Time-Dependent Non-Hermitian Hamiltonians

• Time-dependent non-Hermitian Hamiltonians are quantum operators with explicit time-dependence and a non-self-adjoint structure, crucial for modeling dissipation, gain/loss, and open dynamics.
• Employing Dyson maps and time-dependent metric operators transforms these Hamiltonians into Hermitian counterparts, ensuring consistent probability conservation and observable energy spectra.
• Invariant operator methods and advanced numerical techniques provide efficient strategies to solve dynamic equations, analyze PT-symmetry transitions, and capture non-reciprocal quantum interactions.

A time-dependent non-Hermitian Hamiltonian is a quantum mechanical operator $H(t)$ which is not self-adjoint under the standard Hilbert space inner product and whose coefficients or structure depend explicitly on time. Such Hamiltonians generalize standard Hermitian quantum theory and are particularly relevant for systems exhibiting dissipation, gain/loss, non-reciprocal phenomena, or effective descriptions of open quantum dynamics, especially when subjected to external time-dependent drives or modulation. The explicit time dependence and lack of Hermiticity raise substantial questions about the physical interpretation of evolution, norm (probability) preservation, the construction of observables, and spectral reality, all of which have been at the forefront of non-Hermitian quantum theory during the last two decades.

1. Fundamental Structures and Mathematical Framework

Time-dependent non-Hermitian Hamiltonians $H(t)$, acting on a Hilbert space $\mathcal{H}$, generate evolution through the Schrödinger equation

$$i\frac{\partial}{\partial t}|\Psi(t)\rangle = H(t)|\Psi(t)\rangle.$$

Generically, $H^\dagger(t) \neq H(t)$, and the norm $\langle\Psi(t)|\Psi(t)\rangle$ is not preserved, violating standard quantum probabilistic interpretation. To address physical consistency, several intertwined mathematical structures are used:

• Time-dependent Dyson Map: An invertible, usually nonunitary, operator $n(t)$ that relates a non-Hermitian $H(t)$ to a (time-dependent) Hermitian Hamiltonian $h(t)$ through

$$h(t) = n(t) H(t) n^{-1}(t) + i[\partial_t n(t)]n^{-1}(t),$$

known as the time-dependent Dyson relation. This mapping underpins the quasi-Hermiticity (metric) approach.

• Metric Operator: A positive-definite, generally time-dependent operator $p(t) = n^\dagger(t)n(t)$ which defines a new inner product $\langle\Psi(t)|p(t)|\Psi(t)\rangle$, restoring a generalized form of unitarity or probability conservation.
• Quasi-Hermiticity Relation: For observability (spectrum reality), one may require $H^\dagger(t)p(t) - p(t)H(t) = i\partial_t p(t)$, generalizing the static quasi-Hermiticity to the time-dependent case.
• Invariant (Lewis–Riesenfeld) Operators: Time-dependent operators $I(t)$ or their pseudo-Hermitian counterparts that satisfy $dI/dt = \partial_t I - i[I,H] = 0$ and play a central role in solution theory and spectral analysis for time-dependent non-Hermitian systems.

These structures interconnect through similarity transformations, the evolution of the metric, and the formulation of physically consistent state evolution and measurement theory.

2. Solution Strategies: Dyson Maps, Metrics, and Invariants

The conceptual and practical challenge of time-dependent non-Hermitian Hamiltonians is extracting exact solutions and spectra, constructing dynamically consistent observables, and making sense of non-conserved norms. Two mainstream solution paradigms have emerged:

A. Dyson-Map Based Procedures

One solves the TDSE for $H(t)$ in a basis that diagonalizes $H$ (often taken as time-independent for simplification), relating the physical wavefunction $|\psi(t)\rangle$ of the Hermitian problem to the auxiliary wavefunction $|\Psi(t)\rangle$ of the non-Hermitian system via $|\psi(t)\rangle = n(t)|\Psi(t)\rangle$, and using the time-dependent Dyson relation to reconstruct $h(t)$ and the unitary evolution operator of the Hermitian problem (Fring et al., 2016).

A robust extension (Luiz et al., 2016, Luiz et al., 2017) introduces a time-dependent Dyson map governed by a Schrödinger-like equation

$i\partial_t n(t) = n(t) H(t),$

which guarantees that the corresponding metric operator $p(t) = n^\dagger(t)n(t)$ is time-independent—a necessary condition for the observability of the quasi-Hermitian Hamiltonian. This allows for the consistent definition of observables and expectation values independent of gauge freedom from Dyson-map ambiguity.

B. Invariant Operator Methods

The Lewis–Riesenfeld invariants approach (Khantoul et al., 2016, Maamache et al., 2017, Maamache et al., 2017, Frith, 2020) is generalized to the non-Hermitian context by constructing a time-dependent pseudo-Hermitian invariant $I_{PH}(t)$ satisfying

$$\frac{d I_{PH}(t)}{dt} = \partial_t I_{PH}(t) - i[I_{PH}(t), H(t)] = 0.$$

The eigenstates of $I_{PH}(t)$, together with appropriate generalized phase factors, yield exact solutions of the TDSE for $H(t)$, and real dynamical phases even when $H(t)$ is not quasi-Hermitian. The spectrum of $I_{PH}(t)$ can remain real; the metric operator is adjusted accordingly to ensure unitarity of evolution in the metric-adjusted Hilbert space.

For systems with Lie-algebraic structure (e.g., SU(1,1) or SU(2) generators), the invariant is built as a combination of group generators, and the time-dependent metric follows from the quasi-Hermiticity of the invariant (Maamache et al., 2017).

3. Observability, Unitarity, and the Role of the Metric

A key issue is the connection between time-dependent non-Hermitian Hamiltonians and observable physics. While static non-Hermitian Hamiltonians with suitable metrics can be observable, in the time-dependent framework direct observability of $H(t)$ is lost unless the metric operator is time-independent. In most treatments, $H(t)$ is unobservable, but the associated Hermitian partner $h(t)$ (or other constructed observables) retains physical meaning via the time-dependent Dyson map (Luiz et al., 2017, Luiz et al., 2016). This distinction is essential when formulating a consistent probability interpretation:

• Metric Inner Product: With a time-dependent metric $p(t)$, the inner product $\langle\Psi(t)|p(t)|\Phi(t)\rangle$ is explicitly constructed to be conserved under the evolution generated by $H(t)$, even as the bare norm decays or amplifies.
• Energy Observables: The observable energy operator is generally not $H(t)$, but rather

$\tilde{H}(t) = H(t) + i\hbar n^{-1}(t)\dot{n}(t),$

and its real spectrum is enforced through the symmetries of the Dyson map and the modified metric, even in cases of nominally "broken" PT-symmetry (Fring et al., 2017, Frith, 2020).

• Preservation of Probabilities and Expectation Values: The norm and expectation values are “unitarily” preserved in the sense of the time-evolved metric. In several approaches (e.g., (Maamache, 2017)), explicit conditions are derived to ensure norm preservation under a non-unitary evolution, using suitable operator transformations.

4. PT-Symmetry, Pseudo-Hermiticity, and Symmetry Restoration by Time Dependence

PT-symmetric quantum mechanics extends naturally to time-dependent Hamiltonians. In the time-independent regime, PT-symmetry constrains the spectrum to be real in the unbroken phase. However, time dependence introduces a new structure:

• PT-symmetry breaking in static vs. time-dependent settings: For static $H$, complex spectra signal broken PT-symmetry. When parameters of $H(t)$ gain explicit time dependence, the role of $H(t)$ shifts to that of a generator—not an observable. Instead, the physical observable ("energy operator") acquires its own (possibly time-dependent) PT-symmetry, ensuring real spectra for expectation values at all times, even through regions conventionally considered PT-broken (Fring et al., 2017).
• Pseudo-PT symmetry and Heisenberg evolution: By introducing a symmetry operator $\tilde{\eta}(t) = \mathcal{PT}\,\eta(t)$ (with $\eta(t)$ a time-dependent metric and $\mathcal{PT}$ the combined parity–time operator), one can enforce a combined symmetry that obeys a Heisenberg-type evolution equation. This provides a route to analyze symmetry in higher-complexity, time-dependent, or non-statically PT-symmetric systems (Maamache, 2023).
• Applications to Lie-algebraic Hamiltonians: SU(1,1) and related algebras provide a testbed for these constructions. Both metric and pseudo-PT symmetric approaches have yielded explicit solutions for time-dependent oscillator-like and spin-like systems, clarifying how time dependence may "mend" spectra that are otherwise complex (Maamache et al., 2017, Maamache, 2023).

5. Analytical and Numerical Solution Techniques

Several exact and numerical methods have been tailored to deal with time-dependent non-Hermitian Hamiltonians:

• Closed-form Solution via Riccati Equations or Point Transformations: For two-level and more complex systems, evolution operators may be factorized into products of exponentials of algebra generators, with the time-dependent coefficients constrained by systems of equations (including Riccati-type ODEs). Solvability requires accompanying integrability conditions relating the time-dependence of parameters (Bagchi, 2018).
• Invariant/Darboux-Crum Methods: The time-dependent Darboux or supersymmetric transformation can be extended to the non-Hermitian case, providing a systematic hierarchy of exactly solvable or quasi-exactly-solvable models, organized via intertwining quadruples of Hamiltonians (two Hermitian, two non-Hermitian), with explicit interconnections between their solutions and metrics (Cen et al., 2018, Frith, 2020).
• Variational and Numerical Methods: The multiple Davydov D$_2$ (mD$_2$) Ansatz, combining a variational time-dependent expansion over coherent states for bosonic modes and explicit treatment of non-Hermiticity, has been shown numerically accurate for many-body open quantum systems, dissipative Landau–Zener models, Jaynes–Cummings chains, and complex Holstein–Tavis–Cummings models, capturing phenomena including eigenstate transitions at exceptional points and dissipative population dynamics (Zhang et al., 16 Oct 2024).
• Non-unitary and Unitary Transformations: Both non-unitary (Dyson or gauge) and unitary transformations can be used to simplify the TDSE, in some cases mapping a time-dependent non-Hermitian $H(t)$ to a time-independent PT-symmetric Hamiltonian, with implications for the structure of conserved inner products and guaranteed reality of expectation values (Kecita et al., 2021).

6. Physical Consequences, Spectral Characteristics, and Dynamical Features

The non-Hermitian, time-dependent formalism produces several effects and possibilities not accessible in static Hermitian theory:

• Arbitrary Geometric Phases and Anyonic Statistics: Owing to the time-dependence of the metric and the biorthogonality of left and right eigenstates, non-Hermitian models can generate arbitrary geometric (Berry-type) phases upon parameter evolution, a feature reminiscent of anyonic statistics (Xiong et al., 2017).
• Violation of Lieb–Robinson Bound: The modified TDSE for non-Hermitian Hamiltonians with time-dependent metric derivatives can lead to instantaneous action at a distance, with discontinuous state evolution upon sudden quenches, violating the conventional bounds on information propagation in quantum systems (Xiong et al., 2017).
• Asymmetric and Transitionless Dynamics: Time-dependent perturbation theory generalized to non-Hermitian interactions yields asymmetric (non-reciprocal) transition probabilities between levels, with unidirectional or even transitionless interactions possible by engineering the Fourier spectrum of the temporal perturbation—relevant for control in dynamical encircling of exceptional points (Longhi et al., 2017).
• Entanglement and Exceptional Point Dynamics: Time-dependent metrics mediate the evolution of entanglement in cases such as the non-Hermitian Jaynes–Cummings model, producing transitions from oscillatory to decaying regimes and exotic long-time non-vanishing entanglement in PT-broken phases (Frith, 2020).
• Engineering of Time-dependent PT-symmetry and Hidden Symmetries: By appropriate gauge or Dyson transformations, time-dependent systems can be mapped to effective PT-symmetric forms, even when no explicit gain/loss is present, as in Floquet-free two-level systems—enabling the realization of exceptional points, sensing applications, and dynamical state control without physical gain elements (Ghaemi-Dizicheh et al., 2023).

7. Future Directions and Open Problems

The formalism of time-dependent non-Hermitian Hamiltonians continues to develop across several directions:

• Extension to Infinite-dimensional and Many-body Systems: Ongoing efforts seek to extend exact and variational solution techniques to large-scale systems, including driven quantum fields, spin-boson models, and strongly correlated materials in non-equilibrium environments.
• Quasi-exact and Hierarchical Solubility Structures: The exploitation of the metric picture, quasi-exact solvability, and hierarchies of Darboux-related Hamiltonians provide routes to analytic partial solutions in otherwise intricate time-dependent models (Fring et al., 2018, Frith, 2020).
• Physical Realizations and Experimental Platforms: Advances in photonics, electronics, and atomic platforms are driving experimental investigations of non-reciprocal transport, dynamical encircling of exceptional points, PT-symmetric devices, and the realization of time-dependent non-Hermitian protocols.
• Rigorous Characterization of Observable Quantities: The ongoing challenge is to systematically identify and classify all observables, including time-dependent expectation values and their spectrum, in the broad class of time-dependent non-Hermitian systems, particularly in connection with measurement, quantum statistical mechanics, and open quantum system theory.
• Symmetry-based Classification and Control: Generalizations of PT- and pseudo-Hermiticity, such as pseudo-PT symmetry and combinations with supersymmetry and topological phases, offer rich directions for controlling spectra and engineering desired dynamical properties.

In summary, time-dependent non-Hermitian Hamiltonians form the foundation for the modern analysis of quantum systems with dissipation, driven dynamics, or gain/loss, necessitating the joint evolution of states and metric, the use of Dyson and invariant operator frameworks, and a redefinition of observables and symmetry concepts. These developments have not only clarified fundamental issues around norm preservation and spectral reality but also expanded the landscape of quantum control, sensing, and computational approaches to complex many-body dynamics.