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Annals of Physics

Exact classicalization of N-level systems interacting with a bath: Theory and applications

Daniel Martínez-Gil, Pedro Bargueño, Salvador Miret-Artés

Published May 15, 2026• 0 citations
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Abstract

In this manuscript, starting from a five-step algorithmic procedure for exactly classicalizing the dynamics of N -level quantum systems, we incorporate a classical bath of harmonic oscillators to model environmental interactions. Using the geometry of complex projective spaces CP N − 1 and a Langevin formalism, we obtain N − 1 Hamilton’s equations which encode both the quantum system and the bath degrees of freedom, representing a generalization of the Caldeira–Leggett model in a complex projective space. We demonstrate the efficacy of the method by applying it to two paradigmatic systems: a two-qubit system in CP 3 under entangling interactions, reproducing quantum observables such as state populations, quaternionic population differences and concurrence, and the seven-state Fenna–Matthews–Olson (FMO) complex in CP 6 , reproducing state populations in the picosecond timescale.

DOI
10.1016/j.aop.2026.170527

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In this manuscript, starting from a five-step algorithmic procedure for exactly classicalizing the dynamics of N -level quantum systems, we incorporate a classical bath of harmonic oscillators to model environmental interactions. Using the geometry of complex projective spaces CP N − 1 and a Langevin formalism, we obtain N − 1 Hamilton’s equations which encode both the quantum system and the bath degrees of freedom, representing a generalization of the Caldeira–Leggett model in a complex projective space. We demonstrate the efficacy of the method by applying it to two paradigmatic systems: a two-qubit system in CP 3 under entangling interactions, reproducing quantum observables such as state populations, quaternionic population differences and concurrence, and the seven-state Fenna–Matthews–Olson (FMO) complex in CP 6 , reproducing state populations in the picosecond timescale.
In this manuscript, starting from a five-step algorithmic procedure for exactly classicalizing the dynamics of N-level quantum systems, we incorporate a classical bath of harmonic oscillators to model environmental interactions. Using the geometry of complex projective spaces CP(N-1) and a Langevin formalism, we obtain N-1 Hamilton's equations which encode both the quantum system and the bath degrees of freedom, representing a generalization of the Caldeira-Legget model in a complex projective space. We demonstrate the efficacy of the method by applying it to two paradigmatic systems: a two-qubit system in CP(3) under entangling interactions, reproducing quantum observables such as state populations, quaternionic population differences and concurrence, and the seven-state Fenna-Matthews-Olson (FMO) complex in CP(6), reproducing state populations in the picosecond timescale.
In this manuscript, we present a general and exact method for classicalizing the dynamics of any N -level quantum system, transforming quantum evolution into a classical-like framework using the geometry of complex projective spaces CP N − 1 . The method can be expressed as five-step algorithmic procedure to derive a classical Hamiltonian and a symplectic structure for the Poisson brackets, yielding N − 1 Hamilton’s equations that precisely replicate the quantum dynamics, including complex phenomena like entanglement. We demonstrate the method’s efficacy by classicalizing two interacting qubits in CP 3 , exactly reproducing quantum observables such as quantum probabilities, population differences, and the concurrence, capturing entanglement dynamics via a classical analog.
Solving quantum dynamics is an exponentially difficult problem. Thus, an exact numerical solution is inaccessible for any condensed matter system.\nA promising approach is to divide the system into a quantum subsystem containing degrees of freedom which are of greater interest or those which have more profound quantum character (e.g., have smaller mass) and a classical bath containing the rest of the system.\nImposing such a partition and treating the bath classically results in quantum-classical dynamics. The quantum-classical Liouville equation is a general equation in the Hilbert space of quantum degrees of freedom while it resides in the phase space of the classical degrees of freedom.\nAny numerical solution to this equation requires representation of the quantum subsystem in some basis. Solutions to this equation have been already proposed in the subsystem, adiabatic and force bases, each with its own cons and pros.\n\nIn this work, the quantum-classical equations of motion are cast in the subsystem basis and subsequently mapped to a number of fictitious harmonic oscillators.\nThe result is quantum-classical dynamics in the mapping basis which treats both quantum and classical degrees of freedom on the same footing, i.e., in phase space. Neglecting a portion of the back reaction of the quantum-subsystem to classical bath results in an expression for the time evolution of an operator (density matrix) equal to its Poisson bracket with the Hamiltonian.\nThis equation can be solved in terms of characteristics to provide a computationally tractable method for calculating quantum-classical dynamical properties. The expressions for expectation values and correlation functions in this formalism are derived.\nCalculations on spin-boson system, barrier crossing models---the so called Tully models---and the Fenna-Mathews-Olson pigments show very good agreement between the results of this method and numerical solutions to the Schrödinger equation.
The Caldeira–Leggett (CL) model describes a microscopic quantum system, represented by a harmonic oscillator, in interaction with a heat bath, represented by a large number of harmonic oscillators with a range of frequencies. We consider the case when the system oscillator starts out in the ground state and then thermalizes due to interactions with the heat bath, which is at temperature θ. We calculate the position autocorrelation function 〈x(t')x(t)〉 of the system oscillator at two different times and study its behavior in the small and large time limits. Our results show that the system oscillator thermalizes as expected. We also confirm by explicit calculation that the position autocorrelation function exhibits periodicity for imaginary values of the time difference t'-t = iτ at late-(real) times t.
A recent promising arena for quantum advantage is simulating exponentially large classical systems. Here, we show how this advantage can be used to calculate the dynamics of open classical systems experiencing dissipation, including the effects of non-Markovian baths. This is a particularly interesting class of systems since dissipation plays a key role in contexts ranging from fluid dynamics to thermalization. We adopt the Caldeira-Leggett Hamiltonian, a generic model for dissipation in which the system is coupled to a bath of harmonic oscillators with a large number of degrees of freedom. To date, the most efficient classical algorithms for simulating such systems have a polynomial dependence on the size of the bath. In this work, we give a quantum algorithm with an exponential speedup, capable of simulating ddd system degrees of freedom coupled to N=2n≫dN = 2^n\gg dN=2n≫d bath degrees of freedom, to within error ε\varepsilonε, using O(poly(d,n,t,ε−1))O({\rm poly}(d, n, t, \varepsilon^{-1}))O(poly(d,n,t,ε−1)) quantum gates.
(CPS) has recently been developed to study nonadiabatic dynamics in gas-phase and condensed-phase molecular systems. Although the CPS formulation is exact for describing the discrete (electronic/vibrational/spin) state degrees of freedom (DOFs), when system-bath models in condensed phase are studied, previous works often employ the approximation by discretizing environmental bath DOFs. In this paper, we develop an exact trajectory-based phase space approach by adopting the twin-space (TS) formulation of quantum statistical mechanics, in which the density operator of the reduced system is transformed to the wave function of an expanded system with twice the DOFs. The classical mapping model (CMM) is then used to map the Hamiltonian of the expanded system to its equivalent classical counterpart on CPS. To demonstrate the applicability of the TS-CMM approach, we compare simulated population dynamics and nonlinear spectra for a few benchmark condensed phase system-bath models with those obtained from the hierarchical equations of motion method, which shows that our approach yields accurate dynamics of open quantum systems.