Current studies of non-Hermitian continuous-variable systems heavily revolved around the singularities in the eigen-spectrum by mimicking their discrete-variable counterparts. The growing discussions over the nonunitary features in time evolution, however, are limited in scalability and controllability. We here develop a general theory to control an arbitrary number of bosonic modes under the time-dependent non-Hermitian Hamiltonian. Far beyond the subspace of few excitations, our control theory operates in the full Hilbert space within the Heisenberg framework and exploits the gauge potential underlying the instantaneous frames rather than the eigen-spectrum. In particular, the instantaneous frames are defined by time-dependent ancillary operators as linear combinations of laboratory-frame operators; while the associated gauge potential arises from the unitary transformation connecting the time-dependent and stationary ancillary frames. We find that the upper triangularization condition of non-Hermitian Hamiltonian's coefficient matrix in the stationary ancillary frame gives rise to nonadiabatic passages of two time-dependent ancillary operators, also leading to exact solutions of the time-dependent Schrödinger equation. At the end of these passages, the probability conservation of the system wavefunction can be automatically restored without artificial normalization. Our theory is exemplified with the perfect and nonreciprocal state transfers in a cavity magnonic system. The former holds for arbitrary initial states and is irrelevant to both parity-time symmetry of the coefficient matrix and exceptional points of eigen-spectrum; and the latter is consistent with the coherent perfect absorbtion. We essentially constructs the universal quantum control (UQC) theory for the non-Hermitian continuous-variable systems, promising a powerful and reliable approach for their coherent control.
Current studies of non-Hermitian continuous-variable systems heavily revolved around the singularities in the eigen-spectrum by mimicking their discrete-variable counterparts. The growing discussions over the nonunitary features in time evolution, however, are limited in scalability and controllability. We here develop a general theory to control an arbitrary number of bosonic modes under the time-dependent non-Hermitian Hamiltonian. Far beyond the subspace of few excitations, our control theory operates in the full Hilbert space within the Heisenberg framework and exploits the gauge potential underlying the instantaneous frames rather than the eigen-spectrum. In particular, the instantaneous frames are defined by time-dependent ancillary operators as linear combinations of laboratory-frame operators; while the associated gauge potential arises from the unitary transformation connecting the time-dependent and stationary ancillary frames. We find that the upper triangularization condition of non-Hermitian Hamiltonian's coefficient matrix in the stationary ancillary frame gives rise to nonadiabatic passages of two time-dependent ancillary operators, also leading to exact solutions of the time-dependent Schrödinger equation. At the end of these passages, the probability conservation of the system wavefunction can be automatically restored without artificial normalization. Our theory is exemplified with the perfect and nonreciprocal state transfers in a cavity magnonic system. The former holds for arbitrary initial states and is irrelevant to both parity-time symmetry of the coefficient matrix and exceptional points of eigen-spectrum; and the latter is consistent with the coherent perfect absorbtion. We essentially constructs the universal quantum control (UQC) theory for the non-Hermitian continuous-variable systems, promising a powerful and reliable approach for their coherent control.
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.
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.
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.
Perfect transfer of {\em unknown} states across distinct nodes is a basic function in bosonic quantum networks. Here we develop a general theory to construct an N-node bosonic network governed by the time-dependent Hamiltonian, as the universal quantum control theory for continuous-variable systems. In particular, we can activate nonadiabatic passages superposed of initial and target modes by the commutation condition about the Hamiltonian's coefficient matrix and projection operator in the representation of time-independent ancillary modes, which serves as the necessary and sufficient condition to solve the time-dependent Schrödinger equation of the full Hamiltonian. To exemplify the versatility of our theory on the Heisenberg-picture passages, we perform arbitrary state exchange between two nodes, chiral entanglement transfer among three bosonic nodes, and chiral Fock-state transfer among three of four bosonic nodes. Our work provides a promising avenue toward the universal control of any pair of nodes or modes as well as the entire bosonic network.
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.