Coarse-Grained Dynamics of Open Quantum Systems

Transcription

1 Coarse-Grained Dynamics of Open Quantum Systems Senior Thesis Presented to The Faculty of the School of Arts and Sciences Brandeis University Undergraduate Program in Physics In partial fulfillment of the requirements for the degree of Bachelor of Science Syler Kaso B.S., Physics and Mathematics Brandeis University Advisor: Albion Lawrence May 214

2 Abstract The purpose of this research is to study the dynamical evolution of a system containing entangled degrees of freedom at two separated energy scales. Given a measuring device that is insensitive to the high-frequency oscillations of the high-energy quanta, we are interested in describing the observable dynamics of the low-energy subsystem, which will depend on interactions with the inaccessible degrees of freedom. To do so, we time-average the evolution equation for the reduced density matrix associated with the low-energy subsystem and find an expression for the evolution in second-order perturbation theory. i

3 Contents 1 Introduction 1 2 Evolution Equation General System Setup Perturbation Theory Coupled Spins Setup Evolution Equation Coarse Graining General System Separation of Time Scales Time-Averaged Evolution Equation IR Operators Linear Oscillators Two Linear Oscillators IR System Coupled with UV Oscillators Conclusions 36 A Alternate Calculation of Coupled Spins 37 A.1 State and Energy Corrections A.1.1 First-Order Corrections in λ A.1.2 Second-Order Corrections in λ A.2 Time Evolution Calculation A.2.1 Summary of State and Energy Corrections A.2.2 Time Evolution of the First Excited Spin State ii

4 Bibliography 47 Acnowledgments 48 iii

5 Chapter 1 Introduction Many physical systems contain a certain subset of degrees of freedom that are inaccessible by experiment. When such systems have an interaction term in their Hamiltonian that couples these inaccessible degrees of freedom with the measurable degrees of freedom, the presence of the inaccessible modes causes interesting dynamics for the accessible subsystem when viewed independently. These dynamics are not captured by the usual language of unitary time evolution; in fact, we find that the evolution of the accessible subsystem is nonlocal in time due to an oscillatory effect in which information is transferred from the accessible to the inaccessible modes and then returned at some later time. A successful description of these dynamics requires the development of a new formalism. In this paper we are interested in studying systems in which the inaccessible degrees of freedom are considered to be ultraviolet UV quanta with a characteristic energy beyond experimental resolution, and there is a large energy difference between these and the characteristic energy of the measurable degrees of freedom, which are infrared IR quanta. An example of such a system is that of quantum Brownian motion modeled as a massive harmonic oscillator coupled linearly with an environment consisting of much lighter harmonic oscillators 1. This paper is split into two main sections. In Chapter 2 we develop the aforementioned 1

6 2 Introduction formalism for describing the dynamics of the open quantum systems of interest. We find an evolution equation in terms of a unitary piece involving an effective Hamiltonian for the IR subsystem and a non-unitary, nonlocal piece, both of which are identified to second order in perturbation theory. Finally, we consider an example system of coupled IR and UV spins. In Chapter 3 we coarse-grain our theory in order to tae into account the finite time resolution of our hypothetical measuring apparatus, assumed to be greater than the timescale of the UV modes and less than that of the IR modes. We find an evolution equation for the timeaveraged IR subsystem that is local in time in terms of time-averaged operators, which are found to have constant or trivial time-dependence.

7 Chapter 2 Evolution Equation 2.1 General System For a system whose Hilbert space has a basis { α i }, the density operator is defined by σ i p i α i α i where the p i obey i p i = 1 and could represent the fraction of members of the system which are in the state α i, or the observer s uncertainty about which state the system is in, or a combination of the two. The density operator completely characterizes a system as it contains all measurable information about the system, encapsulated by the formula for the expectation value of an arbitrary operator A in the Hilbert space: A = trσa The systems we will study in this paper can be decomposed into two subsystems, IR and UV, corresponding to a decomposition of the Hilbert space of the form H = H IR H UV. In cases where we have access only to the IR subsystem, all accessible information is contained in the reduced density matrix, given by tracing out the inaccessible U V subsystem: ρ σ IR = tr HUV σ

8 4 Evolution Equation Note that ρ is an operator on the IR subsystem alone. Our goal in this section is to develop general equations for the time evolution of ρ Setup For the Hilbert space described above, we consider the total Hamiltonian H = H IR + H UV + λv IR,UV Here H IR H IR I UV acts only on the IR subsystem, H UV I IR H UV acts only on the UV subsystem, V IR,UV contains the interaction between the two subsystems, and the parameter λ controls the size of the interaction. We denote the basis of the Hilbert space by i, u i u, such that H IR + H UV i, u = E i + E u i, u E iu i, u, where i, j, ī, and j will label the IR degrees of freedom and u, v, ū, and v label the UV ones. We also begin with the assumption that at initial time t = the state of the system is factorized as ψt = = IR ū where IR is an arbitrary state in the IR subsystem. We mae this assumption in order to simplify the calculations below, but since we will find that the system quicly evolves to a non-factorizable state we do not expect this initial condition to qualitatively affect the physics involved. Moreover, the assumption seems reasonable to mae for relevant physical scenarios; in the spin system to be treated below, for example, the ground state is factorized. According to the Schrödinger equation, the evolution of an arbitrary pure state ψt is given by i t ψt = H ψt where t. Using 2.1.6, the time evolution of σ is found to be t i t σt = H, σt 2.1.7

9 2.1 General System is nown as the von Neumann equation. We wish to derive a similar equation for the evolution of the IR subsystem of the form i t ρt = H eff t, ρt + Γt where H eff is the so-called effective Hamiltonian for the IR subsystem and the term Γt isolates the non-hamiltonian piece of the evolution of the IR subsystem. First, we write ρ in component form as ρ ij t i ρt j = u i, u σt j, u Here σt = ψt ψt due to our assumption about the initial state. Next, we differentiate both sides of this equation with respect to t, multiply by a factor of i, and use equation to write i t ρ ij t = u i, u i t σt j, u = u i, u H, σt j, u For brevity we write as H = H +λv where H H IR +H UV and V V IR,UV. Then i t ρ ij t = λ u i, u V, σt j, u + i, u H, σt j, u = λ u i, u V, σt j, u + u E iu E ju i, u σt j, u = λ u i, u V, σt j, u + E i E j ρ ij t Next we define an IR operator H eff by H eff ij = E i + Eūδ ij, which appears to be the zeroth-order Hamiltonian for the observable subsystem. It can be seen that H eff, ρt ij = E i E j ρ ij t such that can be written i t ρ ij t H eff, ρt ij = λ u i, u V, σt j, u If we find an expression for ρt up to first order in perturbation theory we can use to derive second-order results, since σt is already multiplied by λ.

10 6 Evolution Equation Perturbation Theory We wor in the interaction picture in which the state is defined to be ψt I e iht/ ψt and the perturbation is given by V I t e iht/ V e iht/. It follows from these definitions that the Schödinger equation in the interaction picture becomes i t ψt I = V I ψt I This can be written in the integral form ψt I = ψ i t λ dt V I t ψt I where I have used the fact that ψ I = ψ. To zeroth order that is, setting λ = in equation we simply get ψ t I = ψ. We find the next higher-order term by substituting this solution into : ψt I ψ t I + ψ 1 t I = ψ i t λ dt V I t ψ ψ 1 t I = i t λ dt V I t ψ Rewriting the state ets in the Schrödinger picture rather than the interaction picture, ψ t = e ih t/ ψ ψ 1 t = i λe ih t/ t dt V I t e iht/ ψ t = i t λ dt V I t t ψ t Following the same procedure iteratively, we find ψ t = i t λ dt V I t t ψ 1 t We can use this perturbative expansion of ψt to find a perturbative expansion for the reduced density matrix as ρ ij ρ ij + ρ 1 ij. From equation 2.1.9, ρ ij t = u i, u ψ t ψ t j, u = u i, u IR ū IR ū j, u

11 2.1 General System 7 = i, ū ψ t ψ t j, ū = i, ū σ t j, ū ρ 1 ij t = i, u ψ t ψ 1 t j, u + i, u ψ 1 t ψ t j, u u = i, ū σ 1 t j, ū where σ t = ψ t ψ t, σ 1 t = ψ t ψ 1 t + ψ 1 t ψ t Next, we wish to identify from the right-hand side of equation the first- and second-order commutators between the reduced density matrix and the effective Hamiltonian aside from terms involving H eff, which are accounted for in the left-hand side of For that purpose, we will use equations and H, ρ ij = j = j H, ρ 1 ij = j H i j ρ jj ρ i j H jj H i j H i j j, ū σ t j, ū i, ū σ t j, ū H j, ū σ 1 t j, ū i, ū σ 1 t j, ū H jj jj The first-order term of the right-hand side of the evolution equation, , is { λ u } 1 i, u V, σt j, u = λ i, u V, σ t j, u = λ u,v, j u i, u V j, v j, v σ t j, u i, u σ t j, v j, v V j, u = λ j i, ū V j, ū j, ū σ t j, ū i, ū σ t j, ū j, ū V j, ū where in we have included a complete set of states j,v j, v j, v = I IR I UV and in we have used the fact that σ t annihilates any eigenstate of H UV other than ū. Comparing to , we see that we can write { λ u i, u V, σt j, u } 1 = H 1 eff, ρ ij

12 8 Evolution Equation where we have defined H 1 eff ij λ i, ū V j, ū The second-order term of the right-hand side of is given by { λ u } 2 i, u V, σt j, u = λ i, u V, σ 1 t j, u u = λ i, ū V, σ 1 t j, ū + λ u ū i, u V, σ 1 t j, u The last term in the equation above cannot come from the Hamiltonian piece of the evolution equation since neither ρ t nor ρ 1 t involve UV modes other than ū, so it must contribute to Γt. We write the remaining term in as λ i, ū V, σ 1 t j, ū = λ j,u i, ū V j, u j, u σ 1 t j, ū i, ū σ 1 t j, u j, u V j, ū = λ j i, ū V j, ū j, ū σ 1 t j, ū i, ū σ 1 t j, ū j, ū V j, ū + λ u ū, j i, ū V j, u j, u σ 1 t j, ū i, ū σ 1 t j, u j, u V j, ū = H 1 eff ij, ρ 1 t ij + λ i, ū V j, u j, u ψ 1 t ψ t j, ū i, ū ψ t ψ 1 t j, u j, u V j, ū u ū, j where in the first term we identify the commutator using equations and and in the second term we use the fact that u ψ t = for u ū. Next, substituting the expansion of ψ 1 t from equation into , the last term becomes i t λ dt u ū, j i, ū V j, u j, u V I t t ψ t ψ t j, ū + i, ū ψ t ψ t V I t t j, u j, u V j, ū = i t λ dt i, ū V j, u j, u V I t t ī, ū ī, ū σ t j, ū u ū, j,ī

13 2.1 General System 9 + i, ū σ t ī, ū ī, ū V I t t j, u j, u V j, ū = Ltρ ρ L t ij where Lt ij i t λ dt i, ū V j, u j, u V I t t j, ū u ū, j Motivated by the fact that the second-order effective Hamiltonian must be Hermitian, we define H 2 eff 1 2 L + L to be the Hermitian part of L and A L L to be the antihermitian part. Then becomes H 2 eff t, ρ t ij + {A 2 t, ρ t} ij. Plugging all of this bac into 2.1.3, we find the result { λ u } 2 i, u V, σt j, u = λ i, u V, σ 1 t j, u + H 1 eff t, ρ1 t ij u ū + H 2 eff t, ρ t ij + {A 2 t, ρ t} ij Now that we now λ u i, u V, σt j, u up to second order in perturbation theory, we plug our results bac into to find i t ρ ij t = H eff, ρt ij + H 1 eff, ρ ij + H 2 eff t, ρ t ij + H 1 eff t, ρ1 t ij + {A 2 t, ρ t} ij + λ u ū i, u V, σ 1 t j, u We have achieved our goal of writing an evolution equation in the form of 2.1.8, up to second-order in perturbation theory. That is, we have i t ρt = H eff, ρt + Γt where H eff = H eff + H1 eff + H2 eff and Γt gives the non-hamiltonian piece of the evolution. A summary of the terms involved in our evolution equation is given below, written for compactness as IR operators rather than in component form. H eff = H IR + Eū; H 1 eff = λ ū V ū H 2 eff t = i t 2 λ2 dt ū V u u V I t t ū ū V I t t u u V ū u ū

14 1 Evolution Equation Γt = {A 2 t, ρ t} + λ u ū u V, σ 1 t u A 2 t = i 2 λ2 t dt u ū ū V u u V I t t ū + ū V I t t u u V ū Coupled Spins In this section we apply the formalism developed in section 2.1 to a simple example system of coupled spins and derive a result for ρt Setup Consider a system containing one IR spin degree of freedom coupled to M UV spins, in which the UV spins do not interact with one another. The Hamiltonian of this system is M H = H IR + H UV + λv IR,UV = µ IR BSIR z µ UV, BSUV, z + λs M IR S UV, =1 =1 where H IR = µ IR BS z IR and H UV = µ UV,BS z UV,. Here B > and µ UV, µ IR for all, where ranges from 1 to M. The last term in represents the interaction between the low-energy spin and each of the high-energy ones: V = S IR S UV, = SIRS z UV, z + 1 S IR S + UV, 2 + S+ IR S UV, where ranges from 1 to M. 1 S IR S + IR 1 The following proves the right hand side of equation 2.2.2: S + UV, = SIR x is y IR Sx UV, + is y UV, = SIRS x UV, x + S y IR Sy UV, + isx IRS y UV, Sy IR Sx UV, S UV, = SIR x + is y IR Sx UV, is y UV, = SIRS x UV, x + S y IR Sy UV, isx IRS y UV, Sy IR Sx UV, Thus, S IR S+ UV, + S+ IR S UV, = 2 SIRS z UV, z + 1 S IR 2 S+ UV, + S+ IR S UV, = S x IRS x UV, + S y IR Sy UV, SIRS x UV, x + S y IR Sy UV, + Sz IRSUV, z = S IR S UV,

15 2.2 Coupled Spins 11 The eigenstates of H = H IR + H UV are the factorized eigenstates of S z IR, Sz UV, : m IR, m UV m IR m UV,1 m UV,2 m UV,M where m IR ranges from j IR to j IR and m UV, from j UV, to j UV,. The corresponding zeroth-order energies are given by H m IR, m UV = B µ IR m IR + µ UV, m UV, m IR, m UV = E m IR m UV m IR, m UV The ground state of H is E 1 = j IR, j UV where j UV = j UV,1,..., j UV,M, and the first excited state of H is E 2 = j IR 1, j UV It will also be convenient to define an excitation of the th UV mode by E 3, = j IR, j UV ; j UV j UV,1,..., j UV, 1,..., j UV,M Evolution Equation The time evolution of the ground state j IR, j UV is trivial, as it can easily be shown that the reduced density matrix is given by ρ gs t = j IR j IR for all t. Instead, we wish to examine the time evolution of the IR subsystem if we start in the first excited state, ψ t = j IR 1, j UV = E 2. We will use the general expression for the evolution of the IR reduced density matrix found in Section 2.1 to calculate it in this example. Before proceeding, it will be useful to list simplified expressions for certain quantities from section 2.1. The first, second, and third pairs of equations below are calculated from

16 12 Evolution Equation equations , , and , respectively. ψ t = e ie 2 t/ j IR 1, j UV ψ 1 t = i λe ih t/ t e ih E 2 t / V j IR 1, j UV dt ρ ij t = i, j UV σ t j, j UV = i, j UV j IR 1, j UV j IR 1, j UV j, j UV = δ i,jir 1δ jir 1,j ρ 1 ij t = i, j UV σ 1 t j, j UV = e ie 2 t/ δ i,jir 1 ψ 1 t j, j UV + e ie 2 t/ δ jir 1,j i, j UV ψ 1 t H, ρ ij = H i,j IR 1 δ j IR 1,j δ i,jir 1H j IR 1,j H 1 eff, ρ1 ij = λ j i, j UV V j, j UV ρ 1 jj t ρ1 i j t j, j UV V j, j UV = λe ie 2 t/ i, j UV V j IR 1, j UV ψ 1 t j, j UV λe ie 2 t/ i, j UV ψ 1 t j IR 1, j UV V j, j UV + λ j e ie 2 t/ δ jir 1,j i, j UV V j, j UV j, j UV ψ 1 t e ie 2 t/ δ i,jir 1 ψ 1 t j, j UV j, j UV V j, j UV We will first show that all off-diagonal elements of ρ are zero. We begin by considering the two terms that compose Γ: { A 2, ρ t } ij = ī A 2 iī ρ īj t + ρ iī ta 2 īj = δ jir 1,jA 2 i,j IR 1 +δ i,j IR 1A 2 j IR 1,j where I used equation Plugging in the definition of A 2 from 2.1.4, { A 2, ρ t } ij = i 2 λ2 t dt u j UV, j δ jir 1,j i, j UV V j, u j, u V I t t j IR 1, j UV + i, j UV V I t t j, u j, u V j IR 1, j UV + δ i,jir 1 j IR 1, j UV V j, u j, u V I t t j, j UV

17 2.2 Coupled Spins 13 = i 2 λ2 t dt + j IR 1, j UV V I t t j, u j, u V j, j UV δ jir 1,j e ie ju E 2 t t/ i, j UV V j, u j, u V j IR 1, j UV u j UV, j + e ie i, j E jut t/ UV i, j UV V j, u j, u V j IR 1, j UV + e ie ju E j, j t t/ H δ i,jir 1 j IR 1, j UV V j, u j, u V j, j UV + e ie 2 E jut t/ j IR 1, j UV V j, u j, u V j, j UV We need an expression for the matrix elements of V, where V is given by in this example. i, m V j, n = i, m S IRS z UV, j, z n = 2 n jδ m n δ ij i, m S IR S+ UV, j, n i, m S+ IR S UV, j, n j UV, j UV, + 1 n n 1 j IR j IR + 1 jj + 1δ m, n δ i,j+1 j UV, j UV, + 1 n n + 1 j IR j IR + 1 ii 1δ m, n+δ i,j Thus, expressions of the form i, m V j, n j, n V j, m will be proportional to δ ij since the δ m n δ i j terms of the first matrix element will annihilate all but the δ n m δ jj term of the second, the δ m, n δ i, j+1 term of the first will annihilate all but the δ n, m+ δ j,j 1 term of the second, and so forth. Since every term in contains an expression of this form, { A 2 ρ t } ij is nonzero only when i = j. The other term contained in Γ is: λ i, u V, σ 1 t j, u = λ u ū u ū, j i, u V j, j UV j, j UV ψ t ψ 1 t j, u i, u ψ 1 t ψ t j, j UV j, j UV V j, u

18 14 Evolution Equation λ i, u V, σ 1 t j, u = λ u ū u ū e ie 2 t/ i, u V j IR 1, j UV ψ 1 t j, u e ie 2 t/ i, u ψ 1 t j IR 1, j UV V j, u Recall that E 2 E JL 1, j UV. Plugging in the expansion of ψ 1 t from 2.2.9, equation becomes λ 2 u j UV i t e ie 2 t/ i, u V j IR 1, j UV j IR 1, j UV e ih t / V e ih t / e ih t/ j, u + e ie 2 t/ i, u e iht/ e ih t / V e ih t / j IR 1, j UV j IR 1, j UV V j, u dt = λ 2 i t e ie ju E 2 t t/ i, u V j IR 1, j UV j IR 1, j UV V j, u u ū + e ie 2 E iut t/ i, u V j IR 1, j UV j IR 1, j UV V j, u dt Once again, each of these terms is proportional to a product of matrix elements of the form i, m V j, n j, n V j, m, and so is zero when i j. Thus, we have shown that Γ has no off-diagonal elements. Next we consider the commutators with H eff. Setting = 1 in , we find H 1 eff, ρ t ij = λδ jir 1,j i, j UV V j IR 1, j UV λδ i,jir 1 j IR 1, j UV V j, j UV It is clear from the expression for the matrix element of V in that H 1 eff, ρ t ij vanishes when i j. Next, using the fact that i, m V j, m δ ij in this system, we rewrite as H 1 eff, ρ1 ij = λe ie 2 t/ i, j UV V j IR 1, j UV ψ 1 t j, j UV λe ie 2 t/ i, j UV ψ 1 t j IR 1, j UV V j, j UV + λδ jir 1,je ie 2 t/ i, j UV V i, j UV i, j UV ψ 1 t λδ i,jir 1e ie 2 t/ ψ 1 t j, j UV j, j UV V j, j UV

19 2.2 Coupled Spins 15 and, setting = 2 in , we find H 2 eff, ρ t ij = i 2 λ2 t dt u j UV, j δ jir 1,j i, j UV V j, u j, u V I t t j IR 1, j UV i, j UV V I t t j, u j, u V j IR 1, j UV δ i,jir 1 j IR 1, j UV V j, u j, u V I t t j, j UV j IR 1, j UV V I t t j, u j, u V j Once again, plugging the expansion of ψ 1 t from equation into equations and , as in the derivation of , we see that H 1 eff, ρ1 t ij and H 2 eff, ρ t ij also vanish for i j. Finally, as always we have H eff, ρt ij = E i E j ρ ij t. For i j we have shown this is the only nonzero term in the right-hand side of the evolution equation, which becomes i dρ ijt dt = E i E j ρ ij t ρ ij t = Ce ie i E j t/ i j for some constant C depending on the initial conditions of the IR subsystem. In this example our initial state has ρ ij t = = except when i = j = j IR 1. Thus, ρ ij t = = C = ρ ij t = i j Next, we consider the diagonal elements. For i = j, H eff, ρt ii = E i E i ρ ij t = becomes H 1 eff, ρ t ii = λ j IR 1, j UV V j IR 1, j UV λ j IR 1, j UV V j IR 1, j UV = and similarly all four terms in can be seen to cancel, leaving H 2 eff, ρ t ii =. The diagonal elements of the final Hamiltonian commutator are given by see equation λ 2 H 1 eff, ρ1 ii = i λ2 t e ie i, juv E 2 t t / i, j UV V j IR 1, j UV 2 + e ie i, j UV E 2 t t / i, j UV V j IR 1, j UV 2 δ jir 1,i j IR 1, j UV V j IR 1, j UV 2

20 16 Evolution Equation δ i,jir 1 j IR 1, j UV V j IR 1, j UV From , the first two terms are also proportional to δ i,jir 1, such that the exponential factors in those terms disappear and all four components cancel, yielding H 1 eff, ρ1 ii =. Last, we find Γ ii t by considering separately the two terms of which Γ is composed. From , λ i, u V, σ 1 t i, u = λ 2 u ū And from , { A 2,ρ } ii = i 2 λ2 = 2λ 2 u j UV i t e ie iu E 2 t t/ i, u V j IR 1, j UV 2 + e ie 2 E iut t/ i, u V j IR 1, j UV 2 dt u j UV = 2iλ 2 t dt u j UV i u j UV, j t sin cos E 2 E iu t t/ i, u V j IR 1, j UV 2 E 2 E iu t/ δ jir 1,i E 2 E iu i, u V j IR 1, j UV e ie ju E 2t t/ j IR 1, j UV V j, u 2 + e ie 2 E jut t/ j IR 1, j UV V j, u 2 + e ie ju E 2 t t/ j IR 1, j UV V j, u 2 + e ie 2 E jut t/ j IR 1, j UV V j, u 2 = 2i λ2 δ jir 1,i = 2iλ 2 δ jir 1,i u j UV, j u j UV, j t sin cos E 2 E ju t t/ j IR 1, j UV V j, u 2 E 2 E ju t/ E 2 E ju j IR 1, j UV V j, u Using and the fact that S + UV, j UV =, all terms in the sums in equations and are zero except when u = j UV for some. We find sin E Γ ii t = 2iλ 2 2 E 3, t/ δ i,jir j E 2 E IR, j UV V j IR 1, j UV 2 3,

21 2.2 Coupled Spins 17 sin E 2iλ 2 2 E 3, t/ δ jir 1,i j E 2 E IR 1, j UV V j IR, j UV 2 3, = i λ2 4 j IR j IR + 1 j IR 1j IR j UV, j UV, + 1 j UV, j UV, 1 2 sin B µ IR j IR 1 + q µ H,qj UV,q + B µ IR j IR + q µ H,qj UV,q µ UV, t B µ IR j IR 1 + q µ H,qj UV,q + B µ IR j IR + q µ H,qj UV,q µ UV, δ i,jir δ jir 1,i = 2iλ 2 3 j IR j UV, Bµ UV, µ IR sin Bµ UV, µ IR t δ i,jir δ jir 1,i Setting this equal to i t ρ ij t in the evolution equation gives us t ρ ij t = 2λ i = j = j IR j IR j UV, Bµ UV, µ IR sin Bµ UV, µ IR t 1 i = j = j IR 1 otherwise Integrating and applying the initial condition ρ ij = δ i,jir 1δ j,jir 1 yields the final result ρ jir 1,j IR 1t = 1 2λ 2 2 j IR j UV, B 2 µ UV, µ IR 2 1 cos Bµ UV, µ IR t ρ jl,j L t = 2λ 2 2 j IR j UV, B 2 µ UV, µ IR 2 1 cos Bµ UV, µ IR t ρ ij t = for all other i, j In Appendix A we confirm this result by carrying out the calculation of ρt directly in perturbation theory by finding the eigenvalues and eigenstates of H to second order in λ. As can be seen from equations , for times t > ρ becomes a mixed state, meaning the initially factorizable state in the full Hilbert space is no longer factorizable at later times. Since the initial factorization appeared naturally in this example, this lends credence to our prior assumption that an initially factorizable state is a reasonable condition to impose in the general case.

22 Chapter 3 Coarse Graining We have successfully derived an expression for the dynamics of the subsystem of H to which we have access. However, up to this point we have implicitly assumed that our measuring device has infinitely fine resolution in time. In reality, however, any measurement we mae will have some finite time resolution δt. To tae this into account we coarse-grain our theory by calculating the time averages of the quantities found in the previous chapter, which correspond to what we could actually measure in practice. Given an operator Ot, we define its time-average to be Ot = f Eδt t tot dt 3..1 where f Eδt t t is a function centered about t = t with a width corresponding to δt = /E δt ; for example, f might be a Gaussian. For the systems considered in this paper, in eeping with our assumption that the UV quanta are inaccessible to us while the IR quanta are accessible, we assume that δt is greater than the time scale of the UV system but smaller than that of the IR system. In symbols, E IR E δt E UV, where E IR and E UV are the characteristic energies of the corresponding subsystems. In Section 3.1 we derive an evolution equation for the object that describes all of the dynamics actually available to us, namely ρt, the time-averaged reduced density matrix 18

23 3.1 General System 19 for the IR subsystem. In section 3.2 we study the application of this technique to a system of quantum oscillators with linear coupling. 3.1 General System We can time-average ρt according to 3..1: ρt = f Eδt t tρt dt The matrix elements of ρ describe the probability of a given outcome for a series of measurements attempted at time t with a device that has finite time resolution δt. Plugging into the evolution equation, i t ρt = i t f Eδt t tρt dt = i t f Eδt t tρt dt Integrate this expression by parts, the boundary term vanishes since f Eδt t t goes to zero far away from t = t, and we are left with i t ρt = f Eδt t ti t ρt dt = f Eδt t t H eff t, ρt + Γt dt = H eff t, ρt + Γt We would lie to find that H eff tρt H eff t ρt, such that if we can calculate H eff t and Γt then will give us an evolution equation for ρt. Let f Eδt be a Gaussian, defined by f Eδt t t = E δt π e t t 2 E 2 δt / Then the time average of an arbitrary operator Ot is Ot = Eδt π e t t 2 E 2 δt / 2 Ot dt 3.1.5

24 2 Coarse Graining Using the inverse Fourier transform Ot dω 2π e iωt Õω equation becomes Ot = = = E δt dωdt e t t 2 Eδt 2 / 2 +iωt Õω 2π dω dt E δt e Eδt 2 / 2 t 2 2tt iωt 2 /Eδt 2 +t+iω 2 /2Eδt 2 2 iωt+ω 2 /4Eδt 2 Õω 2π dω 2π e ω2 2 /4E 2 δt e iωt Õω The presence of the exponential means that if the function Ot has UV components they will be exponentially suppressed with respect to the IR components by a factor e ω2 UV ω2 IR 2 /4E 2 δt = e E2 UV /4E2 δt e E2 IR /4E2 δt. Since the parameter E UV /E δt 1, the exponential factor e E UV /E δt 2 is extremely small, so we will neglect any terms in our time-averaged expressions containing that factor. Thus, even if Ot has UV components, they will not enter into Ot. Using the Fourier method again we can calculate the time average of a product of arbitrary functions: F tgt = dt E δt e t t 2 Eδt 2 dω1 dω / 2 2 e iω 1+ω 2 t F ω1 π 2π Gω 2 dω1 dω 2 = e ω2 1 +ω2 2 +2ω 1ω 2 2 /4Eδt 2 e iω 1t+iω 2 t F ω1 2π Gω 2 dω1 dω 2 = e iω 1t ω1 2 2 /4Eδt 2 +iω 2t ω2 2 2 /4Eδt 2 e ω 1ω 2 2 /2Eδt 2 F ω1 2π Gω 2 1 n n dω1 = e iω 1t ω 2 2 n 1 2 /4Eδt 2 ω1 F ω1 n! n= 2π E δt n dω2 e iω 2t ω2 2 2 /4Eδt 2 ω2 Gω2 2π = F t Gt + n=1 2n 2 n n!e 2n δt E δt d n F t d n Gt dt n dt n If the variation time scale of the averaged functions is of the order of /E IR as it must be, since we ve just shown that any UV components will not enter into our calculation, the

25 3.1 General System 21 sum is a power series in the small parameter E IR /E δt 2n and to first approximation we find F tgt F t Gt as desired Separation of Time Scales Motivated by the fact that the any purely UV component of the functions to be averaged is strongly suppressed, we wish to rewrite the terms in the evolution equation in such a way as to isolate the UV time dependence. From section 2.1, to second order in perturbation theory the IR reduced density matrix depends on the full density matrix according to ρt = ρ t + ρ 1 t + ρ 2 t = ū σ t ū + ū σ 1 t ū + u u σ 2 t u where, recalling that the initial state of the system is assumed to be ψ = IR ū, ū σ t ū = ū ψ t ψ t ū = IRt IRt ū σ 1 t ū = ū ψ t ψ 1 t ū + ū ψ 1 t ψ t ū t = iλ dt IRt IRt ū V I t t ū ū V I t t ū IRt IRt t = iλ dt ū VI IR t t ū, IRt IRt u σ 2 t u = ū ψ t ψ 2 t ū + ū ψ 2 t ψ t ū + u ψ 1 t ψ 1 t u u u where we have defined the following operators whose time dependence is due entirely to IR subsystem: IRt e ih IRt/ IR ; VI IR t = e ihirt/ V e ih IRt/ Next, we calculate the terms contained in ρ 2 t in : u ψ 1 t = i t λ dt u V I t t ψ t

26 22 Coarse Graining = i λ t = i λe ieut/ t dt e ieu Eūt t/ u VI IR t te ieūt/ ū IRt dt e ieu Eūt / u VI IR t t ū IRt = ie ieut/ O 1 u t IRt where we have defined the operator O u 1 t λ In terms of this operator we have t dt e ieu Eūt / u V IR I t t ū u ψ 1 t ψ 1 t u = u u O 1 u tρ to 1 u t Using with = 2, ψ 2 t = i t λ dt V I t te ih t t / ψ 1 t ū ψ 2 t = λ2 2 = λ2 2 t t t t dt dt ū V I t te ih t t / V I t t ψ t dt dt e ieūt t/ ū VI IR t te ih IRt t e ih UV t t / VI IR t t e ih UV t t / e ih UV t t e ih IRt t ψ t = λ2 2 u t t u e ih IRt t / V IR I = λ2 2 e ieūt/ u dt dt e ieūt t/ ū VI IR t te ieut t / u t t u VI IR t t ū IRt t t e ih IRt t/ e ieūt t e ieūt/ ū IRt dt dt e ieu Eūt / e ieu Eūt / ū VI IR t t u = e ieūt/ O 2 t IRt

27 3.1 General System 23 where we have defined the second-order operator O 2 t λ2 2 u t t dt dt e ieu Eūt / e i Eut / ū V IR I t t u u VI IR t t ū Then the first two terms of ρ 2 t in the right-hand side of will be ū ψ 2 t ψ t ū + ū ψ t ψ 2 t ū = O 2 tρ t + ρ to 2 t And the full reduced density matrix can be written in terms of these operators and ρ t as ρt = ρ t io 1 ū t, ρ t + u O 1 u tρ to u 1 t O 2 tρ t + ρ to 2 t In order to find the time derivative of ρ we compute the derivatives of each of the operators appearing in i t ρ t = H IR, ρ t i t O 1 u t = iλe ieu Eūt/ u V ū + H IR, O 1 u t i t O 2 t = λ u ie ieu Eūt/ ū V u O 1 u t + H IR, O 2 t Each term in equations contains a commutator of the corresponding operator with H IR, which gives the Hamiltonian evolution of ρt. The first order terms are easy to deduce. For the terms that involve u ū, i t ρ 2 t = iλ u e i Eut/ ū V u O u 1 tρ t + iλ u ρ to u 1 t u V ū e i Eut/ iλ u e i Eut/ u V ū ρ to u 1 t iλ u e i Eut/ O u 1 tρ t ū V u + H IR, ρ 2 t where to save space we ve defined E u E u Eū. The second order Hamiltonian comes

28 24 Coarse Graining from the term Lρ ρl as before with L, H 2 eff and A2, given by Lt = iλ u ū H 2 eff t = i 2 λ e ieu Eūt/ ū V u O 1 u ū A 2 t = i 2 λ u ū e ieu Eūt/ ū V u O 1 u t e ieu Eūt/ ū V u O 1 u t e ieu Eūt/ O u 1 u t + e ieu Eūt/ O 1 u t u V ū t u V ū and Γ 2 t = iλ u ū e ieu Eūt/ u V ū ρ to u 1 t + iλ u ū e ieu Eūt/ O u 1 tρ t ū V u + {A 2, ρ t} Time-Averaged Evolution Equation We now wish to coarse-grain the different elements of the evolution equation for the IR reduced density matrix. We first time-average Lt, in terms of which both H 2 eff are defined. We then calculate the product Lt = iλ u ū e ieu Eūt/ O 1 u t = λ e ieu Eūt/ t and A2 e ieu Eūt/ ū V u O 1 u t dt e ieu Eūt / u V IR I t t ū By writing the time dependence of u VI IR t t ū explicitly, we can see the presence of both the IR and UV time scales: t u VI IR t t ū = i,j u, i V ū, j e ie i E j t t/ i j dt e ieu Eūt t/ u VI IR t t ū = ij u, i V ū, j ie u Eū + E i E j i j u, i V ū, j ie u Eū + E i E j e ieu Eū+E i E j t/ i j

29 3.1 General System 25 We neglect the UV-dependent term, which will be exponentially suppressed as discussed above. The remaining term is constant in time, such that Lt = L = λ 2 u ū ij u, i V ū, j ū V u i j E u Eū + E i E j and therefore H 2 eff = λ2 2 A 2 = λ2 2 u, i V ū, j u ū ij u ū ij ū, i V u, j ū V u i j + E u + E ij E u E ij i j u V ū u, i V ū, j ū, i V u, j ū V u i j i j u V ū E u + E ij E u E ij where E u E u Eū and E ij E i E j. Since H 2 eff is constant, its time derivative vanishes such that H effρt = H eff ρt according to equation Thus, we have the desired evolution equation: i t ρt = H eff, ρt + Γt where Γt = λ 2 u ū ij u, i V ū, j ū, i V u, j E u + E ij i j ρ t ū V u E u E ij u V ū ρ t i j + {A 2, ρ t} IR Operators We can write our results in a more convenient form up to second order in λ as pure IR operators. We first calculate L, which contains all of the information in H 2 eff, A2 and Γ 2. L = λ 2 u ū ij u, i V ū, j ū V u i j E u 1 + E ij / E u

30 26 Coarse Graining Now we can write an infinite expansion in the small parameter E ij / E u E IR /E UV using the binomial expansion: L = λ 2 u ū ij = λ 2 u ū ij = λ 2 u ū m= u, i V ū, j E u m m= = = m= Eji E u m ū V u i j m! 1 ū V u m!! E u m+1 i E i u, i V ū, j E m j j m m! 1 ū V u i E m!! E u m+1 i i u V ū i j j E m j j Using the fact that H N IR = i i EN i i, we can write an operator equation not involving the spectrum of H IR : L = λ 2 u ū m m= = m! 1 m!! V u H IR V uh m IR E u m where we have defined V u u V ū. We find H 2 eff = λ2 2 A 2 = λ2 2 m u ū m= = m u ū m= = m! 1 V m!! m! 1 m!! u H IR V uh m IR E u m+1 V u H IR V uh m IR E u m+1 + Hm IR V u HIR V u E u m+1 Hm IR V u HIR V u E u m Γt = {A 2, ρ t} + λ m 2 u ū m= = m! 1 m!! HIR V uh m IR ρ tv u E u m+1 Rewriting these quantities up to second-order in E IR /E UV : H 2 eff = V u 2 λ2 + 1 E u 2 u ū u ū Γt = {A 2, ρ t} + λ 2 u ū V u V u, H IR V u, H IR V u E u 2 Vu, H IR ρ V u + V u ρ V E u 2 V uρ th m IR V u HIR E u m+1 EIR + O u, H IR + O 2 E UV EIR E UV A 2 = λ2 2 u ū V u 2 V u, H IR EIR + O E u 2 E UV

31 3.2 Linear Oscillators 27 Thus, we have derived a double power series in λ and E IR /E UV for the evolution of ρt in terms of quantities that are either constant or local in time. Note that the first-order terms of A 2 and Γt in powers of E IR /E UV are zero, while the corresponding term for H 2 is not. This signifies that up to leading order in E IR /E UV we have unitary time evolution, to any order in λ. The converse of this statement, that to leading order in λ we have unitary evolution to any order in E IR /E UV, was seen implicitly in Chapter Linear Oscillators In this section we present preliminary results for some example systems involving harmonic oscillators with linear coupling. Our motivations for studying these systems are twofold: first, we wish to gain a better understanding of the coarse-graining formalism developed in the previous section by studying the dynamics of simple coarse-grained systems. Our calculations of H eff and Γt should allow us to derive explicit expressions for ρt in these examples, although we have not yet done so. Second, we are interested in comparing the formalism developed here with the path integral treatment of Feynman and Vernon 2, which can be done most simply in the case of linear oscillators 3. In the path integral treatment, one performs an integration over the UV coordinates analogous to taing the partial trace of σ over the UV modes and derives an influence functional that describes the effect of the UV system on the IR system and can be used to derive the dynamics of the IR system. We hope, in particular, to compare our results with Caldeira and Leggett s path integral treatment of linear oscillators Two Linear Oscillators We consider a Hamiltonian of the form H = ˆP 2 2M + M 2 Ω 2 ˆX2 + ˆp2 2m + m 2 ω2ˆx 2 ˆx ˆX 3.2.1

32 28 Coarse Graining which can be written H = I UV H IR + H UV I IR + V where H UV ˆp2 2m + m 2 ω2ˆx 2, H IR ˆP 2 2M + M 2 Ω 2 ˆX2, and V ˆx ˆX. We will consider a state ψt which is initially factorized as ψ = ū IR = n IR, where ū = n is the nth excited state of the UV oscillator. Then we have H 1 eff = ū V ū = n ˆx n ˆX = since x â + â, so ˆx n n 1 + n + 1 and the inner product with n is zero. From equation , H 2 eff = 1 2 m n i,j m, i V n, j n, i V m, j n V m i j + i j m V n E m + E ij E m E ij where E m E m E n = m + 1 ω n E ij E i E j = i + 1 Ω j ω = m n ω Ω = i j Ω Next, m V n = n V m = ˆX n ˆx m and using ˆx = â + 2mω â we find n ˆx m = 2mω nδ m,n 1 + n + 1δ m,n Such that becomes H 2 eff = n n, i V n + 1, j n 2 2mω E i,j n+1 + E ij n n, i V n + 1, j n E n+1 E ij n, i V n 1, j E n 1 + E ij n, i V n 1, j E n 1 E ij ˆX i j i j ˆX 3.2.9

33 3.2 Linear Oscillators 29 Now, n, i V m, j = n ˆx m i ˆX j And similarly to i ˆX j = 2M Ω iδ j,i 1 + i + 1δ j,i So becomes H 2 eff = 2 3/2 i + 1 ˆX i i + 1 i ˆX i i 1 4mω 2M Ω n E i n+1 + E i,i+1 E n+1 + E i,i 1 i + 1 ˆX i i + 1 i ˆX i i 1 i + 1 i i + 1 ˆX i i i 1 ˆX + n + + n E n 1 + E i,i+1 E n 1 + E i,i 1 E n+1 E i,i+1 E n+1 E i,i 1 i + 1 i i + 1 ˆX i i i 1 ˆX + n + E n 1 E i,i+1 E n 1 E i,i Substituting in the values of the denominators, e.g. E n+1 + E i,i+1 = ω Ω, becomes H 2 2 { eff = ˆX 4mω 2M Ωω 2 Ω ω + 2n + 1 Ω i + 1 i i i + ω 2n + 1 Ω i i i 1 + ω 2n + 1 Ω i + 1 i i ω + 2n + 1 Ω } i i i 1 ˆX Which has matrix elements H 2 eff i j = 2 { 4mω 2M Ωω 2 Ω i ˆX i ω + 2n + 1 Ω i + 1δ i+1,j 2 i + ω 2n + 1 Ω iδ i 1,j + δ i i ω 2n + 1 Ω i + 1 i + 1 ˆX j + ω + 2n + 1 Ω } i i 1 ˆX j ˆX

34 3 Coarse Graining Substituting for the matrix elements of ˆX from , { H 2 eff i j = 2 8mωM Ωω 2 Ω ω + 2n + 1 Ω 2i δ i 2 j + i + 1i + 2δ i +2,j + i i 1δ i 2,j + ω 2n + 1 Ω i i 1δ i 2,j + 2i + 1δ i j + i + 1i + 2δ i +2,j } Dropping the primes, we find H 2 2 eff ij = 4mωM Ωω 2 Ω 2 { δ ij 2i + 1ω 2n + 1Ω δ i+2,j ω i + 1i δ i 2,j ω } ii 1 Since A 2 is of the same form as H 2 eff see equation , we follow the derivation of the latter above and find: A 2 2 { = ˆX 4mω 2M Ωω 2 Ω ω + 2n + 1 Ω i + 1 i i i + ω 2n + 1 Ω i i i 1 ω 2n + 1 Ω i + 1 i i ω + 2n + 1 Ω } i i i 1 ˆX For brevity, we define a 2 4mω 2M Ωω 2 Ω ; b 2 2 8mωM Ωω 2 Ω a + ω + 2n + 1 Ω; a ω 2n + 1 Ω where b is dimensionless. Then to calculate the anticommutator in , we find A 2 { ρ = a i,j ˆX a + i + 1ρ i+1,j + a iρ i 1,j 2M Ω 2i + 1ω 2n + 1 Ω ρ ij + a i + 1i + 2ρ i+2,j + a + ii 1ρ i 2,j } i j 3.2.2

35 3.2 Linear Oscillators 31 ρ A 2 = a i,j { i 2M Ω j + 1 a + j + 1ρ i,j+1 + jj + 1ρ i,j 1 + a j 1 jρ i,j 1 + jj + 1ρ i,j+1 which have matrix elements } ρ ij j + 1 a j j 1 a+ j ˆX { A 2 ρ ij = b a + iρ ij + i + 1i + 2ρ i+2,j + a ii 1ρ i 2,j + i + 1ρ ij 2i + 1ω 2n + 1 Ω ρ ij + a i + 1i + 2ρ i+2,j + a } + ii 1ρ i 2,j { ρ A 2 ij = b a + jρ ij + jj 1ρ i,j 2 + a j + 1ρ i,j + j + 1j + 2ρ i,j+2 a jj 1ρ i,j 2 Adding these together, the ρ ij + j + 1ρ ij + a+ jρ ij + j + 1j + 2ρ } i,j+2 terms vanish and we find {A 2, ρ } ij = 2 b 2n + 1 Ω ρ i+2,j i + 1i + 2 ρ i 2,j ii 1 +ρ i,j 2 jj 1 ρ i,j+2 j + 1j Next we calculate the second term in equation for Γ 2, n n + 1, i V n, j n 1, i V n, j n i j ρ ˆX 2mω E i,j n+1 + E ij E n 1 + E ij n n, i V n + 1, j n, i V n 1, j n ˆXρ i j E n+1 E ij E n 1 E ij Following a similar derivation to that of H 2 eff, this becomes { 2a a + i + 1 i i a i i i 1 ρ ˆX i ˆXρ a i + 1 i i a+ i i i 1 } = 2a ij ˆX i ρ i,j { a + i + 1ρ i+1,j + a iρ i 1,j i j ˆX a j + 1 j a+ j j 1 }

36 32 Coarse Graining Which has matrix elements { 2 b a + i + 1j + 1ρ i+1,j+1 + ji + 1ρ i+1,j 1 + a ij + 1ρ i 1,j+1 + ijρ i 1,j 1 a ji + 1ρ i+1,j 1 + ijρ i 1,j 1 + a + i + 1j + 1ρ i+1,j+1 + ij + 1ρ } i 1,j+1 = 2 2n + 1 Ω 2mωM Ωω 2 Ω ρ 2 i+1,j 1 ji + 1 ρ i 1,j+1 ij + 1 Adding to , Γ 2 ij = 2 b 2n + 1 Ω 2ρ i+1,j 1 ji + 1 2ρ i 1,j+1 ij + 1 ρ i+2,j i + 1i ρ i 2,j ii 1 ρ i,j 2 jj 1 + ρ i,j+2 j + 1j + 2 with the dimensionless quantity b defined in equation If we wish to plug this into the evolution equation for ρ, i t ρt = H eff, ρt + Γt = H eff, ρt + H2 eff, ρ t + Γt where the terms proportional to H 1 eff are zero in this example according to equation Now, H eff, ρt ij = E n + H IR, ρt ij = E ij ρ ij t = i j Ωρ ij t and the commutator H 2 eff, ρ t is calculated similarly to the anticommutator in : H 2 eff, ρ t ij = 2 b ω 2ρ ij i j + ρ i+2,j i + 1i ρ i 2,j ii ρ i,j 2 jj 1 ρ i,j+2 j + 1j IR System Coupled with UV Oscillators Next, we consider an arbitrary IR system characterized by phase space coordinates X and P coupled linearly to a set of UV oscillators with coordinates {x i } through the potential

37 3.2 Linear Oscillators 33 V = i C ix i X, where C i are some constants that depend on the ith oscillator. There is no first order contribution to H eff since ū V ū = i C i x i X =. To compute its second-order contribution, we first analyze the terms: u l, i V ū l, j = i X j C l x l C u l x = δ ul,1 i X j where u l 1,, l 1, u l, l+1,, n, i.e., the u excited state of the lth oscillator, and we have defined x l m l ω l. Similarly, ū l V u l = δ ul,1 C 2 lx l X, such that n l=1 u l ū ij u l, i V ū l, j E ul + E ij ū l V u l i j = n l=1 ij C 2 l x2 l X i i X j j 2 E 1l + E ij Following the same procedure as in section with E 1l E u, we find H eff = Eū + H IR n l=1 C 2 l m l ω 2 l n l=1 m m=1 = Defining B H IR XHm IR C 2 l X2 m l ω 2 l m! 1 m!! ω l m XH IRXH m IR + Hm IR XH IRX temporarily dropping the indices, the term of interest taes on the form = XB + B X. This quantity is Hermitian and can be written as XB + B X = {X, C} + X, D, where C = B + B /2 and D = B B /2 are Hermitian and anti-hermitian operators respectively. Additionally, we can write C and D in an interesting way using the following formulae: B = H IRXH m IR = H IR, XH m IR + XHm IR = H IR, X, H m IR = H IR, X, H m IR + Hm IR H IR, X + XHIR m Hm IR XH IR + {X, HIR} m B + B = H IR, X, H m IR + {X, Hm IR} = 2C and similarly: B = H IRXH m IR = H IR, XH m IR + XHm IR

38 34 Coarse Graining = {H IR, X, H m IR = {H IR, X, H m IR } Hm IR H IR, X + XHIR m } + Hm IR XH IR + X, HIR m B B = {H IR, X, H m IR } + X, Hm IR = 2D Hence, in terms of commutators and anti-commutators the expression for the effective Hamiltonian for this linear coupled system is given by H eff = Eū + H IR n l=1 C 2 l m l ω 2 l n l=1 m m=2 = C 2 l X2 m l ω 2 l 1 4 n Cl 2X, X, H IR m l ωl 3 l=1 m! 1 m!! ω l m + X, {HIR, X, H m IR } + X, X, Hm IR {X, HIR, X, H m IR } + {X, {X, Hm IR}} where we have evaluated the first-order contributions in E IR /E UV explicitly. We also see that the terms that are independent of can be added using: m = Finally, our expression for H eff simplifies to: H eff = Eū + H IR n l=1 C 2 l m l ω 2 l m m=2 = n l=1 C 2 l X2 m l ω 2 l m! 1 m!! = 1 1m = δ m, n l=1 m! 1 m!! ω l m C 2 l X, X, H IR m l ω 3 l {X, HIR, X, H m IR }} + X, {H IR, X, H m IR } Similarly, we compute Γ and write down the evolution equation for the reduced density matrix. A 2 = 1 4 n l=1 C 2 l m l ω 2 l m m= = m! 1 XH m!! ω l IRXH m m IR Hm IR XH IRX

39 3.2 Linear Oscillators 35 As above, we can write the term = X, C + {X, D} and find A 2 = 1 8 n l=1 C 2 l m l ω 2 l m m= = m! 1 X, H m!! ω l IR, X, H m m IR Or, writing the first-order terms in E IR /E UV explicitly, A 2 = 1 4 n l=1 C 2 l X2, H IR m l ω 3 l 1 8 n l=1 C 2 l m l ω 2 l m m=2 = + {X, {H IR, X, H m IR }} m! 1 X, H m!! ω l IR, X, H m m IR + X, {X, HIR} m + {X, {HIR, X, H m IR }} + {X, X, Hm IR} and Γt = {A 2, ρ t} n l=1 C 2 l m l ω 2 l m m= = m! 1 m!! ω l m H IRXH m IR ρ tx Xρ H m IR XH IR Evaluating the commutator with A 2, we have to second order Γt = 1 n Cl 2{X2, H IR, ρ t} 4 m l ω 3 l=1 l 1 n C 2 m l m! 1 8 m l ωl 2 m!! ω l m l=1 m= = {X, HIR, X, H m IR, ρ t} + {X, {X, HIR}, m ρ t} + {{X, {HIR, X, H m IR }}, ρ t} + {{X, X, HIR}, m ρ t} + 1 n C 2 l X, HIR ρ tx + Xρ tx, H 2 m l ω 3 IR l=1 l + 1 n Cl 2 m m! 1 H 2 m l ωl 2 m!! ω l IRXH m m IR ρ tx Xρ th m IR XH IR l=1 m=2 =

40 Chapter 4 Conclusions We have developed a formalism that perturbatively solves for the nonlocal, nonunitary evolution of open quantum systems, and have found a return to local behavior when we coursegrain our theory. Our model allows us to separate the dynamics of the accessible subsystem into the usual, unitary piece and the more interesting piece due to the transfer of information between the accessible and inaccessible modes. We also find that to first-order in either the interaction parameter or the ratio of UV to IR energy scales, the dynamics of the IR subsystem appear unitary. Further research will involve connecting our description of open quantum systems, based on the density operator formulation of quantum mechanics, with the description based on the path integral formulation, as outlined in section 3.2. It would also be of interest to extend the examples presented here, for instance by varying the spectrum of UV spins in order to gain a better understanding of the UV system s possible effects on the IR dynamics. Finally, it may prove interesting to study the more general case in which the IR and UV systems begin in an entangled state, and to test our claim that this initial condition should not qualitatively affect the accessible physics. 36

41 Appendix A Alternate Calculation of Coupled Spins We present an alternate calculation of the IR reduced density matrix for the system of coupled spins detailed in section 2.2. A.1 State and Energy Corrections A.1.1 First-Order Corrections in λ The first-order energy correction to the eigenstate whose zeroth-order state is m IR, m UV will be: E 1 m IR m UV = m IR, m UV λv m IR, m UV = λ m IR, m UV SIR z SUV, m z IR, m UV = λ 2 m IR m UV, A.1.1 where the terms involving S + and S disappear because n IR, n UV m IR, m UV = δ nir m IR δ nuv m UV, with δ nuv m UV δ nuv,1 m UV,1 δ nuv,m m UV,M. The 37

42 38 Alternate Calculation of Coupled Spins first-order correction to the ground-state energy eigenvector is E 1 1 = m IR, m UV j IR, j UV m IR, m UV λv j IR, j UV = A.1.2 E E j IR j UV m IR m UV since the S + terms in ill the ground state, and the terms involving S z are zero except when m IR, m UV = j IR, j UV, which is excluded from the sum. The first-order correction to the first excited energy state is E 2 1 = m IR, m UV j IR 1, j UV m IR, m UV λv j IR 1, j UV E m IR, m UV A.1.3 j IR 1 j UV E m IR m UV = λ j IR, j UV S + IR S UV, j IR 1, j UV 2 E j IR, j UV j IR 1 j UV E j IR j UV = λ 2 j IR j IR + 1 j IR 1j IR j UV, j UV, + 1 j UV, j UV, 1 2 B µ IR j IR 1 + q µ H,qj UV,q + B µ IR j IR + q µ H,qj UV,q µ UV, j IR, j UV = λ 2 2 j IR j UV, Bµ UV, µ IR j IR, j UV, 1 = λ K j IR, j UV = λ K E 3, A.1.4 where Similar calculations show that K j IR j UV, Bµ UV, µ IR A.1.5 E 3, 1 = λk j IR 1, j UV = λk E 2 A.1.6 A.1.2 Second-Order Corrections in λ Energy Corrections The second-order correction to the ground state energy E 1 is E 2 1 = A.1.7

43 A.1 State and Energy Corrections 39 for the same reason E 1 1 = in equation A.1.2. The second-order correction to E 2 is E 2 2 = = λ2 4 m IR, m UV j IR 1, j UV = m IR, m UV λv j IR 1, j UV 2 E j IR 1 j UV E m IR m UV j IR, j UV S + IR S UV j IR 1, j UV 2 E j IR 1 j UV E j IR, j UV λ 2 4 4j IR j UV, 4 B µ UV, µ IR = λ2 KB µ 2 UV, µ IR A.1.8 where once again K is given by equation A.1.5. The second-order correction to E 3, is E 2 3, = = λ2 4 m IR, m UV j IR, j UV = λ2 4 4 m IR, m UV λv j IR, j UV 2 E j IR j UV E m IR m UV j IR 1, j UV S + IR S UV, j IR, j UV 2 E j IR j UV E j IR 1 j UV 4j IR j UV, B µ UV, µ IR = λ2 K 2 B µ H µ IR A.1.9 State Corrections The second-order correction to the ground state E 1 is E 1 2 = m IR, m UV L, H m IR, m UV m IR, m UV λv L, H L, H λv j IR, j UV m IR, m UV E E j IR j UV L H E E j IR j UV m IR m UV m IR, m UV λv j IR, j UV j IR, j UV λv j IR, j UV 2 m IR, m UV E E j IR j UV m IR m UV A.1.1 where the primes over the sums indicate that the state E 1 = j IR, j UV is not included in the sum. The first term in equation A.1.1 disappears because L, H λv j IR, j UV will always be zero except when L, H = j IR, j UV, as in equation A.1.2. The same goes for the

44 4 Alternate Calculation of Coupled Spins second term; thus, E 1 2 = A.1.11 The second-order correction to the first excited state is given by E 2 2 = m IR, m UV L, H m IR, m UV m IR, m UV λv L, H L, H λv j IR 1, j UV m IR, m UV E E j IR 1 j UV L H E E j IR 1 j UV m IR m UV m IR, m UV λv j IR 1, j UV j IR 1, j UV λv j IR 1, j UV 2 m IR, m UV E E j IR 1 j UV m IR m UV A.1.12 where the primes over the sums indicate that the state E 2 = j IR 1, j UV is not included in the sum. The first term in equation A.1.12 becomes m IR,m UV λk m IR, m UV λv j IR, j UV m E E IR, m UV j IR 1 j UV m IR m UV = λ 2 K j IR, j UV q Sz IR Sz H,q j IR, j UV j E E IR, j UV j IR 1 j UV j IR j UV = = λ 2 2 K j IR j UV, 1 + q j UV,q j IR, j UV B µ UV, µ IR λ 2 K 2j IR j UV, 1 + q j UV,q j IR, j UV A.1.13 jir j UV, Meanwhile, the second term of equation A.1.12 becomes m IR, m UV λ 2 j IR 1 q j UV,q m IR, m UV λv j IR 1, j UV 2 m IR, m UV E E j IR 1 j UV m IR m UV = λ 2 j IR 1 q j UV,q λ 2 j IR j UV, B 2 2 µ UV, µ IR 2 j IR, j UV

45 A.1 State and Energy Corrections 41 = λ 2 K 2 j IR 1 q j UV,q jir j UV, j IR, j UV A.1.14 Adding equations A.1.13 and A.1.14, equation A.1.12 becomes E 2 2 = λ 2 K 2 j IR j UV, 1 + j UV,q j IR 1 j UV,q j IR, j UV 1 jir j UV, q q q j UV,q j IR = λ 2 K 2 jir j UV, j IR, j UV A.1.15 To normalize E 2 we must adjust E 2 2 by adding to it a factor of λ2 2 m IR, m UV m IR, m UV V j IR 1, j UV 2 E j IR 1 j UV E m IR m UV 2 j IR 1, j UV = λ 2 K 2 2 j IR 1, j UV A.1.16 where the derivation of the right-hand side of equation A.1.16 is similar to that of equation A.1.4. Thus, the normalized form of E 2 2 is E 2 2 = λ 2 K j q IR 1, j UV j UV,q j IR j IR, j UV jir j UV, A.1.17 The second-order correction to the th U V excited state is E 3, 2 = m IR, m UV L, H m IR, m UV m IR, m UV λv L, H L, H λv j IR, j UV m IR, m UV E j IR j UV E L H E j IR j UV E m IR m UV m IR, m UV λv j IR, j UV j IR, j UV λv j IR, j UV 2 m IR, m UV E j IR j UV E m IR m UV where the state E 3, = j IR, j UV is not included in the sums. The first term in equation A.1.18 becomes m IR, m UV λk m IR, m UV λv j IR 1, j UV E m IR, m UV j IR j UV E m IR m UV A.1.18