Dark state role in time-reversal symmetry breaking

Abstract

We investigate the role of the global driving phase $Φ$ in the dynamics of driven few-level quantum systems, a central setting in coherent control of atomic, molecular, and solid-state platforms. In particular, we focus on systems with closed-loop couplings, where external driving fields induce interference effects that strongly influence population transfer and symmetry properties of time-evolution. While full time-reversal symmetry requires $Φ=0,π$, leading to a real Hamiltonian, we focus on a less restrictive transformation, the phase inversion (or complex conjugation of the Hamiltonian), under which population dynamics can remain symmetric even though coherences generally do not. We show that the presence of a dark (spectator) state is a sufficient condition for this population phase symmetry (P$Φ$S), as it constrains the dynamics to reduced subspaces characterized by SU(2) or open-loop SU(3) evolution. We analyze this mechanism in three- and four-level systems and derive general conditions for P$Φ$S that extend to generic $n$-level configurations, with $n$ even. These findings provide practical guidelines for achieving robust control in quantum systems, with potential applications in quantum information processing and quantum computing.

Figures (7)

• Figure 1: For three-level $\Delta$ schemes in the natural basis (a), and in the CPT basis in (b). The four-level scheme is sketched in the natural basis (c) and in the CPT basis in (d). In both cases in the CPT basis the closed loop is transformed into an open one.
• Figure 2: P$\Phi$S violation for different parameters. The results for positive and negative phases spanning the time range $[0,0.5]$ are plotted side-to-side, with the final time values at the plot center. All populations are given in the natural basis $(1,2,3)$. The initial state is represented by a colored dot on the vertical axis. In (a) three-level $\Delta$-D-1 case with parameters $\Omega_{12}=\Omega_{23}=\Omega_{13}=20$, $\Phi=\pm\pi/2$, and $\delta_1=\delta_3=0$. Notice the chirality character associated to the sequence of occupied states. In (b) three-level $\Delta$-D-2 case with $\Omega_{12}=\Omega_{23}=\Omega_{13}=20$, $\Phi=\pm\pi/3$ and $\delta_1=\delta_3=-5$. In (c) four level case with $\Omega_{12}=\Omega_{23}=\Omega_{34}=10$, $\Omega_{41}=10\sqrt{2}$, $\Phi=\pm\pi/4$, $\delta_1=\delta_4=5$, and $\delta_3=0$. The initial states are natural basis states, $|1\rangle$ in (a) and (c), and $|2\rangle$ in (b).
• Figure 3: P$\Phi$S time evolutions within restricted subspaces, in (a) for the three-level SU(2) subspace and in (b) for the four-level SU(3) subspace. Occupations of bright and dark states are plotted, with an initial bright state, $|B_1\rangle_{\Delta}$ in (a), and $|B_1\rangle_{D\Lambda}$ in (b), as defined in Table \ref{['Table:Phi-3Level-123']}. In both cases, the zero line reports the dark state population, $|D\rangle_\Delta$ in (a) and $|D\rangle_{D\Lambda}$ in (b). (a) $\Delta$-D-2 case with $\Omega_{12}=\Omega_{23}=\Omega_{31}=\Omega=20$, $\Phi=\pi/3$, $\delta_1=\delta_3=-5$. (b) $D\Lambda$-D-1 case with $\Omega_{12}=\Omega_{23}= \Omega_{34}=\Omega_{14}=\Omega=16$, $\Phi=\pi/3$, $\delta_1=\delta_3=16$, and $\delta_4=2$.
• Figure 4: Open Loop P$\Phi$S results in three- and four-levels with occupations of the $|B\rangle_\Lambda$ and $|D\rangle_\Lambda$ CPT basis states of Eq. \ref{['eq:3CPTbasis']}. The $]2\rangle$ natural basis state complete the bases, and the $|4\rangle$ stete for the four level case in (c). In (a) three-level $\Delta$-D-1 case with initial state $|D\rangle_\Lambda$ and parameters $\Omega_{12}=\Omega_{23}=\Omega_{31}=20$, $\Phi=\pm\pi/2$, $\delta_1=\delta_3=0$. In (b) three-level $\Delta$-D-2 case with $|2\rangle$ the initial one. Parameters $\Omega_{12}=\Omega_{23}=\Omega_{31}=20$, $\Phi=\pm\pi/3$, $\delta_1=\delta_3=-5$. In (c) four level D$\Lambda$-D-1 case for $|B\rangle_\Lambda$ initial state and $\Omega_{12}=\Omega_{23}=\Omega_{34}=\Omega_{41}=16$, $\Phi=\pm\pi/3$, $\delta_1=\delta_3=16$, $\delta_4=2$.
• Figure 5: Closed-loop P$\Phi$S for four-level $D\Lambda$ configuration at $\delta_1=\delta_3=\delta_4=0$. $|1\rangle$ initial state and occupation of the (1,2,3,4) states. Laser parameters of the D-D$\Lambda$-3 case: $\Omega_{12}=10$, $\Omega_{23}=20$, $\Omega_{34}=30$, $\Omega_{41}=40$, $\Phi=\pi/2$ (left panel) and $\Phi=-\pi/2$ (right panel).