberg gate protocols. (a) $\pi \text{−}2\pi \text{−}\pi$ gate [14] with the pulses marked $c\left(t\right)$ applied to the control (target) qubits. (b) Adiabatic gate with the same pulse applied to both qubits simultaneously with constant detuning.

Figure 3

The infidelity, $1-\mathcal{F}$ in Eq. (27), due to axial momentum kick for the atom in the ground state of a harmonic trap as a function of the trap frequency $f$, using $T=0$ and not including radiative decay of the Rydberg state. The trap potential is on for the full sequence, Eq. (39), for the parameters in the text. This calculation assumes the trapping potential is magic so that the potential is the same for all internal states. The solid red line is the full density matrix calculation, the long dashed blue line is from the approximation, Eq. (D5), and the green dotted line is the approximation that includes the trapping potential, Eq. (E4). The full density matrix calculation and Eq. (E4) are indistinguishable. The horizontal solid black line (66S) and dotted black line (106S) are the contribution to infidelity due to the finite lifetime (130 and 366 $\mu \mathrm{s}$ respectively) of these Rydberg states.

Figure 4

The infidelity, $1-\mathcal{F}$, due to axial momentum kicks for the $\pi \text{−}2\pi \text{−}\pi$ gate with the atom in a thermal distribution of a harmonic trap as a function of the temperature, $T$. The calculation does not include radiative decay of the Rydberg state. The different plots are for different trap frequencies: 10 (red solid), 20 (blue dashed), and 50 kHz (green dotted). The purple short dashed line is the high temperature approximation, Eq. (43), with the second term in the square brackets dropped. The approximation, Eqs. (41) and (D11), are indistinguishable from the full calculation on this scale. The horizontal lines are the same as Fig. 3.

Figure 5

The projection error, ${\varepsilon }^{\left(1\right)}$, due to axial momentum kick for atoms in a thermal distribution of a harmonic trap as a function of the time between $\pi$ pulses. All calculations are for a trap frequency of 10 kHz using Eq. (41). The different plots are for different atom temperatures: 2 (red solid), 5 (blue dashed), and 10 $\mu \mathrm{K}$ (green dotted). The solid (66S) and the dotted (106S) black lines are $\mathrm{\Gamma }{\tau }_1$.

Figure 6

The infidelity, $1-\mathcal{F}$, due to laser focus, Eq. (30), for the $\pi \text{−}2\pi \text{−}\pi$ gate with the atom in a thermal distribution of a harmonic trap as a function of the temperature $T$. The calculation does not include radiative decay of the Rydberg state. The different plots are for different transverse trap frequencies: 10 (red solid), 20 (blue dashed), and 50 kHz (green dotted). This calculation does not include the axial decoherence since it is much smaller than that for the transverse degrees of freedom. The horizontal lines are the same as Fig. 3. A transverse misalignment of $y_0=100\mathrm{nm}$ is assumed.

Figure 7

The infidelity, $1-\mathcal{F}$, due to laser focus, Eq. (30), for the $\pi \text{−}2\pi \text{−}\pi$ gate with the atom in a thermal distribution of a harmonic trap as a function of the displacement, $Y_0$. The calculation does not include radiative decay of the Rydberg state. The different plots are for different transverse trap frequencies and temperatures: 10 kHz, 1 $\mu \mathrm{K}$ (red solid), 20 kHz, 1 $\mu \mathrm{K}$ (blue dashed), and 50 kHz, 1 $\mu \mathrm{K}$ (green dotted), 10 kHz, 5 $\mu \mathrm{K}$ (orange short dash), 20 kHz, 5 $\mu \mathrm{K}$ (purple dash-dot), and 50 kHz, 5 $\mu \mathrm{K}$ (brown dash-dot-dot). The 10 kHz, 5 $\mu \mathrm{K}$ infidelity is divided by 5 to fit on the same graph. This calculation does not include the axial decoherence since it is much smaller than that for the transverse degrees of freedom. The horizontal lines are the same as Fig. 3.

Figure 8

The Bell state infidelity for gate 3, see Table 1, as a function of atom temperature. The trap has a frequency of 50 kHz. The solid red line is from the full density matrix calculation and the dashed blue line is from Eq. (38) added to the intrinsic $1-\mathcal{F}$ from Table 1 for 66S. The orange short dash and green dotted are the same for 106S.