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 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.

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

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). The Hamiltonian. However, this does not imply that the first-order derivative can be obtained by setting t = 0 in eq 2 decoherence time for system and bath are identical, as in eq d 1 7 we have supposed that only the system is pure. In the ρρ̃()tH= [ , (0)]. The second-order derivative is dt t=0 iℏ particular case where both system and bath are initially pure, ff obtained by di erentiating eq 2 with respect to time and setting then τd for system and bath coincide in agreement with the t = 0, Schmidt theorem.28

4290 DOI: 10.1021/acs.jpclett.7b01817 J. Phys. Chem. Lett. 2017, 8, 4289−4294 The Journal of Physical Chemistry Letters Letter

At a qualitative level, eq 7 demonstrates that the initial given by H ϵ0 1 qcxHwhere ϵ and fl =+2 σσzxΔ + 2 0 σ z∑μ μμ+ 0 system and bath state controls the decoherence by in uencing ffi fl Δ are system coe cients, q0 is the system-bath interaction the magnitude of the uctuations in the operators that enter 1 fl strength, σi (i = x, y, z) are the spin- Pauli matrices expressed into H. The larger the crossed uctuations, the faster the 2 decoherence. This essential feature of eq 7 is best appreciated in eigenbasis of σz (σz |±⟩ = ±|±⟩) and ∑μcμxμ is a collective in the specific case where there is only one term contributing to spatial coordinate of the harmonic bath. The bath Hamiltonian fi 2 H = SB⊗ . In this case, eq 7 simpli es to ⎛ p ⎞ μ 1 22 is composed of a collection of H = ∑μ ⎜ + mxμμμω ⎟ 22− 1/2 ⎝ 2mμ 2 ⎠ τd = ℏ(2δδBS ) (8) independent oscillators with frequencies {ωμ}, displacements that directly signals an inverse relation between decoherence {xμ}, momenta {pμ} and associated masses {mμ}. Generally, the time and the magnitude of initial-time physical fluctuations of dynamics of this model is not pure dephasing in nature as the system and bath operators, δ2B ≡ ⟨B2⟩ − ⟨B2⟩ and δ2S ≡ 1 [,]HH 0.IdentifyingB = ∑μcμxμ and , 2 2 ≠ S = 2 q0σz ⟨S ⟩ − ⟨S ⟩, that enter into H. 2 1 −βH Importantly, eq 7 makes the decay of purity at early times an δ Bccexx= ∑ μνTr [ μν] for a bath initially at thermal μν Zβ experimentally and computationally accessible quantity. This is equilibrium of inverse temperature β = 1/kBT and partition because it avoids reconstructing the full many-body density −βH function Zβ = Tr[e ], where we have used ⟨xμ⟩ = 0. The matrix of the system to determine the decoherence dynamics. 31 above expression can be evaluated by taking into account that Instead, it requires measuring Δαβ and Δαβ which are few-body −βHμ quantities accessible in simulations and experiments. The ⟨xeμ|| xμ⟩ fl 1/2 2 crossed uctuations Δ and Δ are determined at initial time ⎛ mmxω ⎞ ⎛ ω ⎞ αβ αβ ⎜ μμ ⎟ ⎜ μμμ ⎟ = ⎜ ⎟ exp⎜− (cosh(βωℏ−μ ) 1)⎟ 2sinh()πβωℏℏ ℏℏsinh(βω ) and require no dynamical propagation. Further, Δαβ (or Δαβ) ⎝ μ ⎠ ⎝ μ ⎠ depends only on the system (or bath) and, thus, to determine 2 c τ it is sufficient to analyze properties of the isolated system and to yield 2B μ coth( /2). In the continuous d δ = ∑μ 2m βωℏ μ bath. Experimentally determining the decoherence time μμω limit, the variance is given by an integral form δ2B = ∫ J(ω) requires identifying the relevant interaction between system c2 and bath, and then measuring the associated fluctuations. coth(βℏω/2) dω, where μ is the J()ωδωω≡ ∑μ (− μ ) A convenient feature of the analysis is that the effects of 2mμμω |ψ independent baths to the decoherence are just additive in τ−2. spectral density. For a general initial state of the system ⟩ = d | | δ2 2| |2| |2 This is because the crossed fluctuations between operators from c+ +⟩ + c− −⟩, S = q0 c+ c− , and different baths vanish. Thus, it is straightforward to consider the ()(2()coth(/2))qc c−−11/2 J d cumulative effect of competing decoherence sources, e.g., the τd = ℏ 0||||+ − ∫ ωβωωℏ (10) effect of solvent and vibrations to electronic decoherence in which is consistent with the results in ref 26 and the basis for an molecules. For a system interacting with multiple baths often-used expression for the electronic decoherence time. , the total purity function is given as a product kN=···1, , As a second example, consider now the Caldeira-Legget N model for quantum Brownian motion in which a harmonic ()k 2 ()k −2 ()ttt==∏ () exp(− ∑ (τd ) ) oscillator with coordinate X couples with harmonic bath k=1 k (9) through . This model is often employed to H = Xcx∑μ μμ ()k capture vibrational dynamics of molecules in condensed phase where ()t is the purity decay due to bath k with 32,33 (k) environments. In this case, we identify S = X and B = decoherence timescale τd . While the early time Gaussian decay of purity does not ∑μcμxμ is identical to the spin-boson case. Thus, necessarily quantify the complete decoherence dynamics of 21/2− 21 (2 ( ) coth( /2)d ) molecular systems, experiments in spin systems, and τd = ℏℏδωβωωXJ∫ (11) computations for electronic22−24 and vibrational25 decoherence in agreement with ref 20. have identified the Gaussian decay to be a dominant event in As a third model, we now focus on the Holstein Hamiltonian the decoherence dynamics in condensed phase environments. 34 1 for molecular crystals. This is a basic model for vibronic In an analysis of a pure-dephasing spin-boson problem, the interactions in molecules and solids.35,36 Adecoherence Gaussian decay is seen to dominate for times t < Ω−1, where Ω timescale due to electron-vibrational interactions has not is the cutoff frequency of the ohmic harmonic bath. In the long been developed for this model. The Hamiltonian consists of time limit, the bath induces an exponential decay of coherence. an electronic tight-binding chain, where each site μ in the chain Thus, the Gaussian decay is expected to be important for non- couples to an independent harmonic oscillator of identical mass Markovian baths as is often the case in chemical dynamics. M and frequency ω . The system−bath interaction in this case Equation 7 makes it straightforward to calculate decoherence 0 is , where x is the μth oscillator timescales for any system-bath model relevant in chemistry in H = −gM2 ω0 ∑μ nxμμ μ the short time regime, and to recover expressions that have displacement, g is the electron−phonon coupling strength, and † † been developed for specific models. For example, consider first nμ = aμaμ (where aμ creates an electron in site μ) is the the well-known spin-boson model describing a two-level system Fermionic number operator in the site basis. Due to the coupled to a harmonic bath.26,29 This model has been widely additive properties of the decoherence time for independent used to study electronic decoherence in condensed phase baths, it suffices to consider the interaction with a single environments,26 and electronic excitations of biomolecules and harmonic mode μ and then add over each of these quantum dots in solvents.30 The Hamiltonian for this model is contributions. For a thermal bath this yields

4291 DOI: 10.1021/acs.jpclett.7b01817 J. Phys. Chem. Lett. 2017, 8, 4289−4294 The Journal of Physical Chemistry Letters Letter

τ−22= 2coth(/2)(gnnβωℏ⟨⟩−⟨⟩22 ) supposed to be pure at initial time, the initial density matrix for d 0 ∑ μμ σ(k) σ μ (12) all trajectories coincide, i.e., (0) = (0). To proceed, we now perform a short time analysis of the which characterizes the temperature dependence of the equations of motion in the interaction picture of H. To decoherence on this many-body molecular system. Equation second-order in time, 12 indicates that for temperatures T < ℏω /2k the 0 B ()k it ()k decoherence is independent of temperature and determined σσ̃ ()t = (0)− [H , σ (0)] fl ℏ by zero-point uctuations of the bath. For T > ℏω0/2kB t 2 ⎛⎡ d ⎤ ⎞ temperature-dependent effects arise because the excited states ⎜ i HHH̃ ()k ,(0) [()kk ,[ () ,(0)]]⎟ − 2 ⎣⎢ ℏ σσ⎦⎥ + of the bath are thermally occupied and lead to additional 2ℏ ⎝ dt t=0 ⎠ sources of decoherence. In fact, the decoherence time decreases We consider interaction forms in which the bath couples to the monotonically with increasing T because of the increased system through its coordinates, i.e., where fl H = ∑α fS()R α uctuations of the xμ in the harmonic bath. At T ≫ ℏω0/2kB, α Sα is a system operator and fα(R) is a general function of the the decoherence time decreases like τd ∼ 1/√T. Further, eq 12 reveals that electronic states for which there is larger fluctuation bath position coordinates. Taking the ensemble average over in the site occupation numbers will exhibit faster decoherence. trajectories and using Interestingly, it also determines the dependence of the fl d decoherence with chain length. If the uctuations of the site H̃ occupations are comparable among sites, τ 1/ N, where N dt d ∼ √ t=0 is the number of sites. This decrease in the decoherence time ̇ with system size N is a consequence of the product properties = ∑ ((0)RR·+∇fSifSHα [(0)]α (1/)[(0)][,ℏ α Rα ]) of the purity. α This analysis can also be used to test the accuracy of we get an equation for the ensemble-averaged reduced approximate quantum dynamics methods to capture system- density matrix analogous to eq 3, bath entanglement. This is useful because, for realistic system- (1) 1 2 (2) σσ∼ =+(/)−ℏit σ+ (/)−ℏ it σ. Here bath problems, exact reference full quantum solutions are often 0 cl 2 cl not available to validate approximate methods. An alternative, (1) σσcl = ∑ fSα [,(0)],α in this context, is to perform a short-time analysis of the α approximate equations of motion and then compare with the (2) exact result in eq 7. This strategy can be used when the σσσcl = ∑ ffαβ([ SSαβ , (0)]+ − 2 S α (0) S β ) equations of motion of the approximate method directly αβ, follows σ(t). ̇ + ∑ ([,(0)][[,],(0)])ifSℏR·+∇ α α σσ fSHα α As a specific case of significant practical importance, consider α quantum molecular dynamic simulations in the condensed (13) phase. A commonly used approach is mixed quantum-classical 37,38 where the bar denotes an ensemble average over a classical methods where the dynamics of the bath is treated c distribution of initial conditions ρ (,RP ). classically by propagating an ensemble of trajectories, while The purity can be calculated in a fashion analogous to eqs the system is described using quantum mechanics. An 4−7, to give important, and currently open, question is whether this class of simulations capture decoherence correctly. cl −1/2 τd,MQC = ℏΔΔ(2∑ αβ αβ ) As a third contribution of this Letter, below we demonstrate αβ (14) that a class of quantum-classical schemes, that includes Ehrenfest dynamics, can offer the correct short-time decoher- where cl . Comparing eq 7 Δαβ ≡−ffαβ()RR () f α () R f β () R ence dynamics. This is so provided that the initial classical with eq 14, it becomes evident that the mixed quantum-classical fi distribution for the quantum-classical trajectories satis es the scheme offers the correct short-time purity dynamics provided fl same crossed uctuations as the true quantum state that it seeks that the initial classical distribution has crossed fluctuations to represent, and that the system couples to the bath through identical to the quantum ones, i.e., when cl . For functions of position or momentum only. Δαβ = Δαβ c To see this, consider a generic quantum-classical method example, if ρ(,RP )is chosen to be the Wigner distribution where the full quantum dynamics is represented by an τd, MQC = τd. More generally, any initial classical distribution ensemble of quantum-classicaltrajectories.Theclassical whose marginals correctly capture the quantum probability dynamics of the kth trajectory for the bath is deterministic distribution in R,i.e.,distributionssuchthat (k) −1 (k) and given by Newton’s equations of motion Ṙ = M P , Ṗ = dPRPRRc (, ) , like the one employed in Bohmian (k) (k) ∫ ρρ= ⟨ ||⟩ −∇V(R ), where R is a vector of the positions of all dynamics,39,40 will suffice. This remarkable result is valid for any particles in the bath with conjugate momentum P(k), V(R(k)) is ff type of bath and system−bath interactions of the form HSB = some e ective potential, and M is a diagonal matrix with the ∑ f (R)S , including anharmonic baths and nonlinear system- masses in the diagonal. In turn, the quantum system responds n n n fi (k) bath couplings. instantaneously to the classical motion and satis es iℏσ̇ (t)= System-bath couplings through positions are adequate to (k) (k) ()kk() [H(R (t)), σ (t)], where H(())RRtHH=+ SB (()) t. describe electrostatic system−bath interactions, such as those of The reduced density matrix for the system is obtained by a molecule in the condensed phase. By symmetry, the quantum- (k) ff averaging over trajectories, σ (t)=∑k σ (t)/Ntraj, where Ntraj classical scheme will also o er the correct purity dynamics for is the total number of trajectories. Because the system is provided that the distribution of classical H = ∑α fSα ()P α

4292 DOI: 10.1021/acs.jpclett.7b01817 J. Phys. Chem. Lett. 2017, 8, 4289−4294 The Journal of Physical Chemistry Letters Letter initial conditions mimics the correct momentum distribution of Notes the quantum state of the bath. These two cases cover system- The authors declare no competing financial interest. bath couplings encountered in problems of chemical interest. For even more general system-bath couplings of the form ■ ACKNOWLEDGMENTS H = ∑ fS(,RP ) the mixed quantum-classical scheme α α α This material is based upon work supported by the National will recover the decoherence dynamics if the distribution of Science Foundation under CHE-1553939. initial conditions correctly captures all the crossed fluctuations between the {fα(R, P)}. This condition is not usually satisfied by the Wigner distribution because the classical distribution ■ REFERENCES function cannot reflect the commutation relation between (1) Breuer, H.-P.; Petruccione, F. 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4293 DOI: 10.1021/acs.jpclett.7b01817 J. Phys. Chem. Lett. 2017, 8, 4289−4294 The Journal of Physical Chemistry Letters Letter

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4294 DOI: 10.1021/acs.jpclett.7b01817 J. Phys. Chem. Lett. 2017, 8, 4289−4294