Phase-Coupled Caldeira–Leggett Model (PCL)

• The PCL model is a generalization of the Caldeira–Leggett framework that employs an exponential phase-type coupling to describe non-Markovian and nonperturbative quantum dynamics.
• It bridges quantum Brownian motion and small-polaron physics by preserving superconducting phase compactness and incorporating a hierarchical dissipaton approach.
• This model reveals new dynamical features, including steady-state coherence and renormalized spectra, with implications for tuning critical behavior in superconducting circuits.

The Phase-Coupled Caldeira–Leggett (PCL) model generalizes quantum dissipative system theory by introducing an exponential, phase-type system–bath coupling. In contrast to the conventional Caldeira–Leggett (CL) model, which employs a linear coupling to a Gaussian bath, the PCL model captures regimes associated with both quantum Brownian motion and non-linear, polaronic physics. As a result, the PCL framework yields a non-Markovian and nonperturbative description that unveils new dynamical features and a significantly richer phase structure, particularly for superconducting circuits where the quantization and compactness of the superconducting phase are preserved (Chang et al., 29 Oct 2025, Kashuba et al., 2023).

1. Model Formulation and Hamiltonian Structure

The total Hamiltonian in the PCL framework incorporates a bare system $H_S$, a harmonic oscillator bath $H_B$, and an exponential system–bath coupling $H_\mathrm{SB}$: $H_\mathrm{tot} = H_S + H_B + H_\mathrm{SB}$

• $H_S$ is the isolated system's Hamiltonian (e.g., for a two-level system, $H_S = \epsilon\sigma_z$).
• $$H_B = \sum_j \bigl(\frac{p_j^2}{2m_j} + \frac12 m_j\omega_j^2 x_j^2\bigr)$$, a sum over harmonic bath modes.
• $H_\mathrm{SB}^{(\mathrm{PCL})} = S\,B$, where $B = e^{i\lambda F} + e^{-i\lambda F}$, $F = \sum_j c_jx_j$, and $S$ is a system operator (e.g., $S=\alpha\sigma_x$).

This exponential (phase-type) coupling contrasts with the linear $H_\mathrm{SB}^{(\mathrm{CL})}=SF$ conventionally used in the CL model. Here, $\lambda\in\mathbb{R}$ sets the strength of the phase-coupling.

In superconducting circuits, the PCL Hamiltonian with compact phase $\phi\in[0,2\pi)$ accounts for charge quantization: $$H_{\mathrm{PCL}} = E_C(N+N_{\rm env})^2 - E_J\cos\phi + H_{\rm env}$$ where $N$ is the Cooper pair number, $E_C$ the charging energy, $E_J$ the Josephson energy, and $N_{\rm env}$ encodes the capacitive environment and possible Andreev-type tunneling (Kashuba et al., 2023).

2. Gaussian Environment and Dissipaton Hierarchy

Despite the highly nonlinear $H_\mathrm{SB}$, the Gaussianity of the bath is preserved due to $F$'s linearity in oscillator coordinates, enabling all bath correlations to be expressed in terms of the two-point function $C(t) = \langle F(t)F(0)\rangle_B$ via Wick's theorem.

A dissipaton decomposition expresses $C(t)$ as a finite sum of exponentials: $C(t) = \sum_{k=1}^K \eta_k e^{-\gamma_k t}$ with complex coefficients $\eta_k$ and decay rates $\gamma_k$ determined by the bath spectral density. Corresponding dissipaton operators $f_k$ satisfy $$\langle f_k(t)f_{k'}(0)\rangle = \delta_{kk'}\eta_ke^{-\gamma_kt}$$.

Through a generalized normal ordering $O(\ldots)$ in Liouville space and closure in Hermite polynomials $H_n(f)$, the exponential bath operator $B = e^{i\lambda F} + e^{-i\lambda F}$ can be contracted exactly, allowing for the systematic generation of a hierarchy of auxiliary density operators (ADOs).

3. Exact Non-Markovian Dynamics

The PCL model leads to an exact, closed-form equation of motion for the ADO hierarchy: $$\dot\rho_{\bf n}^{(n)} = -i[H_S,\rho_{\bf n}^{(n)}] -\sum_k n_k\gamma_k \rho_{\bf n}^{(n)} - i g \sum'_{\bf m,l} [(i\lambda)^m - (-i\lambda)^m] \prod_k \frac{\eta_k^{l_k}(m_k-l_k)!\binom{n_k}{l_k}}{} \, S\rho_{{\bf n}+{\bf m}-2{\bf l}}^{(n+m-2l)} + i g \sum'_{\bf m,l} [...]$$ where $g = e^{-\lambda^2\langle F^2\rangle/2}$, $n=\sum_k n_k$, with sums over multi-indices as defined in (Chang et al., 29 Oct 2025). The zeroth-tier ADO, $\rho_{0...0}^{(0)}$, yields the reduced density matrix of the system. This infinite hierarchy encodes all orders of phase-coupling strength $\lambda$ and non-Markovian memory effects via $\gamma_k$.

For the compact superconducting phase PCL model, the RG flow and dynamical scaling laws reflect the influence of charge quantization and lead to different critical behavior than decompactified (standard CL) treatments (Kashuba et al., 2023).

4. Physical Interpretation and Limiting Regimes

The exponential phase coupling $e^{i\lambda F}$ effects a polaronic phase shift on bath modes, generalizing the displacement (linear) paradigm of quantum Brownian motion. In the weak-coupling ($\lambda\ll1$) regime, expansion of the exponential recovers the CL model. For arbitrary $\lambda$, the PCL model interpolates between Brownian and small-polaron physics, reproducing both linear-response and strong-coupling behavior.

Key physical regimes include:

• Small $\lambda$: Only first-tier dissipaton terms matter; PCL reduces to CL model.
• High temperature: Fast bath decay, hierarchy truncates, and a Markovian limit emerges.
• Short times ($t\lesssim1/\Omega_c$): Coherent oscillations with renormalized frequency $$\Omega_{\mathrm{eff}}\approx2\sqrt{\epsilon^2 + 4\alpha^2g^2}$$.
• Long times ($t\to\infty$):
  • CL: System relaxes to a diagonal thermal Gibbs state.
  • PCL: Steady state is a Gibbs state of a Hamiltonian of mean force $H_{\mathrm{eff}} = H_S + 2gS$, generically non-diagonal in the system basis, allowing residual coherence.

5. Phase Structure and Experimental Implications in Superconducting Circuits

The PCL model with compact phase retains charge quantization and reveals a four-parameter phase diagram, in sharp contrast to the two-parameter structure of decompactified (CL-type) treatments.

Tuning parameters:

• $E_J/E_C$: Junction regime, transmon ($E_J\gg E_C$) vs. Cooper pair box ($E_J\ll E_C$)
• $\lambda^z$: Capacitive (Ohmic dissipation) strength
• $\lambda^\perp$: Electric (Andreev/Cooper-pair exchange) contact strength
• $N_g$: Offset charge parameter, relevant for compact phase

Notable mappings and consequences (Kashuba et al., 2023):

• Transmon regime ($E_J\gg E_C$): RG analysis yields Schmid–Bulgadaev (SB)–like phase transition. Critical line ($\alpha_c=1$, $\alpha=R_Q/R_z$) is controlled exclusively by capacitive coupling $\lambda^z$, insensitive to Andreev exchange $\lambda^\perp$ or offset charge.
• Cooper pair box regime ($E_C\gg E_J$): Effective mapping to the anisotropic Kondo model. No pseudoinsulating Kondo phase can occur for physically allowed capacitive couplings ($\lambda^z>0$).

Regime	Mapping	Criticality Controlled by
Transmon ($E_J\gg E_C$)	Schmid–Bulgadaev	Capacitive ($\lambda^z$)
Cooper pair box ($E_J\ll E_C$)	Kondo Model	Not achievable for $\lambda^z>0$

Preserving phase compactness thus reveals new tuning parameters and critical phenomena inaccessible to traditional CL models.

6. Distinctive Dynamical and Steady-State Signatures

Numerical simulation of two-level systems demonstrates the intrinsic non-Markovian character and enhanced complexity of PCL dynamics:

• Population and coherence exhibit strong high-frequency oscillations and partial revivals in PCL, unlike the monotonic damping in CL.
• On the Bloch sphere, PCL final states are generically non-diagonal in the system Hamiltonian basis, reflecting residual steady-state coherence.
• The steady-state von Neumann entropy is non-monotonic in $\lambda$, minimized for both weak and strong coupling limits.
• As $\alpha$ increases (dissipative coupling), frequency of oscillations rises and entropy decreases.

Novel behaviors in PCL include steady-state coherence, pronounced coherence revivals, non-thermal steady states, and a renormalized ("polaron-shifted") system spectrum. These features are direct consequences of non-linear, nonperturbative, and non-Markovian quantum dissipation (Chang et al., 29 Oct 2025).

7. Broader Impact and Extensions

The PCL model unifies the physics of quantum Brownian motion and small-polaron formation, providing a robust platform for studying phase-mediated decoherence far beyond linear response. In superconducting circuits, incorporating phase compactness demonstrates that only certain environmental couplings dictate critical lines, fundamentally altering phase diagrams and experimental strategies in dissipative quantum circuits.

The PCL approach may be further extended to systems where environmental degrees of freedom are non-Gaussian or where the system–bath coupling is more complex than pure phase-type exponentials. The exact treatment framework—dissipaton algebra, Hermite closure, and infinite ADO hierarchies—provides a general, nonperturbative methodology for open quantum dynamics with complex dissipation (Chang et al., 29 Oct 2025, Kashuba et al., 2023).