Optimized optical tomography of quantum states of a room-temperature alkali-metal vapor

We demonstrate an experimental technique for the quantum-state tomography of the collective qutrit states of a room-temperature alkali-metal vapor. It is based on the measurements of the polarization of light traversing the vapor subjected to a magnetic field. To assess the technique's robustness against errors, experimental investigations are supported with numerical simulations. This not only allows us to determine the fidelity of the reconstructed states, but also to analyze the quality of the reconstruction for specific experimental parameters, such as light tuning and the number of measurements. By utilizing the conditional number, we demonstrate that the reconstruction robustness can be optimized by a proper adjustment of experimental parameters, and further improvement is possible with the repetition of specific measurements. Our results demonstrate the potential of this high-fidelity reconstruction method of quantum states of room-temperature atomic vapors.

Figure 1

Energy-level diagram corresponding to the ${}^{87}\mathrm{Rb}$ $D_1$ line (presented energy gaps are not in scale), with measured qutrit marked with green. Additionally, tunings of all the laser beams are presented; red lines correspond to the pumping and repumping light used for the state preparation, and the blue line shows the probing beam which is blue detuned from the resonance by $\mathrm{\Delta }$ and used for state reconstruction.

Figure 2

(a) Simplified scheme of the experimental setup used for the quantum-state generation and tomography of collective states of an alkali-metal vapor. SAS: Saturation absorption spectroscopy; DAVLL: Dichroic atomic vapor laser lock; AOM: Acousto-optic modulator; PD: Photodiode; Pol: Glan-Thompson polarizer; $\lambda /4$: Quater-wave plate; $\mathrm{Rb}$: Parafin-coated vapor cell filled with ${}^{87}\mathrm{Rb}$; Wol: Wollaston prism; BPD: Balanced photodetector. (b) Experimental sequence used in our method. The initial state of atoms is prepared with the pump light turned on for about 200 ms (red trace). After the preparation period, we apply a sequence of magnetic pulses to modify the state of the atoms (green trace). Here, the CYCLOPS pulses (see Sec. 3d) are first used and then the control pulses are implemented for a total time of about 1.5 ms. Finally, the probe light is turned on, alongside with the longitudinal magnetic field, for about 1 s (blue) and the polarization rotation signal is recorded.

Figure 3

Comparison of two experimentally reconstructed density matrices (blue bars), with simulated ones (red bars), including their amplitude (upper panels) and phase (lower panels). The first state (a) is pumped with circularly polarized light and the second state (b) is pumped with linearly $\mathbf{z}$-polarized pumps, propagating along the $\mathbf{x}$ axis. The fidelity achieved between the experimental results and simulations in both cases exceeds 0.99.

Figure 4

(a) Condition number $\kappa \left(\mathbb{C}\right)$ of a state measured versus the detuning $\mathrm{\Delta }$ of the probing light from the center of the Doppler-broadened $f=1\to F=2$ transition. The red points indicate the values calculated based on experiment (horizontal uncertainty comes from the uncertainty of detuning; the evaluated vertical errors are so small that they are not visible in the plot), while the blue curve shows the theoretical dependence calculated from the absorption measurements. The green dashed line indicates the smallest $\kappa \left(\mathbb{C}\right)=2.25$ achievable using this approach. (b) Relative uncertainty of the linear inversion [see Eq. (13)] as a function of the condition number $\kappa \left(\mathbb{C}\right)$. Red points correspond to the reconstruction of the state pumped with the circularly polarized light that propagated along the $\mathbf{x}$ axis [see Fig. 3] and blue points correspond to the reconstruction of the state generated with the linearly $\mathbf{z}$-polarized pump light, propagating along the $\mathbf{x}$ axis [see Fig. 3]. It is seem that the uncertainty of the condition number is significantly smaller than the size of the data points. Solid lines are added for clarity.

Figure 5

Minimum achievable condition number versus a number of repetitions of specific measurements. Three sets of points correspond to the condition number at different detunings: 300 MHz (green triangles), 200 MHz (red squares), and 60 MHz (blue diamonds). The solid lines are added for eye guidance.

Figure 6

Simplified scheme of the experimental setup used for the measurements of the global scaling factor $\eta \left(\mathrm{\Delta }\right)$. A nonpolarizing beam splitter (BS) with 1-to-$\sigma$ splitting ratio is used to measure the light intensity before the it enters atomic ensemble. The light intensity is measured as a voltage $U_1\left(\mathrm{\Delta }\right)=r_1\sigma I_1\left(\mathrm{\Delta }\right)$, where $r_1$ is the light power-voltage conversion factor in a transimpedance photodetector detector (PD1). The light intensity is written as a function of the light detuning $\mathrm{\Delta }$ to highlight the fact that the changes in the light frequency usually introduce the corresponding changes in the light intensity emitted by the laser. After BS, there are various optical elements, introducing intensity losses $k_{\mathrm{in}}$, and a vapor cell containing atoms whose state is to be reconstructed. Next, the beam experiences losses, $k_{\mathrm{out}}$, due to the elements situated after the cell, and finally the light is split using the balanced polarimeter, consisting of a Wollastone prism and a balanced photodetector. The output signal $U_2$ is the sum of the voltages measured by each of the PDs that constitute the balanced photodetector. Both PDs have the same conversion gain $r_2$, thus $U_2\left(\mathrm{\Delta }\right)=r_2{I}_2\left(\mathrm{\Delta }\right)(1-k_{\mathrm{out}})$.

Figure 7

Detected polarization rotations (red dots) and fitted dependences (blue lines), corresponding to the first [Eq. (D5)] (a) and the second [Eq. (D8)] (b) quadratures of the signal measured without the control pulses. (c) and (d) correspond to the same signals measured with the pulse rotating the state by $\pi /2$ around the $\mathbf{y}$ axis, while (e) and (f) correspond to the pulse rotating the state by $\pi /2$ around the $\mathbf{x}$ axis. The right column shows magnifications of the first 15 ms of the registered signals. The insets in (a), (c), and (e) show the angular-momentum probability surfaces [40] of the reconstructed state after respective rotations (denoted with black dashed arrows). The red axis denotes the axis of the magnetic field applied during detection, which coincides with the probe-light propagation direction.

Figure 8

Results of the measurement of the atomic response when the atoms are polarized along (a) $\mathbf{x}$ and (b) $\mathbf{z}$ axes. Based on the measurements, the local scaling factor was determined as $\zeta \left(\mathrm{\Delta }\right)=-0.4790\left(12\right)$.