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Quantifying Early Time Quantum Decoherence Dynamics Through Letter pubs.acs.org/JPCL Quantifying Early Time Quantum Decoherence Dynamics through Fluctuations Bing Gu† and Ignacio Franco*,†,‡ †Department of Chemistry and ‡Department of Physics, University of Rochester, Rochester, New York 14627, United States 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 modelsthe Holstein chain, spin- boson and Caldeira-Legget modelscommonly 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. he loss of quantum coherence is a fundamental and eigenstate with no net dipole will exhibit zero polarization. T ubiquitous process in nature that occurs in any quantum Other commonly used observables such as transport, line- system coupled to an environment, e.g., electrons coupled to shapes, and interference can only offer a basis-dependent vibrations, or vibrations in the presence of solvent. Under- perspective on the decoherence dynamics that is not necessarily standing decoherence is central to our description of a variety informative of state purity.13−16 Ongoing efforts2,17−19 to of quantum mechanical phenomena that play an important role understand the role of quantum decoherence in molecules and in understanding molecular processes, including interference, materials are currently limited by our ability to directly monitor adiabatic and nonadiabatic molecular dynamics, measurement, state purity. and the quantum-classical transition.1−6 During decoherence, a In this Letter, we introduce a general theory that relates the system changes from a pure state σ = |ψ⟩⟨ψ| to a mixed state σ short-time purity dynamics to the initial-time fluctuations of = ∑iwi |ψi⟩⟨ψi| (ωi > 0). Such a process reduces the ability of a system to fully exhibit its quantum mechanical features. operators that couple a system with a surrounding bath. This In spite of its central role in quantum mechanical processes, theory is based on a perturbative expansion of purity that is determining decoherence timescales remains an open challenge useful in capturing the initial decoherence dynamics for systems because purity and other well-defined basis-independent in interaction with non-Markovian baths. The resulting measures of decoherence7 are based on the full many-body expression permits quantifying short-time decoherence time- density matrix of the system σ(t), which is an experimentally scales through experimentally and computationally accessible and theoretically removed quantity.8,9 Experimental and quantities. It also opens the way to establish rigorous numerical efforts to monitor decoherence dynamics are thus decoherence times for basic chemical processes, and to test forced to do so indirectly via other physical observables that the ability of approximate quantum dynamics methods to reflect off-diagonal elements of σ(t) expressed in a given basis, 2 correctly capture decoherence. The considerations below apply but that are not necessarily indicative of state purity. For to any system-bath Hamiltonian with an initially pure system. instance, in spectroscopic experiments, the decay of laser- To proceed, consider a general quantum system () induced polarization is often used as a measure of decoherence. coupled to an environment (), with Hamiltonian H = The polarization informs about the decay of off-diagonal elements in the system’s density matrix expressed in the energy H++HHwhere H describes the system, H the bath, eigenbasis.10−12 However, as a measure of decoherence, such an and approach is limited by the fact that polarization only reflects coherences between states with nonzero transition dipoles. Received: July 13, 2017 Further, the absence of polarization does not necessarily imply Accepted: August 19, 2017 decoherence as, for instance, a pure state prepared in an energy Published: August 19, 2017 © XXXX American Chemical Society 4289 DOI: 10.1021/acs.jpclett.7b01817 J. Phys. Chem. Lett. 2017, 8, 4289−4294 The Journal of Physical Chemistry Letters Letter 2 H = SB⊗ d 1 1d⎡ ⎤ ∑ α α ρρρ̃()t = [,[,(0)]]HH + ⎢ Ht ̃ () , (0)⎥ (1) dt 2 ()iℏ 2 itℏ ⎣d ⎦ α t=0 t=0 the system−bath interactions. Here Sα (or Bα) is an operator Inserting these two expressions into ρ̃(t) and tracing out the defined in the Hilbert space of (or ). The dynamics of the bath, one obtains ρ composite system with density matrix is governed by the von (1)1 2 (2) d σσ()̃ titit=+ (0) (−ℏ / ) σ+ (/)−ℏ σ Neumann equation i [,]HHH and gener- (3) ℏdt ρρρρ==− 2 ally leads to entanglement between and , and thus to where (1) Tr {[H , (0)]} and decoherence. σρ= fi (2) ⎡ d ⎤ As a well-de ned basis-independent measure of decoherence, σρ=+Tr [H , [H , (0)]]⎢ iℏ H̃ (t ) , ρ (0)⎥ . Such ⎣ dt t=0 ⎦ we focus on the purity () Tr{2 ()} 1which measures { } tt= σ ≤ σ̃(t) can be used to calculate the purity at short times the nonidempotency of the reduced density matrix of the 2(2)(1)2 system of interest σρ()tt= Tr{ ()}, where TrA denotes a ()tit=+ 1 (−ℏ / )Tr{ σσ (0)+ ( σ )} (4) partial trace over the degrees of freedom of . The A = {, } where we have taken into account and that the first- (0)= 1 quantity = 1 for pure states, and < 1 for mixed states. To (1) define a decoherence time, we assume that the system is order contribution vanishes as Tr[ σσ (0) ]= 0 due to the invariance of the trace under cyclic permutation. Equation 4 initially in a pure state. This implies that (0)= 1 and that the 20 initial system-bath state is not entangled, such that recovers the well-known Gaussian decay for the initial purity 2 2 dynamics ()tt= exp(− /τd )where ρ(0)= σρ (0)⊗ (0) where ρ()t is the reduced density matrix for the bath. (2) (1) 2− 1/2 τd = ℏTr [σσ (0)+ ( σ ) ] (5) To capture the short-time purity dynamics, it is natural to is the decoherence time, that has been observed in spin,21 expand the von Neumann equation around initial time t = 0 in 22−24 25 Schrödinger picture as done in ref 20. That procedure leads to a electronic and vibrational decoherence. A remarkably simple expression for τd results by taking formal expression for the decoherence time τd that, while advantage of the form for H in eq 1. Introducing eq 1 into general, is of limited utility because it depends on the many- (1) (2) body reduced density matrix σ and requires performing σ and σ yields operations with the full Hamiltonian H. Below we overcome (1) [,(0)]SB these limitations and derive a universal and simple formula for σσ= ∑ αα⟨⟩ α τd that clearly exposes the basic physics behind the early time decoherence dynamics. To do so, we exploit the general form (2) σσσ= ∑ ([SSαβ , (0)]+ −⟨⟩ 2 S β (0) S α ) BB α β for H in eq 1 and work in the interaction picture of H+ H αβ, to considerably reduce the complexity of the calculation and be + ∑∑[[SHαα ,S ],σσ (0)]⟨⟩ B+ [ Sαα , (0)]⟨⟩ [ BH , ] able to isolate the full consequences of the short time α α expansion. In the interaction picture, the von Neumann (6) equation is given by where [A, B]+ = AB + BA is the anticommutator. Using eq 6 in eq 5 and simplifying yields the decoherence time d i ()tH [̃ , ()] t ℏ ρρ̃ = ̃ (2) −1/2 dt τd = ℏΔ×Δ(2∑ αβ αβ ) αβ (7) where Õ = U†OU denotes the operator O in the interaction picture. In turn, U()tUtUt= ()⊗ () is the evolution where Δαβ≡⟨SS α β ⟩−⟨ S α ⟩⟨ S β ⟩and Δαβ= ⟨BB α β⟩ − ⟨BBαβ⟩⟨ ⟩ fl operator of the noninteracting (i.e., H = 0) composite are the crossed uctuations of the system and bath operator system, where UA(t) denotes the evolution operator for that enter into H. Surprisingly, the decoherence time only subcomponent A. depends on H, with no dependence on the system or bath To determine the purity, it suffices to isolate the reduced Hamiltonian. density matrix for the system in the interaction picture, i.e., Equation 7 is the first result of this Letter. It applies to any σρ̃ = Tr {̃ }. This is because the purity in the interaction and initially pure system provided the system-bath interaction can be written in the form of eq 1 and does not invoke common Schrodinger picture coincide, i.e., 22. ̈ ==Tr {σσ } Tr {̃ } approximations employed in open quantum system dynamics To capture the short-time purity dynamics, we perform a such as harmonic baths, pure dephasing dynamics and rotating- second‑order expansion in time of eq 2 around t =0 ff fi wave approximations. As such, it generalizes previous e orts to and then trace over the bath. Speci cally, develop decoherence times for specific models20,26 and pure d t 22d dephasing dynamics.1,27 Equation 7 is symmetric with respect ρ()tttt=+ρρ (0) ()+ 2 ρ () , where we have ̃ dt ̃ 2 dt ̃ fl t==0 t 0 to ↔ re ecting the inherent system-bath symmetry in the taken into account that at initial time Õ(0) = O(0).
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