diff --git "a/69E0T4oBgHgl3EQfwAF2/content/tmp_files/load_file.txt" "b/69E0T4oBgHgl3EQfwAF2/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/69E0T4oBgHgl3EQfwAF2/content/tmp_files/load_file.txt" @@ -0,0 +1,770 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf,len=769 +page_content='A hierarchical equations of motion (HEOM) analog for systems with delay: illustrated on inter-cavity photon propagation Robert Fuchs1 and Marten Richter1, ∗ 1Institut f¨ur Theoretische Physik, Nichtlineare Optik und Quantenelektronik, Technische Universit¨at Berlin, Hardenbergstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' 36, EW 7-1, 10623 Berlin, Germany (Dated: January 9, 2023) Over the last two decades, the hierarchical equations of motion (HEOM) of Tanimura and Kubo have become the equation of motion-based tool for numerically exact calculations of system-bath problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The HEOM is today generalized to many cases of dissipation and transfer processes through an external bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' In spatially extended photonic systems, the propagation of photons through the bath leads to retardation/delays in the coupling of quantum emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Here, the idea behind the HEOM derivation is generalized to the case of photon retardation and applied to the simple example of two dielectric slabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The derived equations provide a simple reliable framework for describing retardation and may provide an alternative to path integral treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' After the hierarchical equations of motion (HEOM) were initially invented by Tanimura and Kubo [1, 2] to solve numerically exactly the open quantum system prob- lem with a Debye spectral density, the HEOM did not immediately take off, since the limited numeric capabil- ities did not allow for a versatile implementation at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' However, the idea to use the time constant deriva- tive of the Debye spectral density time correlation func- tion stuck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Recently, various implementations [2–7] of HEOM followed after sufficient computing power became available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Soon after its invention, many generalizations using arbitrary spectral densities by decomposition into summed Debye form spectral densities were also devel- oped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' For most system-bath approaches it provides a well-established path to a numerically exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' A different type of system-bath problem is the prop- agation of quantum states e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' through a bath of pho- tons or phonons [8–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' A typical problem is describing quantum interconnects for quantum computing and cryp- tography applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Recently, various applications of these systems with a delay caused by the propagation through the bath were investigated [8–17] including the development of different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' However, the number of propagating photons is still limited, as it was for the open quantum systems approaches until HEOM imple- mentations became widespread, along with other meth- ods such as tensor networks [14, 15, 18–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' In this pa- per, an analysis of the HEOM derivation in the context of delay is carried out and HEOM analog equations for systems with delay are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' We demonstrate that the approach leads to a systematic set of equations ordered by the number of photons propagating through the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' In the future, combinations with, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=', tensor networks or automatic derivation may lead to an additional route to solve problems involving delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The paper starts with a derivation of the HEOM ana- log for open quantum systems with delay and illustrates its potential with a simple photon propagation example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' HEOM derivation: An HEOM analog with delay is derived for an open quantum system with: H = Hs + Hb + Hsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Here, Hs is the Hamiltonian of the system, which consists of quantum emitters in different spatially separated cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Hb is the bath Hamiltonian contain- ing the propagating photon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Finally, Hsb is the system-bath coupling Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' In open quantum systems, only the observables of the system are of in- terest, which can be calculated from the relevant density matrix ρs(t) = trB(ρ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Its calculation is the main objective of HEOM, where we transfer the steps by Tan- imura and Kubo [1] to systems with delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' We assume a factorized initial state ρ(t0) = ρs(t0) ⊗ ρB, where ρB is a harmonic bath state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The system dynamics obey: ρs(t) = trB (T←U(t, t0) exp � − i ℏ � t t0 dτU(t0, τ)Hsb,−(τ)U(τ, t0) � ρs(t0) ⊗ ρB) , (1) where ALρ = Aρ, ARρ = ρA, and A− = AL − AR define the Liouville space operators acting on Liouville operator ρ for any Hilbert space operator A [34] and U(t, t0) = T←exp � − i ℏ � t t0 dτ(Hs,−(τ) + Hb,−(τ)) � with time ordering operator T←.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Hs,−(τ) may also contain Lindblad operators for describing external processes act- ing on the joint system-bath state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Following the HEOM derivation [1] and the path integral derivation from [17], we convert Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (1) to path integral form: ρs(t) = trB ( T← M � i=0 Ui,i−1exp �� t0+εi t0+ε(i−1) dτU † i−1(τ)Hsb,−(τ)Ui−1(τ) � ρs(t0) ⊗ ρB) (2) with ε = (t − t0)/M and M → ∞ (in the fol- lowing equations the limit is always assumed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Fur- thermore, Ui,j = U(t0 + εi, t0 + εj) and Ui(τ) = U(τ, t0 + ε(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' For small ε, the approxima- tion Ui,i−1 exp �� t0+εi t0+ε(i−1) dτU † i−1(τ)Hsb(τ)Ui−1(τ) � ≈ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='02626v1 [quant-ph] 6 Jan 2023 2 Ui,i−1 + ε · Ui,i−1/2Hsb(t0 + ε(i − 1/2))Ui−1/2,i−1 =: Ui,i−1 + ε · U (1) sb (i) holds, yielding: ρs(t) = trB � T← M � i=0 (Ui,i−1 + ε · U (1) sb (i))ρs(t0) ⊗ ρB � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' We assume linear system-bath coupling: Hsb = � ijµ CijµAijBµ with system Aij and linear bath opera- tor Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' For a system A and bath B Liouville operator the relation (AB)− = A+B− + A−B+ holds, so U (1) sb (i) can be written as a sum over products of the system and bath operators U (1) sb (i) = � l A(1) l (i)B(1) l (i), and we define A(0) l = U s i,i−1δl,0 and B(0) l = U b i,i−1δl,0 with the system and bath parts of Ui,i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' With these relations, we write ρS in terms of a system part S and an influence functional (similar form as in [17]), ρs(t) = 1 � k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' kM=0 � l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' lM � M � i=1 εki � S(k1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , kMlM) × I(k1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , kMlM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (3) The system part is still an operator S(k1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , kMlM) = T← �M i=1 A(ki) li (i)ρs(t0), while the influence functional I(k1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , kMlM) = trB(T← �M i=1 B(ki) li (i)ρB) is just a number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Since ρB is assumed to be a harmonic bath equi- librium state, Wick’s theorem allows us to factorize the influence functional I into expectation values of two bath operators B(1) l (·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Furthermore, for small ε, the system propagator is roughly U s i,i−1 ≈ Ids − i ℏεHs,−(t0 + ε(i − 1/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Using the approximations of the time propagators Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (a) Model of two open QNM cavities with dissi- pation rates γµ and effective inter-cavity coupling strength Vµη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (b) 1D model with two slabs of width L = 21 µm with constant permittivity ϵR = π2 serving as QNM cavities, sit- ting against a background ϵB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (c) Scheme of the HEOM depicting a process including inter-cavity transfer and dissi- pation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' and using Wick’s theorem we obtain, ρs(t + ε) ≈ ρs(t) − ε i ℏHs,−(t0 + ε(M + 1/2))ρs(t) + � lM+1 T←εA(1) lM+1 � k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' kM � l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' lM � M � i=1 εki � Aki li (i)ρs(t0) × M � m=1 trB(B(1) lM+1(M + 1)U B M,m+1B(1) lm (m)ρB)δkm,1 I(k1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , km−1lm−1, 00, km+1lm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , kMlM), including only the terms at most linear in ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Collecting the terms linear in ε yields the derivative of ρs[1]: ∂tρs(t) = − i ℏHs,−(t)ρs(t) (4) + � l˜l A(1) l (t) � t t0 dt1⟨B(1) l (t)B(1) ˜l (t1)⟩Bρ(1) s˜l (t, t1), where ⟨A⟩B = trB(AρB) and the bath correlation func- tion is in the interaction picture, and ρ(1) sl (t, ˜t) = δkm1δlml � T← M � i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='i̸=m B(ki) li (i) � B � k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='kM � l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='lM T←A(1) l (m) � � M � i=1,i̸=m εkiA(ki) li (i) � � ρs(t0), with ˜t = mε + t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Here, the derivation deviates from the original recipe of Kubo and Tanimura, since the as- sumption of a spectral density in Debye form (simple exponential e−γt in time) is not compatible with systems including delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Generalizations of HEOM usually rely on a decomposition of the spectral density into a sum of exponential functions to recover the Debye form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' How- ever, an expansion of the correlation function for the de- lay case using e−γ|t−tdelay| does not yield the advantages of Kubo’s and Tanimura’s approach, since the original relies on a time constant derivative of the Debye spec- tral density time correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Instead, a delayed correlation of the above form introduces a sign change at t = tdelay, so that a dependence of ρ(n) on earlier inte- gration times is unavoidable in the case with delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Thus the integration over t1 is not included in the definition of ρ(1) in contrast to the original HEOM [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Keeping the general form of the bath correlation function is more flexible than using a special form, which would simplify the equations of motion in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' ρ(1) sl (·, t1) de- scribes bath disturbances to the system density matrix, which are initially caused by an interaction with A(1) ˜l at time t1 (similar to the auxiliary dimensions in extended TCL [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Of course, the additional time argument pre- vents direct numerical implementations for increasing n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' But specific bath correlation functions together with an- alytic calculation or tensor network methods [14, 15, 29– 33, 36–40] will allow solutions nevertheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Using the ER ER3 same technique as for ∂tρs(t) yields: ∂tρ(1) sl1(t, t1) = − i ℏHs,−(t)ρ(1) sl1(t, t1) + � l2˜l2 A(1) l2 (t) � t t0 dt2⟨B(1) l2 (t)B(1) ˜l2 (t2)⟩Bρ(2) sl1˜l2(t, t2, t1) + δ(t − t1)A(1) l1 (t1)ρs(t1 − 0+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (5) where we use the interaction picture for the bath correla- tion function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Instead of an initial condition ρ(1) sl1(t1, t1) = Al1(t1)ρs(t1 − 0+), the δ term at the time of the initial condition is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=', ρ(1) sl1(·, t1) is equal to zero (in the delta case) or not defined (in the initial condition case) before time t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Note that t1, t2 of ρ(2)(t, t2, t1) are not time ordered since different delay/retardation times can occur in open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The form of ρ(2) points to a general definition of ρ(n) starting with ρ(0)(t) = ρs(t): ρ(n) s˜l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' ˜ln(t, ˜tn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , ˜t1) = � k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='kM � l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='lM T← � � n � j=1 A(1) ˜lj (mj)δ˜lj,lmj δ˜kmj 1 � � � � M � j=1,∧n i=1j̸=mi εkjA(kj) lj (j) � � ρs(t0) � T← M � j=1,∧n i=1j̸=mi B(kj) lj (j) � B , (6) with ˜ti = miε + t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Analogous to ρ(1), this yields: ∂tρ(n) sl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' ln(t, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , tn) = − i ℏHs,−(t)ρ(n) sl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' ln(t, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , tn) + � ln+1˜ln+1 A(1) ln+1(t) � t t0 dtn+1⟨B(1) ln+1(t)B(1) ˜ln+1(tn+1)⟩B ρ(n+1) sl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='ln˜ln+1(t, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , tn+1) + n � p=1 δ(t − tp)A(1) lp (tp) (7) × ρ(n−1) sl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' lp−1lp+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' ln(tp − 0+, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , tp−1, tp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' tn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The last term is again a replacement to an initial condi- tion: ρ(n) sl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' ln(tp, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , tn) = Alp(tp)ρ(n−1) sl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' lp−1lp+1ln(tp − 0+, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , tp−1, tp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , tn) with tp = maxi(ti), and it is clear that ρ(n) sl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' ln(t, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , tn) = 0 for t < tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' So for the last term only p with the largest time tp contributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Fur- thermore ρ(n) s is invariant under permutations of t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , tn including their corresponding l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' , ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The physics behind Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (7) is very accessible: n corre- sponds to the maximum number of photons traveling in the bath at a given time t, so an exact truncation of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (a) Single-photon occupations of the two 1D dielec- tric slabs from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The dotted lines show the full reference wave function solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (b) Absolute values of the two-photon coherences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' For both cases, the QNM frequencies of the slabs are identical ˜ω1 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='06 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='0124i) eV, with coupling strength VBA = VAB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='0062 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' the equations based on the traveling photons is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Note, the photons on the left and right side states of the density matrix count accumulating, so a transfer of a single photon density requires two traveling photons (left and right side of density matrix), as opposed to one trav- eling photon for a single photon coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' For other open quantum system equations of motion techniques such as Nakajima-Zwanzig [41] or time convolution less (TCL) equations [41], the generators K in the equations of motion contain the system-bath coupling in any order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' A calculation of higher-order contributions from K is gen- erally cumbersome involving higher products of system- bath correlation functions as well as a truncation at a given photon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' For the HEOM analog, only one system-bath correlation function appears in the second term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (7) cleanly separating on photon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The first term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (7) describes the system dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The second term represents the absorption of a bath photon, which entered the bath at time tn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The last term describes photon emission into the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Two photon propagation: As a benchmark for the new approach, we consider two spatially separated quasinor- mal mode (QNM) cavities, coupled to a common pho- tonic bath (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The QNMs ˜fµ are an open sys- tem analog to normal modes, which solve the Helmholtz equation under an outgoing radiation condition [42–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' QNMs have complex eigenfrequencies ˜ωµ = ωµ−iγµ with photon decay rate γµ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Here, two dielectric slabs serve as QNM cavities as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' We assume an effective 1D problem with homogenous continuation in the y, z direction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The model allows the analytical calculation of the modes (assuming a constant real permittivity ϵR) and coupling elements (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' We include only the lowest energy QNM, assuming that all other modes are off-resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Since the slabs are identical, both have the same frequency ˜ωA = ˜ωB = ˜ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' However, we keep the indices for generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The slabs are separated by the distance R, which is large enough for a separate quanti- zation of the modes without inter-cavity coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' As a first step, we consider a population with initially 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='00 (a) slab A (b) (20ps/00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='75 slab B Occupation (02/ps/00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='75 Amplitude (11/ps/00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='00 0 100 200 300 0 100 200 300 t [ps] t [ps]4 one excitation (one photon on each side of the density matrix) in slab A and a vacuum bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Therefore, the hierarchy truncates at n = 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' ρ(n) = 0, n > 2, and: ρ(2) s,l1,l2(t, t1, t2) = Θ(t1 − t2)U s(t, t1)A(1) l1 (t1)ρ(1) s,l2(t1 − 0+, t2) + Θ(t2 − t1)U s(t, t2)A(1) l2 (t2)ρ(1) s,l1(t2 − 0+, t1), (8) using the initial conditions for ρ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The truncation is exact since the maximal number of propagating photons at any time is set by the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (8) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (5), we obtain: ∂tρ(1) sl1(t, t1) = − i ℏHs,−(t)ρ(1) sl1(t, t1) + � l2˜l2 A(1) ˜l2 (t) � t1 t0 dt2⟨B(1) ˜l2 (t)B(1) l2 (t2)⟩B × U s(t, t1)A(1) l1 (t1)ρ(1) sl2(t1 − 0+, t2) + � l2˜l2 A(1) ˜l2 (t) � t t1 dt2⟨B(1) ˜l2 (t)B(1) l2 (t2)⟩B × U s(t, t2)A(1) l2 (t2)ρ(1) sl1(t2 − 0+, t1) + δ(t − t1)A(1) l1 (t1)ρs(t1 − 0+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (9) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (9) and (4) form a closed set of equations of motion for the system density matrix that are exactly solvable (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' [49]) for at most two traveling photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' 1(c) illustrates connections between the equations with one photon traveling from time t1 = t − τ until t through the bath, requiring the calculation of ρ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Intermittently a second photon is emitted into the bath at t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The dynamics of a specific system are deter- mined by the system-bath correlation function trB(B(1) ˜l2 (t)B(1) l2 (t1)ρB), which describes the emission of a photon into the bath at time t1 and reabsorption at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The correlation function results from the system-bath coupling and reads (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' [49]): Cµη(t − t′) ≈ 2Vµηℏ2 (Θ(t − t′)δ(t − t′ − τ) + Θ(t′ − t)δ(t − t′ + τ)) , (10) where µ, η are cavity indices (A or B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The coupling strength is given by Vµη = (1 + δµη)γ1/2 with the cavity decay rate γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Due to topology, the inter-cavity coupling is exactly half the dissipation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' For the 1D case, a photon emitted away from the other cavity will not return, while a photon emitted towards the other cavity can be transferred into that cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' In higher dimensions, the inter-cavity coupling will be much smaller than the dissipation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The delay time τ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (10) depends implicitly on the involved cavities, with τ = R/c, µ ̸= η, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' For one initial excitation, three system states |A⟩ = |10⟩, |B⟩ = |01⟩, |0⟩ = |00⟩ contribute, with the excitation in slab A or B, or both slabs in the ground state, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (10) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (4) yields the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' As an example, the occupation in slab A ⟨A|ρs(t)|A⟩ evolves as (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' [49]): ∂t⟨A|ρs(t)|A⟩ = −2γA⟨A|ρs(t)|A⟩ − 2V ∗ BAe−iωBτ⟨0|ρ(1)L s,0B(t, t − τ)|A⟩ + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (11) For the auxiliary density matrix ρ(1), starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (9) results in (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' [49]): ∂t⟨0|ρ(1)L s,0B(t, t1)|A⟩ = δ(t − t1)⟨B|ρs(t1)|A⟩ = −γA⟨0|ρ(1)L s,0B(t, t1)|A⟩ − 2VBAeiωBτ⟨B|ρ(1)R s,B0(t1, t − τ)|0⟩ − 2VBAeiωAτ⟨0|ρ(1)L s,0B(t − τ, t1)|A⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (12) The remaining equations for the occupation in B, the coherences, and matrix elements for ρ(1) are calculated analogously (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' For time-non-local interactions, the system density matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' (11) only couples to the first auxiliary density matrix ρ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Time-local processes such as the cavity photon dissipation are included in the zeroth step of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' 2(a) shows the time dynamics of the single-photon occupations in slabs A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The model system allows a calculation using the wave function (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' [49]) as a bench- mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The HEOM (solid lines) and exact wave func- tion (dotted lines) results agree perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Note, that the HEOM allows in principle the inclusion of Lindblad terms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' for pumping), which the wave function does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Over time the single excitation in slab A will dissipate into the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' However, some photons are transferred to the QNM of slab B with delay τ ≈ 44 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' For the used parameters, the occupation in B is even larger than the occupation in A after some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Eventually, the system arrives at a trapped state [14, 50–55] due to constructive interference from the inter-cavity transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Note, that the HEOM allows in principle the inclu- sion of Lindblad terms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' for pumping), which the wave function does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Also an extension to two pho- ton processes is feasible for the HEOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' 2(b) shows the two-photon coherences (two photons on one side of the density matrix, none on the other) for the two slabs from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' 1(b), which includes at most two trav- eling photons, resulting in a calculation analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The amplitudes of the intra-cavity coherences ⟨20|ρs|00⟩/⟨02|ρs|00⟩ resemble the dynamics of the den- sities in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' 2(a), since in principle the same indepen- dent processes are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The inter-cavity coherence ⟨11|ρs|00⟩ requires the transfer of just one photon and thus shows a rapid increase after t = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' In the final equilibrium state the probability (coherence squared) of an inter-cavity contribution matches the sum of the two intra-cavity probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' 5 A feasible calculation of the exact solution as shown here is limited to a small number of photons by the ex- ponential scaling of the numerical complexity with the number of excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' For systems requiring a higher number of traveling photons, a calculation of the higher steps in the hierarchy via matrix product states or other tensor networks [14, 15, 29–33, 36–40] may be possible as well as analytic calculations in special setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Further- more, the HEOM allows a perturbative truncation of the hierarchy for systems with a small system-bath coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Thus, at least an approximate solution may be possible for higher excitation numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' In conclusion, we analyzed the derivation of hierar- chical equations of motion and transferred the idea to open quantum systems with delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The resulting equa- tions allow a natural, easy truncation on the number of excitations in the bath, which is otherwise cumbersome for Nakajima-Zwanzig or time convolution-less equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The first implementation for single- and multi-photon transfer between two cavities demonstrated the feasibility of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' We expect that in the future more de- manding implementations including tensor network ap- proaches may allow the simulation of several photons traveling through complex quantum networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' ∗ marten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='richter@tu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content='de [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Tanimura and R.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' A 53, 1818 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' [60] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' Mukamel, Principles of nonlinear optical spectroscopy, 6 (Oxford University Press on Demand, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' S1 Analytic coupling elements We use analytic expressions of the mode frequencies, decay constants, and coupling elements for numeric eval- uation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' For linearly polarized waves and assuming a ho- mogeneous continuation in y, z-direction, the problem re- duces to the 1D model from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQfwAF2/content/2301.02626v1.pdf'} +page_content=' The QNM within each slab is given by [48, 56] ˜fµ(x) ��� |x|