Relaxation and Dephasing in Open Quantum Systems Time-Dependent Density Functional Theory: Properties of Exact Functionals from an Exactly-Solvable Model System The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Tempel, David Gabriel, and Alán Aspuru-Guzik. 2011. Relaxation and dephasing in open quantum systems time-dependent density functional theory: Properties of exact functionals from an exactly- solvable model system. Chemical Physics 391(1): 130–142. Published Version doi:10.1016/j.chemphys.2011.03.014 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:8438167 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#OAP Relaxation and dephasing in open quantum systems time-dependent density functional theory: Properties of exact functionals from an exactly-solvable model system. David G. Tempel1 and Alán Aspuru-Guzik2 1Department of Physics, Harvard University, 17 Oxford Street, 02138, Cambridge, MA 2Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, 02138, Cambridge, MA∗ The dissipative dynamics of many-electron systems interacting with a thermal environment has remained a long-standing challenge within time-dependent density functional theory (TDDFT). Recently, the formal foun- dations of open quantum systems time-dependent density functional theory (OQS-TDDFT) within the master equation approach were established. It was proven that the exact time-dependent density of a many-electron open quantum system evolving under a master equation can be reproduced with a closed (unitarily evolving) and non-interacting Kohn-Sham system. This potentially offers a great advantage over previous approaches to OQS-TDDFT, since with suitable functionals one could obtain the dissipative open-systems dynamics by simply propagating a set of Kohn-Sham orbitals as in usual TDDFT. However, the properties and exact conditions of such open-systems functionals are largely unknown. In the present article, we examine a simple and exactly- solvable model open quantum system: one electron in a harmonic well evolving under the Lindblad master equation. We examine two different representitive limits of the Lindblad equation (relaxation and pure dephas- ing) and are able to deduce a number of properties of the exact OQS-TDDFT functional. Challenges associated with developing approximate functionals for many-electron open quantum systems are also discussed. PACS numbers: I. INTRODUCTION OQS Runge-Gross theorem was established for Markovian master equations of the Lindblad form. A scheme in which the many-body master equation is mapped onto a non-interacting Due to its attractive balance between accuracy and effi- Kohn-Sham master equation was proposed for application to ciency, time-dependent density functional theory (TDDFT) single-molecule transport. A Kohn-Sham master equation is an ideal method for computing the real-time dynamics of was also used in formulating the linear-response version of many-electron systems [1–16]. In its original formulation by OQS-TDDFT, giving access to linewidths of environmentally Runge and Gross [17], TDDFT addresses the isolated dynam- broadened spectra [28]. In [28], OQS Casida-type equations ics of electronic systems evolving unitarily under the time- were derived and used to calculate the spectrum of an atom in- dependent Schrödinger equation. However, there exist many teracting with a photon bath. A different formulation of OQS- situations in which the electronic degrees of freedom are not TDDFT based on the stochastic Schrodinger equation rather isolated, but must be treated as a subsystem imbedded in a than the master equation has also been developed [47–49]. much larger thermal bath. Several important examples in- clude vibrational relaxation of molecules in liquids and solid Recently, in [50, 51], the OQS Runge-Gross theorem was matrices [18–20], photo-absorption of chromophores in a pro- extended to arbitrary non-Markovian master equations and a tein bath [21–24], nonlinear spectroscopy in the condensed Van Leeuwen construction was established, thereby proving phase [25, 26], electron-phonon coupling in single-molecule the existence of an OQS-TDDFT Kohn-Sham scheme [52]. transport [27, 30–33] and exciton and energy transfer [35– We showed that the time-dependent density of an interact- 37, 39]. ing OQS can be reproduced with a non-interacting and closed In the above examples, the theory of open quantum systems (unitarily evolving) Kohn-Sham system. In principle, the (OQS) within the master equation approach is often used to closed Kohn-Sham scheme is remarkably useful for real-time model the dissipative electron dynamics [40–43, 45, 46]. In dynamics, since it allows one to calculate any property of a the master equation approach, one traces over the bath degrees many-body OQS by unitarily propagating a set of one-particle of freedom, arriving at a simpler description in terms of the orbital equations evolving in a local potential. With suitable reduced density matrix of the electrons only. The price paid functionals, such a scheme could readily be implemented in is that the resulting dynamics are non-unitary, and in general existing real-time TDDFT codes [1–3]. In practice, the OQS- the interaction of the system with it’s environment must be TDDFT exchange-correlation potential is very complicated modeled in an approximate way. Even with simple system- object. Not only does it have initial-state and memory de- bath models, the exact solution of the master equation for the pendence as in usual TDDFT [53–56], but it must also be a reduced dynamics of an interacting many-electron system is functional of bath quantities such as the bath spectral density. computationally intractable. Therefore, open quantum sys- In the present manuscript, we study exact features of the tems TDDFT (OQS-TDDFT) offers an attractive approach to OQS-TDDFT closed Kohn-Sham scheme using an exactly- the many-body open-systems problem. solvable one-electron model system. By focusing on a single Several different formulations of OQS-TDDFT have been electron, we are able to isolate the part of the exact functional proposed in the last few years [27–29, 47–51]. In [27], an arising solely from interaction with the bath, without the need 2 to describe electron-electron interaction effects within the sys- to the system. The density of states of HˆR determines the de- tem. cay rate of reservoir correlation functions, whose time-scale The paper is organized as follows. In Section II, we review in turn determines the reduced system dynamics. the theory of OQS for a many-electron system and the closed Defining the reduced density operator for the electronic sys- Kohn-Sham scheme presented in previous work [50, 51]. Sec- tem alone by tracing over the reservoir degrees of freedom, tion III presents the model system to be analyzed and dis- cusses the procedure used to obtain exact Kohn-Sham quan- tities. Section IV presents results and an analysis. The paper rˆS(t) = TrRfrˆ(t)g; (4) concludes with an outlook and discussion of challenges for OQS-TDDFT in Section V. Atomic units in which e = h¯ = one arrives at the formally-exact quantum master equation, me = 1 are used throughout. d Z t rˆ (t) = −ı[Hˆ (t);rˆ (t)] + dtX˘ (t − t)rˆ (t) + Y(t): dt S S S S II. OPEN QUANTUM SYSTEMS TDDFT USING UNITARY t0 PROPAGATION (5) Here, X˘ (t − t) is the memory kernel and Y(t) arises from initial correlations between the system and its environment. In this section, we begin by briefly reviewing the mas- It is referred to as the inhomogeneous term. The above equa- ter equation approach for many-body OQS. We first discuss tion is still formally exact, as rˆ (t) gives the exact expectation the most general master equation including non-Markovian S value of any observable depending only on the electronic de- effects and initial correlations [62]. We then introduce the grees of freedom. In practice, however, approximations to X˘ Markov approximation and the widely used Lindblad master and Y are required. Of particular importance in TDDFT is the equation [42–45, 65, 66]. Lastly, we review the construction time-dependent electronic density, discussed in [50], where the time-evolving density of an in- teracting OQS is reproduced with a noninteracting and closed Kohn-Sham system. n(r;t) = TrSfrˆS(t)nˆ(r)g; (6) N wheren ˆ(r) = ∑i d(r − rˆi) is the number density operator A. The formally exact many-body master equation for the electronic system. For OQS, the continuity equation is not strictly satisfied and is modified to The starting point of many-body OQS is the unitary evo- lution for the full density matrix of the system and the reser- voir (we use the terms "reservoir" and "bath" interchangeably ¶ n(r;t) = −∇ · TrSfrˆS(t)jˆ(r)g throughout) , ¶t Z t + Trfnˆ(r) dtX˘ (t − t)rˆS(t) + Y(t) g: (7) d t0 rˆ(t) = −ı[Hˆ (t);rˆ(t)]: (1) dt Here, the first term is the divergence of the usual current The full Hamiltonian is given by arising in closed systems, and is referred to as the "Hamilto- nian current". The second term is a contribution to the current due to scattering of electrons with particles in the bath and Hˆ (t) = HˆS(t) + HˆR +Vˆ : (2) arises from the non-unitary part of the evolution. One may define a "Dissipative current" by [31, 70], Here, Z t − · ( ;t) = Trfn( ) d ˘ (t − ) ˆ ( ) + (t) g: 1 N N 1 ∇ jdisp r ˆ r tX t rS t Y Hˆ (t) = − ∇2 + + v (r ;t); (3) t0 S ∑ i ∑ ∑ ext i (8) 2 i=1 i< j jri − r jj i It will be seen later that the functional dependence of the is the Hamiltonian of the electronic system of interest in an Kohn-Sham potential on jdisp(r;t) plays an important role in external potential vext (r;t).