Decoherence due to Elastic Rayleigh Scattering

We present theoretical and experimental studies of the decoherence of hyperfine ground-state superpositions due to elastic Rayleigh scattering of light off resonant with higher lying excited states. We demonstrate that under appropriate conditions, elastic Rayleigh scattering can be the dominant source of decoherence, contrary to previous discussions in the literature. We show that the elastic-scattering decoherence rate of a two-level system is given by the square of the difference between the elastic-scattering amplitudes for the two levels, and that for certain detunings of the light, the amplitudes can interfere constructively even when the elastic-scattering rates from the two levels are equal. We confirm this prediction through calculations and measurements of the total decoherence rate for a superposition of the valence electron spin levels in the ground state of ${}^9\mathrm{Be}^{+}$ in a 4.5 T magnetic field.

Figure 1

Relevant energy level structure of ${}^9\mathrm{Be}^{+}$ at 4.5 T. We only show the $m_I=+\frac{3}{2}$ levels which are prepared experimentally though optical pumping. The off-resonant laser is shown with a frequency ${\omega }_o$ appropriate for the experimental results presented in Fig. 3.

Figure 2

(a)–(c) Schematic pulse sequences used to measure (a) total decoherence and (b), (c) Raman scattering rates. Raman scatter rates ${\Gamma }_{du}$ and ${\Gamma }_{ud}$ (used to calibrate laser intensity, see Fig. 3) obtained by initializing the ions in $|d⟩$ or $|u⟩$ respectively and measuring the $|u⟩$ state population vs off-resonant laser interaction time. (d) Example measurements of the total decoherence and Raman scatter rates. Left axis: Total decoherence (solid markers); Right axis: Population measured in $|u⟩$ to determine Raman scattering rates (open markers). Solid lines are fits to the data from which the Raman scattering and decoherence rates were extracted.

Figure 3

(a) Comparison of the total decoherence rate with different theoretical models as a function of light detuning from the $|u⟩\to |3/2,3/2⟩$ cycling transition for two independent sets of measurements. The light intensity was calibrated through fits to (b) light shift measurements of the qubit transition vs the polarization angle (experiment A) and (c) Raman scatter rates (experiment B). The full theory developed in this Letter agrees with the experimental data for all detunings while a theory accounting only for decoherence due to Raman transitions significantly underestimates the decoherence rate and an estimate based on scattering rate differences (${\Gamma }_{\mathrm{el},\mathrm{diff}}$) agrees with experiment only near the $|u⟩\to |3/2,1/2⟩$ and $|d⟩\to |3/2,-1/2⟩$ resonances at $-79.4$ and $-37.7\mathrm{GHz}$, respectively. The difference in the two full theory curves is due to uncertainty in the laser polarization and absolute laser intensity calibration.