Theory of open quantum systems with bath of electrons and phonons and spins: Many-dissipaton density matrixes approach YiJing Yan
Citation: The Journal of Chemical Physics 140, 054105 (2014); doi: 10.1063/1.4863379 View online: http://dx.doi.org/10.1063/1.4863379 View Table of Contents: http://aip.scitation.org/toc/jcp/140/5 Published by the American Institute of Physics
THE JOURNAL OF CHEMICAL PHYSICS 140, 054105 (2014)
Theory of open quantum systems with bath of electrons and phonons and spins: Many-dissipaton density matrixes approach YiJing Yan Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China and Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong
(Received 11 December 2013; accepted 15 January 2014; published online 3 February 2014) This work establishes a strongly correlated system-and-bath dynamics theory, the many-dissipaton density operators formalism. It puts forward a quasi-particle picture for environmental influences. This picture unifies the physical descriptions and algebraic treatments on three distinct classes of quantum environments, electron bath, phonon bath, and two-level spin or exciton bath, as their participating in quantum dissipation processes. Dynamical variables for theoretical description are no longer just the reduced density matrix for system, but remarkably also those for quasi-particles of bath. The present theoretical formalism offers efficient and accurate means for the study of steadystate (nonequilibrium and equilibrium) and real-time dynamical properties of both systems and hybridizing environments. It further provides universal evaluations, exact in principle, on various correlation functions, including even those of environmental degrees of freedom in coupling with systems. Induced environmental dynamics could be reflected directly in experimentally measurable quantities, such as Fano resonances and quantum transport current shot noise statistics. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4863379] I. INTRODUCTION
Modern research activities, both theoretical and experimental, inevitably deal with open quantum systems. Examples are quantum transport through molecular wires and mesoscopic devices, biological or artificial light-harvesting complexes, controlling quantum entanglement in realistic spins and/or photonic devices, quantum measurement and quantum information, just name a few. Advancing theoretical tools for open quantum systems remains very active in more than six decades.1–10 Recent intensive development focuses mainly on nonperturbative theories and methods for open quantum systems coupling with phonon bath. Among them, the hierarchical equations of motion (HEOM) formalism has emerged as a standard theory.11–16 Bosonic HEOM is mathematically equivalent to the Feynman-Vernon influence functional path integral theory,3, 4, 14, 15 but operationally and numerically much more tractable. It has been applied to various systems, including evaluation of coherent two-dimensional electronic spectra on a model of light-harvesting Fenna-Matthews-Olson proteins.17–19 Fermionic HEOM has also been constructed via path integral approach with Grassmannian algebra.20 It has been proved efficient in characterizing various equilibrium and nonequilibrium strongly correlated properties of quantum impurity systems,21–23 including the dynamical Coulomb blockade,24 and dynamical Kondo transitions in quantum dots.25, 26 Nevertheless, nonperturbative treatment of open quantum systems remains challenging, especially for strongly correlated systems such as d- or f-electronic materials below Kondo temperature.22, 23
0021-9606/2014/140(5)/054105/10/$30.00
For a general theory of open quantum systems, one would like the bath, which is modeled as a collection of noninteracting particles, can be of any one of the three universal classes, phonons or electrons or spins. One would also like to address correlated system and environment dynamics. Conventional theories can only evaluate mean-value evolutions for system Oˆ S -operators. The established HEOM formalism has further supported evaluation on correlation functions of any system operators.16, 17, 19, 22, 23 However, dynamics for bath Oˆ B -operators and strongly correlated (Oˆ S Oˆ B )-operators are yet to be addressed. The correlated system-and-bath dynamics could also be important. They are often reflected directly in experimentally measurable quantities, such as coherent transport current shot noise spectrum.27 The celebrated Fano resonance phenomenon is another example that has been observed in a variety of systems. In recent experiments by exploiting electrical gating control on bilayer graphene bandgap, Fano resonance arises as strong coupling between a discrete phonon mode and continuous gap excitons.28, 29 To address excitons (two-level spins) bath is another objective of the unified theory to be developed in this work. This paper is organized as follows. In Sec. II, a quasiparticle (“dissipaton”) picture is introduced and used to decompose the interacting bath operators. It will be used further in the unified theoretical development on many-dissipaton density operators (MDDOs) dynamics. The resulting formalism will be called Dissipatons Equation of Motion (DEOM), for its close relation to HEOM that will be evident in due course. No longer just the reduced density matrix for system, but remarkably also those for quasi-particles of bath are explicit dynamics variables. We exploit an elegant algebraic approach and construct the unified DEOM formalism
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for fermionic bath in Sec. III, and further for bosonic and exitonic baths in Sec. IV. The capability of addressing correlated system and bath dynamics is exemplified in Sec. V, with the DEOM-space prescription of quantum transport current correlation functions. Concluding remarks presented in Sec. V are also on other key features of the unified DEOM formalism. II. DISSIPATONS DECOMPOSITION OF SYSTEM-BATH INTERACTION A. Prelude
We shall be interested in a unified theory for arbitrary quantum systems (HS ) in the presence of bath (hB ). The latter is modeled as a collection of noninteracting particles that can be electrons or phonons or excitons (two-level spins). The total system-and-bath composite Hamiltonian assumes HT = HS + hB + HSB .
(2.1)
Throughout this paper, we set the system-bath coupling to have HSB B ≡ trB (HSB ρBst ) = 0. Here, ρBst = e−βhB /trB e−βhB denotes the equilibrium or nonequilibrium steady-state bath ensemble, with β ≡ 1/(kB T) being the inverse temperature, scaled by the Boltzmann constant kB . Adopt the unit of ¯ = e = 1 for the Planck constant and the electron charge. For convenience of the logic flow, this section and Sec. III focus on the theory for electron bath. The algebraic approach proposed here will be readily transferable to the cases of phonon and spin baths in Sec. IV. In contact with electronic current transport studies, con † sider hB = α hα = αk (αk + μα )dˆαk dˆαk , for the reser† voirs (electrodes) bath Hamiltonian. Here, dˆαk (dˆαk ) denotes the creation (annihilation) operator of electron in the specified α-reservoir spin-orbital state |k of energy αk . Nonequilibrium chemical potential, μα with α = L or R, will also arise, in the presence of applied electric bias of μL − μR . The energy zero is set to be the equilibrium chemical potential, eq μα = 0, of all leads in contact. The system-bath coupling assumes the form of − + − HSB = aˆ μ . aˆ μ+ Fˆαμ (2.2) + Fˆαμ αμ
aˆ μ+
aˆ μ†
(aˆ μ−
Here, ≡ ≡ aˆ μ ) denotes the creation (annihilation) operator of electron in the specified impurity system spin-orbital state, |μ. The hybridizing bath opera− = k tαμk dˆαk tors in Eq. (2.2) assume a linear form, Fˆαμ + † = (Fˆαμ ) . Therefore, their influence can be characterized by the hybridization bath spectral density functions, Jαμν (ω) ∗ − δ(ω − αk ). By setting Jαμν (ω) ≡ Jαμν (ω) = π k tαμk tανk + ≡ Jανμ (ω) we can define the bath spectral density function via anticommutators as16, 20 σ
1 ∞ σ σ¯ Jαμν (ω) ≡ dt eiωt Fˆαμ (t), Fˆαν (0) B . (2.3) 2 −∞ σ σ −ihα t Here, Fˆαμ (t) ≡ eihα t Fˆαμ e is a linear stochastic bath operator, with σ = +, −, and σ¯ being the opposite sign; i.e., σ¯ σ = −1. Equation (2.3) together with the detailed-balance relation lead to the fermionic fluctuation-dissipation theorem:16, 20
σ
1 σ¯ Fˆαμ (t)Fˆαν (0) B = π
∞
dω
σ (ω) eσ i(ω+μα)t Jαμν
1 + eσβω
−∞
.
(2.4)
In this work, we formally recast the bath correlation function of Eq. (2.4) at finite temperature as16 σ
σ σ¯ σ σ Fˆαμ (t)Fˆαν (0) B ≡ ηαμνk e−γαμνk t + δCαμν (t). (2.5) k
It exploits a certain Lorentz series parametrization form16, 20 σ for the bath spectral density function Jαμν (ω), together 2, 3 or its optimized with a truncated Matsubara expansion version30, 31 for the Fermi function. Both components of the Fourier integrant in Eq. (2.4) are now in sum-overpoles decompositions. The Cauthy’s contour integration is then applied. The aforementioned sum-over-poles decompositions not just contribute directly to the exponential series in σ (t) comEq. (2.5), but also formally identify its residue δCαμν ponent, as Eq. (2.5) in total follows Eq. (2.4) that satisfies σ (t) → 0 if the detailed-balance relation. In principle, δCαμν the number of exponential terms becomes sufficiently large. The to-be-developed DEOM formalism will treat the effects of exponential bath correlation function components in an exact and nonperturbative manner. However, it will be evident that numerical cost of this exact treatment grows exponentially with the number of such components. Thus, an optimized formalism would require an accuracy controllable σ (t). resum treatment on the effect of nonzero residue δCαμν One such resum scheme will also be considered, in connection to the formulation development in this work. B. Dissipatons decomposition of bath influence
It is well known that in quantum statistical mechanics linσ σ (t)} with all Fˆαμ (t)B = 0, ear stochastic bath operators, {Fˆαμ are completely characterized by their correlation functions. This property enables the so-called dissipatons decomposition of bath operators, the following algebra that preserves σ σ¯ (t)Fˆαν (0)B of the form of Eq. (2.5). the composite Fˆαμ For convenience in later formulations, we construct the dissipatons decomposition on σ σ σ αμ αμ σ¯ Fˆαμ ≡F + δF .
(2.6)
+ − + aˆ μ = −aˆ μ− Fˆαμ , due to the underlying fermionic naAs Fˆαμ ture, Eq. (2.6) leads to Eq. (2.2) the expression, σ σ αμ αμ HSB = aˆ μσ¯ F aˆ μσ¯ δ F + . (2.7) σ αμ
σ αμ
σ αμ ασ μ (0)B = 0 and (note that σ¯ σ = −1), We set F (t)δ F σ
σ σ¯ σ αμ (t)F αν F (0) B = − ηαμνk e−γαμνk t , (2.8)
k
σ σ¯ σ αν αμ (t)δ F (0) B = −δCαμν (t). δF
(2.9)
While Eq. (2.5) is preserved, now the two terms there characterize two statistically independent hybridizing bath operaσ σ αμ αμ tors, {F } and {δ F }, respectively. σ αμ Let us first treat {F }, with the exponential part of correlation function by Eq. (2.8). Assume in general that all
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σ exponents there, {γαμνk }, are distinct. Introduce hereafter the abbreviations of
j ≡ {σ αμνk}, j¯ ≡ {σ¯ αμνk}, j˜ ≡ {σ¯ ανμk}.
(2.10)
σ σ σ Therefore, γj ≡ γαμνk and ηj ≡ ηαμνk , whereas aˆ j ≡ aˆ αμνk σ¯ σ σ¯ = aˆ μ and aˆ j˜ ≡ aˆ ανμk = aˆ ν , and further the so-called dissipaσ }, that are statistically independent ton operators, {fˆj ≡ fˆαμνk σ αμ of Eq. (2.8) as quasi-particles of bath. They decompose F σ σ αμ fˆαμνk fˆj . = ≡ (2.11) F νk
νk∈j
∗j¯ = j .
∗j¯ = − j ,
(2.17)
The residue correlation functions, Eq. (2.16), read in terms of σ j ≡ δ F αμν δF as
The underlying statistical independency goes with fˆj (t)fˆj (0)B = −δj j˜ ηj e−γj t ,
(2.12)
δj j˜ = δσ σ¯ δαα δμν δνμ δkk = δj j˜ .
(2.13)
where
More explicitly, the nonzero components are only those σ
σ σ¯ σ fˆαμνk (t)fˆανμk (0) B = −ηαμνk e−γαμνk t . (2.14) Apparently, Eq. (2.8) is strictly preserved. The exponential form of statistical independency here is crucially important [cf. Eq. (3.9)], in conjunction with the construction of dynamical formulations. We shall also take care of the discontinuity at t = 0, for correlation functions in exponential form of Eq. (2.5) and the resultant Eq. (2.12). The issue here ˆ ˆ is concerned with the values of fˆj fˆj > B = fj (0+)fj B < ˆ ˆ ˆ ˆ and fj fjr B = fj (0−)fjr B , for the onset of KeldyshLiouville-space forward (>) and backward ( B = −δj j˜ ηj and fj fj B = −δj j˜ ηj¯ .
is generally achievable.30–32 In particular, if bath spectral denσ (ω) has the asymptotical (1/ω2 )-behaviors, Eq. (2.16) sity Jαμν would arise in conjunction with the (ω3 )-divergence of [(N + 1)/N] Padé spectrum decomposition on Fermi function.31 Let j¯ ≡ {σ¯ αμν} ∈ j, j¯ ≡ {σ¯ αμν}, and j˜ ≡ {σ¯ ανμ}, to label WNR dissipatons. These abbreviations are associated with those in Eq. (2.10). The aforementioned symmetry relations in WNR parameters read now
(2.15)
Both relations will be used directly in the later construction of the DEOM formalism; see Eqs. (3.4) and (3.5). C. White-noise residue ansatz σ αμ Turn now to {δ F }, with the residue bath correlation σ function, δCαμν (t) in Eq. (2.9). This is the leftover part from finite exponential series. The simplest way is to let it be negliσ (t) ≈ 0, by increasing the numgibly small, i.e., setting δCαμν ber of exponential terms, if the exponential growth in numerical cost could still be acceptable. Alternatively, one would like to have a partial resum scheme to efficiently account for residue effect on quantum dissipative dynamics. For a viable and also likely accuracycontrollable re-sum treatment,32 we consider in this work the white-noise-residue (WNR) ansatz,
σ ˙ . σαμν δ(t) δCαμν (t) ≈ 2 σαμν δ(t) + i (2.16) ¯∗ ¯∗ σαμν σαμν , as inferred from Note that σαμν = − σαμν and = σ¯ ∗ σ symmetry relation of [Jαμν (ω)] = Jαμν (ω). The WNR ansatz
˙ j (t)δ F j (0)B ≈ −2δj j˜ j δ(t) + i j δ(t) δ F , followed by [cf. Eqs. (2.6) and (2.11)] σ σ αμ αμν j . δF = δF ≡ δF ν
(2.18)
(2.19)
ν∈j
III. DISSIPATONS EQUATION OF MOTION VIA ALGEBRAIC CONSTRUCTION A. Many-dissipaton density operators and the related contraction theorem
The dissipatons decomposition of system-bath coupling Hamiltonian, Eq. (2.2) or (2.7) with Eqs. (2.11) and (2.19), is now completed. With the abbreviated indexes it reads j ≡ HSB aˆ j¯ fˆj + aˆ j¯ δ F HSB = + δHSB . (3.1) j
j
Many-dissipaton density operators (MDDOs), the dynamical quantities in this paper are defined as ρj(n) (t) ≡ ρj(n) (t) ≡ trB [(fˆjn · · ·fˆj1 )◦ ρT (t)]. 1 ···jn
(3.2)
Here, (fˆjn · · ·fˆj1 )◦ specifies an ordered set of n irreducible dissipatons. A swap of any two irreducible fermionic dissipatons causes a minus sign, such that (fˆj fˆj )◦ = −(fˆj fˆj )◦ .
(3.3)
Apparently, ρ S (t) = trB ρ T (t), the reduced system density operator, is just the special one, with n = 0 by Eq. (3.2). Remarkably, MDDOs provide further powerful means to address strongly correlated system and bath dynamics, governed by the DEOM formalism to be developed below. Theoretical construction will made use of the following Wick’s-like contraction relations. ◦
> ρj>j ≡ trB fˆjn · · ·fˆj1 fˆj ρT (t) = ρj(n+1) + j
n
> (−)r−1 fjr fj B ρj(n−1) − r
r=1
= (−)n ρjj(n+1) +
n (−)r δjr j˜ ηjr ρj(n−1) , − r
r=1
(3.4)
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where Eq. (2.15) was used, and
= (−)n trB [fˆj (fˆjn · · ·fˆj1 )◦ ρT (t)]< =
j
j
n (n−1) + (−) (−)n−r fj fjr < B ρj− n
=
r
r=1
r
(3.5)
r=1
ρT (t0 ) = ρS (t0 )ρBst .
(3.6)
The ansatz is justifiable with setting t0 → −∞. We will discuss this issue in Sec. V.
B. The DEOM formalism with electron bath
To obtain the DEOM formalism that governs the time evolution of MDDOs, we apply for ρj(n) of Eq. (3.2) the Liouville-von Neumann equation, HSB
+ δHSB , ρT (t)].
(3.7)
We evaluate, one-by-one, the specified four components in total composite Hamiltonian, for their contributions. (a) The [HS , ρ T ]-contribution: Apparently,
≡ Lρj(n) . (3.8) trB {(fˆjn · · · fˆj1 )◦ [HS , ρT ]} = HS , ρ (n) j This is the coherent dynamics contribution to ρ˙j(n) . (b) The [hB , ρ T ]-contribution: A remarkable feature arises from the exponential form of dissipaton correlation function, Eq. (2.12), where Re γ j > 0. That is, each dissipaton is subject a sort of “diffusive” motion in bare-bath hB environment, satisfying ∂ ∂t
fˆj
ρ (t) B T
[(−)n aˆ j¯ ρj>j − ρjj< aˆ j¯ ]
=
n
aˆ j¯ ρjj(n+1) − (−)n ρjj(n+1) aˆ j¯ + (−)n−r j
The second identity above is a relation for bath-subspace trace cyclic permutation on fermionic operator; see Eq. (A.12) and comments there. The above contraction relations are valid, assuming the σ } in Eq. (2.2), are Gaussianhybridizing bath operators, {Fˆαμ Wick’s stochastic processes in the bath of ρBst . Assumed is also
trB
j
n = (−)n ρjj(n+1) − (−)r δjr j˜ ηj∗¯r ρj(n−1) . −
ρ˙T (t) = −i[HS + hB +
aˆ j¯ fˆj as in Eq. (3.1),
trB {(fˆjn · · · fˆj1 )◦ [HSB , ρT ]} = trB {(fˆjn · · · fˆj1 )◦ [aˆ j¯ fˆj , ρT ]}
ρjj< ≡ trB [(fˆjn · · ·fˆj1 )◦ ρT (t)fˆj ]
B = δj j˜ ηj and
∗ fˆj fˆj < B = δj j˜ ηj .
(4.9)
B. The DEOM formalism with phonon bath
Consider now the form of MDDOs with bosonic bath. Differing from fermionic bath where each dissipaton can have at most one occupation, the occupation number for individual bosonic dissipaton can be any nonnegative integer; i.e., njr = 0, 1, 2, · · · ; with r = 1, · · · , K, where K denotes the total number of distinct dissipatons (exponential terms). Therefore, the bosonic counterpart of Eq. (3.2) reads
nj nj ◦ (t) ≡ trB fˆjK K · · · fˆj1 1 ρT (t) . (4.10) ρn(n) (t) ≡ ρn(n) j ···nj 1
K
Here, n = nj1 + · · · + njK . Unlike the fermionic case, the order of dissipatons in Eq. (4.10) does not matter. Irreducible bosonic dissipatons are commutable [cf. Eq. (3.3)]. (fˆj fˆj )◦ = (fˆj fˆj )◦ .
(4.11)
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Denote for use soon n± jr that replaces the specified njr in n ≡ (nj1 · · · njK ) with njr ± 1. The DEOM formalism with phonon (bosonic) bath can be readily constructed, following the basic algorithm presented in Sec. III and AppendixA. Involved quantities being dealt with by Wick’s contraction are now [cf. Eqs. (3.4), (3.5) and (4.9)]
The final results on the DEOM formalism with spin bath are summarized as follows, n
(n) Qj , ρjj(n+1) γjr ρj(n) − i ρ˙j = − iL + −i
nj
> nj ◦ + nj˜r ηj˜r ρn(n−1) trB fˆjK K · · · fˆj1 1 fˆjr ρT (t) = ρn(n+1) + − and
< nj nj ◦ trB fˆjr fˆjK K · · · fˆj1 1 ρT (t) = ρn(n+1) + nj˜r ηj∗˜r ρn(n−1) . + − jr
jr
K K
(n) Qjr , ρn(n+1) = − iL + njr γjr ρn − i + −i
K r=1
−
jr
r=1
r
r
(4.15) ˙ a δ(t), bewith the residue term, arising from δCaa (t) = 2 i ing given by [cf. Eq. (3.17)]
a Qa , Qa , HS , ρj(n)
j(n) = a
a
+
a Qa , Qa , Qj ρjj(n+1)
j
n
a (ηjr + ηj∗ ) Qa , Qa , Qj˜ ρ (n−1)
. r r j− r
a
r=1
(4.16)
njr ηjr Qj˜r ρn(n−1) − ηj∗r ρn(n−1) Qj˜r − − jr
ηjr Qj ρj(n−1) − ηj∗r ρj(n−1) Qj + j(n) , − −
r=1
+2
We obtain the final result of
r=1
n
jr
jr
ρ˙n(n)
j
r=1
The derivation on this WNR contribution is presented in Appendix B.
jr
a Qa , Qa , ρn(n) .
(4.12)
a
V. CONCLUDING REMARKS AND SUMMARY
The last term is just n(n) ; see Eqs. (3.13) and (3.17), arising from δCaa (t) ≈ 2 a δ(t). As discussed earlier, this is the only survival type of WNR for bosonic bath in study.
C. The DEOM formalism with spin bath
Consider the form of excitonic MDDOs. Formally, it can go with Eq. (4.10), but now the occupation number njr can only be 0 or 1, for each specified excitonic dissipaton. Consequently [cf. Eq. (3.2)], ρj(n) (t) ≡ ρj(n) (t) ≡ trB [(fˆjn · · · fˆj1 )◦ ρT (t)]. 1 ···jn
(4.13)
The spin counterpart to Eqs. (3.3) and (4.11) reads (fˆj fˆj )◦ = (1 − δjj )(fˆj fˆj )◦ .
(4.14)
This is consistent with the fact that two-level spins or excitons are hard-core bosons, each individual is fermion-like, can only have at most single occupation, but different spins are commutable. The DEOM formalism with spin bath can now be readily obtained. Involving quantities to be contracted are now [cf. Eqs. (3.4), (3.5) and (4.9)] trB [(fˆjn · · · fˆj1 )◦ fˆj ρT (t)]> = ρjj(n+1) +
n
δjr j˜ ηjr ρj(n−1) , − r
r=1
trB [fˆj (fˆjn · · · fˆj1 )◦ ρT (t)]< = ρjj(n+1) +
n
δjr j˜ ηj∗r ρj(n−1) . − r
r=1
Concluding remarks and comments on some key features of the unified DEOM formalism, Eq. (3.13) with Eq. (3.17) for electron bath, Eq. (4.12) for phonon bath, and Eq. (4.15) for spin bath, are as follows. (i) Nonperturbative nature: The DEOM formalism would be exact, in principle, assuming Gaussian stochastic processes of dissipatons that can be treated as linear operators of noninteracting bath. All involving dynamical variables, MDDOs of Eq. (3.2), are irreducible. This validates even the simplest truncation that sets all {ρ (n > L) = 0}, where L = ntrun is the truncated level. The resulting closed DEOM formalism fully accounts for all n ≤ L numbers of dissipatons, or up to (2L)thorder system-bath coupling effect. The higher-order effects are partially included via the mean field contraction approximation. The above observations conclude the nonperturbative nature of the DEOM formalism. (ii) The DEOM space for quantum mechanism of open systems: The linearity of the formalism defines the DEOM space and its propagator. Each element in the DEOM space is a collection of all involving MDDOs, {ρj(n) ; n = 0, 1, · · · , L}, with individual ρj(n) being operators in system subspace. The Schrödinger picture, the Heisenberg picture, and the interaction picture for the DEOM-space dynamics can therefore be defined accordingly, following the standard quantum mechanics linear algebra, with extended dimensionality.7, 19 (iii) Correlated initial system and bath steady states at nonequilibrium or equilibrium: The initial factorization of Eq. (3.6) implies that {ρj(n>0) (t0 ) = 0}, which would physically be valid only when t0 → −∞. However, the DEOM formalism dictates its own steady states {ρj(n);st }, in the absence of time-dependent external fields. They are obtained by setting {ρ˙j(n) = 0}, together with the normalization, trS ρ (0) = 1,
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that acts effectively as an inhomogeneous term in the steadystate DEOM. The resultants ρj(n>0;st) = 0, in general, which do reflect the reality of ρTst = ρSst ρBst , rather than Eq. (3.6). In other words, the initial factorization practically plays no roles for the DEOM evaluation on such as correlation functions and transient dynamics driven by time-dependent external fields from the initial steady states. (iv) Correlated system and bath dynamics: The DEOM formalism provides means to evaluate both the expectation values and correlation functions for correlated system-andbath operators {Oˆ S Oˆ B }. While the system operators Oˆ S here are arbitrary (including Oˆ S = 1ˆ S ), the bath operators Oˆ B can have any dissipaton operators in Taylor series. We illustrate this feature of the DEOM formalism with transport current and its correlation function, respectively, in (v) and (vi), as follows. (v) The time-dependent transport current: The current operator for electron transfer from bath α-reservoir to † system is given by Iˆα ≡ − dtd ( k dˆαk dˆαk ). With the total composite defined in Sec. II A, we have Iˆα Hamiltonian † ˆ † ˆ = −i μ (aˆ μ Fαμ − Fαμ aˆ μ ). It reads in the dissipatons decomposition form as [cf. Eq. (3.1)] σ σ αμ Iˆα = −i σ¯ aˆ μσ¯ fˆαμνk σ¯ aˆ μσ¯ δ F −i σ μνk
≡ −i
a˜ j¯ fˆj − i
jα ∈j
σμ
j . a˜ j¯ δ F
(5.1)
jα ∈j
Here, a˜ μσ ≡ σ aμσ , while jα ≡ {σ μνk} ∈ j ≡ {σ αμνk} and jα ≡ {σ μν} ∈ j ≡ {σ αμν}. With the MDDOs of Eqs. (3.2) and (3.15), current be evaluated as Iˆα can now the mean (1) (0) j ). With Eq. (3.16) for = −i jα trS (a˜ j¯ ρj ) − i jα trS (a˜ j¯ i j(0) , we obtain
j [HS , aˆ j ] , ρ (0) Iˆα = − trS a˜ j¯ j aˆ j + + jα
−i
j ) trS a˜ j¯ ρj(1) . (1 + i2
(5.2)
jα
It is valid for both steady-steady and time-dependent transient current. (vi) Current correlation functions: Let us now examine Cαα (t) = Iˆα (t)Iˆα (0) ≡ TrT Iˆα e−iLT t Iˆα ρTst . (5.3) This is just Cαα (t) = TrT [Iˆα ρT (t; α )], the total composite space description, with ρT (t; α ) ≡ e−iLT t (Iˆα ρTst ), the Liouville-von Neumann dynamics, on ρT (0; α ) ≡ Iˆα ρTst , which can be expressed as [cf. Eq. (5.1)] j ρTst . (5.4) ρT (0; α ) = −i a˜ j¯ fˆj ρTst − i a˜ j¯ δ F jα ∈j
jα ∈j
Together with Eq. (5.2) the above observation leads immediately to the DEOM-space equivalence of j [HS , aˆ j ]), ρ (0) (t; α )]+ ) Cαα (t) = − trS (a˜ j¯ [( j aˆ j + jα
−i
jα
j ) trS a˜ j¯ ρj(1) (t; α ) . (1 + i2
(5.5)
The underlying initial MDDOs of Eq. (3.2) are now ◦
(0; α ) = trB fˆjn · · · fˆj1 ρT (0; α ) . ρj(n) (0; α ) ≡ ρj(n) 1 ···jn Applying ρ T (0; α ) of Eq. (5.4), followed by Eqs. (3.15) and (3.4), we have (n);st ρj(n) (0; α ) = −i a˜ j¯ a˜ j¯ ρjj(n+1);st jj −i jα ∈j n
−i
jα
(−)n−r δαr α ηjr a˜ j¯r ρj(n−1);st . − r
r=1 (n);st above, we obtain With Eq. (3.16) for i jj
j [HS , aˆ j ] , ρj(n);st ρj(n) (0; α ) = − a˜ j¯ j aˆ j + ∓ jα ∈j
−i
j )a˜ j¯ ρjj(n+1);st (1 + i2 jα
n jr (ηjr − η∗¯ )] −i (−)n−r δαr α [ηjr + i jr r=1
×a˜ j¯r ρj(n−1);st . − r
(5.6)
Thus, the DEOM evaluation on current correlation function comprises the following four steps: solving the steadystate solutions {ρj(n);st } to Eq. (3.13); determining the initial {ρj(n) (0; α )} via Eq. (5.6); propagating {ρj(n) (t; α )} with Eq. (3.13); and evaluating Cαα (t) via Eq. (5.5). (vii) Extension of the previous HEOM formalism: Remarkably, the fermionic DEOM reduces to the HEOM formalism,20 by setting j(n) = 0 in Eq. (3.13); whereas the bosonic DEOM, Eq. (4.12), is identical to the HEOM result.12, 15, 16, 32 Therefore, the so-called auxiliary density operators in the HEOM formalism are just the MDDOs of the present work. It is noticed that there is no spin HEOM counterpart to Eq. (4.15). It is known that the Boson bath with linear coupling gives linear response. This is crucial in obtaining the Feynman-Vernon influence functional,4 and thus the HEOM formalism.11–15 In the case of the Fermion bath, the required linear response and Grassmann algebra are both crucial in identifying the corresponding influence functional and the HEOM formalism.20 It is anticipated that the spin bath, as treated in this work, should also be of linear response, but associated with the underlying hard-core Fermi-Grassmann nature. Evidently, the present unified algebraic construction for the DEOM formalism is an elegant alternative and generalization to the path integral influence functional approach to the existing HEOM frameworks. Moreover, it also enables readily ˙ the general treatment of the δ(t)-type white noise residue that is challenging with path integral approach. The most distinct feature of the unified DEOM formalism is that its dynamical quantities, the MDDOs, are all of transparent physical meanings. This feature renders its capability in treating strongly correlated system and bath dynamics. (viii) In relation to optimized formalism: Including the white noise residue effects is closely related to the further development of an optimized formalism. It will go with an efficient and accuracy controllable means to numerical
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characterization of complex dynamics of open quantum systems, as what have been comprehensively established for the optimized bosonic HEOM/DEOM formalism.32 The optimized fermionic and spin DEOM constructions, which will be published elsewhere, can be actually carried readily out in ˙ a very similar manner, despite there are now also the δ(t)-type white noise residues. In summary, this work establishes a unified theory of open quantum systems for electron, phonon, and spin baths. It also offers accuracy controllable means to characterize both system and hybridizing bath dynamics in strong correlation. This novel feature enables its broad range of applications, including Fano resonances and quantum transport current shot noise statistics. Work along this direction will be published elsewhere.
where Eq. (2.18) was used. Similarly, j δ F j )< = −δj j˜ ( ∗j¯ aˆ j + i ∗j¯ a˙ˆ j ) aˆ j¯ (δ F j a˙ˆ j ), = δj j˜ ( j aˆ j − i
(A.3)
where used were the time-reversal of Eq. (2.18) and then Eq. (2.17). The parameter accounts for not only the unit in Eqs. (A.2) and (A.3), but also the fact that WNR dissipatons are broadband limit of -colored dissipatons. We will return to this issue later; see Eq. (A.8) and thereafter. With jj(n) = ϒjj(n) , these identifications, Eq. (A.1) reads rather as ϒ where j a˙ˆ j ), ρT }]. (A.4) ϒjj(n) = (−)n trB [(fˆjn · · · fˆj1 )◦ {( j aˆ j − i
ACKNOWLEDGMENTS
The support from the National Natural Science Foundation of China (Grant No. 21033008), the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB01020000), and the Hong Kong University Grants Committee (AoE/P-04/08-2) and Research Grants Council (No. 605012), is gratefully acknowledged.
Here, a˙ˆ j should be evaluated via the Heisenberg equation of motion withoutδ HSB , as its performing effective contraction for the double-white-noise dissipatons in study, −i a˙ˆ j = [HS , aˆ j ] −
fˆj .
(A.5)
k∈j
APPENDIX A: TREATMENT OF WHITE-NOISE RESIDUE CONTRIBUTION: FERMIONIC BATH
The derivations below will go with the underlying physics of white noises: There are no more than singleirreducible white-noise-dissipatons that can physically participate in dynamics. As results, any pseudo-MDDO that contains two WNR dissipatons could be reduced to ordinary MDDO of Eq. (3.2) via the Wick’s contraction theorem. This suggests the following algebraic approach to the desired rela(n) of Eq. (3.16), in terms of MDDOs. tion of SW-MDDO, ˜ jj j and examine To proceed, we denote δHSB = j aˆ j¯ δ F (n) ˙ jj , with the focus first on its pseudo-MDDO contribution:
To have the final result on ϒjj(n) , we exploit the contraction relations, Eqs. (3.4) and (3.5), to evaluate the following quantity, trB [(fˆjn · · · fˆj1 )◦ {fˆj , ρT }] = (−)n 2ρjj(n+1) +
j )> aˆ j¯ : = j δ F (δ F
∞
j [HS , aˆ j ])ρj(n) ϒjj(n) = ( j aˆ j + j [HS , aˆ j ]) + (−)n ρj(n) ( j aˆ j + j ρjj(n+1)
−2 −
n
j (ηjr − η∗¯ )ρ (n−1) (−)n−r δj j˜r . (A.7) jr j− r
k∈j r=1
(n) To obtain Eq. (3.16) that is just i (n) jj = ϒjj , we treat
SW-MDDO, { (n) jj } of Eq. (3.15), via the asymptotical equivalence of Eq. (2.18),
j (0) = −δj j˜ lim ( j − i j (t)δ F j )e−t . δF B
0
j a˙ˆ j ), = −δj j˜ ( j aˆ j − i
k∈j
(A.1)
j (τ )δ F j B aˆ j¯ (·) dτ δ F
(A.6)
We obtain
j
While the first identity of Eq. (A.1) is just definition, the second one is subject to scrutiny that will be carried out to the end of this appendix. Equation (A.1) contains double white noise dissipaton quantities that are however mathematically ill-defined. Their contraction cannot be carried out directly, but be rather via certain integration form, as follows,
r
r=1
j (fˆjn · · · fˆj1 )◦ [δHSB , ρT ]] jj(n) ≡ trB [δ F ϒ j δ F j )> aˆ j¯ ]ρT trB {(fˆjn · · · fˆj1 )◦ [(δ F = (−)n+1 j δ F j )< ]}. − (fˆjn · · · fˆj1 )◦ ρT [aˆ j¯ (δ F
n (−)r δj j˜r (ηjr − ηj∗¯r )ρj(n−1) . −
→∞
(A.2)
(A.8)
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The HEOM for the -colored counterparts of Eqs. (3.15) behaves asymptotically as
(n) jj + iϒjj(n) . (A.9) ˙ (n) jj = {Eq. (3.13) analogue} − The first part contains just those of Eq. (3.13) that do not explicitly depend on . The second part has two terms. One (n) is from the hB -action on -dissipaton, resulting in − jj , as inferred from Eq. (3.10). Another term is from the δHSB jj(n) = ϒjj(n) , via the asymptotical identity to action or ϒ Eq. (A.1). Since ˙ (n) jj should not diverge in the second part of Eq. (A.9) must vanish. We have therefore (n) (n) jj = −iϒjj .
now the expression of ◦ ˙ j ), ρT }], ϒjj(n) = −trB [(fˆjn · · · fˆj1 {(i Q
which goes with the Heisenberg equation of motion of ˙ j = [Qj , HS ] + [Qj , Qj ]fˆj . (B.3) iQ j
From the Wick’s contraction relations presented right before Eq. (4.15), we have trB [(fˆjn · · · fˆj1 )◦ {fˆj , ρT }]
(A.10)
= 2ρjj(n+1) +
ϒjj(n)
This completes the derivation of Eq. (3.16), since had been given in Eq. (A.7). To close this appendix, it is worth to scrutinize the starting relation in Eq. (A.1), the second identity there, jj(n) . (a) For the first term inside after the definition of ϒ the curly brackets, we comment on the physical posi j )> aˆ j¯ ] there. It is consistent with the un j δ F tion of [(δ F derlying algebra in Eqs. (3.4) and (A.2). Furthermore, it is related to the later involvement fˆj -dissipatons, as in˙ and treated in Eq. (A.6). Note duced from a˙ˆ j via δ(t) j = −δ F j aˆ j¯ . Thus, δ F j (fˆjn · · · fˆj1 )◦ aˆ j¯ δ F j also that aˆ j¯ δ F n+1 ˆ ◦ j δ F j aˆ j¯ , which was used in writing = (−) (fjn · · · fˆj1 ) δ F the middle line of Eq. (A.1); (b) For the second term, the last line of Eq. (A.1), we focus on the sign of ( − )n + 1 . It is associated with the bath-subspace trace cyclic permutation on a fermionic bath operator of (FˆB> )-type, {n} {n} (A.11) trB FˆB> Oˆ S+B = (−)n+1 trB Oˆ S+B FˆB> . {n}
Concerned for Oˆ S+B here, an operator truly in the total system-plus-bath composite space, is only about its odd/even (n = 2m + 1 or 2m) parity. Equation (A.11) was exploited {n} j and Oˆ S+B in the last line of Eq. (A.1), with FˆB> = δ F j . On the other hand, the (FˆB< )-type = (fˆjn · · · fˆj1 )◦ ρT aˆ j¯ δ F counterpart of Eq. (A.11) reads {n} {n} (A.12) trB Oˆ S+B FˆB< = (−)n trB FˆB< Oˆ S+B . This had been used in other places, such as Eq. (3.5) and Eq. (A.6). The additional minus sign in Eq. (A.11) is due ˆ to an additional hidden system a-operator, which arises from the forward path and thus physically locates ahead the bathsubspace trace (trB ). APPENDIX B: TREATMENT OF WHITE-NOISE RESIDUE CONTRIBUTION: SPIN BATH
Let us start with the spin-bath counterpart of Eq. (3.14), which together with Eq. (A.10) can be written as
j Qj , ϒjj(n) . (B.1)
Qj , (n) j(n) = −i jj = − j
j
˙ a δ(t), As the coefficients in δCaa (t) ≈ i 2 had been explic a there, ϒjj(n) in j ≡ itly specified in the last identity, with Eq. (B.1) and below will no longer depend on these parameters. Following the steps of derivation to Eq. (A.4) we have
(B.2)
n
δj j˜r (ηjr + ηj∗r )ρj(n−1) . − r
(B.4)
r=1
This together with Eq. (B.3) lead to Eq. (B.2) the expression, [Qj , Qj ]ρjj(n+1) ϒjj(n) = − [Qj , H ], ρj(n) − 2 j
−
n (ηjr + ηj∗r )[Qj , Qj˜r ]ρj(n−1) . − r
(B.5)
r=1
Substituting it to Eq. (B.1) and noticing that j = a in the present study, we obtain Eq. (3.17). 1 H.
P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, New York, 2002). 2 U. Weiss, Quantum Dissipative Systems, 3rd ed., Series in Modern Condensed Matter Physics Vol. 13 (World Scientific, Singapore, 2008). 3 H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed. (World Scientific, Singapore, 2009). 4 R. P. Feynman and F. L. Vernon, Jr., Ann. Phys. 24, 118 (1963). 5 A. O. Caldeira and A. J. Leggett, Ann. Phys. 149, 374 (1983); 153, 445 (1984) (Erratum). 6 A. G. Redfield, Adv. Magn. Reson. 1, 1 (1965). 7 Y. J. Yan and R. X. Xu, Annu. Rev. Phys. Chem. 56, 187 (2005). 8 N. V. Prokof’ev and P. C. E. Stamp, Rep. Prog. Phys. 63, 669 (2000). 9 J. T. Stockburger and H. Grabert, Phys. Rev. Lett. 88, 170407 (2002). 10 J. S. Shao, J. Chem. Phys. 120, 5053 (2004). 11 Y. Tanimura, Phys. Rev. A 41, 6676 (1990). 12 Y. Tanimura, J. Phys. Soc. Jpn. 75, 082001 (2006). 13 Y. A. Yan, F. Yang, Y. Liu, and J. S. Shao, Chem. Phys. Lett. 395, 216 (2004). 14 R. X. Xu, P. Cui, X. Q. Li, Y. Mo, and Y. J. Yan, J. Chem. Phys. 122, 041103 (2005). 15 R. X. Xu and Y. J. Yan, Phys. Rev. E 75, 031107 (2007). 16 X. Zheng, R. X. Xu, J. Xu, J. S. Jin, J. Hu, and Y. J. Yan, Prog. Chem. 24, 1129 (2012); http://www.progchem.ac.cn/EN/abstract/abstract10858. shtml. 17 L. P. Chen, R. H. Zheng, Y. Y. Jing, and Q. Shi, J. Chem. Phys. 134, 194508 (2011). 18 B. Hein, C. Kreisbeck, T. Kramer, and M. Rodríguez, New J. Phys. 14, 023018 (2012). 19 J. Xu, H. D. Zhang, R. X. Xu, and Y. J. Yan, J. Chem. Phys. 138, 024106 (2013). 20 J. S. Jin, X. Zheng, and Y. J. Yan, J. Chem. Phys. 128, 234703 (2008). 21 X. Zheng, J. S. Jin, and Y. J. Yan, J. Chem. Phys. 129, 184112 (2008). 22 Z. H. Li, N. H. Tong, X. Zheng, D. Hou, J. H. Wei, J. Hu, and Y. J. Yan, Phys. Rev. Lett. 109, 266403 (2012). 23 S. K. Wang, X. Zheng, J. S. Jin, and Y. J. Yan, Phys. Rev. B 88, 035129 (2013). 24 X. Zheng, J. S. Jin, and Y. J. Yan, New J. Phys. 10, 093016 (2008).
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Citation: The Journal of Chemical Physics 140, 054105 (2014); doi: 10.1063/1.4863379 View online: http://dx.doi.org/10.1063/1.4863379 View Table of Contents: http://aip.scitation.org/toc/jcp/140/5 Published by the American Institute of Physics
THE JOURNAL OF CHEMICAL PHYSICS 140, 054105 (2014)
Theory of open quantum systems with bath of electrons and phonons and spins: Many-dissipaton density matrixes approach YiJing Yan Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China and Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong
(Received 11 December 2013; accepted 15 January 2014; published online 3 February 2014) This work establishes a strongly correlated system-and-bath dynamics theory, the many-dissipaton density operators formalism. It puts forward a quasi-particle picture for environmental influences. This picture unifies the physical descriptions and algebraic treatments on three distinct classes of quantum environments, electron bath, phonon bath, and two-level spin or exciton bath, as their participating in quantum dissipation processes. Dynamical variables for theoretical description are no longer just the reduced density matrix for system, but remarkably also those for quasi-particles of bath. The present theoretical formalism offers efficient and accurate means for the study of steadystate (nonequilibrium and equilibrium) and real-time dynamical properties of both systems and hybridizing environments. It further provides universal evaluations, exact in principle, on various correlation functions, including even those of environmental degrees of freedom in coupling with systems. Induced environmental dynamics could be reflected directly in experimentally measurable quantities, such as Fano resonances and quantum transport current shot noise statistics. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4863379] I. INTRODUCTION
Modern research activities, both theoretical and experimental, inevitably deal with open quantum systems. Examples are quantum transport through molecular wires and mesoscopic devices, biological or artificial light-harvesting complexes, controlling quantum entanglement in realistic spins and/or photonic devices, quantum measurement and quantum information, just name a few. Advancing theoretical tools for open quantum systems remains very active in more than six decades.1–10 Recent intensive development focuses mainly on nonperturbative theories and methods for open quantum systems coupling with phonon bath. Among them, the hierarchical equations of motion (HEOM) formalism has emerged as a standard theory.11–16 Bosonic HEOM is mathematically equivalent to the Feynman-Vernon influence functional path integral theory,3, 4, 14, 15 but operationally and numerically much more tractable. It has been applied to various systems, including evaluation of coherent two-dimensional electronic spectra on a model of light-harvesting Fenna-Matthews-Olson proteins.17–19 Fermionic HEOM has also been constructed via path integral approach with Grassmannian algebra.20 It has been proved efficient in characterizing various equilibrium and nonequilibrium strongly correlated properties of quantum impurity systems,21–23 including the dynamical Coulomb blockade,24 and dynamical Kondo transitions in quantum dots.25, 26 Nevertheless, nonperturbative treatment of open quantum systems remains challenging, especially for strongly correlated systems such as d- or f-electronic materials below Kondo temperature.22, 23
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For a general theory of open quantum systems, one would like the bath, which is modeled as a collection of noninteracting particles, can be of any one of the three universal classes, phonons or electrons or spins. One would also like to address correlated system and environment dynamics. Conventional theories can only evaluate mean-value evolutions for system Oˆ S -operators. The established HEOM formalism has further supported evaluation on correlation functions of any system operators.16, 17, 19, 22, 23 However, dynamics for bath Oˆ B -operators and strongly correlated (Oˆ S Oˆ B )-operators are yet to be addressed. The correlated system-and-bath dynamics could also be important. They are often reflected directly in experimentally measurable quantities, such as coherent transport current shot noise spectrum.27 The celebrated Fano resonance phenomenon is another example that has been observed in a variety of systems. In recent experiments by exploiting electrical gating control on bilayer graphene bandgap, Fano resonance arises as strong coupling between a discrete phonon mode and continuous gap excitons.28, 29 To address excitons (two-level spins) bath is another objective of the unified theory to be developed in this work. This paper is organized as follows. In Sec. II, a quasiparticle (“dissipaton”) picture is introduced and used to decompose the interacting bath operators. It will be used further in the unified theoretical development on many-dissipaton density operators (MDDOs) dynamics. The resulting formalism will be called Dissipatons Equation of Motion (DEOM), for its close relation to HEOM that will be evident in due course. No longer just the reduced density matrix for system, but remarkably also those for quasi-particles of bath are explicit dynamics variables. We exploit an elegant algebraic approach and construct the unified DEOM formalism
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for fermionic bath in Sec. III, and further for bosonic and exitonic baths in Sec. IV. The capability of addressing correlated system and bath dynamics is exemplified in Sec. V, with the DEOM-space prescription of quantum transport current correlation functions. Concluding remarks presented in Sec. V are also on other key features of the unified DEOM formalism. II. DISSIPATONS DECOMPOSITION OF SYSTEM-BATH INTERACTION A. Prelude
We shall be interested in a unified theory for arbitrary quantum systems (HS ) in the presence of bath (hB ). The latter is modeled as a collection of noninteracting particles that can be electrons or phonons or excitons (two-level spins). The total system-and-bath composite Hamiltonian assumes HT = HS + hB + HSB .
(2.1)
Throughout this paper, we set the system-bath coupling to have HSB B ≡ trB (HSB ρBst ) = 0. Here, ρBst = e−βhB /trB e−βhB denotes the equilibrium or nonequilibrium steady-state bath ensemble, with β ≡ 1/(kB T) being the inverse temperature, scaled by the Boltzmann constant kB . Adopt the unit of ¯ = e = 1 for the Planck constant and the electron charge. For convenience of the logic flow, this section and Sec. III focus on the theory for electron bath. The algebraic approach proposed here will be readily transferable to the cases of phonon and spin baths in Sec. IV. In contact with electronic current transport studies, con † sider hB = α hα = αk (αk + μα )dˆαk dˆαk , for the reser† voirs (electrodes) bath Hamiltonian. Here, dˆαk (dˆαk ) denotes the creation (annihilation) operator of electron in the specified α-reservoir spin-orbital state |k of energy αk . Nonequilibrium chemical potential, μα with α = L or R, will also arise, in the presence of applied electric bias of μL − μR . The energy zero is set to be the equilibrium chemical potential, eq μα = 0, of all leads in contact. The system-bath coupling assumes the form of − + − HSB = aˆ μ . aˆ μ+ Fˆαμ (2.2) + Fˆαμ αμ
aˆ μ+
aˆ μ†
(aˆ μ−
Here, ≡ ≡ aˆ μ ) denotes the creation (annihilation) operator of electron in the specified impurity system spin-orbital state, |μ. The hybridizing bath opera− = k tαμk dˆαk tors in Eq. (2.2) assume a linear form, Fˆαμ + † = (Fˆαμ ) . Therefore, their influence can be characterized by the hybridization bath spectral density functions, Jαμν (ω) ∗ − δ(ω − αk ). By setting Jαμν (ω) ≡ Jαμν (ω) = π k tαμk tανk + ≡ Jανμ (ω) we can define the bath spectral density function via anticommutators as16, 20 σ
1 ∞ σ σ¯ Jαμν (ω) ≡ dt eiωt Fˆαμ (t), Fˆαν (0) B . (2.3) 2 −∞ σ σ −ihα t Here, Fˆαμ (t) ≡ eihα t Fˆαμ e is a linear stochastic bath operator, with σ = +, −, and σ¯ being the opposite sign; i.e., σ¯ σ = −1. Equation (2.3) together with the detailed-balance relation lead to the fermionic fluctuation-dissipation theorem:16, 20
σ
1 σ¯ Fˆαμ (t)Fˆαν (0) B = π
∞
dω
σ (ω) eσ i(ω+μα)t Jαμν
1 + eσβω
−∞
.
(2.4)
In this work, we formally recast the bath correlation function of Eq. (2.4) at finite temperature as16 σ
σ σ¯ σ σ Fˆαμ (t)Fˆαν (0) B ≡ ηαμνk e−γαμνk t + δCαμν (t). (2.5) k
It exploits a certain Lorentz series parametrization form16, 20 σ for the bath spectral density function Jαμν (ω), together 2, 3 or its optimized with a truncated Matsubara expansion version30, 31 for the Fermi function. Both components of the Fourier integrant in Eq. (2.4) are now in sum-overpoles decompositions. The Cauthy’s contour integration is then applied. The aforementioned sum-over-poles decompositions not just contribute directly to the exponential series in σ (t) comEq. (2.5), but also formally identify its residue δCαμν ponent, as Eq. (2.5) in total follows Eq. (2.4) that satisfies σ (t) → 0 if the detailed-balance relation. In principle, δCαμν the number of exponential terms becomes sufficiently large. The to-be-developed DEOM formalism will treat the effects of exponential bath correlation function components in an exact and nonperturbative manner. However, it will be evident that numerical cost of this exact treatment grows exponentially with the number of such components. Thus, an optimized formalism would require an accuracy controllable σ (t). resum treatment on the effect of nonzero residue δCαμν One such resum scheme will also be considered, in connection to the formulation development in this work. B. Dissipatons decomposition of bath influence
It is well known that in quantum statistical mechanics linσ σ (t)} with all Fˆαμ (t)B = 0, ear stochastic bath operators, {Fˆαμ are completely characterized by their correlation functions. This property enables the so-called dissipatons decomposition of bath operators, the following algebra that preserves σ σ¯ (t)Fˆαν (0)B of the form of Eq. (2.5). the composite Fˆαμ For convenience in later formulations, we construct the dissipatons decomposition on σ σ σ αμ αμ σ¯ Fˆαμ ≡F + δF .
(2.6)
+ − + aˆ μ = −aˆ μ− Fˆαμ , due to the underlying fermionic naAs Fˆαμ ture, Eq. (2.6) leads to Eq. (2.2) the expression, σ σ αμ αμ HSB = aˆ μσ¯ F aˆ μσ¯ δ F + . (2.7) σ αμ
σ αμ
σ αμ ασ μ (0)B = 0 and (note that σ¯ σ = −1), We set F (t)δ F σ
σ σ¯ σ αμ (t)F αν F (0) B = − ηαμνk e−γαμνk t , (2.8)
k
σ σ¯ σ αν αμ (t)δ F (0) B = −δCαμν (t). δF
(2.9)
While Eq. (2.5) is preserved, now the two terms there characterize two statistically independent hybridizing bath operaσ σ αμ αμ tors, {F } and {δ F }, respectively. σ αμ Let us first treat {F }, with the exponential part of correlation function by Eq. (2.8). Assume in general that all
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σ exponents there, {γαμνk }, are distinct. Introduce hereafter the abbreviations of
j ≡ {σ αμνk}, j¯ ≡ {σ¯ αμνk}, j˜ ≡ {σ¯ ανμk}.
(2.10)
σ σ σ Therefore, γj ≡ γαμνk and ηj ≡ ηαμνk , whereas aˆ j ≡ aˆ αμνk σ¯ σ σ¯ = aˆ μ and aˆ j˜ ≡ aˆ ανμk = aˆ ν , and further the so-called dissipaσ }, that are statistically independent ton operators, {fˆj ≡ fˆαμνk σ αμ of Eq. (2.8) as quasi-particles of bath. They decompose F σ σ αμ fˆαμνk fˆj . = ≡ (2.11) F νk
νk∈j
∗j¯ = j .
∗j¯ = − j ,
(2.17)
The residue correlation functions, Eq. (2.16), read in terms of σ j ≡ δ F αμν δF as
The underlying statistical independency goes with fˆj (t)fˆj (0)B = −δj j˜ ηj e−γj t ,
(2.12)
δj j˜ = δσ σ¯ δαα δμν δνμ δkk = δj j˜ .
(2.13)
where
More explicitly, the nonzero components are only those σ
σ σ¯ σ fˆαμνk (t)fˆανμk (0) B = −ηαμνk e−γαμνk t . (2.14) Apparently, Eq. (2.8) is strictly preserved. The exponential form of statistical independency here is crucially important [cf. Eq. (3.9)], in conjunction with the construction of dynamical formulations. We shall also take care of the discontinuity at t = 0, for correlation functions in exponential form of Eq. (2.5) and the resultant Eq. (2.12). The issue here ˆ ˆ is concerned with the values of fˆj fˆj > B = fj (0+)fj B < ˆ ˆ ˆ ˆ and fj fjr B = fj (0−)fjr B , for the onset of KeldyshLiouville-space forward (>) and backward ( B = −δj j˜ ηj and fj fj B = −δj j˜ ηj¯ .
is generally achievable.30–32 In particular, if bath spectral denσ (ω) has the asymptotical (1/ω2 )-behaviors, Eq. (2.16) sity Jαμν would arise in conjunction with the (ω3 )-divergence of [(N + 1)/N] Padé spectrum decomposition on Fermi function.31 Let j¯ ≡ {σ¯ αμν} ∈ j, j¯ ≡ {σ¯ αμν}, and j˜ ≡ {σ¯ ανμ}, to label WNR dissipatons. These abbreviations are associated with those in Eq. (2.10). The aforementioned symmetry relations in WNR parameters read now
(2.15)
Both relations will be used directly in the later construction of the DEOM formalism; see Eqs. (3.4) and (3.5). C. White-noise residue ansatz σ αμ Turn now to {δ F }, with the residue bath correlation σ function, δCαμν (t) in Eq. (2.9). This is the leftover part from finite exponential series. The simplest way is to let it be negliσ (t) ≈ 0, by increasing the numgibly small, i.e., setting δCαμν ber of exponential terms, if the exponential growth in numerical cost could still be acceptable. Alternatively, one would like to have a partial resum scheme to efficiently account for residue effect on quantum dissipative dynamics. For a viable and also likely accuracycontrollable re-sum treatment,32 we consider in this work the white-noise-residue (WNR) ansatz,
σ ˙ . σαμν δ(t) δCαμν (t) ≈ 2 σαμν δ(t) + i (2.16) ¯∗ ¯∗ σαμν σαμν , as inferred from Note that σαμν = − σαμν and = σ¯ ∗ σ symmetry relation of [Jαμν (ω)] = Jαμν (ω). The WNR ansatz
˙ j (t)δ F j (0)B ≈ −2δj j˜ j δ(t) + i j δ(t) δ F , followed by [cf. Eqs. (2.6) and (2.11)] σ σ αμ αμν j . δF = δF ≡ δF ν
(2.18)
(2.19)
ν∈j
III. DISSIPATONS EQUATION OF MOTION VIA ALGEBRAIC CONSTRUCTION A. Many-dissipaton density operators and the related contraction theorem
The dissipatons decomposition of system-bath coupling Hamiltonian, Eq. (2.2) or (2.7) with Eqs. (2.11) and (2.19), is now completed. With the abbreviated indexes it reads j ≡ HSB aˆ j¯ fˆj + aˆ j¯ δ F HSB = + δHSB . (3.1) j
j
Many-dissipaton density operators (MDDOs), the dynamical quantities in this paper are defined as ρj(n) (t) ≡ ρj(n) (t) ≡ trB [(fˆjn · · ·fˆj1 )◦ ρT (t)]. 1 ···jn
(3.2)
Here, (fˆjn · · ·fˆj1 )◦ specifies an ordered set of n irreducible dissipatons. A swap of any two irreducible fermionic dissipatons causes a minus sign, such that (fˆj fˆj )◦ = −(fˆj fˆj )◦ .
(3.3)
Apparently, ρ S (t) = trB ρ T (t), the reduced system density operator, is just the special one, with n = 0 by Eq. (3.2). Remarkably, MDDOs provide further powerful means to address strongly correlated system and bath dynamics, governed by the DEOM formalism to be developed below. Theoretical construction will made use of the following Wick’s-like contraction relations. ◦
> ρj>j ≡ trB fˆjn · · ·fˆj1 fˆj ρT (t) = ρj(n+1) + j
n
> (−)r−1 fjr fj B ρj(n−1) − r
r=1
= (−)n ρjj(n+1) +
n (−)r δjr j˜ ηjr ρj(n−1) , − r
r=1
(3.4)
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where Eq. (2.15) was used, and
= (−)n trB [fˆj (fˆjn · · ·fˆj1 )◦ ρT (t)]< =
j
j
n (n−1) + (−) (−)n−r fj fjr < B ρj− n
=
r
r=1
r
(3.5)
r=1
ρT (t0 ) = ρS (t0 )ρBst .
(3.6)
The ansatz is justifiable with setting t0 → −∞. We will discuss this issue in Sec. V.
B. The DEOM formalism with electron bath
To obtain the DEOM formalism that governs the time evolution of MDDOs, we apply for ρj(n) of Eq. (3.2) the Liouville-von Neumann equation, HSB
+ δHSB , ρT (t)].
(3.7)
We evaluate, one-by-one, the specified four components in total composite Hamiltonian, for their contributions. (a) The [HS , ρ T ]-contribution: Apparently,
≡ Lρj(n) . (3.8) trB {(fˆjn · · · fˆj1 )◦ [HS , ρT ]} = HS , ρ (n) j This is the coherent dynamics contribution to ρ˙j(n) . (b) The [hB , ρ T ]-contribution: A remarkable feature arises from the exponential form of dissipaton correlation function, Eq. (2.12), where Re γ j > 0. That is, each dissipaton is subject a sort of “diffusive” motion in bare-bath hB environment, satisfying ∂ ∂t
fˆj
ρ (t) B T
[(−)n aˆ j¯ ρj>j − ρjj< aˆ j¯ ]
=
n
aˆ j¯ ρjj(n+1) − (−)n ρjj(n+1) aˆ j¯ + (−)n−r j
The second identity above is a relation for bath-subspace trace cyclic permutation on fermionic operator; see Eq. (A.12) and comments there. The above contraction relations are valid, assuming the σ } in Eq. (2.2), are Gaussianhybridizing bath operators, {Fˆαμ Wick’s stochastic processes in the bath of ρBst . Assumed is also
trB
j
n = (−)n ρjj(n+1) − (−)r δjr j˜ ηj∗¯r ρj(n−1) . −
ρ˙T (t) = −i[HS + hB +
aˆ j¯ fˆj as in Eq. (3.1),
trB {(fˆjn · · · fˆj1 )◦ [HSB , ρT ]} = trB {(fˆjn · · · fˆj1 )◦ [aˆ j¯ fˆj , ρT ]}
ρjj< ≡ trB [(fˆjn · · ·fˆj1 )◦ ρT (t)fˆj ]
B = δj j˜ ηj and
∗ fˆj fˆj < B = δj j˜ ηj .
(4.9)
B. The DEOM formalism with phonon bath
Consider now the form of MDDOs with bosonic bath. Differing from fermionic bath where each dissipaton can have at most one occupation, the occupation number for individual bosonic dissipaton can be any nonnegative integer; i.e., njr = 0, 1, 2, · · · ; with r = 1, · · · , K, where K denotes the total number of distinct dissipatons (exponential terms). Therefore, the bosonic counterpart of Eq. (3.2) reads
nj nj ◦ (t) ≡ trB fˆjK K · · · fˆj1 1 ρT (t) . (4.10) ρn(n) (t) ≡ ρn(n) j ···nj 1
K
Here, n = nj1 + · · · + njK . Unlike the fermionic case, the order of dissipatons in Eq. (4.10) does not matter. Irreducible bosonic dissipatons are commutable [cf. Eq. (3.3)]. (fˆj fˆj )◦ = (fˆj fˆj )◦ .
(4.11)
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Denote for use soon n± jr that replaces the specified njr in n ≡ (nj1 · · · njK ) with njr ± 1. The DEOM formalism with phonon (bosonic) bath can be readily constructed, following the basic algorithm presented in Sec. III and AppendixA. Involved quantities being dealt with by Wick’s contraction are now [cf. Eqs. (3.4), (3.5) and (4.9)]
The final results on the DEOM formalism with spin bath are summarized as follows, n
(n) Qj , ρjj(n+1) γjr ρj(n) − i ρ˙j = − iL + −i
nj
> nj ◦ + nj˜r ηj˜r ρn(n−1) trB fˆjK K · · · fˆj1 1 fˆjr ρT (t) = ρn(n+1) + − and
< nj nj ◦ trB fˆjr fˆjK K · · · fˆj1 1 ρT (t) = ρn(n+1) + nj˜r ηj∗˜r ρn(n−1) . + − jr
jr
K K
(n) Qjr , ρn(n+1) = − iL + njr γjr ρn − i + −i
K r=1
−
jr
r=1
r
r
(4.15) ˙ a δ(t), bewith the residue term, arising from δCaa (t) = 2 i ing given by [cf. Eq. (3.17)]
a Qa , Qa , HS , ρj(n)
j(n) = a
a
+
a Qa , Qa , Qj ρjj(n+1)
j
n
a (ηjr + ηj∗ ) Qa , Qa , Qj˜ ρ (n−1)
. r r j− r
a
r=1
(4.16)
njr ηjr Qj˜r ρn(n−1) − ηj∗r ρn(n−1) Qj˜r − − jr
ηjr Qj ρj(n−1) − ηj∗r ρj(n−1) Qj + j(n) , − −
r=1
+2
We obtain the final result of
r=1
n
jr
jr
ρ˙n(n)
j
r=1
The derivation on this WNR contribution is presented in Appendix B.
jr
a Qa , Qa , ρn(n) .
(4.12)
a
V. CONCLUDING REMARKS AND SUMMARY
The last term is just n(n) ; see Eqs. (3.13) and (3.17), arising from δCaa (t) ≈ 2 a δ(t). As discussed earlier, this is the only survival type of WNR for bosonic bath in study.
C. The DEOM formalism with spin bath
Consider the form of excitonic MDDOs. Formally, it can go with Eq. (4.10), but now the occupation number njr can only be 0 or 1, for each specified excitonic dissipaton. Consequently [cf. Eq. (3.2)], ρj(n) (t) ≡ ρj(n) (t) ≡ trB [(fˆjn · · · fˆj1 )◦ ρT (t)]. 1 ···jn
(4.13)
The spin counterpart to Eqs. (3.3) and (4.11) reads (fˆj fˆj )◦ = (1 − δjj )(fˆj fˆj )◦ .
(4.14)
This is consistent with the fact that two-level spins or excitons are hard-core bosons, each individual is fermion-like, can only have at most single occupation, but different spins are commutable. The DEOM formalism with spin bath can now be readily obtained. Involving quantities to be contracted are now [cf. Eqs. (3.4), (3.5) and (4.9)] trB [(fˆjn · · · fˆj1 )◦ fˆj ρT (t)]> = ρjj(n+1) +
n
δjr j˜ ηjr ρj(n−1) , − r
r=1
trB [fˆj (fˆjn · · · fˆj1 )◦ ρT (t)]< = ρjj(n+1) +
n
δjr j˜ ηj∗r ρj(n−1) . − r
r=1
Concluding remarks and comments on some key features of the unified DEOM formalism, Eq. (3.13) with Eq. (3.17) for electron bath, Eq. (4.12) for phonon bath, and Eq. (4.15) for spin bath, are as follows. (i) Nonperturbative nature: The DEOM formalism would be exact, in principle, assuming Gaussian stochastic processes of dissipatons that can be treated as linear operators of noninteracting bath. All involving dynamical variables, MDDOs of Eq. (3.2), are irreducible. This validates even the simplest truncation that sets all {ρ (n > L) = 0}, where L = ntrun is the truncated level. The resulting closed DEOM formalism fully accounts for all n ≤ L numbers of dissipatons, or up to (2L)thorder system-bath coupling effect. The higher-order effects are partially included via the mean field contraction approximation. The above observations conclude the nonperturbative nature of the DEOM formalism. (ii) The DEOM space for quantum mechanism of open systems: The linearity of the formalism defines the DEOM space and its propagator. Each element in the DEOM space is a collection of all involving MDDOs, {ρj(n) ; n = 0, 1, · · · , L}, with individual ρj(n) being operators in system subspace. The Schrödinger picture, the Heisenberg picture, and the interaction picture for the DEOM-space dynamics can therefore be defined accordingly, following the standard quantum mechanics linear algebra, with extended dimensionality.7, 19 (iii) Correlated initial system and bath steady states at nonequilibrium or equilibrium: The initial factorization of Eq. (3.6) implies that {ρj(n>0) (t0 ) = 0}, which would physically be valid only when t0 → −∞. However, the DEOM formalism dictates its own steady states {ρj(n);st }, in the absence of time-dependent external fields. They are obtained by setting {ρ˙j(n) = 0}, together with the normalization, trS ρ (0) = 1,
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that acts effectively as an inhomogeneous term in the steadystate DEOM. The resultants ρj(n>0;st) = 0, in general, which do reflect the reality of ρTst = ρSst ρBst , rather than Eq. (3.6). In other words, the initial factorization practically plays no roles for the DEOM evaluation on such as correlation functions and transient dynamics driven by time-dependent external fields from the initial steady states. (iv) Correlated system and bath dynamics: The DEOM formalism provides means to evaluate both the expectation values and correlation functions for correlated system-andbath operators {Oˆ S Oˆ B }. While the system operators Oˆ S here are arbitrary (including Oˆ S = 1ˆ S ), the bath operators Oˆ B can have any dissipaton operators in Taylor series. We illustrate this feature of the DEOM formalism with transport current and its correlation function, respectively, in (v) and (vi), as follows. (v) The time-dependent transport current: The current operator for electron transfer from bath α-reservoir to † system is given by Iˆα ≡ − dtd ( k dˆαk dˆαk ). With the total composite defined in Sec. II A, we have Iˆα Hamiltonian † ˆ † ˆ = −i μ (aˆ μ Fαμ − Fαμ aˆ μ ). It reads in the dissipatons decomposition form as [cf. Eq. (3.1)] σ σ αμ Iˆα = −i σ¯ aˆ μσ¯ fˆαμνk σ¯ aˆ μσ¯ δ F −i σ μνk
≡ −i
a˜ j¯ fˆj − i
jα ∈j
σμ
j . a˜ j¯ δ F
(5.1)
jα ∈j
Here, a˜ μσ ≡ σ aμσ , while jα ≡ {σ μνk} ∈ j ≡ {σ αμνk} and jα ≡ {σ μν} ∈ j ≡ {σ αμν}. With the MDDOs of Eqs. (3.2) and (3.15), current be evaluated as Iˆα can now the mean (1) (0) j ). With Eq. (3.16) for = −i jα trS (a˜ j¯ ρj ) − i jα trS (a˜ j¯ i j(0) , we obtain
j [HS , aˆ j ] , ρ (0) Iˆα = − trS a˜ j¯ j aˆ j + + jα
−i
j ) trS a˜ j¯ ρj(1) . (1 + i2
(5.2)
jα
It is valid for both steady-steady and time-dependent transient current. (vi) Current correlation functions: Let us now examine Cαα (t) = Iˆα (t)Iˆα (0) ≡ TrT Iˆα e−iLT t Iˆα ρTst . (5.3) This is just Cαα (t) = TrT [Iˆα ρT (t; α )], the total composite space description, with ρT (t; α ) ≡ e−iLT t (Iˆα ρTst ), the Liouville-von Neumann dynamics, on ρT (0; α ) ≡ Iˆα ρTst , which can be expressed as [cf. Eq. (5.1)] j ρTst . (5.4) ρT (0; α ) = −i a˜ j¯ fˆj ρTst − i a˜ j¯ δ F jα ∈j
jα ∈j
Together with Eq. (5.2) the above observation leads immediately to the DEOM-space equivalence of j [HS , aˆ j ]), ρ (0) (t; α )]+ ) Cαα (t) = − trS (a˜ j¯ [( j aˆ j + jα
−i
jα
j ) trS a˜ j¯ ρj(1) (t; α ) . (1 + i2
(5.5)
The underlying initial MDDOs of Eq. (3.2) are now ◦
(0; α ) = trB fˆjn · · · fˆj1 ρT (0; α ) . ρj(n) (0; α ) ≡ ρj(n) 1 ···jn Applying ρ T (0; α ) of Eq. (5.4), followed by Eqs. (3.15) and (3.4), we have (n);st ρj(n) (0; α ) = −i a˜ j¯ a˜ j¯ ρjj(n+1);st jj −i jα ∈j n
−i
jα
(−)n−r δαr α ηjr a˜ j¯r ρj(n−1);st . − r
r=1 (n);st above, we obtain With Eq. (3.16) for i jj
j [HS , aˆ j ] , ρj(n);st ρj(n) (0; α ) = − a˜ j¯ j aˆ j + ∓ jα ∈j
−i
j )a˜ j¯ ρjj(n+1);st (1 + i2 jα
n jr (ηjr − η∗¯ )] −i (−)n−r δαr α [ηjr + i jr r=1
×a˜ j¯r ρj(n−1);st . − r
(5.6)
Thus, the DEOM evaluation on current correlation function comprises the following four steps: solving the steadystate solutions {ρj(n);st } to Eq. (3.13); determining the initial {ρj(n) (0; α )} via Eq. (5.6); propagating {ρj(n) (t; α )} with Eq. (3.13); and evaluating Cαα (t) via Eq. (5.5). (vii) Extension of the previous HEOM formalism: Remarkably, the fermionic DEOM reduces to the HEOM formalism,20 by setting j(n) = 0 in Eq. (3.13); whereas the bosonic DEOM, Eq. (4.12), is identical to the HEOM result.12, 15, 16, 32 Therefore, the so-called auxiliary density operators in the HEOM formalism are just the MDDOs of the present work. It is noticed that there is no spin HEOM counterpart to Eq. (4.15). It is known that the Boson bath with linear coupling gives linear response. This is crucial in obtaining the Feynman-Vernon influence functional,4 and thus the HEOM formalism.11–15 In the case of the Fermion bath, the required linear response and Grassmann algebra are both crucial in identifying the corresponding influence functional and the HEOM formalism.20 It is anticipated that the spin bath, as treated in this work, should also be of linear response, but associated with the underlying hard-core Fermi-Grassmann nature. Evidently, the present unified algebraic construction for the DEOM formalism is an elegant alternative and generalization to the path integral influence functional approach to the existing HEOM frameworks. Moreover, it also enables readily ˙ the general treatment of the δ(t)-type white noise residue that is challenging with path integral approach. The most distinct feature of the unified DEOM formalism is that its dynamical quantities, the MDDOs, are all of transparent physical meanings. This feature renders its capability in treating strongly correlated system and bath dynamics. (viii) In relation to optimized formalism: Including the white noise residue effects is closely related to the further development of an optimized formalism. It will go with an efficient and accuracy controllable means to numerical
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characterization of complex dynamics of open quantum systems, as what have been comprehensively established for the optimized bosonic HEOM/DEOM formalism.32 The optimized fermionic and spin DEOM constructions, which will be published elsewhere, can be actually carried readily out in ˙ a very similar manner, despite there are now also the δ(t)-type white noise residues. In summary, this work establishes a unified theory of open quantum systems for electron, phonon, and spin baths. It also offers accuracy controllable means to characterize both system and hybridizing bath dynamics in strong correlation. This novel feature enables its broad range of applications, including Fano resonances and quantum transport current shot noise statistics. Work along this direction will be published elsewhere.
where Eq. (2.18) was used. Similarly, j δ F j )< = −δj j˜ ( ∗j¯ aˆ j + i ∗j¯ a˙ˆ j ) aˆ j¯ (δ F j a˙ˆ j ), = δj j˜ ( j aˆ j − i
(A.3)
where used were the time-reversal of Eq. (2.18) and then Eq. (2.17). The parameter accounts for not only the unit in Eqs. (A.2) and (A.3), but also the fact that WNR dissipatons are broadband limit of -colored dissipatons. We will return to this issue later; see Eq. (A.8) and thereafter. With jj(n) = ϒjj(n) , these identifications, Eq. (A.1) reads rather as ϒ where j a˙ˆ j ), ρT }]. (A.4) ϒjj(n) = (−)n trB [(fˆjn · · · fˆj1 )◦ {( j aˆ j − i
ACKNOWLEDGMENTS
The support from the National Natural Science Foundation of China (Grant No. 21033008), the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB01020000), and the Hong Kong University Grants Committee (AoE/P-04/08-2) and Research Grants Council (No. 605012), is gratefully acknowledged.
Here, a˙ˆ j should be evaluated via the Heisenberg equation of motion withoutδ HSB , as its performing effective contraction for the double-white-noise dissipatons in study, −i a˙ˆ j = [HS , aˆ j ] −
fˆj .
(A.5)
k∈j
APPENDIX A: TREATMENT OF WHITE-NOISE RESIDUE CONTRIBUTION: FERMIONIC BATH
The derivations below will go with the underlying physics of white noises: There are no more than singleirreducible white-noise-dissipatons that can physically participate in dynamics. As results, any pseudo-MDDO that contains two WNR dissipatons could be reduced to ordinary MDDO of Eq. (3.2) via the Wick’s contraction theorem. This suggests the following algebraic approach to the desired rela(n) of Eq. (3.16), in terms of MDDOs. tion of SW-MDDO, ˜ jj j and examine To proceed, we denote δHSB = j aˆ j¯ δ F (n) ˙ jj , with the focus first on its pseudo-MDDO contribution:
To have the final result on ϒjj(n) , we exploit the contraction relations, Eqs. (3.4) and (3.5), to evaluate the following quantity, trB [(fˆjn · · · fˆj1 )◦ {fˆj , ρT }] = (−)n 2ρjj(n+1) +
j )> aˆ j¯ : = j δ F (δ F
∞
j [HS , aˆ j ])ρj(n) ϒjj(n) = ( j aˆ j + j [HS , aˆ j ]) + (−)n ρj(n) ( j aˆ j + j ρjj(n+1)
−2 −
n
j (ηjr − η∗¯ )ρ (n−1) (−)n−r δj j˜r . (A.7) jr j− r
k∈j r=1
(n) To obtain Eq. (3.16) that is just i (n) jj = ϒjj , we treat
SW-MDDO, { (n) jj } of Eq. (3.15), via the asymptotical equivalence of Eq. (2.18),
j (0) = −δj j˜ lim ( j − i j (t)δ F j )e−t . δF B
0
j a˙ˆ j ), = −δj j˜ ( j aˆ j − i
k∈j
(A.1)
j (τ )δ F j B aˆ j¯ (·) dτ δ F
(A.6)
We obtain
j
While the first identity of Eq. (A.1) is just definition, the second one is subject to scrutiny that will be carried out to the end of this appendix. Equation (A.1) contains double white noise dissipaton quantities that are however mathematically ill-defined. Their contraction cannot be carried out directly, but be rather via certain integration form, as follows,
r
r=1
j (fˆjn · · · fˆj1 )◦ [δHSB , ρT ]] jj(n) ≡ trB [δ F ϒ j δ F j )> aˆ j¯ ]ρT trB {(fˆjn · · · fˆj1 )◦ [(δ F = (−)n+1 j δ F j )< ]}. − (fˆjn · · · fˆj1 )◦ ρT [aˆ j¯ (δ F
n (−)r δj j˜r (ηjr − ηj∗¯r )ρj(n−1) . −
→∞
(A.2)
(A.8)
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J. Chem. Phys. 140, 054105 (2014)
The HEOM for the -colored counterparts of Eqs. (3.15) behaves asymptotically as
(n) jj + iϒjj(n) . (A.9) ˙ (n) jj = {Eq. (3.13) analogue} − The first part contains just those of Eq. (3.13) that do not explicitly depend on . The second part has two terms. One (n) is from the hB -action on -dissipaton, resulting in − jj , as inferred from Eq. (3.10). Another term is from the δHSB jj(n) = ϒjj(n) , via the asymptotical identity to action or ϒ Eq. (A.1). Since ˙ (n) jj should not diverge in the second part of Eq. (A.9) must vanish. We have therefore (n) (n) jj = −iϒjj .
now the expression of ◦ ˙ j ), ρT }], ϒjj(n) = −trB [(fˆjn · · · fˆj1 {(i Q
which goes with the Heisenberg equation of motion of ˙ j = [Qj , HS ] + [Qj , Qj ]fˆj . (B.3) iQ j
From the Wick’s contraction relations presented right before Eq. (4.15), we have trB [(fˆjn · · · fˆj1 )◦ {fˆj , ρT }]
(A.10)
= 2ρjj(n+1) +
ϒjj(n)
This completes the derivation of Eq. (3.16), since had been given in Eq. (A.7). To close this appendix, it is worth to scrutinize the starting relation in Eq. (A.1), the second identity there, jj(n) . (a) For the first term inside after the definition of ϒ the curly brackets, we comment on the physical posi j )> aˆ j¯ ] there. It is consistent with the un j δ F tion of [(δ F derlying algebra in Eqs. (3.4) and (A.2). Furthermore, it is related to the later involvement fˆj -dissipatons, as in˙ and treated in Eq. (A.6). Note duced from a˙ˆ j via δ(t) j = −δ F j aˆ j¯ . Thus, δ F j (fˆjn · · · fˆj1 )◦ aˆ j¯ δ F j also that aˆ j¯ δ F n+1 ˆ ◦ j δ F j aˆ j¯ , which was used in writing = (−) (fjn · · · fˆj1 ) δ F the middle line of Eq. (A.1); (b) For the second term, the last line of Eq. (A.1), we focus on the sign of ( − )n + 1 . It is associated with the bath-subspace trace cyclic permutation on a fermionic bath operator of (FˆB> )-type, {n} {n} (A.11) trB FˆB> Oˆ S+B = (−)n+1 trB Oˆ S+B FˆB> . {n}
Concerned for Oˆ S+B here, an operator truly in the total system-plus-bath composite space, is only about its odd/even (n = 2m + 1 or 2m) parity. Equation (A.11) was exploited {n} j and Oˆ S+B in the last line of Eq. (A.1), with FˆB> = δ F j . On the other hand, the (FˆB< )-type = (fˆjn · · · fˆj1 )◦ ρT aˆ j¯ δ F counterpart of Eq. (A.11) reads {n} {n} (A.12) trB Oˆ S+B FˆB< = (−)n trB FˆB< Oˆ S+B . This had been used in other places, such as Eq. (3.5) and Eq. (A.6). The additional minus sign in Eq. (A.11) is due ˆ to an additional hidden system a-operator, which arises from the forward path and thus physically locates ahead the bathsubspace trace (trB ). APPENDIX B: TREATMENT OF WHITE-NOISE RESIDUE CONTRIBUTION: SPIN BATH
Let us start with the spin-bath counterpart of Eq. (3.14), which together with Eq. (A.10) can be written as
j Qj , ϒjj(n) . (B.1)
Qj , (n) j(n) = −i jj = − j
j
˙ a δ(t), As the coefficients in δCaa (t) ≈ i 2 had been explic a there, ϒjj(n) in j ≡ itly specified in the last identity, with Eq. (B.1) and below will no longer depend on these parameters. Following the steps of derivation to Eq. (A.4) we have
(B.2)
n
δj j˜r (ηjr + ηj∗r )ρj(n−1) . − r
(B.4)
r=1
This together with Eq. (B.3) lead to Eq. (B.2) the expression, [Qj , Qj ]ρjj(n+1) ϒjj(n) = − [Qj , H ], ρj(n) − 2 j
−
n (ηjr + ηj∗r )[Qj , Qj˜r ]ρj(n−1) . − r
(B.5)
r=1
Substituting it to Eq. (B.1) and noticing that j = a in the present study, we obtain Eq. (3.17). 1 H.
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