Time-Dependent Dyson Map in Quantum Mechanics

• The time-dependent Dyson map is an invertible operator that implements similarity transformations on time-dependent, non-Hermitian Hamiltonians to recover a physically meaningful, Hermitian counterpart.
• Construction methodologies include Lie-algebraic ansatz, direct Dyson equation solutions, invariant-based approaches, and perturbative expansions to ensure a positive-definite metric.
• Applications span PT-symmetry restoration, fractional-time quantum dynamics, and modeling coupled oscillators and optical systems, thereby enabling unitary evolution in non-Hermitian frameworks.

A time-dependent Dyson map is an invertible (in general, non-unitary) operator-valued function of time that implements a similarity transformation between a non-Hermitian, possibly explicitly time-dependent, Hamiltonian and a Hermitian counterpart. The primary purpose of such a transformation is to restore unitarity and ensure a consistent quantum theoretic interpretation of systems whose natural time evolution, due to non-Hermitian generators or fractional-time dynamics, is otherwise non-unitary. This construction is central to modern developments in pseudo-Hermitian and PT-symmetric quantum mechanics, especially for systems that are dynamically deformed, exhibit broken PT-symmetry, or arise from fractional calculus generalizations of the Schrödinger equation.

1. Formal Definition and Mathematical Structure

Let $H(t)$ be a non-Hermitian, explicitly time-dependent Hamiltonian acting on a Hilbert space $\mathcal{H}_\psi$, with states evolving according to

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

A time-dependent Dyson map $\eta(t)$ (or $\Omega(t)$) is an invertible linear operator such that, under the transformation

$|\phi(t)\rangle = \eta(t)|\psi(t)\rangle,$

the state $|\phi(t)\rangle$ satisfies a Hermitian Schrödinger equation,

$$i\hbar \frac{\partial}{\partial t}|\phi(t)\rangle = h(t)|\phi(t)\rangle,$$

where $h(t)$ is Hermitian. The two Hamiltonians are related by the time-dependent Dyson equation,

$$h(t) = \eta(t) H(t) \eta^{-1}(t) + i\hbar\, \bigl[\partial_t \eta(t)\bigr] \eta^{-1}(t).$$

This nonunitary similarity transformation is, for finite-dimensional systems, often constructed via parametrized exponential operators using decompositions (e.g., Gauss–Bruhat for SU(2)), or for Lie-algebraic models via factorized exponentials of algebra generators.

The time-dependent metric operator

$\rho(t) = \eta^\dagger(t) \eta(t)$

induces a dynamically deformed Hilbert space with inner product $\langle \psi|\rho(t)|\psi\rangle$, ensuring a conserved, positive-definite norm even when $H(t)$ is not self-adjoint in the standard sense.

2. Connection to Unitary Evolution and Observability

The time-dependent Dyson map framework provides a rigorous reinterpretation of quantum evolution in non-Hermitian settings by isolating the generator of evolution (often non-observable due to lack of quasi-Hermiticity when $\rho$ is time-dependent) from the observable physical Hamiltonian. The evolution operator in the deformed Hilbert space,

$$U_{\text{phys}}(t,0) = \eta(t) U_H(t,0) \eta^{-1}(0),$$

is strictly unitary with respect to $\langle\cdot|\cdot\rangle_{\rho(t)}$. It follows that

$\langle\psi(t)|\rho(t)|\psi(t)\rangle$

is time-invariant. In this structure, the operator $H(t)$ is the generator of motion, but the true "energy observable" is

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

guaranteed to have a real spectrum as long as $\rho(t)$ remains positive-definite.

A crucial distinction arises: while $h(t)$ is observable and governs physical predictions in the mapped (Hermitian) system, $H(t)$ ceases to be an observable operator if the metric is time-dependent, as the quasi-Hermiticity relation $H^\dagger \rho = \rho H$ does not hold for $\dot\rho \neq 0$ (Fring et al., 2015).

3. Construction Methodologies and Algorithmic Workflow

There exist several systematic approaches for constructing time-dependent Dyson maps:

• Lie-Algebraic and Factorized Ansatz: For finite-dimensional systems (e.g., two-level systems, coupled oscillators) one uses factorized exponentials of algebra generators with time-dependent coefficients determined by the requirement that $h(t)$ is Hermitian. For the two-level (SU(2)) case, a general ansatz is

$$\Omega(t) = e^{\kappa(t)} e^{\lambda(t) \sigma_+} e^{\frac{1}{2} \ln \Lambda(t)\, \sigma_3} e^{\lambda^*(t) \sigma_-},$$

with $\kappa, \ln \Lambda \in \mathbb{R}$, $\lambda \in \mathbb{C}$.

• Direct Solution of the Dyson Equation: Insert an operator-ansatz for $\eta(t)$ and solve the corresponding system of ODEs for the parameters by enforcing Hermiticity of $h(t)$. For two-level or oscillator systems, this implies solving coupled nonlinear ODEs, frequently reducible to (generalized) Ermakov–Pinney equations (Fring et al., 2017, Fring et al., 2018).
• Invariant-Based (Lewis–Riesenfeld) Approach: One first constructs dynamical invariants $I_H(t)$ (for $H(t)$) and $I_h(t)$ (for $h(t)$), requiring $I_h(t) = \eta(t) I_H(t) \eta^{-1}(t)$ to be Hermitian. This yields an algebraic problem for $\eta(t)$, advantageous in Lie-algebraic models or when point-transformations map between known reference and target invariants (Fring et al., 2018, Fring et al., 2021).
• Point-Transformation Techniques: Non-Hermitian $H(t)$ is constructed via point transformations from a Hermitian reference system, ensuring that the same transformation maps their invariants and hence facilitating the construction of $\eta(t)$ (Fring et al., 2021).
• Perturbative and Recursive Expansion: For systems with a small (or large) non-Hermitian coupling parameter, one expands $\eta(t) = \exp[\frac{1}{2}\sum_{n} g^n q^{(n)}(t)]$ and solves recursively at each order, bypassing the need for guessing a global ansatz (Fring et al., 2020).

These methods yield a local, algorithmic workflow:

1. Select ansatz for $\eta(t)$ (factorized or expanded form) appropriate to the algebraic structure.
2. Impose Hermiticity of $h(t)$ and solve for coupled parameter ODEs.
3. Reconstruct $\rho(t)$, verify positivity, and extract $h(t)$ and physical energy observable.

4. Applications: Fractional-Time and Non-Hermitian Quantum Dynamics

The time-dependent Dyson map is indispensable in the following contexts:

• Fractional-Time Schrödinger Dynamics: In the Caputo-fractional Schrödinger equation (FTSE), the formal evolution operator,

$$U_{FT}(t) = E_\alpha\Bigl( \frac{\hat{\mathcal H}_0^\alpha\, i^\alpha\, t^\alpha}{\hbar_\alpha} \Bigr),$$

is generally non-unitary. By constructing a time-dependent Dyson map $\Omega(t)$ (often in Gauss–Bruhat SU(2) form for two-level systems), one embeds the non-unitary evolution into a dynamically deformed Hilbert space, ensuring $$\langle\Psi^\alpha(t)|\rho(t)|\Psi^\alpha(t)\rangle =$$ const. Explicit examples include longitudinal field spin-1/2, Lee–Yang field, and PT-symmetric optical Hamiltonian, all mapped to strictly unitary Hermitian evolution via this method (Cius et al., 2022).

• PT-Symmetry Breaking and Restoration: For systems where the time-independent Hamiltonian $H$ is in the broken PT-symmetric regime (complex eigenvalues, unphysical), time-dependence in $H(t)$ and $\eta(t)$ enables restoration of a real, physically meaningful energy spectrum. The energy observable constructed via the time-dependent Dyson map remains real for all $t$, and decoherence can be controlled (e.g., the sudden death of entanglement entropy is replaced by a plateau) (Fring et al., 2017, Fring et al., 2018, Fring, 2022).
• Non-Hermitian Coupled Oscillators, Spin Chains, and Optical Systems: The technique has been systematically extended to two-dimensional oscillators, non-Hermitian Lee–Yang models, PT-symmetric optical couplers, and bosonic models associated with $\mathfrak{su}(2)$ or $\mathfrak{su}(1,1)$ algebras. In these, the time-dependent Dyson map (often in multiple-parameter exponential or product form) ensures the existence of a positive-definite metric and unitary evolution, independent of PT-symmetry regime (Fring et al., 2018, Cius et al., 2022).

5. Ambiguity, Infinite Series, and Gauge Structure

A salient feature of the time-dependent Dyson map is its non-uniqueness for a given $H(t)$:

• If $\eta_1(t)$ and $\eta_2(t)$ are two distinct Dyson maps for the same $H(t)$, their ratio

$A(t) = \eta_2^{-1}(t)\eta_1(t)$

intertwines the corresponding Hermitian partners,

$$h_2(t) = A(t)\, h_1(t)\,A^{-1}(t) + i\hbar\, (\partial_t A)\,A^{-1}(t).$$

• Under broad conditions, this leads to an infinite series (tower) of physically inequivalent Hermitian Hamiltonians and Dyson maps, labeled by integer powers of $A(t)$ (Fring et al., 2021). These structures are physically relevant in systems with dynamical symmetries or multiple Lewis–Riesenfeld invariants.
• In contrast, requiring that $H(t)$ itself be observable in the deformed inner product (i.e., enforcing a time-independent metric) collapses all gauge freedom, resulting in a unique, physically distinguished Dyson map (Luiz et al., 2017, Luiz et al., 2016).

Gauge-linked (local or global) transformations between Dyson maps allow for the construction of chains of non-Hermitian Hamiltonians, where the matrix elements of observables can be mapped across the chain, but the observability of all elements restricts the chain severely.

6. Limitations, Physical Interpretation, and Domains of Applicability

• The existence and explicit construction of a positive-definite metric $\rho(t)$ (and hence of $\eta(t)$) can fail if the system encounters a dynamical singularity or fatal instability. For most models appearing in the literature (finite-dimensional, Lie-algebraic, or weakly coupled), this does not pose an obstacle unless the system crosses a dynamical phase transition.
• In the context of fractional or PT-broken dynamics, the time-dependent Dyson map allows extension of the physical regime into parameter domains previously considered unphysical and enables control over entanglement and other time-dependent observables (Cius et al., 2022, Cius et al., 2022).
• At a foundational level, the separation between the generator of time evolution and the observable Hamiltonian in the non-Hermitian picture is a defining paradigm shift; only in the mapped Hermitian system are standard quantum mechanical interpretations directly applicable.

7. Summary Table: Structural Elements of Time-Dependent Dyson Map Construction

Object	Definition/Role	Typical Form / Construction
$H(t)$	Non-Hermitian Hamiltonian	$H(t)$ (explicit in model; e.g., $H_{FT}(t)$ in FTSE)
$\eta(t)$ (Dyson map)	Similarity map to Hermitian sector	e.g., $\exp[\sum_j \epsilon_j(t) K_j]$ (factorized exponentials)
$\rho(t) = \eta^\dagger\eta$	Dynamical metric operator	Ensures $\langle \psi|\rho(t)|\psi\rangle>0$
$h(t)$	Hermitian partner Hamiltonian	$$h(t) = \eta H \eta^{-1} + i\hbar \dot\eta \eta^{-1}$$
$\tilde H(t)$	Physical energy operator in $H$-picture	$$\tilde H(t) = \eta^{-1} h \eta = H(t) + i\eta^{-1} \dot\eta$$
Evolution Operator	Unitary in deformed Hilbert space	$u(t) = \eta(t) U_H(t) \eta^{-1}(0)$

The time-dependent Dyson map is thus a foundational mathematical and conceptual tool underlying modern approaches to non-Hermitian and fractional quantum evolution, enabling the restoration of unitarity, preservation of physical observables, and mapping of dynamics across Hermitian and non-Hermitian sectors, even well beyond the regime of conventional PT-symmetric quantum mechanics.