Collective strong coupling between ion Coulomb crystals and an optical cavity field: Theory and experiment

A detailed description and theoretical analysis of experiments achieving coherent coupling between an ion Coulomb crystal and an optical cavity field are presented. The various methods used to measure the coherent coupling rate between large ion Coulomb crystals in a linear quadrupole radiofrequency ion trap and a single-field mode of a moderately high-finesse cavity are described in detail. Theoretical models based on a semiclassical approach are applied in assessment of the experimental results of P. F. Herskind et al.[Nature Phys. 5, 494 (2009)] and of complementary new measurements. Generally, a very good agreement between theory and experiments is obtained.

Figure 1

Scheme considered for the description of the interaction between an ion Coulomb crystals and the cavity field. The cavity is formed by the two mirrors $M_1$ and $M_2$. $a_{\mathrm{in}}$ is the input light field, $a$ the intracavity field, and $a_r$ is the reflected field. $\kappa$ denotes the cavity field decay rate. The spontaneous dipole decay rate of the ions is denoted by $\gamma$.

Figure 2

(a) Calculated probe reflectivity spectrum as a function of cavity detuning ${\Delta }_c$ for an empty cavity (red line) and for $N=500$ ions interacting with the cavity field (blue line) [see Eq. (13)] for a probe detuning of $\Delta =\gamma$. The coupling of the ions and the cavity mode lead to a broadening and a shift of the resonance dip. The values of the parameters are set to typical values used in the experiments presented in Secs. 3 and 4: $\kappa =2\pi \times 2.1\mathrm{MHz}$, ${\kappa }_1=2\pi \times 1.5\mathrm{MHz}$, $\gamma =2\pi \times 11.2\mathrm{MHz}$, $g=2\pi \times 0.53\mathrm{MHz}$. (b) Expected effective cavity decay rate ${\kappa }^{\prime }$ and shift of the cavity resonance ${\Delta }_c^{\prime }-{\Delta }_c$ as a function of probe detuning $\Delta$ calculated for the same set of parameters.

Figure 3

Schematic experimental setup. The abbreviations are: polarizing beam splitter (PBS), single mode fiber (SMF), acousto-optical modulator (AOM), dichroic mirror (DM), Pound-Drever-Hall lock (PDHL), avalanche photodiode (APD), second harmonic generation (SHG). The photoionization laser is not shown.

Figure 4

(a) Experimental sequence used to measure the collective coupling rate. (b) Energy levels of ${}^{40}{\mathrm{Ca}}^{+}$ including the relevant transitions addressed in the three parts of the experimental sequence. The acronyms are: laser cooling beam (LC), repumping beam (RP), optical pumping beam (OP), probe beam (PB), avalanche photodiode (APD).

Figure 5

(a) Typical projection image of a crystal used in collective strong coupling measurements. All ions are exposed to cooling and repumping light. The solid blue line indicates the outline of the ellipsoid. (b) Same crystal, but only the ions in the cavity mode are exposed to repumping light, now injected into the cavity. The ions outside the cavity mode volume are not visible, as they are shelved into the metastable $3d{}^2{D}_{3/2}$ level. The crystal contains $N_{\mathrm{tot}}=8780\pm 180$ ions, of which $N={489}_{-27}^{+18}$ effectively interact with the cavity field.

Figure 6

Typical probe reflectivity for various values of the atomic detuning $\Delta$. The probe detunings were (a) $\Delta \approx 2\pi \times 54.3\mathrm{MHz}$, (b) $\Delta \approx 2\pi \times 24.3\mathrm{MHz}$, (c) $\Delta \approx 2\pi \times 8.3\mathrm{MHz}$, and (d) $\Delta \approx 2\pi \times 0.3\mathrm{MHz}$. Solid lines are Lorentzian fits to the data; the effective cavity field decay rate ${\kappa }^{\prime }$ is deduced from the fit.

Figure 7

(a) Measured cavity field effective decay rate ${\kappa }^{\prime }$ versus probe detuning $\Delta$ for a crystal with $N={520}_{-32}^{+24}$ ions interacting with the cavity field. The blue solid line is a fit to the data. (b) Measured shift of the cavity resonance frequency ${\Delta }_c^{\prime }-{\Delta }_c$ versus atomic detuning $\Delta$ for the same crystal. The blue line is a fit to the data.

Figure 8

Probe reflectivity signal as a function of $\Delta ={\Delta }_c$ for the empty cavity (blue triangles) and with a crystal with $N={520}_{-32}^{+24}$ effectively interacting ions present in the cavity mode volume (red circles). The solid lines are fits to the theory.

Figure 9

Cooperativity as a function of the effective number of ions. The solid line is a linear fit to the data and yields a scaling parameter of $\frac{C}{N}=(5.1_{-0.2}^{+0.4})\times {10}^{-3}$. The shaded area indicates the confidence region for the upper and lower limit of $C$ determined by the (systematic) uncertainties on the effective number of ions, $N$. The dashed line indicates the strong collective coupling limit $g_N>(\kappa ,\gamma )$.

Figure 10

Vacuum Rabi splitting spectra ($\Delta ={\Delta }_c$) obtained for increasing effective number of ions [0 (blue squares), 243 (lilac circles), 601 (dark red triangles), 914 (red diamonds)], the lines are presented to guide the eye.

Figure 11

Collective coupling rate $g_N$ versus effective number of ions $N$ deduced from reflectivity spectra, such as shown in Fig. 10, obtained with crystals of different shape and density. The blue line is a fit to the data and gives a single ion coupling rate $g=2\pi \times (0.53\pm 0.01)$ MHz. The shaded area indicates the lower and upper bound of the collective coupling rate $g_N$ within the uncertainties of $N$. The horizontal error bars are calculated according to Eq. (24)

Figure 12

(a) Experimental sequence used to measure the coherence time of collective Zeeman substate coherences in the $3d{}^2{D}_{3/2}$ level. (b) Energy levels of ${}^{40}{\mathrm{Ca}}^{+}$, including the relevant transitions and their polarization for the four phases of the experimental cycle. In the third phase, all lasers are turned off for a variable delay $\tau$, and the system evolves freely in presence of a transverse magnetic field component $B_x$.

Figure 13

Calibration of the Larmor frequency for different currents of the $B_x$ coils. Shown is the cooperativity as a function of delay time $\tau$ for different transverse $B$-fields: $I_x=10\mathrm{mA}$ (blue squares), $I_x=16\mathrm{mA}$ (lilac stars), $I_x=26\mathrm{mA}$ (dark red circles), and $I_x=36\mathrm{mA}$ (red triangles). The solid lines are fits according to Eq. (30).

Figure 14

Larmor frequency as a function of current through the $B_x$ coils. The solid line is a fit of the form ${\omega }_L=\sqrt{{\omega }_z^2+a^2{I}_x^2}$ and we deduce ${\omega }_z=2\pi \times (0.150\pm 0.002)\mathrm{MHz}$ and ${\omega }_x=2\pi \times (5.5\pm 0.1)\frac{\mathrm{kHz}}{\mathrm{mA}}\times I_x$.

Figure 15

(a) Normalized cooperativity paramter as a function of delay $\tau$. Due to the presence of a nonzero $B$-field component orthogonal to the quantization axis ($B_z=B_x=0.15\mathrm{G}$), coherent Larmor precessions are observed. Long term drifts are compensated by normalizing to the mean of one oscillation period. The solid line corresponds to a fit, assuming an exponential decay and yields a coherence time of ${\tau }_e=1.7_{-0.8}^{100}\mathrm{ms}$. (b) Cooperativity as a function of delay with a $B$-field present only along the quantization axis ($B_x=B_y=0,B_z=0.15\mathrm{G}$). The data points are normalized to the mean cooperativity of $\langle C\rangle =1.43\pm 0.02$.