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The Journal of Physical Chemistry Letters

Quantifying Early Time Quantum Decoherence Dynamics through Fluctuations

Bing Gu, Ignacio Franco

Published Aug 19, 2017• 43 citations
Publisher

Abstract

We introduce a general but simple relation between the timescale for quantum coherence loss and the initial fluctuations of operators that couple a quantum system with a surrounding bath. The relation allows the prediction and measurement of early time decoherence dynamics for any open quantum system, through purity, without reconstructing the system's many-body density matrix. It is applied to predict the decoherence time for basic models-the Holstein chain, spin-boson and Caldeira-Legget models-commonly employed to capture electronic, vibrational, and vibronic dynamics in molecules. Such development also offers a practical platform to test the ability of approximate quantum dynamics methods to capture decoherence. In particular, a class of mixed quantum-classical schemes for molecular dynamics where the bath is treated classically, such as Ehrenfest dynamics, are shown to correctly capture short-time decoherence when the initial conditions are sampled from the Wigner distribution. These advances provide a useful platform to develop decoherence times for molecular processes and to test approximate molecular dynamics methods.

DOI
10.1021/acs.jpclett.7b01817

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We introduce a general but simple relation between the timescale for quantum coherence loss and the initial fluctuations of operators that couple a quantum system with a surrounding bath. The relation allows the prediction and measurement of early time decoherence dynamics for any open quantum system, through purity, without reconstructing the system's many-body density matrix. It is applied to predict the decoherence time for basic models-the Holstein chain, spin-boson and Caldeira-Legget models-commonly employed to capture electronic, vibrational, and vibronic dynamics in molecules. Such development also offers a practical platform to test the ability of approximate quantum dynamics methods to capture decoherence. In particular, a class of mixed quantum-classical schemes for molecular dynamics where the bath is treated classically, such as Ehrenfest dynamics, are shown to correctly capture short-time decoherence when the initial conditions are sampled from the Wigner distribution. These advances provide a useful platform to develop decoherence times for molecular processes and to test approximate molecular dynamics methods.
Electronic decoherence processes in molecules and materials are usually thought and modeled via schemes for the system-bath evolution in which the bath is treated either implicitly or approximately. Here we present computations of the electronic decoherence dynamics of a model many-body molecular system described by the Su-Schrieffer-Heeger Hamiltonian with Hubbard electron-electron interactions using an exact method in which both electronic and nuclear degrees of freedom are taken into account explicitly and fully quantum mechanically. To represent the electron-nuclear Hamiltonian in matrix form and propagate the dynamics, the computations employ the Jordan-Wigner transformation for the fermionic creation/annihilation operators and the discrete variable representation for the nuclear operators. The simulations offer a standard for electronic decoherence that can be used to test approximations. They also provide a useful platform to answer fundamental questions about electronic decoherence that cannot be addressed through approximate or implicit schemes. Specifically, through simulations, we isolate basic mechanisms for electronic coherence loss and demonstrate that electronic decoherence is possible even for one-dimensional nuclear bath. Furthermore, we show that (i) decreasing the mass of the bath generally leads to faster electronic decoherence; (ii) electron-electron interactions strongly affect the electronic decoherence when the electron-nuclear dynamics is not pure-dephasing; (iii) classical bath models with initial conditions sampled from the Wigner distribution accurately capture the short-time electronic decoherence dynamics; (iv) model separable initial superpositions often used to understand decoherence after photoexcitation are only relevant in experiments that employ delta-like laser pulses to initiate the dynamics. These insights can be employed to interpret and properly model coherence phenomena in molecules.
We introduce a general theory of electronic decoherence for molecules in the condensed phase that captures contributions coming from pure dephasing effects, electronic transitions among diabatic states, and their interference. The theory is constructed by taking advantage of a recently developed [ J. Phys. Chem. Lett. 2017 , 8 , 4289 - 4294 ] general expression for decoherence times that is based on an early time expansion of the purity dynamics and extends early electronic decoherence models based on pure dephasing ideas. Using this theory, we construct the decoherence time for the displaced harmonic oscillator model amended with constant and linear diabatic couplings, which is a widely used model of the photoexcited dynamics of molecules. The validity of the short-time expansion is demonstrated by the quantitative agreement of the theory with exact numerical computations of the decoherence dynamics obtained using the hierarchical equation of motion method. These developments offer a rigorous understanding of early time electronic decoherence processes that accompany basic molecular events and demonstrate that electronic transitions among diabatic states play a major role in the decoherence dynamics.
In this work, we examine how the structure of system-bath interactions can determine commonly encountered temporal decoherence patterns, such as Gaussian and exponential decay, in molecular and other qubits coupled to a thermal bosonic bath. The analysis, based on a pure dephasing picture that admits analytical treatment, shows that decoherence, in general, is neither purely Gaussian nor exponential but rather the exponential of oscillatory functions, with periods determined by the bath's frequencies. For initially unentangled qubit-bath states, Gaussian decay is always present at early times. It becomes increasingly dominant with increasing temperature, qubit-bath interaction strength, and bath correlation time. Initial system-bath entanglement that arises due to displacement in the position of the bath states preserves the Gaussian decay. By contrast, strict exponential decay arises only in very specific models that we isolate. However, it becomes dominant for times longer than the bath correlation time or for early times when there is initial entanglement due to momentum displacement of the bath states. For molecular electronic decoherence, the long-time exponential regime plays a limited role as it emerges after most coherence is lost. Thus, the Gaussian decay provides a more suitable (albeit imperfect) model of such decoherence. Furthermore, we discuss the connection between electronic decoherence dynamics and electronic spectroscopic line shape theory, where Gaussian spectral peaks correspond to Gaussian coherence decay and Lorentzian peaks correspond to exponential coherence decay. We find that Gaussian spectral peaks, usually associated with inhomogeneous broadening, can emerge from the entangling unitary system-bath dynamics even when there is no inhomogeneity in the initial conditions.
A relationship between the time scales of quantum coherence loss and short-time solvent response for a solute/bath system is derived for a Gaussian wave packet approximation for the bath. Decoherence and solvent response times are shown to be directly proportional to each other, with the proportionality coefficient given by the ratio of the thermal energy fluctuations to the fluctuations in the system-bath coupling. The relationship allows the prediction of decoherence times for condensed phase chemical systems from well-developed experimental methods.
We discuss the evolution of purity in mixed quantum/classical approaches to electronic nonadiabatic dynamics in the context of the Ehrenfest model. As it is impossible to exactly determine initial conditions for a realistic system, we choose to work in the statistical Ehrenfest formalism that we introduced in Alonso et al. [J. Phys. A: Math. Theor. 44, 396004 (2011)]. From it, we develop a new framework to determine exactly the change in the purity of the quantum subsystem along with the evolution of a statistical Ehrenfest system. In a simple case, we verify how and to which extent Ehrenfest statistical dynamics makes a system with more than one classical trajectory, and an initial quantum pure state become a quantum mixed one. We prove this numerically showing how the evolution of purity depends on time, on the dimension of the quantum state space D, and on the number of classical trajectories N of the initial distribution. The results in this work open new perspectives for studying decoherence with Ehrenfest dynamics.