Exact dynamics and qubit inversion of non-Hermitian driven two-level systems

TL;DR

Using supersymmetric quantum mechanics, the paper constructs complex time-dependent couplings $f(t)$ that render the non-Hermitian, time-dependent two-level (Rabi) system exactly solvable. A unitary rotation maps the dynamics to a pair of SUSY partner Schrödinger-like equations with potentials $V_{1,2}(t)=f^2(t)\pm \dot f(t) -1$ linked by $V_2=V_1-2\dot f(t)$ and a fixed factorization energy $\epsilon=-1$, enabling explicit $a_1,a_2$ and full evolution. The results show that purely imaginary drivings suppress transitions, while nonzero imaginary parts enable controlled population transfer, including periodic and nonperiodic transitions in both hyperbolic and trigonometric driving regimes, demonstrated through decaying and oscillatory solutions. The work discusses experimental realizations in NMR and optics with gain/loss, and suggests further exploration of additional SUSY partner potentials for richer non-Hermitian dynamics. Overall, the paper broadens the class of analytically tractable non-Hermitian driven quantum systems and connects mathematical structure to experimentally accessible platforms.

Abstract

The supersymmetric structure of a generalized non-Hermitian driven two-level system is demonstrated. A unitary rotation turns the Hamiltonian into a more convenient form. After decoupling a set of differential equations, the supersymmetric structure of the problem can be unequivocally ap- appreciated. Performing a spectral analysis of an auxiliary stationary Schrödinger-like equation, complex time-dependent driving functions are obtained for which the corresponding (time-dependent) Schrödinger equation can be straightforwardly solved. Such complex drivings are seen to produce transitions in the qubit state in different, however interesting, manners. We believe that the results reported here will be of interest for designing and carrying out various experiments in laboratories specializing in nuclear magnetic resonance or in optics with gain and loss materials.

Figures (6)

• Figure 1: Schematic of the non-Hermitian system under consideration. In $A$, a qubit (two-level system) is represented by the Bloch sphere. In $B$, the qubit, with energy split $\Delta$, is subjected to a mechanism that produces transitions in it (blue shadow) and another that represents gain/loss of the non-Hermitian system (red shadow).
• Figure 2: Graphical representation of the SUSY transformation. Left panel: constant potential $V_1(t) = 1$ (black line), plus the energy levels $E = 0$ and $\epsilon = -1$, indicated by horizontal red and blue lines, respectively. Right panel: SUSY partner $V_2(t)$ in (\ref{['PT']}), for $k = 1$ and $\eta = i/4$. For the chosen values of the parameters, the SUSY partner $V_2(t)$ is a complex function of $t$. Its real (imaginary) part is given by the solid black line (dashed purple). Besides, the mapping (\ref{['propto']}) between solutions of the Schrödinger equations associated with $V_1(t)$ and $V_2(t)$, performed by the SUSY operators $\hat{A}$ and $\hat{B}$, is indicated by green horizontal arrows. The complex function $f(t)$ connecting both potentials $V_{1,2}(t)$ is given in (\ref{['fv1k']}).
• Figure 3: Spectral analysis of the solutions associated with the constant potential. For fixed values of both $E = 0$ (red horizontal line) and $\epsilon = -1$ (blue horizontal line), three representative cases of $k$ (black horizontal line) have been chosen. These are $k=1$ (left), $k = -0.5$ (middle), and $k = -2$ (right). The relative position of $k$ with respect to $E$ defines the nature of the solutions $a_{1,2}(t)$. For $E>k$ (middle and right panels), the solution $a_1(t)$ associated with $E$, is oscillatory, which is schematically represented by a dashed-red sinusoidal line. For $E < k$ (left panel), the solution $a_1(t)$ is of exponential nature. In turn, the relative position of $k$ with respect to $\epsilon$ determines the form of the superpotential $f(t)$: it is either hyperbolic ($k>\epsilon$, left and middle panels) or trigonometric ($k<\epsilon$, right panel).
• Figure 4: Non-Hermitian dynamics of decaying solutions. Top left: Complex driving defined by (\ref{['f1']}). Top right: The atomic inversion (\ref{['inv']}) shows a transition from a point in the lower hemisphere ($W(0)<0$) of the Bloch sphere to the ground state $|g\rangle$ ($W(t_1)=-1$). This value is marked with a green star. Bottom left: square moduli of the solutions $a_{1,2}(t)$. The exponential decay of $a_1(t)$ can be appreciated (blue). Bottom right: The quantity $P(t)= |c_1(t)|^2 + |c_2(t)|^2$ is shown as evidence that unitarity is not preserved in the non-Hermitian system defined by (\ref{['piti']}). For all figures the parameter values are $k = 1.566$, $\theta = -0.83$ and $\varphi = 3.14$.
• Figure 5: Transitions in the qubit state for the case of oscillatory solutions. Top figure: Transition between two particular states of the qubit. From $|\phi(0)\rangle = \frac{1}{\sqrt 2}(|e\rangle + |g\rangle)$ (top left) to $|\phi(t^\star) = |g\rangle$ (top right), for a fixed instant of time $t = t^\star$. Middle panels: hyperbolic driving $w(t)/\Delta =if(t)$ (middle left), with $f(t)$ as given in (\ref{['f2']}), and the atomic inversion $W(t)$ (middle right). Bottom panels: trigonometric driving $w(t)/\Delta =if(t)$ (bottom left), with $f(t)$ as given in (\ref{['f3']}), and the atomic inversion $W(t)$ (bottom right). The time $t=t^\star$ at which $|\phi(t^\star) = |g\rangle$ is different for the hyperbolic (middle right) and trigonometric (bottom right) drivings. However, in both cases, the value $W = -1$ is marked with a green star. The parameter values for the middle panels are: $k=-0.49$, $\theta = -25$, $\varphi=-0.54$, with initial condition $c_1(0) = c_2(0) = 1/\sqrt 2$. Those for the lower panels are: $k=-5.8$, $\theta = -25$ and $\varphi=0.455$, for the initial condition $c_1(0) = -c_2(0) = - 1/\sqrt 2$.