Phase control and diagnostic of quantum mechanical superposition states

We report experiments on the control of the phase $\eta$ in quantum mechanical superposition states which emerge in electromagnetically induced transparency, $|\psi \rangle =\left(\right|1\rangle +e^{i\eta }|2\rangle )/\sqrt{2}$. We interpret our findings in terms of the measurement role that spontaneous emission and the light fields play in selecting the optically dark and bright superpositions. A phase-switching tool is introduced which enables the rapid measurement of the absolute depth of the dark-state minimum and its absolute position on the frequency scale without detuning the lasers. The phase-switching technique allows one to determine the relative phase $\eta$ in an ensemble of quantum mechanical superposition states of a $\Lambda$ system in a minimally invasive way.

Figure 1

Scheme of experimental frequency and phase control using the simplified $\Lambda$ system.

Figure 2

Transient response of absorption to phase switching with the EOM by $\varphi =\pm \pi /2$ and by $\pm \pi$ when the lasers are on two-photon resonance. A common photodiode signal of $-0.3$ V was subtracted. Such a subtraction is done for all experimental photodiode signals shown in this paper in order to avoid excessive digits in the $y$ label. For referencing the absorption scale a frequency jump of $\Delta f=5$ kHz is implemented in the time window from $0.3<t<2.2$ ms.

Figure 3

Transient response of absorption to phase switching by $\pm \pi /2,\pm \pi$ and by $\pm 3\pi /2$ at two-photon detunings of $\Delta f=\pm 400$ Hz.

Figure 4

The dimensionless quantity brightness immediately following a phase change at two-photon resonance. The open and full markers refer to measurement of $b^{-}$ and $b^{+}$, respectively. Squares and circles refer to different days of measurement. The fitted ${\sin }^2$ function has a prefactor of $2.08\pm 0.02$, quite in line with the amplitude predicted by Eq. (19).

Figure 5

Experimental transients for phase shifting by $\frac{\pi }{2}$ are shown in the left column for various values of the detuning. For referencing the absorption scale, a frequency jump by $\Delta f=5$ kHz is implemented in the time window from $0.3<t<2.2$ ms. The simulation in the right column uses the parameters $g_1=2\pi \times 210$ kHz, $g_2=-2\pi \times 420$ kHz, and $\gamma =2\pi \times 20$ Hz. These Rabi frequencies correspond to intensities which are 10$\%$ higher than using Eq. (35) in [3] with the experimental intensities $I_1=210$ and $I_2=155\mu$W/cm${}^2$. In view of the uncertainty of laser beam profiles this is considered as quite consistent.

Figure 6

The dimensionless brightness ratio $B^{\pm }=b^{\pm }/(b^{+}+b^{-})$ as a function of detuning for two values of the phase shift. Open circles refer to $B^{+}$, full circles refer to $B^{-}$. The full lines represent Eq. (35), see text.

Figure 7

Response of transient absorption to a sudden phase change by $\varphi =\pi$ over a time duration of $\Delta t$ at two-photon resonance.

Figure 8

Two measurements of the phase of a superposition state prepared at ${\eta }_{\mathrm{set}}=0.3\pi$ and ${\eta }_{\mathrm{set}}=0$ using a burst of 12 short pulses of differing phase shift $\varphi$. The traces at the left give the spiked response of the two samples in transmission. The curves shown on the right are least-squares fits of the function $a+b{\sin }^2(\varphi -{\eta }_{\mathrm{fit}})$ to the data. The fitted phases agree well with the phase at which the state was prepared.