Non-Perturbative Time-Dependent Hamiltonian

• Non-Perturbative Time-Dependent Hamiltonian Models are frameworks that solve the full time-dependent Schrödinger equation without relying on small-parameter expansions, capturing complete quantum dynamics.
• They decompose the Hamiltonian into time-independent and time-dependent parts and employ configuration-interaction spectral methods to simulate complex multiband transitions.
• These models enable precise simulation of quantum phenomena such as non-adiabatic transitions and interband coupling, validated by numerical results that align with analytic benchmarks.

A non-perturbative time-dependent Hamiltonian model is a theoretical and computational framework for describing the real-time evolution of quantum systems under time-dependent external perturbations without relying on perturbative expansions in a small parameter. Such models are crucial for accurately capturing the dynamics of interacting particles—such as atoms in optical lattices, electrons in strong laser fields, or more general quantum field systems—under external drives or rapidly varying environments, especially in regimes where perturbation theory fails.

1. Definition and Core Principles

A non-perturbative time-dependent Hamiltonian model treats quantum dynamics by solving the full time-dependent Schrödinger equation,

$$i\hbar \frac{\partial}{\partial t} \Psi(t) = H(t) \Psi(t),$$

where the Hamiltonian $H(t)$ has explicit time dependence, often due to external drives or modulations. "Non-perturbative" indicates that no approximation in terms of a weak-coupling or small-parameter expansion is made; instead, all orders of the dynamics induced by $H(t)$ are captured, typically by projecting onto a complete set of stationary eigenstates of an unperturbed, time-independent part of the Hamiltonian. Time-dependent perturbations are incorporated exactly within the subspace considered, often up to a high order in a Taylor or functional expansion of the perturbing operator.

2. Mathematical Framework

Hamiltonian Decomposition and Basis Construction

The general strategy is to separate the Hamiltonian into a time-independent component $H_0$ and a time-dependent perturbation $W(t)$: $H(t) = H_0 + W(t).$ For two interacting atoms in an optical lattice (1209.0162), $H_0$ includes kinetic energy, a (Taylor-expanded) lattice potential, and interatomic interactions. The system is described in terms of center-of-mass ($R$) and relative ($\rho$) coordinates, allowing for an efficient configuration interaction construction of the many-body stationary eigenstates.

The stationary eigenvalue problem is

$$H_0 \Phi_i^{(\sigma)} = E^{(\sigma)}_i \Phi^{(\sigma)}_i,$$

where $\Phi^{(\sigma)}_i$ are symmetry-adapted basis functions (via, e.g., point-group $D_{2h}$ irreps), constructed from tensor products of B-spline radial functions and spherical harmonics for $R$ and $\rho$, combined via a configuration interaction (CI) procedure.

Non-Perturbative Time Propagation

The time-dependent wavefunction is expanded in the complete set of $H_0$ eigenstates: $$\Psi(t) = \sum_{\sigma,i} B_{(\sigma,i)}(t) \Phi^{(\sigma)}_i,$$ with $B_{(\sigma,i)}(t)$ time-dependent coefficients. Insertion into the TDSE yields a coupled set of ordinary differential equations: $$i\hbar \frac{d}{dt} B_{(\kappa,j)}(t) = E_j^{(\kappa)} B_{(\kappa,j)}(t) + \sum_{\sigma,i} B_{(\sigma,i)}(t) \langle \Phi_j^{(\kappa)} | W(t) | \Phi_i^{(\sigma)} \rangle.$$ Thus, all complexity arising from the external perturbation $W(t)$ is subsumed in the time-dependent matrix elements $P_{mn}^{(\tau,\sigma)}(t)$.

This CI-based, spectral propagation allows arbitrary (non-perturbative) time dependence, so long as $W(t)$ can be expressed as a Taylor expansion in system coordinates to some finite order.

Taylor Expansion of the Perturbation

For typical lattice problems, $W(t)$ is Taylor-expanded up to quadratic order in $\rho$ and $R$ (i.e., terms like $$f_{01}(t)\rho_x + f_{10}(t)R_x + f_{11}(t)\rho_x R_x + f_{02}(t)\rho_x^2 + f_{20}(t) R_x^2$$, where the $f_{ij}(t)$ are time-dependent envelopes). This form encompasses both linear “forces” (acceleration, tilt) and quadratic “traps” (harmonic confinement).

3. Treatment of Common Physical Scenarios

Lattice Acceleration (Linear Perturbation)

A lattice acceleration corresponds to a term $W(t) = f_{10}(t) R_x$. In this case, the perturbation acts only on the center-of-mass coordinate, with angular integrals simplifying considerably. The expected center-of-mass motion $\langle R_x \rangle (t)$ displays dynamics that, in the harmonic limit, reproduce analytically derived driven oscillator trajectories.

Fast, non-adiabatic acceleration manifests as a breakdown of the single-band (lowest Bloch band) approximation, inducing population transfer to higher bands—captured naturally in the non-perturbative framework, as the full stationary eigenbasis supports such transitions.

Additional Harmonic Confinement (Quadratic Perturbation)

A time-dependent quadratic term, $W(t) = f_{20}(t) R_x^2$, models the addition of a global harmonic trap. In the adiabatic limit, the trap frequency shifts dynamically, with wavefunction broadening/compression encoded in the width evolution

$$\sigma(t) = \frac{A(0)}{\sqrt{2[1 + 2 C_{\text{harm}} \omega t]}},$$

where $C_{\text{harm}}$ parameterizes the trap strength. Non-adiabatic switching excites breathing (width) oscillations, manifest in local uncertainties, which are again accessible in the exact numerical solution.

Realistic Two-Atom Systems and Inter-Band Physics

Application to heteronuclear ($^6$Li-$^7$Li) dimers in multi-well lattices illustrates correlated quantum migration, tunneling, and redistribution under both slow and fast drives. For strong/abrupt driving, significant amplitude appears in excited (non-ground) Bloch bands or “repulsively bound” states, emphasizing a key feature: physics beyond standard Hubbard (single-band) approximations requires a non-perturbative approach.

4. Numerical and Analytical Agreement

The spectral expansion and time stepping yield results agreeing to within $10^{-10}$ with analytically solvable limits (e.g., single-particle, harmonic driving). The method preserves unitarity and supports observables calculation at all time slices, including non-adiabatic and highly excited regimes. Non-perturbative effects, such as the breakdown of adiabatic following or coherent population transfer among symmetry-adapted sectors, are accurately captured.

5. Computational Implementation and Resources

The central computational bottleneck lies in the diagonalization of $H_0$ for construction of the stationary eigenbasis. For inhomogeneous situations (e.g., high-order Taylor lattice expansion), the B-spline plus spherical harmonics basis offers flexibility. The CI matrix dimension is set by desired energy resolution and the number of symmetry sectors included.

Propagation of the $B_{(\sigma,i)}(t)$ coefficients involves time-integrating a set of coupled ODEs, the dimension of which scales with the product of retained $R$ and $\rho$ basis states. Matrix elements of quadratic-in-position operators (e.g., $R^2$, $\rho R$) are efficiently evaluated via analytic angular integration and sparse matrix techniques. For rapid time-varying $W(t)$, small time steps are used; conservation of probability and consistent physical observables serve as numerical checks.

Scaling to more particles or higher spatial dimension is limited primarily by basis size and associated matrix sizes, but no aspect of the numerical propagation is perturbative—time dependence enters only through the matrix elements and their explicit time dependence.

6. Physical Implications and Scope

This non-perturbative, time-dependent Hamiltonian model provides a robust treatment of spectral and dynamical properties in driven lattice systems beyond traditional perturbative or adiabatic approaches. It is directly applicable to:

• Quantum quenches and rapid ramps in ultracold atomic lattices.
• Breakdown of simple Hubbard-type models under strong driving or interaction-induced band mixing.
• Correlated tunneling, repulsively bound states, and real-time emergence of interband excitations.
• Diagnostics of quantum simulation or quantum computation protocols relying on control of external fields over interaction timescales.

The flexibility in $W(t)$ permits simulations of a broad class of experimental sequences, including time-dependent disorder, shaking, and global or local potential modifications up to quadratic order. However, more singular or higher-order terms in $W(t)$, or systems requiring full field-theoretic treatments, may require different frameworks or supplementary methods.

7. Summary Table: Key Model Features

Aspect	Technique	Capability
Basis	$H_0$ eigenstates in CM and relative coords (CI over B-splines, spherical harmonics, point-group adapted)	Supports arbitrary time dependence, symmetry reduction
Time propagation	ODEs for expansion coefficients	Exact, non-perturbative in $W(t)$
Perturbation form	Taylor expansion to quadratic order	Captures linear, quadratic, and mixed external fields
Observable calculation	Full time-dependent state	Dynamical correlation functions, occupation dynamics
Non-adiabatic effects	Naturally included	Interband transitions, tunneling, breakdown of adiabaticity

This non-perturbative, configuration-interaction spectral method is established as a powerful tool for simulating the time-dependent dynamics of few-body quantum systems in optical lattices under arbitrary external drives, facilitating both numerical precision and insight into fundamentally non-perturbative quantum phenomena (1209.0162).