Magnetic coherent population trapping in a single ion

Magnetically induced coherent population trapping has been studied in a single trapped laser cooled ion. The magnetic-field-dependent narrow spectral feature is found to be a useful tool in determining the null point of magnetic field at the ion position. In particular, we use a double $\mathrm{\Lambda }$ scheme that allows us to measure the null magnetic-field point limited by the detector shot noise. We analyzed the system theoretically and found certain long-lived bright states as the dark state is generated under steady-state condition.

Figure 1

Relevant energy levels of ${}^{138}\mathrm{Ba}^{+}$ along with the wavelengths of transitions. It shows two $\mathrm{\Lambda }$-type energy levels.

Figure 2

Schematic diagram of a simple model that exhibits the same feature as the experimental scheme. As long as the two states have the same energy, i.e., $\delta =0$, they form a perfect two-photon resonant $\mathrm{\Lambda }$ system. In the situation where the two lasers have different detunings (${\mathrm{\Delta }}_L\ne {\mathrm{\Delta }}_P$), the four level system exhibits one single dark state $\left(\right|+\rangle -|-\rangle )/\sqrt{2}$, i.e., fully independent of the different laser parameters. In the stationary regime, the atom is shelved in this dark state and, therefore, is not scattering any photons.

Figure 3

Schematic diagram of the experimental setup. The laser frequency and amplitudes are controlled by acousto-optical modulators (AOM). All the laser frequencies are monitored by a wave meter with 5 MHz resolution. A typical temporal pulse sequence generated by switching the AOMs via field-programmable gate arrays (FPGA) and direct digital synthesizers (DDS) for all the lasers used for the measurement is also presented.

Figure 4

Fluorescence at $455\mathrm{nm}$ as a function of the magnitude of the applied magnetic field. The dashed curve is obtained when 493 nm laser is also applied on the ion while the solid line is when it is off. The right-side inset shows the fluorescence rate as a function of the magnetic field as calculated from theory using the same experimental parameters. The left-side inset shows the derivative of the dashed curve which determines the sensitivity of the rate of fluorescence as a function of magnetic field.

Figure 5

Numerical value of the slope of the derivative of the fluorescence at $455\mathrm{nm}$ as a function of the saturation parameter of the 614 nm laser and for different values of the saturation of the 650 nm laser. As one can see, lowering the repumper intensities is increasing the slope of the dispersion curve and, thereby, the sensitivity of the experiment.

Figure 6

Fluorescence at $493\mathrm{nm}$ as a function of the magnetic-field strength for three different saturation parameters of the repumps. The zoomed portion of the graph is shown in the inset above while a derivative of the black curve with respect to magnetic field is plotted on the inset to the left. The slope obtained at the zero magnetic field is about $1.61\times {10}^6$ counts/s/gauss for the lowest saturation parameters of the repumpers.

Figure 7

Measured sensitivity as function of the repump laser power for the background free detection of $493\mathrm{nm}$ photons. Both the repump lasers are kept at the same power during a measurement. It was not possible to detect any signal below a power of $200\mathrm{nW}$.

Figure 8

Decay rates of numerically obtained intermediate states are plotted as a function of applied magnetic field. The size of the circles is proportional to the overlap with the initial density matrix. The two insets show the fluorescence decay rate for two specific magnetic fields as examples. The data fits well with the numerical decay rate (light green) for the same parameters as used in the experiment.