Driven Caldeira-Leggett Model Overview

• The driven Caldeira-Leggett model is an open quantum system framework where external forcing acts on the system coordinate, the bath, or via non-equilibrium reservoir preparation, introducing non-Markovian dynamics and memory effects.
• It encompasses various methodologies such as explicit driving of the system, forcing of bath oscillators, and engineered equilibrium states, each yielding distinct deterministic and stochastic responses.
• The model's predictive accuracy hinges on factors like the validity of harmonic approximations, the treatment of initial system-bath correlations, and challenges in representing anharmonic or nonlinear dynamics.

The driven Caldeira-Leggett model denotes a family of open-quantum-system constructions built on the Caldeira-Leggett system-plus-bath framework, in which time dependence enters through direct forcing of the distinguished system coordinate, explicit driving of the bath coordinates, or non-equilibrium preparation of the reservoir. Taken together, the literature suggests that the term does not identify a single universally fixed formalism: in some works it means an externally driven system coupled to a harmonic bath, in others a bath that is itself driven by the external field, and in condensed-phase spectroscopy it can refer more loosely to the dissipative, noise-driven dynamics generated by the Caldeira-Leggett Hamiltonian and its generalized Langevin reduction (Grabert et al., 2018, 1712.06397, Gottwald et al., 2016).

1. Standard framework and the meaning of “driving”

The original Caldeira-Leggett construction describes a distinguished degree of freedom coupled linearly to many harmonic oscillators. In the formulation emphasized in the historical review of Caldeira’s work, the system-plus-environment Lagrangian is

$$L = \frac{1}{2}M\dot q^2 - V(q) + \frac{1}{2}\sum_j\left(m_j\dot x_j^2-m_j\omega_j^2x_j^2\right)-\sum_jF_j(q)x_j+\sum_j\frac{F_j^2(q)}{2m_j\omega_j^2},$$

with bilinear coupling $F_j(q)=c_j q$ in the standard model. The last term is the counter-term, whose stated role is to cancel the bath-induced renormalization of the bare system potential so that the dressed potential in the Hamiltonian is the intended physical one (Bonança et al., 29 Apr 2026).

A Hamiltonian form used in driven-bath studies is

$H_{\rm tot}(t)=H_S+H_R+H_{\rm int}+H_{\rm ext}(t),$

with

$$H_S=\frac{p^2}{2M}+V(q), \qquad H_R=\sum_n \left(\frac{p_n^2}{2m_n}+\frac{1}{2}m_n\omega_n^2 x_n^2\right),$$

$$H_{\rm int}=-q\sum_n c_n x_n+q^2\sum_n \frac{c_n^2}{2m_n\omega_n^2},$$

and spectral density

$$J(\omega)=\frac{\pi}{2}\sum_n \frac{c_n^2}{m_n\omega_n}\delta(\omega-\omega_n).$$

Within this architecture, “driving” can be introduced in structurally distinct ways: by adding an explicit external term acting on the system coordinate, by forcing the bath coordinates themselves, or by engineering the bath state so that it acts as a source of deterministic or stochastic work (Grabert et al., 2018, Cavina et al., 2024).

This distinction matters because the reduced system does not merely feel a local force. Once the bath is eliminated, the effective dynamics contains friction, fluctuations, and, in general, nonlocal memory kernels. A recurrent theme across the literature is that driving the environment is not equivalent to simply increasing the direct drive on the system; the bath has its own dynamics and mediates retarded response (Grabert et al., 2018).

2. Explicitly driven baths and retarded effective forcing

A direct extension of the standard model is obtained by coupling the same external force $F(t)$ both to the system coordinate and to each bath mode: $$H_{\rm ext}(t)= -\left(d_0 q+\sum_n d_n x_n\right)F(t).$$ The bath equations of motion then read

$$\dot x_n(t)=\frac{p_n(t)}{m_n},\qquad \dot p_n(t)=-m_n\omega_n^2 x_n(t)+c_n q(t)+d_nF(t),$$

so the drive enters the bath oscillator dynamics as a direct force term (Grabert et al., 2018).

Eliminating the bath yields the exact operator equation

$$M\ddot q(t)+M\int_0^t ds\,\gamma(t-s)\dot q(s)+\frac{\partial V(q(t))}{\partial q(t)} = \xi(t)+d_0F(t)+\int_0^t ds\,\lambda(t-s)F(s),$$

with the usual friction kernel

$F_j(q)=c_j q$0

and a bath-driving delay kernel

$F_j(q)=c_j q$1

The entire effect of bath driving is therefore captured by the effective force

$F_j(q)=c_j q$2

The bath-induced part is, in principle, non-Markovian, because it depends on the full history of the applied field (Grabert et al., 2018).

For a harmonic central potential,

$F_j(q)=c_j q$3

the long-time response to $F_j(q)=c_j q$4 is governed by the dynamic susceptibility

$F_j(q)=c_j q$5

The reported qualitative effect of bath driving is a low-frequency enhancement of the dispersive part, a maximum at zero frequency in $F_j(q)=c_j q$6, a shoulder-like feature in the absorptive part, and a slight shift of spectral weight toward lower frequencies (Grabert et al., 2018).

The Rubin chain provides an explicit microscopic realization. A heavy central particle coupled to two semi-infinite harmonic chains can be mapped to a driven Caldeira-Leggett bath when the chain particles are also driven. In that mapping the Rubin spectral density is

$F_j(q)=c_j q$7

and the damping kernel is

$F_j(q)=c_j q$8

which decays oscillatory and algebraically, making the non-Markovian character explicit (Grabert et al., 2018).

3. Driven spin-boson dynamics from coupled equilibrium

A second major realization of driven Caldeira-Leggett physics is the spin-boson model driven from an initially equilibrated coupled state. In the normal-mode representation the Hamiltonian is

$F_j(q)=c_j q$9

with Ohmic spectral density

$H_{\rm tot}(t)=H_S+H_R+H_{\rm int}+H_{\rm ext}(t),$0

The central conceptual point is that the initial condition is not factorized; instead, the system and bath start in the thermal equilibrium state of the full interacting Hamiltonian (1712.06397).

The exact treatment is based on the Extended Stochastic Liouville Equation. Equilibrium preparation is performed in imaginary time through

$H_{\rm tot}(t)=H_S+H_R+H_{\rm int}+H_{\rm ext}(t),$1

and the subsequent real-time evolution is

$H_{\rm tot}(t)=H_S+H_R+H_{\rm int}+H_{\rm ext}(t),$2

The reduced density matrix is obtained by averaging over the stochastic noises, and the imaginary-time endpoint supplies the real-time initial condition (1712.06397).

Two regimes were studied. For constant bias, the exact partition-free dynamics remains stationary on average, confirming that the imaginary-time preparation reproduces the coupled thermal equilibrium. For a Landau-Zener sweep,

$H_{\rm tot}(t)=H_S+H_R+H_{\rm int}+H_{\rm ext}(t),$3

the transient dynamics and asymptotic state differ substantially from partitioned approximations. The reported effects include suppression of short-time oscillations by cross-time correlations between imaginary-time preparation and real-time evolution, movement of the asymptote toward the zero-temperature Landau-Zener limit as temperature is lowered, and suppression of coherent oscillations with increasing bath coupling $H_{\rm tot}(t)=H_S+H_R+H_{\rm int}+H_{\rm ext}(t),$4 (1712.06397).

This formulation shows that in driven Caldeira-Leggett dynamics the choice of initial condition is not a technical detail. Initial system-bath correlations can affect both transients and late-time observables.

4. Non-equilibrium Gaussian reservoirs as deterministic and stochastic drives

A more recent non-equilibrium version of the model prepares the bath modes in squeezed and displaced thermal states before system-bath coupling is switched on. The total Hamiltonian is

$H_{\rm tot}(t)=H_S+H_R+H_{\rm int}+H_{\rm ext}(t),$5

with

$H_{\rm tot}(t)=H_S+H_R+H_{\rm int}+H_{\rm ext}(t),$6

The non-equilibrium ingredient lies in the reservoir preparation, not in an externally imposed field acting during the evolution (Cavina et al., 2024).

Displacement and squeezing play different dynamical roles. For displacement,

$H_{\rm tot}(t)=H_S+H_R+H_{\rm int}+H_{\rm ext}(t),$7

and, in the weak-coupling, strong-displacement limit, a single displaced mode yields approximately unitary reduced dynamics,

$H_{\rm tot}(t)=H_S+H_R+H_{\rm int}+H_{\rm ext}(t),$8

More generally,

$H_{\rm tot}(t)=H_S+H_R+H_{\rm int}+H_{\rm ext}(t),$9

with

$$H_S=\frac{p^2}{2M}+V(q), \qquad H_R=\sum_n \left(\frac{p_n^2}{2m_n}+\frac{1}{2}m_n\omega_n^2 x_n^2\right),$$0

The displaced reservoir therefore acts as a source of effective deterministic time dependence in the system Hamiltonian (Cavina et al., 2024).

Squeezing modifies fluctuations instead of shifting the mean. In the weak-coupling, strong-squeezing limit, the system experiences stochastic driving,

$$H_S=\frac{p^2}{2M}+V(q), \qquad H_R=\sum_n \left(\frac{p_n^2}{2m_n}+\frac{1}{2}m_n\omega_n^2 x_n^2\right),$$1

with Gaussian $$H_S=\frac{p^2}{2M}+V(q), \qquad H_R=\sum_n \left(\frac{p_n^2}{2m_n}+\frac{1}{2}m_n\omega_n^2 x_n^2\right),$$2. The same work shows that squeezing breaks the equilibrium fluctuation-dissipation relation, whereas displacement does not alter the fluctuation kernel in that way (Cavina et al., 2024).

The thermodynamic interpretation is correspondingly modified. The work invested in preparing a non-equilibrium reservoir is

$$H_S=\frac{p^2}{2M}+V(q), \qquad H_R=\sum_n \left(\frac{p_n^2}{2m_n}+\frac{1}{2}m_n\omega_n^2 x_n^2\right),$$3

and the total reservoir energy change, identified as heat, is

$$H_S=\frac{p^2}{2M}+V(q), \qquad H_R=\sum_n \left(\frac{p_n^2}{2m_n}+\frac{1}{2}m_n\omega_n^2 x_n^2\right),$$4

The paper’s stated conclusion is that the apparent breakdown of equilibrium thermodynamic relations is reconciled with the second law once the energy spent to generate the squeezed or displaced initial state is included (Cavina et al., 2024).

5. Spectroscopy, generalized Langevin descriptions, and the “noise-driven” interpretation

In condensed-phase spectroscopy, the expression “driven Caldeira-Leggett model” is used in a broader sense. One detailed study states that the term is not a separate formalism so much as the open-system, dissipative, noise-driven dynamics implied by the Caldeira-Leggett Hamiltonian and its generalized Langevin reduction. In that reduction the system momentum obeys

$$H_S=\frac{p^2}{2M}+V(q), \qquad H_R=\sum_n \left(\frac{p_n^2}{2m_n}+\frac{1}{2}m_n\omega_n^2 x_n^2\right),$$5

with memory kernel

$$H_S=\frac{p^2}{2M}+V(q), \qquad H_R=\sum_n \left(\frac{p_n^2}{2m_n}+\frac{1}{2}m_n\omega_n^2 x_n^2\right),$$6

and noise satisfying

$$H_S=\frac{p^2}{2M}+V(q), \qquad H_R=\sum_n \left(\frac{p_n^2}{2m_n}+\frac{1}{2}m_n\omega_n^2 x_n^2\right),$$7

The bath thus appears through deterministic forces, non-Markovian friction, and stochastic fluctuations consistent with thermal equilibrium (Gottwald et al., 2016).

For anharmonic system potentials, parameterization from molecular dynamics requires both the momentum autocorrelation function

$$H_S=\frac{p^2}{2M}+V(q), \qquad H_R=\sum_n \left(\frac{p_n^2}{2m_n}+\frac{1}{2}m_n\omega_n^2 x_n^2\right),$$8

and the momentum-force correlation function

$$H_S=\frac{p^2}{2M}+V(q), \qquad H_R=\sum_n \left(\frac{p_n^2}{2m_n}+\frac{1}{2}m_n\omega_n^2 x_n^2\right),$$9

linked by the Volterra equation

$$H_{\rm int}=-q\sum_n c_n x_n+q^2\sum_n \frac{c_n^2}{2m_n\omega_n^2},$$0

The extracted spectral density is

$$H_{\rm int}=-q\sum_n c_n x_n+q^2\sum_n \frac{c_n^2}{2m_n\omega_n^2},$$1

This framework is technically attractive for vibrational spectroscopy, but its applicability is limited by the invertibility problem: in general there is no one-to-one mapping between a real many-body system and a unique Caldeira-Leggett or generalized Langevin representation (Gottwald et al., 2016, Ivanov et al., 2014).

The sharpest statement of this limitation is that the mapping can be established self-consistently only when the system part is effectively harmonic. For a harmonic effective potential,

$$H_{\rm int}=-q\sum_n c_n x_n+q^2\sum_n \frac{c_n^2}{2m_n\omega_n^2},$$2

one has

$$H_{\rm int}=-q\sum_n c_n x_n+q^2\sum_n \frac{c_n^2}{2m_n\omega_n^2},$$3

so the two correlations are not independent. For anharmonic systems they remain genuinely independent, and compressing them into a single memory kernel loses information (Ivanov et al., 2014).

The same literature proposes empirical criteria for when the reduced driven/open-system description is likely to work: Gaussianity of the fluctuating force distribution, independence of the extracted spectral density on the choice of system potential, and, to a lesser extent, linearity of the system-bath coupling on the system side. A further stated implication is that an effectively harmonic mapping may reproduce linear spectra but cannot faithfully represent nonlinear spectroscopic dynamics, because higher-order response functions vanish for a purely harmonic system (Gottwald et al., 2016, Ivanov et al., 2014).

6. Related nonequilibrium bath engineering: damped baths and thermal gradients

Some extensions modify the environment itself rather than applying an explicit periodic drive. One example replaces the ordinary bath oscillators by damped Caldirola-Kanai oscillators,

$$H_{\rm int}=-q\sum_n c_n x_n+q^2\sum_n \frac{c_n^2}{2m_n\omega_n^2},$$4

which leads, in the high-temperature Ohmic limit, to the master equation

$$H_{\rm int}=-q\sum_n c_n x_n+q^2\sum_n \frac{c_n^2}{2m_n\omega_n^2},$$5

with modified potential

$$H_{\rm int}=-q\sum_n c_n x_n+q^2\sum_n \frac{c_n^2}{2m_n\omega_n^2},$$6

The new term is an inverted harmonic oscillator contribution induced by damping of the bath oscillators. In the double-well example reported in that work, the modification changes both short-time decoherence and long-time well-transfer probability, and a rescaling

$$H_{\rm int}=-q\sum_n c_n x_n+q^2\sum_n \frac{c_n^2}{2m_n\omega_n^2},$$7

keeps $$H_{\rm int}=-q\sum_n c_n x_n+q^2\sum_n \frac{c_n^2}{2m_n\omega_n^2},$$8 fixed and preserves the full potential while changing bath damping (Buxton et al., 2023).

A different nonequilibrium extension addresses thermal gradients. In one construction, bath oscillators are driven by an external force proportional to the local gradient,

$$H_{\rm int}=-q\sum_n c_n x_n+q^2\sum_n \frac{c_n^2}{2m_n\omega_n^2},$$9

leading, in the Ohmic limit and for $$J(\omega)=\frac{\pi}{2}\sum_n \frac{c_n^2}{m_n\omega_n}\delta(\omega-\omega_n).$$0, to

$$J(\omega)=\frac{\pi}{2}\sum_n \frac{c_n^2}{m_n\omega_n}\delta(\omega-\omega_n).$$1

with

$$J(\omega)=\frac{\pi}{2}\sum_n \frac{c_n^2}{m_n\omega_n}\delta(\omega-\omega_n).$$2

This model produces an explicit thermophoretic force $$J(\omega)=\frac{\pi}{2}\sum_n \frac{c_n^2}{m_n\omega_n}\delta(\omega-\omega_n).$$3 and, for constant gradient, admits a clean Hamiltonian formulation. A second construction replaces the single bath by a continuum of local baths at temperatures $$J(\omega)=\frac{\pi}{2}\sum_n \frac{c_n^2}{m_n\omega_n}\delta(\omega-\omega_n).$$4; in that case the gradient enters primarily through a space-dependent diffusion coefficient. Both models yield transport toward colder regions (Valente et al., 27 Mar 2026).

These variants are not “driven” in exactly the same sense as periodic forcing, but they show that within the Caldeira-Leggett logic environmental control parameters can enter through bath dynamics, bath statistics, or spatial bath structure.

7. Conceptual limits, non-Markovianity, and baseline equilibrium behavior

The driven Caldeira-Leggett model inherits the conceptual limitations of the undriven framework. The high-temperature Caldeira-Leggett master equation is historically central, but it is not exact and, as emphasized in the historical review, can violate complete positivity at low temperatures and short times. For a harmonic oscillator linearly coupled to a bosonic bath, the exact reduced dynamics is instead given by the Hu-Paz-Zhang master equation (Bonança et al., 29 Apr 2026).

A stronger claim has been advanced for the microscopic position-position model: dissipation is a genuinely non-Markovian feature. In that analysis, the standard Caldeira-Leggett master equation

$$J(\omega)=\frac{\pi}{2}\sum_n \frac{c_n^2}{m_n\omega_n}\delta(\omega-\omega_n).$$5

should be interpreted as an approximation to non-Markovian dynamics rather than as a true dissipative Markovian generator. In the strict Markov limit, a position-coupled bath yields only decoherence,

$$J(\omega)=\frac{\pi}{2}\sum_n \frac{c_n^2}{m_n\omega_n}\delta(\omega-\omega_n).$$6

while the friction term originates from the imaginary part of the bath correlation function and therefore from bath memory (Ferialdi, 2017).

For driven settings, this point is especially consequential. A plausible implication is that effective friction-like responses generated by bath driving, delayed forcing, or non-equilibrium reservoir engineering should be understood against a non-Markovian background rather than as purely local-in-time damping. The solvable equilibrium oscillator provides the reference case: an initially pure system coupled to a thermal bath approaches a thermal Gaussian state, its two-time correlation function tends to the thermal oscillator correlator, and late-time imaginary-time periodicity emerges only after thermalization (Ayyar et al., 2012).

Taken together, these results place the driven Caldeira-Leggett model in a precise niche. It is a flexible microscopic framework for driven dissipative quantum dynamics, response theory, decoherence, and engineered reservoir physics, but its predictive status depends on the bath model, the initial state, the presence or absence of memory, and the validity of the harmonic-bath reduction for the problem at hand.