Lewis-Ermakov approach for the time-dependent two-level system

Technical Summary: Lewis–Ermakov Approach for Time–Dependent Two–Level Systems

Problem Statement
Time-dependent two-level systems (TLS) are fundamental to quantum control, yet they are typically analyzed as distinct cases from the time-dependent harmonic oscillator (THO). While the Lewis–Ermakov method provides a robust framework for the THO—yielding dynamical invariants, closed-form evolution operators, and exact propagators for arbitrary time-dependent parameters—a unified, fully developed Lewis–Ermakov framework for TLS has been lacking. Specifically, a formulation featuring a dynamical invariant, a factorized time-dependent Hamiltonian, and a closed-form propagator valid for arbitrary coupling $a(t)$ and detuning $b(t)$ has not been established in the same unified language used for the THO. This gap is particularly significant in open or effectively non-Hermitian settings, where exceptional points and state-dependent gain complicate the concept of adiabatic following.

Methodology
The authors construct an explicit Lewis–Ermakov-type dynamical invariant for TLS by exploiting the algebraic correspondence between the TLS and the THO. The methodology relies on the fact that both systems realize representations of the same complexified Lie algebra, $sl(2, \mathbb{C})$:

• The THO utilizes the non-compact $su(1, 1)$ algebra generated by $\{\hat{x}^2, \hat{p}^2, \hat{x}\hat{p} + \hat{p}\hat{x}\}$.
• The TLS utilizes the compact $su(2)$ algebra generated by $\{\hat{\sigma}_-, \hat{\sigma}_+, \hat{\sigma}_z\}$.

The core procedure involves mapping the time-dependent TLS Hamiltonian, $\hat{H}_{TLS}(t) = a(t)(\hat{\sigma}_+ + \hat{\sigma}_-) + b(t)\hat{\sigma}_z$, onto a stationary form via a pair of time-dependent transformations, $\hat{G}(t)$ and $\hat{T}(t)$.

1. Transformation: The transformation $\hat{G}(t)$ incorporates the detuning and scaling, while $\hat{T}(t)$ removes residual terms. Notably, $\hat{G}(t)$ is non-unitary due to the nilpotent nature of $\hat{\sigma}_-$, necessitating an intermediate complexified state space.
2. Ermakov-Pinney Equation: The mapping requires an auxiliary scaling function $\rho(t)$ to satisfy a complex Ermakov-Pinney equation:
   $$\ddot{\rho} + \theta^2(t)\rho = \frac{\theta_0^2}{\rho^3}$$
   Here, the effective frequency $\theta^2(t)$ encodes the physical parameters $a(t)$, $b(t)$, and their time derivatives:
   $$\theta^2(t) = a^2 + b^2 + i\dot{b} - i\frac{\dot{a}b}{a} + \frac{\ddot{a}}{2a} - \frac{3\dot{a}^2}{4a^2}$$
3. Invariant Construction: The dynamical invariant $\hat{I}_{TLS}(t)$ is constructed by acting with the composite transformation $\hat{S}_{TLS}(t) = \hat{G}(t)\hat{T}(t)$ on the stationary operator $(\hat{\sigma}_+ + \hat{\sigma}_-)$. This invariant satisfies the Lewis condition $\frac{d\hat{I}}{dt} = \frac{\partial \hat{I}}{\partial t} - i[\hat{I}, \hat{H}] = 0$.
4. Evolution Operator: Because the transformed Hamiltonian commutes with itself at all times, the time-ordering operator becomes trivial. The exact propagator is derived in closed form, expressed in terms of the scaling function $\rho(t)$ and an accumulated phase $f(t) = \int dt'/\rho^2(t')$.

Key Contributions and Results
The paper presents the following specific results:

• General Solution: An exact, closed-form evolution operator and propagator for arbitrary $a(t)$ and $b(t)$, provided the complex Ermakov-Pinney equation for $\rho(t)$ admits a solution.
• Landau-Zener-Majorana-Stückelberg (LZMS) Transitions: The framework is applied to the standard LZMS model ($a(t)=g, b(t)=\alpha t$). The method recovers the known non-adiabatic dynamics and transition probabilities, with the complex frequency $\theta^2(t)$ capturing the sweep rate effects via an imaginary component $i\alpha$.
• Non-Hermitian Dynamics: The approach is extended to a dissipative LZMS model with imaginary detuning $b(t) = i\alpha t$. The authors demonstrate that while the static exceptional point (where eigenvalues coalesce) is determined by the instantaneous spectrum, the dynamical exceptional point (where exponential amplification begins) is shifted. This shift is governed by the zero of the effective frequency $\theta^2(t)$, highlighting that finite sweep rates alter the onset of symmetry breaking.
• Adiabatic Rapid Passage (ARP): The method is applied to the Allen-Eberly model (hyperbolic secant coupling and hyperbolic tangent detuning). The invariant-based approach provides an exact, approximation-free description of population transfer, accurately capturing deviations from perfect adiabaticity due to finite sweep rates.
• Inverse Engineering (Shortcuts to Adiabaticity): The authors develop a protocol to design control fields $a(t)$ and $b(t)$ that drive the system along a specified trajectory without residual excitations (quantum friction).
  • By promoting the integration constant $\theta_0$ to a free geometric parameter $\vartheta_0$, they derive boundary conditions for $\rho(t)$ that ensure the invariant commutes with the Hamiltonian at the start and end of the protocol.
  • For a real scaling function $\rho(t)$, the inverse engineering yields fully explicit, closed-form control fields: $a(t) = \vartheta_0 \cos\phi / \rho^2(t)$ and $b(t) = \vartheta_0 \sin\phi / \rho^2(t)$, where $\phi$ is a fixed angle. This restricts the protocol to proportional modulation of coupling and detuning but guarantees analytical solvability.
  • For a complex $\rho(t)$, the fields are determined by a system of coupled non-linear integro-differential equations, offering greater flexibility in shaping $a(t)$ and $b(t)$ independently but requiring numerical solution.

Significance and Claims
The paper claims that its primary contribution is the explicit construction and practical application of the Lewis–Ermakov dynamical invariant for time-dependent two-level systems. The significance lies in:

1. Unification: It establishes a conceptual and mathematical link between continuous-variable (oscillator) and discrete-variable (two-level) realizations of the $sl(2, \mathbb{C})$ algebra, showing that the same sequence of factorization, auxiliary equations, and invariant construction applies to both.
2. Computational Utility: It replaces problem-specific integrations with a single auxiliary equation (the complex Ermakov-Pinney equation), providing exact propagators for arbitrary driving fields, including non-unitary regimes.
3. Control Design: It elevates the invariant from a theoretical tool to a practical resource for inverse engineering. It provides a systematic method to design "shortcuts to adiabaticity" with fully explicit control fields (in the real-$\rho$ case) or numerically determined fields (in the complex-$\rho$ case), eliminating the need for arbitrary constraints on the driving fields.
4. Generalizability: The authors note that the algebraic framework naturally extends to arbitrary $N$-level systems (spin-$j$ representations), where the dynamics remain governed by an Ermakov-Pinney equation with modified coefficients, suggesting a universal approach for multilevel quantum dynamics.

The work does not propose new experimental setups but rather provides the theoretical machinery to analyze existing scenarios (trapped ions, superconducting circuits, etc.) and design control protocols for them.