Conventional manipulations over quantum systems for, such as coherent population trapping and unidirectional transfer, focus on Hamiltonian engineering while regarding the system's manifold geometry and constraint equation as secondary causes. Here, we treat them on equal footing in controlling a finite-dimensional quantum system under a time-dependent non-Hermitian Hamiltonian, which is inspired by the D'Alembert principle of regarding active force, constraint force, and inertial force in an unbiased way. Under the biorthogonal condition, the non-Hermitian Hamiltonian could be triangularized in a constraint picture spanned by a set of completed and orthonormal basis states, which is found to be a sufficient condition to construct at least one universal nonadiabatic passage in both bra and ket spaces. The passage ends up with a desired target state that is automatically normalized without artificial normalization in the existing treatments for non-Hermitian quantum systems. Moreover, the passage is found to be robust against the parametric deviation when the real part of its global phase is rapidly varying with time. Our protocol is explicitly verified for the perfect population transfer in the two-level system and the chiral population transfer in the three-level system. It generalizes our framework of universal quantum control to the field of the biorthogonal quantum mechanics.
Conventional manipulations over quantum systems for, such as coherent population trapping and unidirectional transfer, focus on Hamiltonian engineering while regarding the system's manifold geometry and constraint equation as secondary causes. Here, we treat them on equal footing in controlling a finite-dimensional quantum system under a time-dependent non-Hermitian Hamiltonian, which is inspired by the D'Alembert principle of regarding active force, constraint force, and inertial force in an unbiased way. Under the biorthogonal condition, the non-Hermitian Hamiltonian could be triangularized in a constraint picture spanned by a set of completed and orthonormal basis states, which is found to be a sufficient condition to construct at least one universal nonadiabatic passage in both bra and ket spaces. The passage ends up with a desired target state that is automatically normalized without artificial normalization in the existing treatments for non-Hermitian quantum systems. Moreover, the passage is found to be robust against the parametric deviation when the real part of its global phase is rapidly varying with time. Our protocol is explicitly verified for the perfect population transfer in the two-level system and the chiral population transfer in the three-level system. It generalizes our framework of universal quantum control to the field of the biorthogonal quantum mechanics.
As a paradigm for describing open-quantum-system dynamics, the non-Hermitian Schrödinger equation generally modifies state norms, complicating coherent control over the system's Hilbert space. With a general instantaneous frame spanned by biorthogonal ancillary bases, we develop a biorthogonal quantum-control theory for discrete-variable systems governed by time-dependent non-Hermitian Hamiltonian. In particular, we derive a necessary and sufficient condition for extracting the transitionless pathways that transport parallel in ket and bra states. When this condition is satisfied by a set of projectors constructed with biorthogonal basis states, the non-Hermitian Hamiltonian becomes diagonal in the relevant biorthogonal subspace, enabling arbitrary initial state to evolve as a superposition of controlled transitionless passages. We further identify the parametric constraints under which the transformation linking the initial and final states becomes unitary. As a result, the final system state is automatically normalized even though the transitional dynamics is nonunitary. Our theory is justified by the bidirectional population transfer and a holomonic quantum gate implemented by a non-Hermitian three-level system in Λ type. For the population transfer, controlled gain and loss reduce the required coherent pulse area, enabling shorter transfer times than the Hermitian benchmark with the same average Rabi frequency. Our work integrates the universal quantum control (UQC) into the biorthogonal representation, displaying full-rank passage engineering with coherent operations in non-Hermitian systems.
Current studies about the continuous-variable systems in non-Hermitian quantum mechanics heavily revolved around the singularities in the eigenspectrum by mimicking their discrete-variable counterparts. Discussions over the nonunitary features in time evolution are growing and yet limited in scalability and controllability. We develop here a general theory to control an arbitrary number of bosonic modes under time-dependent non-Hermitian Hamiltonian. Far beyond the subspace of few excitations, our control theory operates in the Heisenberg picture and exploits the gauge potential underlying the instantaneous frames rather than the eigenspectrum. In particular, instantaneous frames are defined by time-dependent ancillary operators as linear combinations of the laboratory-frame operators, while the gauge potential arises from the unitary transformation between the time-dependent and stationary ancillary frames. We find that upper triangularization condition of the non-Hermitian Hamiltonian's coefficient matrix in the stationary ancillary frame yields two nonadiabatic passages in both bra and ket spaces and also the exact solutions of the time-dependent Schrödinger equation. At the end of these passages, probability conservation of wave function is automatically restored without brute-force normalization. Our theory is exemplified by perfect and nonreciprocal state transfers in a cavity magnonic system under non-Hermitian Hamiltonian rigorously derived from the Lindblad master equation with all quantum-jump terms retained. Under certain conditions, perfect state transfer holds for arbitrary initial states and is irrelevant to both parity-time symmetry of coefficient matrix and exceptional points of eigenspectrum. The nonreciprocal transfer is consistent with coherent perfect absorption, providing a first-principles route to coherent control of non-Hermitian continuous-variable systems.
A stable and fast path linking two arbitrary states of a quantum system is generally required for state-engineering protocols, such as stimulated Raman adiabatic passage, shortcuts to adiabaticity, and holonomic transformation. Such a path is also fundamental to the exact solution of the time-dependent Schr\"odinger equation. We construct a universal control framework using an ancillary picture, in which the time-dependent Hamiltonian can be diagonalized. Multiple desired paths can be derived by the von Neumann equation for parametric ancillary projection operators. No transition exists among the ancillary basis states during the time evolution. Under various conditions, our control framework reduces to the nonadiabatic holonomic transformation, the Lewis-Riesenfeld theory for invariants, and counterdiabatic driving methods. In addition, it is applicable to the cyclic transfer of system populations that could be a hard problem for existing methods. Our work can provide a full-rank time-evolution operator for a time-dependent quantum system with a finite number of dimensions.
The ability to control matter at the quantum level is a central requirement for the continued advance of quantum technologies. Recent experimental progress has enabled continuous monitoring of quantum systems in time, giving rise to dynamics conditioned on measurement outcomes. Such measurement-conditioned evolution is naturally described by non-Hermitian Hamiltonians, whose complex eigenvalues encode gain and loss of population associated with the measurement record. Non-Hermitian quantum dynamics exhibit geometric features without Hermitian counterparts, including non-orthogonal eigenspaces, exceptional points, and complex geometric phases. This thesis investigates geometric structures underlying non-Hermitian quantum dynamics from two complementary perspectives: the geometry of quantum states under measurement-conditioned evolution, and the geometry of Hamiltonians and parameter spaces relevant for control in experimentally accessible platforms. First, measurement-conditioned non-Hermitian dynamics are formulated within quantum information geometry. The normalized evolution preserves the rank of the density operator, confining the dynamics to fixed-rank manifolds with a natural geometric structure. This framework enables optimization of the success probability of prescribed non-Hermitian trajectories and, via a purification-bundle formulation, leads to a non-Hermitian quantum speed limit expressed in terms of the Bures geometry. Explicit shortest geodesics and their associated non-Hermitian generators are derived, including results on the existence of time-independent generators in finite dimensions. The thesis then addresses control of non-Hermitian dynamics from a complementary geometric viewpoint focused on Hamiltonians and parameter spaces. A complex mixing angle is introduced for the detuned parity--time-symmetric dimer, providing a transparent mapping between parameter space and eigenspaces and enabling implementable counter-diabatic control on a superconducting transmon qubit. This approach is validated experimentally through accurate eigenstate tracking and exceptional-point encircling, and is subsequently generalized to arbitrary two-level systems, bipartite lattice models, and multilevel transmons. Finally, the geometric origin of complex Berry phases is identified, with predictions indicating their observability on the transmon platform and their role in purity stabilization and noise-assisted chiral transfer. Overall, the thesis establishes geometry as a unifying and operational language for the analysis and control of non-Hermitian quantum systems, addressing both foundational theory and experimentally relevant protocols.
We find that the seemingly disparate control approaches for classical and quantum continuous-variable systems can be unified via differential manifolds of the ancillary representations. For classical systems, the ancillary representation is defined by the time-dependent ancillary canonical variables resulting from symplectic transformation over the original canonical variables. Under the Hamilton-Jacobi theory, the ancillary canonical variables act as dynamical invariants to guide the system nonadiabatically through the entire phase space. The second quantization of the Liouville equation for the dynamical invariants leads to the Heisenberg equation for the relevant ancillary operators, which is found to be a sufficient condition to activate nonadiabatic passages towards arbitrary target states in both Hermitian and non-Hermitian systems and yield constrained exact solutions of the time-dependent Schroedinger equation. Using the non-Hermitian Hamiltonian rigorously derived from the Lindblad master equation, our theory is exemplified by the generation of single-mode squeezed states with a squeezing level about 29.3 dB and double-mode squeezed states with 20.6 dB, respectively.
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.