Experimental observation of partial parity-time symmetry and its phase transition with a laser-driven cesium atomic gas

Realizing and manipulating parity-time ($\mathcal{PT}$) symmetry in multidimensional systems are highly desirable for the exploration of nontrivial physics and the discovery of exotic phenomena in non-Hermitian systems. Finding non-Hermitian systems that still have all-real spectra even if their Hamiltonians possess only partial $\mathcal{PT}$ symmetry has also attracted tremendous attention in recent years. Here, we report the experimental observation of partial $\mathcal{PT}$ symmetry in a cesium atomic gas coupled with laser fields, where a two-dimensional partially $\mathcal{PT}$-symmetric optical potential for a probe laser beam is created. A transition of the partial $\mathcal{PT}$ symmetry from an unbroken phase to a broken one is observed through changing the beam-waist ratio of the control and probe laser beams, and the domains of unbroken, broken, and nonpartial $\mathcal{PT}$ phases are also discriminated unambiguously. Moreover, we develop a technique to precisely determine the exceptional point location of the partial $\mathcal{PT}$ symmetry breaking by measuring the asymmetry degree of the probe-beam intensity distribution. The findings reported here pave the way for controlling multidimensional laser beams in non-Hermitian systems via laser-induced atomic coherence, and have potential applications for designing light amplifiers and attenuators in different parts of laser beams.

Figure 1

Experimental design for observing the partial $\mathcal{PT}$ symmetry and its phase transition. (a) Three-level excitation scheme of cesium atoms. The left and right panels show the loss and gain configurations, respectively; ${\mathrm{\Omega }}_{\alpha }$ ($\alpha =p,c,r$) are laser beam Rabi frequencies and ${\mathrm{\Delta }}_{\beta }$ ($\beta =2,3$) are frequency detunings. (b) Sketch of the experimental setup. The control beam covers the whole probe beam whereas the repumping beam covers the right-half region of the probe beam (see the inset), which creates a 2D partially $\mathcal{PT}$-symmetric potential for the probe beam. The output probe beam is detected with a photodiode and a charge coupled device camera.

Figure 2

Measurements of the partial $\mathcal{PT}$ symmetry and its phase transition obtained by changing the atomic-cell temperature and the beam-waist ratio $\sigma \equiv w_c/w_p$. (a) The measured result of the probe intensity distribution for $\sigma =2.14$: the probe beam displays uniform (nonuniform). intensity distribution for the temperature $T<{29}^{\circ }\mathrm{C}$ ($\gtrsim {29}^{\circ }\mathrm{C}$) as the system works with the partial $\mathcal{PT}$ symmetry (nonpartial $\mathcal{PT}$ symmetry). No partial $\mathcal{PT}$ phase transition occurs in this case. (a1) Measured result of the probe absorption as a function of $T$ (atomic density ${\mathcal{N}}_a$) with (orange circles) and without (blue squares) the repumping beam. $S_{GL}$ ($S_L$): the PD signal in the presence (absence) of the repumping beam. Shaded region: the domain where the system works with the partial $\mathcal{PT}$ symmetry. (b) and (b1) show similar measurements to (a) and (a1) but for $\sigma =4.55$; the probe beam displays nonuniform intensity distribution for all temperatures as the system works in the broken phase of partial $\mathcal{PT}$ symmetry (see the text for details).

Figure 3

Discrimination of the unbroken, broken, and nonpartial $\mathcal{PT}$ phases. (a) Measurements (samples) and fittings (lines) of the asymmetry degree $D_{\mathrm{asym}}$ as a function of the temperature $T$ for different beam-waist ratios (indicated in the figure). (b) Measurements (samples) and calculations (lines) of $D_{\mathrm{asym}}$ as a function of $\sigma$ for different temperatures (indicated in the figure), where the EP locates at $\sigma ={\sigma }_{cr}\simeq 3.8$. (c) Phase diagram by taking $D_{\mathrm{asym}}$ as a function of $T$ and $\sigma$, where domains of the unbroken, broken, and nonpartial $\mathcal{PT}$ phases are shown. The solid (dashed) line indicates the boundary between domains of unbroken and broken (non)partial $\mathcal{PT}$ phases.

Figure 4

The control of partial $\mathcal{PT}$ symmetry and light amplification and attenuation. [(a)–(c)] Measured result [for different $(\sigma ,T)$] of the probe intensity distribution, which is uniform in (a) due to the unbroken partial $\mathcal{PT}$ symmetry, nonuniform in (b) due to the broken partial $\mathcal{PT}$ symmetry, and nonuniform in (c) due to the nonpartial $\mathcal{PT}$ symmetry. [(a1)–(c1)] Numerical results corresponding to (a)–(c). (d) Red circles (blue squares): The gain and loss of output probe intensity observed in the left (right) part of the probe beam as a function of the cell length $L$ for $(\sigma ,T)=(4.55,{28}^{\circ }\mathrm{C}$). The gain (loss) of the probe beam in the right (left) part arrives at 7 dB ($-7$ dB) at $L=10$ cm.

Figure 5

The measurements of PD signals for monitoring the gain and loss properties of the probe laser. The signal $p$ corresponds to the case that the probe laser propagates in the cell with the frequency far detuned from the resonance; $p+c$ corresponds to the case that the probe laser propagates in the atoms driven only by the control laser; $p+c+r$ corresponds to the case that the probe laser propagates in the atoms driven by both control and repumping lasers. Particularly, the repumping field is initially fully overlapped with the probe beam, in which the signals $p+c+r$ and $p+c$ are located symmetrically about the signal $p$, as shown in (a). The repumping field is then moved adiabatically to be half overlapped with the probe beam, in which the signal $p+c+r$ coincides with the signal $p$, as shown in (b).

Figure 6

The heating unit in the experiment. (a) The heating unit. The heating tape, the cesium cell, and the copper chamber in the unit are indicated, respectively. (b) The atomic density ${\mathcal{N}}_a$ as a function of the temperature $T$.

Figure 7

(a) The real and imaginary parts of the linear dispersion relation with the repumping field (the gain-channel dispersion relation), $\mathrm{Re}(k_{\mathrm{G}}$) (blue solid line) and $\mathrm{Im}(k_{\mathrm{G}}$) (red dashed line), as functions of $\omega$. Inset: The real and imaginary parts of the linear dispersion relation without the repumping field (the loss-channel dispersion relation), $\mathrm{Re}(k_{\mathrm{L}}$) (blue solid line) and $\mathrm{Im}(k_{\mathrm{L}}$) (red dashed line), as functions of $\omega$. (b) The details of the linear dispersion relation around $\omega =2.8$ GHz. $\mathrm{Im}\left(k_G\right)$ (red solid line) and $\mathrm{Im}\left(k_L\right)$ (red dashed line) are plotted as functions of $\omega$, enlarged by 100 times. $-\mathrm{Im}\left(k_G\right)\approx \mathrm{Im}\left(k_L\right)$ at $\omega \approx 2.87$ GHz. Inset: $\mathrm{Re}\left(k_G\right)$ (red solid line) and $\mathrm{Re}\left(k_L\right)$ (red dashed line) are plotted as functions of $\omega$, enlarged by 10 times. $\mathrm{Re}\left(k_G\right)\approx \mathrm{Re}\left(k_L\right)$ at $\omega \approx 2.87$ GHz.