We demonstrate the supersymmetric (SUSY) structure of a generalized non-Hermitian driven two-level system. By applying a unitary rotation and decoupling the resulting differential equations, we map the time-dependent dynamics onto a SUSY pair of stationary-like Schrödinger equations. This framework allows us to derive exactly solvable, complex time-dependent driving functions from the spectral analysis of auxiliary stationary systems. We show that these complex couplings induce a variety of non-trivial qubit state transitions, offering precise control mechanisms in non-conservative regimes. Our results provide a versatile analytical tool for state engineering in platforms such as nuclear magnetic resonance and non-Hermitian optics with gain and loss profiles.
We demonstrate the supersymmetric (SUSY) structure of a generalized non-Hermitian driven two-level system. By applying a unitary rotation and decoupling the resulting differential equations, we map the time-dependent dynamics onto a SUSY pair of stationary-like Schrödinger equations. This framework allows us to derive exactly solvable, complex time-dependent driving functions from the spectral analysis of auxiliary stationary systems. We show that these complex couplings induce a variety of non-trivial qubit state transitions, offering precise control mechanisms in non-conservative regimes. Our results provide a versatile analytical tool for state engineering in platforms such as nuclear magnetic resonance and non-Hermitian optics with gain and loss profiles.
The conventional time-dependent perturbation theory in quantum mechanics is established within the framework of Hermitian Hamiltonians, applicable for describing quantum transitions and associated energy level responses in such systems. However, this theory has fundamental limitations when applied to non-Hermitian systems. Consequently, researchers have systematically extended time-dependent perturbation theory to non-Hermitian systems, establishing a corresponding mature framework. Building on this foundation, this study extends the theory to investigate the transition dynamics induced by non-Hermitian interactions in non-Hermitian Hamiltonian systems. We employ a biorthogonal basis representation for a three-level non-Hermitian system. This work investigates a system comprising an unperturbed static non-Hermitian Hamiltonian and a periodically driven time-dependent perturbation Hamiltonian. Taking the three-level system as a concrete example, we combine analytical methods with numerical simulations to solve and analyze its dynamical evolution equations. These complementary approaches reveal that when system parameters complete a full cycle around an exceptional point, the transitional behavior exhibits specific evolutionary patterns. In this system, quantum transition probabilities exhibit significant asymmetry and non-conservation that depend on the initial and final states, revealing inherent directional characteristics in the dynamical process. Furthermore, for a three-level, periodically driven non-Hermitian system with time-dependent perturbations, this asymmetry is even more pronounced, manifesting as a distinct disparity between forward and reverse transition probabilities. The periodic driving actively amplifies the asymmetry in the transition process. By designing the perturbation spectrum, selective manipulation of specific quantum states can be achieved. Moreover, transition probabilities can be significantly enhanced under resonance conditions, while non-Hermiticity further breaks the system's inherent symmetry, leading to substantial amplification of transitions in a single direction. By precisely tuning the drive frequency, interactions between specific coupling channels can be selectively enhanced or suppressed. The amplification of channel asymmetry by non-Hermitian properties provides a novel mechanism for directional control of quantum states and opens new pathways for realizing related quantum technologies.
The non-Hermitian models, which are symmetric under parity ( P ) and time-reversal ( T ) operators, are the cornerstone for the fabrication of new ultra-sensitive optoelectronic devices. However, providing the gain in such systems usually demands precise control of nonlinear processes, limiting their application. In this paper, to bypass this obstacle, we introduce a class of time-dependent non-Hermitian Hamiltonians (not necessarily Floquet) that can describe a two-level system with temporally modulated on-site potential and couplings. We show that implementing an appropriate non-Unitary gauge transformation converts the original system to an effective one with a balanced gain and loss. This will allow us to derive the evolution of states analytically. Our proposed class of Hamiltonians can be employed in different platforms such as electronic circuits, acoustics, and photonics to design structures with hidden PT -symmetry potentially without imaginary onsite amplification and absorption mechanism to obtain an exceptional point.
Relations between propagators and Green functions of Hamiltonians which are SUSY partners have been obtained. New exact propagators for the family of multi-well, time-dependent and non-hermitian potentials have been calculated. Non-conservative SUSY transformation has been studied in the case of multichannel Schrodinger equation with different thresholds. Spectrum (bound/virtual states and resonances) of the non-conservative SUSY partner of zero potential has been founded. Exactly solvable model of the magnetic induced Feshbach resonance has been constructed. This model was tested in the case of Rb85. Conservative SUSY transformations of the first and the second order have been studied in the case of multichannel Schrodinger equation with equal thresholds. Transformations which introduce non-trivial coupling between scattering channels have been founded. The first order SUSY transformation which preserves S-matrix eigen-phase shifts and modifies mixing parameter has been founded in the case of two channel scattering with partial waves of different parities. In the case of coinciding parities we have found the second order SUSY transformation which preserves S-matrix eigen-phase shifts and modifies mixing parameter. Phenomenological two channel 3S1−3D1 neutron-proton potential has been obtained by using single channel, phase equivalent and coupling SUSY transformations applied to zero potential.
Abstract It is well known that the time-dependent Schrrödinger equation can only be solved exactly in very rare cases, even for two-level quantum systems. Thus, finding the exact quantum dynamics under a time-dependent Hamiltonian is not only fundamentally important in quantum physics but also facilitates active quantum manipulations for quantum information processing. In this work, we present a method for generating nearly infinite analytically assisted solutions to the Schrödinger equation for a qubit under time-dependent driving. These analytically assisted solutions feature free parameters with only boundary restrictions, making them applicable in a variety of precise quantum manipulations. Due to the general form of the time-dependent Hamiltonian in our approach, it can be readily implemented in various experimental setups involving qubits. Consequently, our scheme offers new solutions to the Schrödinger equation, providing an alternative analytical framework for precise control over qubits.
Pure quantum states, as described in quantum mechanics textbooks, are ideal representations inevitably deteriorated in real systems by any dissipative connection to the environment. In this work we derive a unified theoretical frame to study the non-Hermitian dynamics of a driven-dissipative quantum system and validate it by comparison to experimental data of matter-wave diffraction. To highlight the role of dissipation, we perform the comparison in a near-resonant regime of light-matter interaction, where perturbative approaches fail. Here, we show that the developed formalism enables both exact simulations and an intuitive interpretation based on modal analysis. The theoretical analysis is based on the general master equation description of driven-dissipative systems, from which we derive an effective complex potential Veff(Ω,Δ) that depends on control parameters such as the amplitude and detuning of the light matter interaction. Experimentally, we drive a 87Rb BEC near resonance using periodic excited-state engineering, mapping the spatially modulated Veff[Ω,Δ(x)] into momentum space for precise quantification. The experimental results agree with numerical simulations of the master equation and demonstrate the interplay between coherent drive and dissipation, with an exceptional signature: a reduced decay rate with increasing drive. Such dynamics is then interpreted by a modal analysis of the non-Hermitian Hamiltonian Heff=p2/2m+Veff which provide an intuitive and qualitative explanation for such driven-dissipative system.
Analytical solutions to the time-dependent Schrödinger equation describing a driven two-level system are invaluable to many areas of physics, but they are also extremely rare. Here, we present a simple algorithm that generates an unlimited number of exact analytical solutions. We show that a general single-axis driving term and its corresponding evolution operator are determined by a single real function which is constrained only by a certain inequality and initial conditions. Any function satisfying these constraints yields an exact analytical solution. We demonstrate this method by presenting several new exact solutions to the time-dependent Schrödinger equation. Our general method and many of the new solutions we present are particularly relevant to qubit control in quantum computing applications.