Population trapping in three-state quantum loops revealed by Householder reflections

Population trapping occurs when a particular quantum-state superposition is immune to action by a specific interaction, such as the well-known dark state in a three-state $\Lambda$ system. We here show that in a three-state loop linkage, a Hilbert-space Householder reflection breaks the loop and presents the linkage as a single chain. With certain conditions on the interaction parameters, this chain can break into a simple two-state system and an additional spectator state. Alternatively, a two-photon resonance condition in this Householder-basis chain can be enforced, which heralds the existence of another spectator state. These spectator states generalize the usual dark state to include contributions from all three bare basis states and disclose hidden population trapping effects and hence hidden constants of motion. Insofar as a spectator state simplifies the overall dynamics, its existence facilitates the derivation of analytic solutions and the design of recipes for quantum-state engineering in the loop system. Moreover, it is shown that a suitable sequence of Householder transformations can cast an arbitrary $N$-dimensional Hermitian Hamiltonian in a tridiagonal form. The implication is that a general $N$-state system, with arbitrary linkage patterns where each state connects to any other state, can be reduced to an equivalent chainwise-connected system, with nearest-neighbor interactions only, with ensuing possibilities for discovering hidden multidimensional spectator states and constants of motion.

Figure 1

(Color online) RWA linkage pattern for a loop, showing linkages: states ${\psi }_1$ and ${\psi }_2$ by Rabi frequency ${\Omega }_P$, states ${\psi }_2$ and ${\psi }_3$ by ${\Omega }_S$, and states ${\psi }_1$ and ${\psi }_3$ by ${\Omega }_C$. The energy levels are shown with an ordering appropriate to state ${\psi }_1$ as ground state and state ${\psi }_2$ as excited state, but the symmetry of the loop linkage allows initial population in any state.

Figure 2

(Color online) The Householder reflection leaves state ${\psi }_1$ unchanged. Initial population might be (a) in this state or (b) in one of the altered states. Relative energies of the original bare states are not relevant, only the couplings.

Figure 3

(Color online) Creation of an equal superposition of states ${\psi }_1$, ${\psi }_2$, and ${\psi }_3$ for Gaussian pulses: ${\Omega }_P\left(t\right)={\Omega }_{P0}{e}^{-t^2/T^2}$, ${\Omega }_C\left(t\right)={\Omega }_{C0}{e}^{-t^2/T^2}$, ${\Delta }_2\left(t\right)={\Delta }_3\left(t\right)=-{\Omega }_S\left(t\right)/2$, and ${\Omega }_S\left(t\right)={\Omega }_{S0}{e}^{-{\left(t-\tau \right)}^2/T^2}$, with the parameters ${\Omega }_{P0}={\Omega }_{C0}=0.76/T$, ${\Omega }_{S0}=1/T$, and $\tau =0.5T$.

Figure 4

(Color online) Creation of an equal superposition of states ${\psi }_2$ and ${\psi }_3$, with the couplings and detunings ${\Omega }_P\left(t\right)={\Omega }_0\text{cos}\theta e^{-{\left(t-\tau \right)}^2/T^2}$, ${\Omega }_C\left(t\right)={\Omega }_0\text{sin}\theta e^{-{\left(t-\tau \right)}^2/T^2}$, ${\Omega }_S\left(t\right)={\Omega }_0{e}^{-{\left(t+\tau \right)}^2/T^2}$, and ${\Delta }_2\left(t\right)={\Delta }_3\left(t\right)=0$, where the parameters are $\theta =\pi /4$, ${\Omega }_0=30/T$, and $\tau =0.5T$.