Quantum control of qubits and atomic motion using ultrafast laser pulses

Abstract

Pulsed lasers offer significant advantages over continuous wave (CW) lasers in the coherent control of qubits. Here we review the theoretical and experimental aspects of controlling the internal and external states of individual trapped atoms with pulse trains. Two distinct regimes of laser intensity are identified. When the pulses are sufficiently weak that the Rabi frequency Ω is much smaller than the trap frequency ω _trap, sideband transitions can be addressed and atom-atom entanglement can be accomplished in much the same way as with CW lasers. By contrast, if the pulses are very strong Ω ≫ ω _trap, impulsive spin-dependent kicks can be combined to create entangling gates which are much faster than a trap period. These fast entangling gates should work outside of the Lamb-Dicke regime and be insensitive to thermal atomic motion.

Appendix: Motional evolution operator with nonzero pulse duration

In Sect. 3, Eq.  35 was derived by approximating the pulse as a δ-function. This section examines the validity of that approximation. The pulse duration is of order 10 ps, meaning it is several orders of magnitude faster than the trap frequency or the AOM frequency. Therefore, the Rosen-Zener solution in Sect. 2 can be used, with $\theta\rightarrow\theta\sin\left(\Updelta k \hat{x} + \phi\right)$ in Eqs. 5 and 6:

$$A = \frac{\varGamma^2\left(\xi\right)}{\varGamma\left(\xi-\frac{\theta}{2\pi}\sin\left(\Updelta k \hat{x} + \phi\right)\right)\varGamma\left(\xi+\frac{\theta}{2\pi}\sin\left(\Updelta k \hat{x} + \phi\right)\right)}$$

(62)

$$B = -\sin\left(\frac{\theta}{2}\sin\left(\Updelta k \hat{x} + \phi\right)\right) \hbox{sech}\left(\frac{\omega_q T_p}{2}\right)$$

(63)

The $\hat{\sigma}_x$ term in part of Eq. 4 is given by iB, which can be expanded using the Jacobi-Anger expansion as:

$$iB = \hbox{sech}\left(\frac{\omega_q T_p}{2}\right)\sum_{\hbox{odd}n = -\infty}^{\infty} e^{in\phi} J_n(\theta)D\left[in\eta\right]$$

(64)

This is nearly identical to the $\hat{\sigma}_x$ term in Eq. 35, but with an overall $\hbox{sech}\left(\omega_q T_p/2\right)$ term modifying the populations. The even-order diffraction terms are of order θ ² or higher, which were assumed to be negligible in Sect. 3. Nonzero pulse duration can thus be accounted for by replacing $\theta\rightarrow\theta \hbox{sech}\left(\omega_q T_p/2\right)$. This will correspond to a slight reduction in the effective pulse area as compared to a δ-function pulse.