Non-unitary evolution control of open quantum systems

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(1)Non-Unitary Evolution Control of Open Quantum Systems. Graduation Project Sebastián Restrepo Adviser: Luis Quiroga Puello. Universidad de Los Andes Faculty of Science Department of Physics. November 2012.

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(3) Contents List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. v. 1 Introduction. 2. 2 Open Quantum Systems. 4. 2.1. Reduced System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2.2. Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2.3. 2-Level Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 3 General Problem of an Open Quantum System with a Time-Dependent Hamiltonian 3.1. Master Equation for our General Problem . . . . . . . . . . . . . . . . . . . . . . . .. 3.2. Time Dependent Qubit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 Time Periodic Systems. 9 13 18. 4.1. Floquet Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18. 4.2. Floquet Decomposition of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21. 5 Treatment of the Environment 5.1. 23. Thermal Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6 Master Equation. 23 27. 6.1. Master Equation for Periodic Driving . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27. 6.2. Master Equation No Driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 30. 6.3. 2-Level System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31. 6.3.1. 31. No Driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii. 8.

(4) 6.3.2. Driving (approx.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7 Master equation for Periodic Driving - Schrodinger Picture. 33 37. 7.1. Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37. 7.2. 2-Level System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40. 7.3. Harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49. 8 Summary and Outlook. 53. Bibliography. 56. iv.

(5) List of Figures 2.1. 2 level atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 6.1. Population inversion for the driven 2-level system - Approx. . . . . . . . . . . . . . .. 34. 6.2. Population inversion for the driven 2-level system when Ω → 0 - Approx. . . . . . . .. 35. 7.1. Mollows triplet for Sodium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42. 7.2. Population inversion for the driven 2-level system - Floquet basis Ω/! = 0.1 . . . . .. 45. 7.3. Population Inversion for the Driven 2-level system - Floquet Basis Ω/! = 1. . . . . .. 46. 7.4. Population Inversion for the Driven 2-level system - Floquet Basis Ω/! = 0.5. . . . .. 47. 7.5. Energy level Diagram Dressed States . . . . . . . . . . . . . . . . . . . . . . . . . .. 48. 7.6. Energy level Spectrum and Allowed Transitions for the Driven 2-Level Atom . . . .. 49. v.

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(7) Chapter 1. Introduction Normally when one learns quantum mechanics, it is taught that the evolution of the state of a quantum system is described by the Schrodinger equation. If this equation is solved for a particular Hamiltonian, the state of the system is determined at every time t and all predictions of any observables can be made. This is only true if the system of study is considered as a closed system and it is described only by a pure state. An important issue that can not be ignored is the fact that the system of study is usually in contact with a much larger system (the environment), with which the system may exchange matter or energy. When this is taken into account, the evolution of the system is no longer described by the Schrodinger equation but a density operator approach must be used. This leads to an equation of motion for the reduced (system of study only) density operator called master equation. Examples of open quantum systems are the interaction of matter with light or the one-dimensional movement of a particle interacting with an environment described by harmonic oscillators in thermal equilibrium also known as quantum Brownian motion.. Open quantum systems have been widely studied [1,2,3], but the majority of work has focused on the case where there is no time dependence in the system of interest. The main focus of this work is to study open quantum system in which there is time dependence in our system, especially periodic time dependence. This time dependence can appear as a driving applied to the system for gaining control or on the system itself. We will center on simple cases such as qubits (two-level systems) and harmonic oscillators.. 2.

(8) It is important to note that the master equation approach is not the only one for the study of open quantum system(see [1,4]) but it would be the main line we will follow in this monograph. An overview of the outline of this thesis is as follows:. We start in chapter 2 with a rapid introduction to the dynamics of an open quantum system and the main approximations yielding to the master equation approach.. In chapter 3 a master equation for the general problem of an open quantum system is derived and applied to a 2-level system (qubit) with a time dependent Hamiltonian.. Chapter 4 contains all the theory needed to study open time-periodic quantum systems.. Chapter 5 explains in detail how to describe the thermal bath and to treat its coupling to the system.. Chapter 6 & 7 contain the principal results of this work. A master equation for periodic driving is derived in both Schrodinger and interaction picture and simple quantum open systems are studied. The main system is a 2-level atom driven by a continuous-wave field.. Chapter 8 presents a summary of the main results obtained in this work and conclusions. Finally, suggestions for further study are made.. 3.

(9) Chapter 2. Open Quantum Systems A quantum system that exchanges energy o matter with its surrounding environment is an open quantum system. One of the simplest models of an open system is an atom embedded in an electromagnetic environment and subject to an externally applied time dependent interaction as a classical radiation field. The dynamics of the system (atom only) can be studied by use of the Schrodinger equation but the external influence of the environment (electromagnetic field) is left out. A better description of an open quantum system requires a quantum consideration of the external influence. That is to consider the environment as a new quantum system, so the problem now involves the dynamics of two (or more) quantum systems in which one of them is the one of interest. We therefore need the ability to extract information of the system of interest from the bigger quantum system. This prevents us of using a pure state description; an open system must be characterized in terms of a density matrix.. The characterization of a system in terms of its density matrix reflects our ignorance about the system itself. Really, there are only a few cases where we can describe a system by just one pure state. Usually, we don’t know the state of the system and neither the probability in which that state comes.. 4.

(10) 2.1. Reduced System. We have a large quantum system, and we are only interested in a specific part (system of interest) of it, the rest is the environment, from now on called reservoir or thermal bath. Measurements of interest only concern us with observables of our quantum central system. To obtain these observables one can reduce the whole description of the large system to a reduced density operator characterizing only the system of interest.. Consider an observable OS of our system of interest. If ρSB is the density operator of the large system (system of interest + bath) the expectation value of the operator would be given by. !OS " = T rSB {ρSB OS }. (2.1). Defining a reduced density operator as ρS = T rB {ρSB }, the expectation value of operator OS is !OS " = T rS {ρS OS }. 2.2. (2.2). Master Equation. The equation of motion of the reduced density operator ρS is called master equation and it is usually written in what is known as Lindblad form # !" 1 1 † d † † ρS (t) = −i [HS , ρS (t)] + Li ρS Li − Li Li ρS − ρS Li Li dt 2 2. (2.3). i. where Li denotes a jump operator associated to our system of interest S. The second term in equation (2.3) is the dissipative part and is commonly written in terms of superoperator D as DρS . Equation (2.3) will not be derived here, the reader may refer to [2,5,6] for its derivation. Only the main approximations made in its derivation will be discussed.. 5.

(11) • Born Approx: The bath is considered to be a huge system (many degrees of freedom) in comparison to our system of interest. Thus, we assume it remains stationary and that the density operator can be written like ρSB (t) ≈ ρS (t) ⊗ ρB (0) at any time. • Markov Approx: The future state of the system is determined only by the current state of the system and not by its past or history. We ignore all memory effects. With these two approximations and starting from Liouville’s equation, (2.3) is finally obtained.. 2.3. 2-Level Atom. Throughout this monograph the main system of study will be a two level atom or qubit driven by an electromagnetic field. Before we start with all the theory needed to treat this problem we specify what we mean when we talk about a two level atom.. An atom has an infinite number of energy levels, but for our purposes we are going to approximate it as if it has only two: an exited state |e$ and a ground state |g$. The energy difference between the two levels is Ee − Eg = !.. Figure 2.1: Two level atom with electromagnetic field. When the atom and an electromagnetic field interact, only the two atomic levels resonant or nearly resonant with the frequency of the field react to this interaction. If the field with frequency ωP is nearly resonant with the two levels |e$ and |g$ , the atom can make transitions from the ground 6.

(12) state to the exited state by absorbing a quantum of energy ! or from the exited state to the ground state by releasing a quantum of energy ! (see Figure 2.1). This quantum of energy is called a photon. Therefore, as long as the electromagnetic field is nearly resonant with the energy difference between two levels of the atom, we can consider the atom as a 2 level system and treating it, in an analogous way, as a problem of a fictitious spin -1/2 system with a static magnetic field that has only 2 eigenenergy states.. 7.

(13) Chapter 3. General Problem of an Open Quantum System with a Time-Dependent Hamiltonian We study the general problem of a quantum open system with a time dependent Hamiltonian. The time dependence will be in our system of interest described by Hamiltonian ĤS (t). The system is in contact with a heat bath at temperature T (βT = 1/KB T ) and represented by the Hamiltonian ĤB . The interaction between the system and the bath will be described by ĤSB . Then,. Ĥ(t) = ĤS (t) + ĤB + ĤSB = Ĥ0 (t) + ĤSB. (3.1). where Ĥ0 (t) = ĤS (t) + ĤB . We follow a procedure similar to the one used by Maurice Goldman in [7] to find the equation of motion for the density operator ρ̂S of the central quantum system with Hamiltonian ĤS (t) from the density operator of the super-system ρ̂SB described by the complete Hamiltonian Ĥ. Finally we apply this equation to a 2-level system with a time varying energy splitting.. 8.

(14) 3.1. Master Equation for our General Problem. We assume that the density operator of the complete system can be written as ρ̂SB = ρ̂S (t) ⊗ ρ̂B , where the density operator of the bath at temperature T is given by. ρ̂B =. eβT ĤB T r{eβT ĤB }. ,. T rĤB {ρ̂B } = 1. (3.2). We express the interaction Hamiltonian in the form. ĤSB =. !. V̂α Fˆα =. α. !. V̂α† F̂α†. (3.3). α. In the interaction picture Õ(t) = U † (t)ÔU (t) where d U (t) = −iĤ0 (t)U (t) dt d † U (t) = iU † (t)Ĥ0 (t) dt. (3.4). Throughout the whole document we will take ! = 1. The equation for the density operator in this picture is d ρ̃SB (t) = −i[H̃SB (t), ρ̃SB (t)] dt. (3.5). By formally integrating this last equation, we obtain d i 1 ρ̃SB (t) = − [H̃SB (t), ρ̃SB (0)] − 2 dt ! !. ". t. [H̃SB (t), [H̃SB (τ ), ρ̃SB (τ )]] dτ. (3.6). 0. Averaging the first term to zero and taking s = t − τ d ρ̃SB (t) = − dt. ". 0. t. [H̃SB (t), [H̃SB (t − s), ρ̃SB (t − s)]] ds. (3.7). # $ The average we talk about refers to T rB H̃SB ρSB . It is not really an assumption to set it to zero as we can always absorb terms into the system’s Hamiltonian H̃S to ensure that the mean value of the interaction Hamiltonian, averaged over the density matrix of the environment, is 9.

(15) zero. Later we will be taking this type of trace in this equation so we will drop this term from now on.. Going back to the Schrödinger’s picture, approximating ρ̂SB (t − s) −→ ρ̂SB (t) and extending the upper limit of the integral to infinity we get. d ρ̂SB (t) = −i[Ĥ0 (t), ρ̂SB (t)] − dt. ". ∞. [ĤSB (t), [HSB (t; s), ρ̂SB (t)]] ds. (3.8). 0. where HSB (t; s) = U (t)H̃SB (t, s)U † (t). (3.9). = U (t)U † (t − s)ĤSB U (t − s)U † (t) = U (t; t − s)ĤSB U † (t; t − s) As ĤS (t) and ĤB act over two different Hilbert spaces we can write U (t; t − s) = U (t)U † (t − s) % & ' & " t−s '( " i t i = T exp − Ĥ0 (u) du exp Ĥ0 (u) du ! 0 ! 0 = UB (s)US (t, t − s). (3.10). (3.11). T is the time ordering operator and # $ UB (s) = exp −iĤB s & " t ' US (t, t − s) = T exp −i ĤS (u) du. (3.12). t−s. This allows us to write HSB (t; s) = U (t; t − s)ĤSB U † (t; t − s) =. !. U (t; t − s)V̂α Fˆα U † (t; t − s). (3.13). α $# $ !# † = US (t, t − s)V̂α US (t, t − s) UB (s)Fˆα UB† (s) α. =. !. V̂α (t; s)Fˆα (s). (3.14). α. 10.

(16) Now, in order to study the system described by the Hamiltonian ĤS (t) we take the trace over the bath in equation (3.8) d ρ̂S (t) = −i[ĤS (t), ρ̂S (t)] − dt. ". ∞ 0. ) * T rB [ĤSB (t), [HSB (t; s), ρ̂SB (t)]] ds. (3.15). and using eq. (3.13) gives ) * ! ) * T rB [ĤSB (t), [HSB (t; s), ρ̂SB (t)]] = T rB [V̂α Fˆα , [V̂β† (t; s)F̂β† (s), ρ̂S (t)ρB ]]. (3.16). α, β. Using. [V F, ρS ρB ] = V [F, ρS ρB ] + [V, ρS ρB ]F. (3.17). "0 ! "0 ! = V ρS [F, ρB ] + V ! [F,!ρ! [V,!ρ! S ]ρ1 + ρS! B ]F + [V, ρS ]ρB F. = V ρS [F, ρB ] + [V, ρS ]ρB F ) * ) * and the fact that T rB [Fˆα , [F̂β† , ρ̂S (t)]] = T rB [Fˆα , ρ̂S (t)F̂β† ] = 0. We have d ρ̂S (t) = −i[ĤS (t), ρ̂S (t)] (3.18) dt " ∞!) # $ # $* [V̂α , [V̂β† (t; s), ρ̂S (t)]]T rB ρB F̂β† (s)Fˆα − [V̂α , V̂β† (t; s)ρ̂S (t)]T rB [ρB , F̂β† (s)]Fˆα ds − 0. α, β. We express the operators V̂β† (t; s) as V̂β† (t; s) =. !. ηβδ (t; s)V̂δ†. (3.19). δ. So that the time dependence is only on the functions η that we decompose like ηβδ (t; s) =. ". ∞. −∞. λδβ (t; ω)exp (iωs) dω. (3.20). That is nothing but a Fourier transform where λδβ (t; ω). 1 = 2π. ". ∞. −∞. ηβδ (t; s)exp (−iωs) ds 11. (3.21).

(17) This restricts the calculation because depending on the time dependence of the functions ηβδ (t, s) the integrals of equations (3.20) and (3.21) can not always be found. However, when this is possible, the equation (3.18) acquires a simpler form. # $ First we work on the term T rB [ρB , F̂β† (s)]Fˆα . Let |f $ be an eigenstate of the hamiltonian ĤB and using the notation &f | ĤB |f $ = !ωf we have + , + , + , &f | F̂β† (s) +f # = &f | e−iĤB s/! F̂β† eiĤB s/! +f # = e−i(ωf −ωf ! )s &f | F̂β† +f #. (3.22). −βT !ωf !. &f # | ρ̂B |f # $ &f | ρ̂B |f $. e = eβT !(ωf −ωf ! ) e−βT !ωf. =. (3.23). So we can write # $ ! = T rĤB [ρB , F̂β† (s)]Fˆα &f | [ρB , F̂β† (s)]Fˆα |f $. (3.24). f. ' &f # | ρ̂B |f # $ = 1− &f | ρ̂B |f $ ! f,f # $ ! + ,- + = e−i(ωf −ωf ! )s 1 − eβT !(ωf −ωf ! ) &f | ρ̂B |f $ &f | F̂β† +f # f # + F̂α |f $ !. + ,- + &f | ρ̂B |f $ &f | F̂β† (s) +f # f # + F̂α |f $. &. f,f !. Using this result and the expansion of V̂β† (t, s) given by (3.19) is easy to see that the third term of equation (3.18) has the integral ". ∞ 0. . / exp i(ω − (ωf − ωf ! ))s ds ∝ δ(ω − (ωf − ωf ! )). (3.25). # $ 0 1 That allows 1 − eβT !(ωf −ωf ! ) (−→ 1 − eβT !ω such that the equation of motion for the density. operator acquires the form. d ρ̂S (t) = −i[ĤS (t), ρ̂S (t)] (3.26) dt " ) # $* ! ∞ − dω λδβ (t; ω)Jαβ (ω) [V̂α , [V̂δ† , ρS (t)]] − [V̂α , V̂δ† ρS (t)] 1 − eβT !ω α,β,δ. −∞. with. Jαβ (ω) =. ". 0. ∞. # $ ds eiωs T rB ρ̂B F̂β† (s)F̂α 12. (3.27).

(18) The function Jαβ (ω) is commonly known as spectral density and carries all the information about the bath in the master equation. Later (section 6.1) we will define correlation functions Cαβ (t) for the bath and see that the spectral density is just the Fourier transform of these. If the system is now in contact with two baths:. Ĥ = ĤS (t) + ĤB1 + ĤB2 + ĤSB. ĤSB =. 2 ! !. V̂j,α Fˆj,α. (3.28). j=1 α. The master equation has the form d i ρ̂(t) = − [Q̂(t), ρ̂(t)] (3.29) dt ! 2 ! " ∞ ) # $* ! δ (j) (j) β !ω † † dω λβ (t; ω)Jαβ (ω) [V̂j,α , [V̂j,δ , ρ(t)]] − [V̂j,α , V̂j,δ ρ(t)] 1 − e Tj − j=1 α,β,δ. −∞. To obtain this equation it is important to take into account that the baths are described by an infinite set of oscillators in thermal equilibrium with a Hamiltonian in terms of bosonic operators. Therefore when the trace of a term that contains products like Fj,α Fl,β is taken, only the terms j = l make a contribution (j and l refer to the baths).. 3.2. Time Dependent Qubit. We apply equation (3.26) to the particular case of a Qubit in equilibrium with a bath at temperature T. The time dependence of the system will be in the gap of energy between the excited and ground state. This case can be considered as a spinning charged particle interacting with a time dependent magnetic field.. ĤS (t) = ω(t)Ŝz. (3.30). So that the time dependence is only in the frequency ω. The system-bath interaction is ĤSB = Ŝ+ F̂ + Ŝ− F̂ †. 13. (3.31).

(19) that is V̂α = Ŝα with α = ± in equation (3.3). The evolution of the system (a single qubit) is given by eq. (3.26) so we need to find λδβ (t; ω) 2 (eq. (3.21)) where V̂β† (t; s) = δ ηβδ (t; s)V̂δ† . V̂β† (t; s) = Ŝβ† (t; s) = UĤS (t; t − s)Ŝβ† U † (t; t − s) ĤS 4 3 " t 4 3 " t i i † ĤS (u) du Ŝβ exp ĤS (u) du = exp − ! t−s ! t−s. (3.32) (3.33). with ". t. ĤS (u) du =. t−s. ". t. ω(u)Ŝz du = Ω(t, s)Ŝz ;. Ω(t, s) =. t−s. ". t. ω(u) du. (3.34). t−s. We have Ŝβ† (t; s). =. 5. &. and using the fact that Ŝz ,. & ' i Ŝβ† 1 + Ω(t, s)Ŝz + · · · ! 5 6 i = Ŝβ† − Ω(t, s) Ŝz , Ŝβ† + · · · !. i 1 − Ω(t, s)Ŝz + · · · !. Ŝβ†. 6. '. (3.35) (3.36). = −β!Ŝβ† with β = ± it is easy to see that Ŝβ† (t; s) = exp (iβΩ(t, s)) Ŝβ†. (3.37). This result implies in eq. (3.19) that δ = 1 and ηβδ (t; s) = exp (iβΩ(t, s)). Using the equation (3.21) λδβ (t; ω). 1 = 2π. ". ∞. exp (iβΩ(t, s)) exp (−iωs) ds. (3.38). −∞. The behavior of λδβ (t; ω) is determined by the time dependence in Ω. First, just to verify eq.(3.26) we consider the case where ω(t) = ω0 . With this. Ω(t, s) =. ". t. ω(u) du = sω0 t−s. =⇒. λδβ (t; ω). 1 = 2π. 14. ". ∞. −∞. ηβδ (t; s)exp (−iωs) ds = δ(ω − βω0 )(3.39).

(20) The bath is in a thermal equilibrium state, so we expect that Jα,β (ω) = 0 for α *= β. This gives us. # $ d ρ̂(t) = −iω0 [Ŝz , ρ̂(t)] − J(ω0 ) [Ŝ+ , [Ŝ− , ρ̂(t)]] − (1 − eβT ω0 )[Ŝ+ , [Ŝ− , ρ̂(t)]] dt # $. (3.40). J(−ω0 ) [Ŝ− , [Ŝ+ , ρ̂(t)]] − (1 − eβT ω0 )[Ŝ− , [Ŝ+ , ρ̂(t)]]. Using the standard matrix representation for the S operators in the basis {|+$ , |−$} and rememd bering that ρ21 = −ρ∗12 , ρ11 + ρ22 = 1, we look for the stationary solution ρ̂(t) = 0. Therefore, dt we get. ρ11 =. e−βT ω0 /2 ; e−βT ω0 /2 + eβT ω0 /2. ρ22 =. eβT ω0 /2 e−βT ω0 /2 + eβT ω0 /2. (3.41). with ρ12 = ρ21 = 0 and we have used J(−ω0 ) = eβT ω0 J(ω0 ). We see that the density operator is a diagonal operator that does not depend on the spectral density J(ω) and with elements corresponding to the Boltzmann distribution at temperature T.. Now we consider the case where ω(t) is not constant, but it depends harmonically on time: ω(t) = −A sin(ω0 t), where A is a constant and the Hamiltonian of the system is ĤS (t) = −A sin(ω0 t)Iˆz . A In this case, from (3.34), we get Ω(t, s) = {cos(ω0 t) − cos(ω0 (t − s))} ω0 3. 4 iβA 3 4 exp cos(ω0 t) " ∞ −iβA ω0 δ exp cos(ω0 (t − s)) exp (−iωs) ds λβ (t; ω) = 2π ω0 ∞ This last integral can be very complicated to evaluate so we consider the case in which. (3.42) A + 1 so ω0. we can make the approximation: ". ∞. ∞. 4 ' " ∞& −iβA iβA exp cos(ω0 (t − s)) exp (−iωs) ds = 1− cos(ω0 (t − s)) + · · · exp (−iωs) ds (3.43) ω0 ω0 ∞ " " ∞ iβA ∞ cos(ω0 (t − s))exp (−iωs) ds exp (−iωs) ds − ≈ ω0 ∞ ∞ 3. 15.

(21) Solving these integrals is the same as finding the inverse Fourier transform of 1 and cos(ω0 (t−s)). After some calculation we get λδβ (t; ω) ≈ exp. 3. 4% ( / iβA iβA . −iω0 t cos(ω0 t) δ(ω) − e δ(ω − ω0 ) + eiω0 t δ(ω + ω0 ) ω0 2ω0. (3.44). So our master equation for a system with a hamiltonian given by ĤS (t) = −A sin(ω0 t)Iˆz is 6 ! iβA cos(ω0 t) d iA sin(ω0 t) 5 ω0 (3.45) Ŝz , ρ̂(t) − e ρ̂(t) = dt ! α,β,δ & ' 1 iβA 0 −iω0 t † iω0 t ×{[Ŝα , [Ŝβ , ρ(t)]] J(0) − e J(ω0 ) + e J(−ω0 ) 2ω0 # # $ # $$ iβA [Ŝα , Ŝβ† ρ(t)] e−iω0 t J(ω0 ) 1 − eβT !ω + eiω0 t J(−ω0 ) 1 − e−βT !ω } + 2ω0 Later in section 5 the function J(ω) would be calculated for a collection of oscillators that describe the thermal bath. Its frequency dependency ends up being given by Planck‘s distribution (eωβ − 1)−1 that clearly diverges for ω = 0, one of the values appearing in equation (3.45). In order to avoid this type of divergences a bunch of phenomenological spectral densities have been introduced. Some proportional to ω other to ωe−ω . It is of no interest to discuss these spectral densities in this monograph, but it is important to note that, for example if an Ohmic type spectral density J ∝ ω is used to solve this equation, complex values for the populations are obtained. This is a huge error since the populations are probabilities of finding the system in a particular state, which are of course always real. We already tested equation (3.45) for the case when there is no time dependence so we know the error was not introduced in our derivation. The problem arises when the Fourier transforms are calculated. Since the integrals that appeared for this case were very difficult to calculate we gave an approximate result, which turns out to be (in my point of view) the main cause for the complex value of the populations. Another cause of error might be the introduction of the phenomenological spectral densities.. Due to these results we proceed to reduce our main goal (derive a master equation for an open quantum system with a time dependent Hamiltonian) to a simpler one. We study only systems. 16.

(22) whose time dependence is only periodic. To do this we study first Floquet‘s theory, that describes how to treat time periodic problems and then proceed to obtain our master equation.. 17.

(23) Chapter 4. Time Periodic Systems We saw in the previous chapter that treating the problem of an open quantum system with arbitrary time dependence can be extremely difficult to treat. The difficulty arises when we make the Fourier decompositions of the system operators (3.19) and (3.21. To simplify the problem we now study only time periodic systems, which is actually of practical importance. Many of real quantum systems under different control protocols are subject to a periodic time dependence, for example all systems driven by continuous wave radiation fields. In this chapter we study time periodic systems by the use of Floquet theory and obtain a decomposition for the system operators that replaces the Fourier decomposition made earlier, thus making calculations easier.. 4.1. Floquet Theory. Consider a Hamiltonian periodic in time H(t) = H(t + P ) with P its period and |ψ(t)$ = U (t, t0 ) |ψ(t0 )$ a state that is a solution to the Schrödinger’s equation. Since H is periodic it is clear that the evolution operator satisfies the condition U (t + P, t0 + P ) = U (t, t0 ). Note that H and U are commuting operators:. H(t)U (t + P, t) = H(t + P )U (t + P, t) = U (t + P, t)H(t)U † (t + P, t)U (t + P, t) = U (t + P, t)H(t). 18. (4.1).

(24) Therefore it is possible to find a set of common eigenstates for these operators. We are interested in finding the eigenvalues of the operator U (t + P, t) that we expect, as the operator is unitary, only introduces a phase when it is applied to a state. U (t + P, t) |ψ(t)$ = eiθ(P ) |ψ(t)$. (4.2). Notice that U (t + nP, t) = U (t + nP, t + (n − 1)P )U (t + (n − 2)T ) . . . U (t + P, t) = (U (t + P, t))n ,. n ∈ N(4.3). So iθ(nP ). e. #. iθ(P ). |ψ(t)$ = e. $n. |ψ(t)$. (4.4). from which we can deduce: θ(P ) ∝ P . Thus, we propose: U (t + T, t) |ψ(t)$ = e−iεP |ψ(t)$. (4.5). Where ε is to be determined.. We split the time dependence in |ψ(t)$ as |ψ(t)$ = e−iε(t−t0 ) |φ(t)$ such that the state |φ(t)$ is periodic in time |φ(t + P )$ = eiε(t+P −t0 ) |ψ(t + P )$ = eiε(t+P −t0 ) U (t + T, t) |ψ(t)$ = eiε(t−t0 ) |ψ(t)$ = |φ(t)$ (4.6) Now consider the Hamiltonian H # (t) = H(t) − i∂t. 19. (4.7).

(25) which is known as Floquet Hamiltonian. Clearly, it is also periodic in time with the same period as H(t) and if |ψ(t)$ satisfies the Schrödinger equation, then H # (t) |ψ(t)$ = 0. Therefore we have 0 1 H # |ψ(t)$ = (H(t) − i∂t ) e−iε(t−t0 ) |φ(t)$ = e−iε(t−t0 ) H # − ε |φ(t)$ = 0. ⇒. H # |φ(t)$ = ε |φ(t)$. In conclusion, we have that the solution to the Schrödinger equation for a Hamiltonian periodic in time can be expressed in the form |ψ(t)$ = e−iε(t−t0 ) |φ(t)$. (4.9). where |φ(t)$ = |φ(t + P )$ and H # |φ(t)$ = ε |φ(t)$ Note that for a state |φj (t)$ and its exponent εj there exists an infinite set of states and equivalent exponents. εj ! = εj + nωP n ∈ Z , ωP = 2π/P + , +φj ! (t) = einωP (t−t0 ) |φj (t)$. (4.10). + + , , +ψj ! (t) = eiεj! (t−t0 ) +φj ! (t) = e−i(εj +nωP )(t−t0 ) einωP (t−t0 ) |φj (t)$ = |ψj (t)$. (4.12). (4.11). because. The most general solution to Schrödinger equation with a Hamiltonian periodic in time is then. |ψj (t)$ =. ! j. aj e−iεj (t−t0 ) |φj (t)$ ;. aj = &φj (t0 ) |ψ(t0 )$. (4.13). Applying a discrete Fourier transform to the periodic state |φj (t)$ |φj (t)$ =. ! n. e−inωP (t−t0 ) |φj (n)$. 20. (4.14). (4.8).

(26) where |φj (n)$ is 1 |φj (n)$ = P. t" 0 +P t0. einωP (t−t0 ) |φj (t)$ dt. (4.15). that allows us to write the solution to to Schrödinger equation as !!. |ψ(t)$ =. 4.2. n. j. aj e−i(εj +nωP )(t−t0 ) |φj (n)$ dt. (4.16). Floquet Decomposition of Operators. We know that the evolution operator satisfices ∂t Û (t, t# ) = −iĤS (t)Û (t, t# ). (4.17). and that the Floquet states have the completeness relation ! j. |φj (t)$ &φj (t)| = 1. # $ ĤS (t) − i∂t |φj (t)$ = εj |φj (t)$. with. (4.18). With this, it is easy to see that we can write Û (t, t# ) =. ! j. + 7 8 |φj (t)$ φj (t# )+ exp −iεj (t − t# ). (4.19). which gives the decomposition of the evolution operator in terms of the Floquet states. Now we use this to find the decomposition of an arbitrary operator.. Consider operator Â in the interaction picture and use the decomposition of the evolution operator to get Ã(t) = Û † (t, 0)ÂÛ (t, 0) =. ! j,j !. + + ,7 8 |φj (0)$ &φj (t)| Â +φj ! (t) φj ! (0)+ exp −i(εj ! − εj )t. 21. (4.20).

(27) Floquet states are periodic in time so we can make a Fourier decomposition of the matrix element in the equation above. (See eq. (4.14)) + + , ! , &φj (t)| Â +φj ! (t) = &φj (n)| Â +φj ! (k) eiωP (n−k)t. (4.21). + + , ! , &φj (t)| Â +φj ! (t) = &φj (q + k)| Â +φj ! (k) eiqωP t. (4.22). n,k. Setting q = n − k. q,k. We finally have. Ã(t) =. !!. Â(n, ω) ei(ω+nωP )t. (4.23). n∈Z {ω}. with. Â(n, ω) =. !! k. j,j !. + , - + &φj (n + k)| Â +φj ! (k) |φj $ φj ! +. (4.24). and ωp = 2π/P , |φj (0)$ = |φj $ and ω = εj − εj ! represents the Floquet frequency spectrum. We now have a way to study time periodic systems and have obtained a decomposition for the system’s operators based on Floquet theory. Before trying to obtain the master equation for a time periodic system, in the next chapter we see how to treat the coupling of the system of interest with the bath.. 22.

(28) Chapter 5. Treatment of the Environment When we derived the master equation for on open quantum system with a time dependent Hamiltonian in section 3.1, we reduced the complete system to our system of interest by taking the trace over the environment degrees of freedom and reduced all the information of the environment to a function J(ω) called the spectral density. In this chapter we study more deeply the effects of the environment to the system and obtain an expression for the spectral density when the environment is considered as a collection of harmonic oscillators called a thermal bath or reservoir.. 5.1. Thermal Bath. In general and for all the systems studied in this monograph we will consider the environment (described by Hamiltonian ĤB ) as a collection of harmonic oscillators of frequencies ωk in thermal equilibrium.. ĤB =. !. ω'k b̂†λ (0k)b̂λ (0k). (5.1). 'k. ρ̂B =. eβT ĤB T r{eβT ĤB }. This Hamiltonian describes a quantized radiation field, where 0k is the wave vector and λ refers to one of the two possible polarization transverse directions given by vectors 0eλ . We now show how to calculate the Fourier transform of the correlation functions Cαβ in order to obtain the final 23.

(29) master equation. Basically we want to find a specific form for the spectral density of the bath.. 5 6 The bath operators are annihilation and creation operators so they satisfy b̂λ (0k), b̂†λ! (0k# ) = δλ,λ! δ'k,'k! . This implies:. ) * ) * T rB b̂λ (0k)b̂λ! (0k# )ρ̂B = T rB b̂†λ (0k)b̂†λ! (0k# )ρ̂B = 0 ) * T rB b̂†λ (0k)b̂λ! (0k# )ρ̂B = δλ,λ! δ'k,'k! n(ω'k ) ) * 0 1 T rB b̂λ (0k)b̂†λ! (0k# )ρ̂B = δλ,λ! δ'k,'k! n(ω'k ) + 1. (5.2). Where n(ω'k ) is the average number of photons in a mode with frequency ω'k . Planck’s distribution is:. n(ω'k ) = exp. &. 1 ' ω'k −1 KB T. (5.3). 0 E 0 where We will consider the interaction in the limits of the dipole approximation ĤSB = −D· the electric field in the Schrodinger picture looks like. 0 =i E. 2 !! 'k λ=1. # $ E'k b̂λ (0k) − b̂†λ (0k) 0eλ (0k) ;. E'k =. 9. 2πω'k V. (5.4). 0 is the dipole operator. The form of this operator depends on the specific system of interest. and D Notice that the interaction is in the form of equation (3.3). In the interaction picture we have. 0 =i E. 2 !! 'k λ=1. E'k. # $ b̂λ (0k)e−iω#k t − b̂†λ (0k)eiω#k t 0eλ (0k). 24. (5.5).

(30) So we can now calculate Γαβ (ω) = now calling the spectral density Γ.. :∞ 0. eiωx Cαβ (x) dx. Compare with equation (3.27). We are. Using equations (5.2) and (5.5). Γαβ (ω) =. !! 'k,'k ! λ,λ#. ". 0. ∞). E'k E'k# 0eλ,α (0k) 0eλ#,β (0k# ) ×. (5.6). ) * ) * * T rB b̂λ (0k)b̂†λ! (0k# )ρ̂B ei(ω#k! −ω#k )t−i(ω#k! −ω)τ + T rB b̂†λ (0k)b̂λ! (0k# )ρ̂B e−i(ω#k! −ω#k )t+i(ω#k! −ω)τ dτ. Making the approximation to the continuum we get. Γαβ (ω) =. 2 δα,β 3πc3. ". ∞ 0. % " dωk ωk3 n(ωk ). ∞. ei(ωk +ω)τ dτ + (n(ωk ) + 1). 0. ". ∞. e−i(ωk −ω)τ dτ. 0. (. (5.7). where the continuum limit is. " " " ∞ d3 k 1 1 ! 2 ωk dωk dΩ ≈ = V (2π)3 (2π)3 c3 0. (5.8). 'k. We used. :. # dΩ δα,β −. kα kβ k2. $. =. 8π 3 δα,β. The integrals can be solved using. :∞ 0. to integrate the term that comes from. 22. 0. 0. eλ,α (k) 0eλ,β (k). λ=1 0. e±iax dx = πδ(a)±iP (1/a), where P refers to Cauchy principal. value. Equation (5.7) takes the form:. 2 Γαβ (ω) = δα,β 3πc3. % " 3 3 n(−ω)π(−ω) + (n(ω) + 1)πω +. 0. Taking the decomposition Γαβ (ω) =. ∞. dωk ωk3. 3. 1 1 i P −P ωk + ω ωk − ω. 4(. (5.9). γαβ (ω) + iGαβ , we have 2. γαβ (ω) = γ̃(ω)(1 + n(ω)) ;. 25. γ̃(ω) =. 4ω 3 3c3. (5.10).

(31) This function contains all the information from the bath in the master equation for the system’s density operator ρ̂S (t). Now that we know how to put all the bath information in just γ(ω) and understand Floquet theory, we are ready to obtain the equation of motion for the density operator of our system described by a periodic time-dependent Hamiltonian.. 26.

(32) Chapter 6. Master Equation In this chapter we start by deriving a master equation for periodic driving in the interaction picture, then we use this result to obtain the commonly known master equation when there is no time dependence in the Hamiltonian. We apply our results to the cases of a 2 level atom without driving and a two level atom driven by circularly polarized light.. 6.1. Master Equation for Periodic Driving. Consider the Hamiltonian describing the open system (system +bath):. Ĥ(t) = ĤS (t) + ĤB + ĤSB ,. Ĥ0 = ĤS (t) + ĤB. (6.1). We start by integrating the equation of motion for the density operator in the interaction picture eq. (3.4). d d ρ̃SB (t) = −i[H̃SB (t), ρ̃SB (t)] =⇒ ρ̃SB (t) = −i[S̃1 (t), ρ̃SB (0)] − dt dt. 27. ". t. [H̃SB (t), [H̃SB (τ ), ρ̃SB (τ )]] dτ(6.2) 0.

(33) Making the change of variable s = t − τ , averaging the first term to zero, approximating ρ̃SB (t − s) (−→ ρ̃SB (t), extending the integral to infinity and renaming the variable s (−→ τ we have d ρ̃SB (t) = − dt where H̃SB (t) =. 2. α S̃α F̃α. =. 2. ". ∞. 0. † † α S̃α F̃α. [H̃SB (t), [H̃SB (t − τ ), ρ̃SB (t)]] dτ. (6.3). and ρ̃SB (t) = ρ̃S (t)ρ̃B .. We are interested in the evolution of system S so we take the partial trace of eq (6.3) with respect to the bath. d ρ̃S (t) = − dt. ". ∞ 0. ) * T rB [H̃SB (t), [H̃SB (t − τ ), ρ̃SB (t)]] dτ. (6.4). which, using the expressions of H̃SB (t) and ρ̃SB (t), can be simplified to:. ! d ρ̃S (t) = − dt α,β. −. ∞. 0. # $ {[S̃α (t), [S̃β† (t − τ ), ρ̃S (t)]]T rB ρ̃B F̃β† (t − τ )F̃α (t). (6.5). # $ † − τ ), ρ̃S (t)]T rB [ρ̃B F̃β (t − τ )]F̃α (t) }dτ # $# $ {T rB F̃α (t)F̃β† (t − τ )ρ̃B S̃α (t)S̃β† (t − τ )ρ̃S (t) − S̃β† (t − τ )ρ̃S (t)S̃α (t). [S̃α (t), S̃β† (t !" ∞. = − +. ". 0. α,β. # $# $ T rB F̃β† (t − τ )F̃α (t)ρ̃B ρ̃S (t)S̃β† (t − τ )S̃α (t) − S̃α (t)ρ̃S (t)S̃β† (t − τ ) }dτ. # $ Defining Cαβ (t, t − τ ) = T rB F̃α (t)F̃β† (t − τ )ρ̃B we can write. ! d ρ̃S (t) = − dt α,β. ". 0. ∞). Cαβ (τ )[S̃α (t), S̃β† (t. − τ )ρ̃S (t)] +. ∗ Cβα (τ )[ρ̃S (t)S̃β† (t. * − τ ), S̃α (t)] dτ. (6.6). ∗ (τ ) = C (−τ ). with Cαβ (t, t − τ ) = Cαβ (t − (t − τ )) and Cβα βα. The reason for the correlation functions C to behave like C(t, τ ) = C(t − τ ) is that we have assumed that ρB is a stationary state of the bath (that is to say ρB commutes with the bath’s Hamiltonian) 28.

(34) so that functions C are homogeneous in time.. Now we assume that the Hamiltonian of the system is periodic in time:. H̃S (t + P ) = H̃S (t) ;. P is the period and ωP the associated frequency.. (6.7). With this assumption we can use a Floquet decomposition (sec. 4.2) for the system operators: S̃α (t) = Û † (t)Ŝα Û (t) =. !!. Ŝα (k, ω) ei(ω+kωP )t. (6.8). !!. Ŝβ† (q, ω # ) e−i(ω +qωP )(t−τ ). (6.9). k∈Z {ω}. S̃β† (t − τ ) = Û † (t)Ŝβ† Û (t) =. q∈Z. !. {ω ! }. to obtain !! ! " ∞ d ! ) ρ̃S (t) = − {Cαβ (τ ) [Sα (k, ω), Sβ† (q, ω # ), ρ̃S (t)] ei(ω+kωP )t e−i(ω +qωP )(t−τ (6.10) dt 0 ! α,β q,k {ω},{ω }. ∗ Cβα (τ ) [ρ̃S (t)Sβ† (q, ω # ), Sα (k, ω)] e−i(ω+kωP )t ei(ω +qωP )(t−τ ) } dτ !. +. Notice that !. !. !. ei(ω+kωP )t e−i(ω +qωP )(t−τ ) = ei(ω +qωP )τ ei[(ω+kωP )−(ω +qωP )]t. (6.11). where the last term in this equation does not depend on τ and can be put outside the integral in eq (6.10) and also we can consider a secular approximation in this term taking (ω + kωP ) = (ω # + qωP ), that is to take ω = ω # and k = q, yielding to. * ! ! !) d ρ̃S (t) = − Γαβ (ω + nωP )[Sα (n, ω), Sβ† (n, ω), ρ̃S (t)] + Γ∗βα (ω + nωP )[ρ̃S (t)Sβ† (n, ω), Sα (n, ω)] (6.12) dt α,β n∈Z {ω}. with Γαβ (ω) =. :∞ 0. eiωx Cαβ (x) dx.. Now we decompose the functions Γ as follows 29.

(35) γαβ (ω) + iGαβ Γαβ (ω) = 2. =⇒. γαβ (ω) = Γαβ (ω). + Γ∗βα (ω). =. ". ∞. eiωs Cαβ (s) ds. (6.13). −∞. Ignoring the imaginary part G we obtain the master equation for periodic driving in the interaction picture. $ ! ! ! γαβ (ω + nωP ) # † d [Sβ (n, ω)ρ̃S (t), Sα (n, ω)] + [Sβ† (n, ω), ρ̃S (t)Sα (n, ω)] (6.14) ρ̃S (t) = − dt 2 α,β n∈Z {ω}. where {ω} is the Floquet spectrum and Ŝα (n, ω) =. !! q. j,j !. + , - + &φj (n + q)| Sˆα +φj ! (q) |φj $ φj ! + ;. 1 |φj (q)$ = P. t" 0 +P t0. eiqωP (t−t0 ) |φj (t)$ dt (6.15). Thus, in order to use equation (6.14) the Floquet problem for the system’s Hamiltonian must be solved first. The equation for periodic driving (6.14) agrees with the equation previously obtained by Alicki in [8].. 6.2. Master Equation No Driving. We consider the case now that there is no driving in the system so that the Hamiltonian HS is not a time dependent function. We expect to obtain a typical master equation in Lindblad form.. If there is not periodic time dependence in the Hamiltonian of the system all the Floquet eigenvalues are zero and the Floquet states become the solutions to the Schrödinger equation. The Floquet decomposition of operators (6.8) will now look as:. Sα (t) =. !. Sα (ω)eiωt. (6.16). ω. where there is no sum over any integer because ωP = 0 and the frequency spectrum is now given by the commutation relations between the Hamiltonian of the system and its operators. [HS , Sα (ω)] = 30.

(36) ωSα (ω). Equation (6.14) now takes the form $ ! ! γαβ (ω) # † d ρ̃S (t) = − [Sβ (ω)ρ̃S (t), Sα (ω)] + [Sβ† (ω), ρ̃S (t)Sα (ω)] dt 2 ω. (6.17). α,β. This is the master equation for a system in which its Hamiltonian does not have time dependence. When this equation is applied to a two level system it is known as the quantum optical master equation. We will study this system in the next section and find an approximate solution for the driving problem using (6.17).. 6.3. 2-Level System. As mentioned before we study this system twice. One without driving and other one with driving, but both studies will be made using equation (6.17) that works only for no time dependent Hamiltonians.. 6.3.1. No Driving. Consider a two level system described by hamiltonian. HS =. ! σz 2. (6.18). where ! is the separation between the 2 levels. Interaction is of the usual form in the dipole 0 + and S− (ω) = d0∗ σ− satisfy 0 E. 0 System operators S+ (ω) = dσ approximation. HSB = −D· [HS , σ± ] = ± ! S± so in the sum of equation (6.17) we only consider two frequencies. With this and the results of chapter (5) we have % ( d 1 1 ρ̃S (t) = γ0 (n(ω0 ) + 1) σ− ρS (t)σ+ − σ + σ− ρS (t) − ρS (t)σ+ σ+ dt 2 2 ( % 1 1 + γ0 n(ω0 ) σ+ ρS (t)σ− − σ− σ+ ρS (t) − ρS (t)σ− σ+ 2 2 with γ0 =. 4!3 0 2 |d| . 3c3. 31.

(37) If we choose basis {|e$ , |g$} for the two level system, the master equation in the interaction 0 1 picture US (t) = e−iHS t gives d ρee = γ0 (n + 1)ρee + γ0 nρgg dt d ρgg = −γ0 nρgg + γ0 (n + 1)ρee dt #γ $ d 0 ρeg = − (2n + 1) + i! ρeg dt 2. (6.19) (6.20) (6.21). with n = n(ω0 ). We have omitted the other equation for the matrix element ρge because it can be obtained from the one of ρeg . The last term in the last equation comes from changing to the Schrödinger picture. These equations can be written in a simpler form d &σz $ = −γ0 ((2n + 1) &σz $ + 1) dt #γ $ d 0 &σ+ $ = − (2n + 1) − i! &σ+ $ dt 2 #γ $ d 0 &σ− $ = − (2n + 1) + i! &σ− $ dt 2. (6.22) (6.23) (6.24). where &σz $ = ρee − ρgg = T r {σz ρ} and &σ− $ = ρeg = T r {σ− ρ} These equations are easily solved. We are interested in σz that gives the difference between the populations of the two states of the system. We get. &σz $ (t) =. e−γ0 t (1 + &σz $t=0 (1 + 2n)) − 1 1 + 2n. (6.25). For the particular case ρ(t = 0) = |g$ &g| ⇒ &σz $t=0 = −1 &σz $ (t) = −. e−γ0 t 2n − 1 1 + 2n. (6.26). We see that the difference of populations shows an exponential decrease tending to a stationary −1 . value 2n + 1. 32.

(38) 6.3.2. Driving (approx.). Now we include driving in the problem but we will not be using the master equation for periodic driving derived at the beginning of this chapter. We will find first an approximate solution of the problem. When we add driving to the system we are expecting to be able to apply some control to it. Driving will be made by a a circularly polarized monochromatic resonant continuous-wave field so that the system is described by the Hamiltonian Ω ! HS (t) = σz + (σx cos(ωP t) + σy sin(ωP t)) 2 2. (6.27). Where ! . Ω so that we can make the approximation % " t ( ) ! * US (t) = exp −i HS (u) du ≈ exp −i σz t 2 0. (6.28). Note that even though we have a different Hamiltonian than before, we have approximated the transformation to the interaction picture as if we had no driving. Derivation of the master equation proceeds as before, in the Schrödinger picture we get 5! 6 d Ω ρS (t) = −i σz , ρS (t) − i [σx cos(ωP t) + σy sin(ωP t), ρS (t)] dt 2 2 ( % 1 1 + γ0 (n(ω0 ) + 1) σ− ρS (t)σ+ − σ + σ− ρS (t) − ρS (t)σ+ σ+ 2 2 % ( 1 1 + γ0 n(ω0 ) σ+ ρS (t)σ− − σ− σ+ ρS (t) − ρS (t)σ− σ+ 2 2. (6.29). To solve this equation it results convenient to transform to a rotating frame U = eiωP σz t/2 such that ρ#S = U ρS U † . / / ∆. Ω. d # ρS (t) = −i σz , ρ#S (t) − i σx , ρ#S (t) dt 2 2 % ( 1 1 # # # + γ0 (n(ω0 ) + 1) σ− ρS (t)σ+ − σ + σ− + ρS (t) − ρS (t)σ+ σ+ 2 2 % ( 1 1 # # # + γ0 n(ω0 ) σ+ ρS (t)σ− − σ− σ+ ρS (t) − ρS (t)σ− σ+ 2 2 with ∆ = ! − ωP . 33. (6.30).

(39) That can again be written as: d &σz $ = −γ0 ((2n + 1) &σz $ + 1) + iΩ (&σ− $ − &σ+ $) dt #γ $ d Ω 0 &σ− $ = − (2n + 1) + i∆ &σ− $ + i &σz $ dt 2 2 #γ $ Ω d 0 &σ+ $ = − (2n + 1) − i∆ &σ+ $ − i &σz $ dt 2 2. (6.31) (6.32) (6.33). These equations are commonly known as the Optical Bloch Equations. In the simplest case were ∆ = 0 and zero temperature (n = 0) they look like d &σz $ = −γ0 (&σz $ + 1) + iΩ (&σ− $ − &σ+ $) dt d γ0 Ω &σ− $ = − &σ− $ + i &σz $ dt 2 2 γ0 Ω d &σ+ $ = − &σ+ $ − i &σz $ dt 2 2. (6.34) (6.35) (6.36). !Σz". 1.0. 0.5. !t 5. 10. 15. 20. 25. 30. "0.5. "1.0. Figure 6.1: Population Inversion for the Driven 2-level system. The 3 different curves correspond to the parameters γΩ0 = 1/10 (Blue), γΩ0 = 1/2 (Purple) and γΩ0 = 1 (Yellow).. 34.

(40) The initial conditions are taken as &σz $ (t = 0) = −1 and &σ+ $ (t = 0) = &σ− $ (t = 0) = 0. The solution for the difference of populations is   3γ0 t & '  2 − 2Ω 3γ0 sin(ζt) 1+ 2 e 4 cos(ζt) + &σz $ (t) = − 2  4ζ γ0 + 2Ω2  γ0 γ02. with ζ 2 = Ω2 −. (6.37). # γ $4 0. 4. Figure 6.1 shows results for different cases of driving intensity in order to understand what is happening to the driven 2-level system.. !Σz" "0.980. "0.985. "0.990. "0.995. !t 50. 100. 150. 200. Figure 6.2: Population Inversion for the Driven 2-level system when Ω → 0. The 3 different curves correspond to the parameters γΩ0 = 1/10 (Blue), γΩ0 = 1/2 (Purple) and γΩ0 = 1 (Yellow).. Form figure 6.1 we see that the system performs Rabi oscillations between the excited and ground states as expected. Oscillations approach a stationary value for long times. &σz $ (t → ∞) = γ2 − 2 0 2. γ0 + 2Ω It is also clear that if the intensity of the light, that is to say the driving, is increased so is the amplitude of the oscillations and the time for them to reach its stationary value. If the driving is 35.

(41) very strong compared to γ0 but still smaller than !, in the long time limit populations are divided equally between the two states. &σz $ (t → ∞) → 0. We can find the atom in either of the two states with the same probability.. When we make Ω → 0 figure 6.2 shows that no oscillations occur and we obtain the initial condition &σz $ = −1 as was expected from the case studied before. See equation (6.26) when we have zero temperature.. Now that we have found results for a 2 level system with no driving and an approximate solution for the driven 2-level system it is time to use equation 6.14.. It turns out that the utility of equation its very restricted. The 2 infinite sums appearing, one in the master equation and the other one in the decomposition of operators Sα (n, ω) make calculations extremely complicated for this problem. Just for the very bad approximation of considering only the first two terms in the series the master equation becomes too complicated to be solved. In order to obtain results in next section we make a derivation similar to the one presented here but with subtle changes that allow us to obtain the master equation for periodic driving but this time in the Schrodinger picture. This change of picture will make calculations easier and allow us to obtain results for the driven 2-level problem.. 36.

(42) Chapter 7. Master equation for Periodic Driving - Schrodinger Picture Following a similar procedure as that shown in the chapter 6 we use the Floquet decomposition of the evolution operator to derive the master equation for periodic driving in the Schrodinger picture, obtaining explicit time dependence in the dissipative part. We then apply this equation to a driven harmonic oscillator obtaining the typical form of the optical master equation but with a renormalized Hamiltonian.. We also study a two level atom driven by a monochromatic resonant continuous-wave field concentrating on the case of zero temperature and show that there is a correspondence between the Floquet states of the driven two level system and the dressed state of an atom.. 7.1. Derivation. The master equation derived in section 6.1 was obtained in the interaction picture, but as we have discuss before, obtaining results from equation (6.14) becomes a very complicated task.. One could try to change equation (6.14) to the Schrödinger picture but calculation becomes a bit complex. Therefore we follow a similar procedure to the one showed in section 6.1 but using a particular decomposition of the system operators in the interaction picture, allowing us to obtain 37.

(43) a master equation for periodic driving that is in the Schrödinger picture.. 0 E 0 where D 0 is the dipole system operator and We consider an interaction of the form ĤSB = −D· seek a decomposition of the form 0 (t) = U † (t)DU 0 S (t) = D S. !. −iωt 0 A(ω)e =. !. 0 † (ω)e+iωt A. (7.1). ω. ω. with the property A† (ω) = A(−ω). From now on we will drop the vector problem and consider only one direction. Generalization of the problem is straightforward. Using the Floquet decomposition of the evolution operator. D(t) =. ! j,j !. + ,- + |φj $ &φj (t)| D +φj ! (t) φj ! + e−i(εj! −εj )t. (7.2). + + , , 2 Expanding the time periodic matrix elements in the following form &φj (t)| D +φj ! (t) = n &φj | D +φj ! n einωP t + , + , 1 :∞ with &φj | D +φj ! n = &φj (t)| D +φj ! (t) e−inωP t . The dipole operator in the interaction picture 0 P takes the form. D(t) =. !. j,j ! ,n. + , - + &φj | D +φj ! n |φj $ φj ! + e−i(εj! −εj −nωP )t. Comparing with the dipole operator decomposition A(ω) =. 2. j,j ! ,n &φj | D. (7.3). + , - + +φj ! |φj $ φj ! + where n. the numbers (j, j # , n) satisfy εj ! − εj − nωP = ω. Note the difference between this decomposition and the one given in the previous section for the system operators.. From now on the procedure to obtain the master equation in the interaction picture is the same as in section 6.1 where the secular approximation is made for two frequencies ω and ω # . Starting from equation (6.6) we get % ( ! 1 † 1 d † † ρ̃S (t) = γ(ω) A(ω)ρ̃S (t)A (ω) − A (ω)A(ω)ρ̃S (t) − ρ̃S (t)A (ω)A(ω) dt 2 2 ω. (7.4). Changing this equation to the Schrödinger picture ρ̃S (t) = US† ρ̂S (t)US involves treating terms like A(ω)US† ρ̂S (t)US A† (ω) , A† (ω)A(ω)US† ρ̂S (t)US and US† ρ̂S (t)US A† (ω)A(ω). We will consider only 38.

(44) the first one, the other two are treated in the same way A(ω)US† ρ̂S (t)US A† (ω) =. !. j,j ! ,n. =. !. j,j ! ,n. =. !. j,j ! ,n. + , - + &φj | D +φj ! n |φj $ φj ! + US† ρ̂S (t)US A† (ω). + + , &φj | D +φj ! n |φj $ φj ! (t)+ eiεj! t US† ρ̂S (t)US A† (ω). (7.5) (7.6). + , + ei(εj! −εj )t &φj | D +φj ! n |φj (t)$ φj ! (t)+ US† ρ̂S (t)US A† (ω) (7.7). Doing the same for the operator A† A(ω)US† ρ̂S (t)US A† (ω) ! ! =. (7.8) + , + ei(εj! −εj +εi −εi! )t US† &φj | D +φj ! n |φj (t)$ φj ! (t)+ ρ̂S (t) &φi! | D † |φi $q |φi! (t)$ &φi (t)| US. j,j !,n i,i! ,q. Remember that εj ! − εj = ω + nωP , εi! − εi = ω # + qωP and that we made the approximation ω = ω # . So we finally have A(ω)US† ρ̂S (t)US A† (ω) = US† (t)A(ω, t)ρS (t)A† US (t). (7.9). with. A(ω, t) =. !. j,j ! ,n. + , - + = &φj | D +φj ! n |φj,n (t)$ φj ! + ;. |φj,n (t)$ = einωP t |φj (t)$. (7.10). Treating the other terms in the same way leads to the equation 6 5 6* ! γ(ω) )5 d ρS (t) = −i [HS (t), ρS (t)] + A(ω, t)ρS (t), A† (ω, t) + A(ω, t), ρS (t)A† (ω, t) (7.11) dt 2 ω This is the master equation for periodic driving in the Schrödinger picture. In contrast to the typical master equation where HS does not depend on time the driving introduces a time dependence in the dissipative part.. 39.

(45) 7.2. 2-Level System. Now, we apply equation (7.11) to the case of an open system consisting of a two level atom driven by a monochromatic resonant continuous-wave field, which interacts with a quantized radiation field. We assume that the intensity of the monochromatic light is of the same order as the two level atom splitting and obtain the Floquet decomposition of the dipole operator obtaining the frequencies that correspond to Mollow‘s triplet and its corresponding transitions that can be viewed as transition between the dressed states of an atom. The system is described by the Hamiltonian Ω ! HS (t) = σz + (σx cos(ωP t) + σy sin(ωP t)) 2 2. (7.12). We wish to find the master equation for this system in the Schrödinger picture so we need to solve the Floquet problem first, in order to obtain the dipole operator decomposition made in section 7.1.. For simplicity we assume ωP = !. Floquet states are given by $ 1 # |φ± (t)$ = √ ±e−i- t/2 |e$ + ei- t/2 |g$ 2. (7.13). Ω with a Floquet spectrum given by ε = ± , and |e$ and |g$ referring to the excited and ground 2 states of the two level system. 0 + + d0∗ σ− . This is the same interaction we In this case we have that the dipole operator is D = dσ took before in equation (3.31) but now the operator F has a particular form given by the electric field operator (see section 5). D(t, t0 ) = US† (t, t0 )DUS (t, t0 ) =. !. A(ω, t0 )e−iωτ. (7.14). ω. where we have made τ = t − t0 . Since for this problem we know all the Floquet states and energies, calculations are straightforward.. 40.

(46) D(t) = + +. 1 # 0∗ i-(t0 +τ ) 0 −i-(t0 +τ ) $ d e + de {|φ+ (t0 )$ &φ+ (t0 )| − |φ− (t0 )$ &φ− (t0 )|} 2# $ 1 0∗ i-(t0 +τ ) 0 −i-(t0 +τ ) iΩτ d e − de e |φ+ (t0 )$ &φ− (t0 )| 2# $ 1 0 −i-(t0 +τ ) 0∗ i-(t0 +τ ) −iΩτ de −d e e |φ− (t0 )$ &φ+ (t0 )| 2. (7.15). This gives us 6 frequencies in the decomposition of the dipole operator {±!, ±(! + Ω), ±(! − Ω)} and 6 operators A(ω, t). A(!, t) = A(! + Ω, t) = A(! − Ω, t) = and their adjoint operators.. d0 {|φ+ (t)$ &φ+ (t)| − |φ− (t)$ &φ− (t)|} e−i- t 2 d0 |φ− (t)$ &φ+ (t)| e−i- t 2 d0 |φ+ (t)$ &φ− (t)| e−i- t 2. (7.16). With these operators we now have the master equation in the. Schrdinger picture for driven two-level system but lets analyze them first.. The A(ω, t) operators give the spectrum of frequencies of the problem. We see three main frequencies ! = ωP , ! + Ω and ! − Ω. These are the same frequencies find in Mollow‘s triplet spectrum of resonance fluorescence. If we would like to plot the spectrum of frequencies for the system we are studying, we would get 3 high peaks in the frequencies given by operators A(ω, t) (see Figure 7.1).. Figure 7.1 shows experimental results for the spectrum of resonance for Sodium atoms in a monochromatic field [9]. Although up till now, from our results we have not determined the exact form of the spectrum, we see that there are actually three peaks at the frequencies appearing in the Floquet decomposition of the system operator. Note that the graph is plotted as a function of ω − ! and thats why the central peak corresponding to ω = ! appears at zero. To obtain the master equation we could proceed by just replacing operators (7.16) in equation (7.11). On the other hand, it turns out to be easier to look at the representation of these operators. 41.

(47) Figure 7.1: Experimental results for the spectrum of resonance for Sodium obtained in [9]. Ω = 78M Hz and γ0 = 10M Hz.. in the Floquet basis {|φ+ (t)$ , |φ− (t)$}. 0 z; 2A(!, t) = dσ. 0 −; 2A(! + Ω, t) = dσ. 0 + 2A(! − Ω, t) = −dσ. (7.17). We have dropped the exponential term in the operators due to the fact that in the master equation, only products of the form AA† are involved, and the exponential terms cancel. In the Floquet basis the dipole operator decomposition ends up made of just simple Pauli matrices.. In order to write the master equation for the system in the Floquet basis the Hamiltonian HS (t) in terms of the states {|φ+ (t)$ , |φ− (t)$} must be found.. HS (t) =. . . −! + iΩ sin [t(! − ωp )] 1  Ω cos [t(! − ωp )]   2 −! − iΩ sin [t(! − ω )] Ω cos [t(! − ωp )] p. 42. (7.18) (7.19).

(48) But we are at resonance ! = ωp so .  1  Ω −! HS (t) =   2 −! Ω. (7.20) (7.21). With this it is straightforward to write the master equation for a driven two level system in the Floquet basis. d ρ++ dt d ρ+− dt d ρ−+ dt d ρ−− dt with γ(ω) =. ! 1 = −i (ρ−+ − ρ+− ) + (γ(! + Ω)ρ−− + γ(! − Ω)ρ++ ) 2 4 i ρ+− = − (2Ωρ+− + ! (ρ++ − ρ−− )) − (16γ(!) + γ(! − Ω) + γ(! − Ω)) 2 8 i ρ−+ = − (2Ωρ−+ + ! (ρ++ − ρ−− )) − (16γ(!) + γ(! − Ω) + γ(! − Ω)) 2 8 ! 1 = i (ρ−+ − ρ+− ) − (γ(! + Ω)ρ−− + γ(! − Ω)ρ++ ) 2 4. (7.22) (7.23) (7.24) (7.25). 4!3 0 2 |d| (1 + n(ω)). 3c3. We already studied the case when the separation between the excited and ground atom energies is bigger than the intensity of the driving ! . Ω. The system was found to make exponentially damped Rabi oscillations (Figure 6.1). Now we consider the case where the intensity of the light (driving) is of the same order of the energy difference in the two levels.. If Ω ∼ ! then as γ ∝ ω 3 we can drop the terms that have γ(! − Ω) because they are too small as compared with the ones that have γ(! + Ω). We also eliminate the dependence of Ω from γ by approximating γ(! + Ω) ≈ γ(2!) = 8γ(!).. 43.

(49) With this, the Bloch type equations for the driven two level system in the Schrodinger picture and in the Floquet basis look like d ρ++ dt d ρ+− dt d ρ−+ dt d ρ−− dt. ! = −i (ρ+− − ρ−+ ) + 2γ(!)ρ−− 2 i = − (2Ωρ+− − !(ρ++ − ρ−− )) − 3γ(!)ρ+− 2 i = (2Ωρ−+ − !(ρ++ − ρ−− )) − 3γ(!)ρ−+ 2 ! = i (ρ+− − ρ−+ ) − 2γ(!)ρ−− 2. (7.26) (7.27) (7.28) (7.29). We solve these equations numerically for the zero temperature case (n(ω) = 0) assuming that initially the system is in its ground state ρ = |g$ &g|. That is in the Floquet bases equal to ρ = 1/2. We consider three main cases for the solutions: Ω/! = 1 (approximation turns out to be exact), Ω/! = 1/2 (range of validity of our approximation) and Ω/! = 1/10 (approximation is not so reliable). Even though the results of the last case of study are not valid, we have already studied this situation before (section 6.3) and know what is expected for the behavior of the system.. It is important to note that the population inversion &σz $F = ρ++ − ρ−− gives the difference between the probabilities of finding the system in state |φ+ (t)$ and |φ− (t)$ and not |e$ and |g$. All three graphs (Figures 7.2, 7.3, 7.4) show that the system starts from an equal probability of finding it in any of its two states and the driving alters it creating an amplitude that depending on the relation γ/Ω, thus it forces exponentially damped oscillations. As the driving gets more intense γ/Ω → 1 populations oscillate even more. It is also clear that as the driving Ω gets smaller with respect to the difference between the two levels !, the systems tend to oscillate more but decreasing its amplitude. For example, figure 7.2 shows high frequency oscillations but a very low amplitude. Its stationary value (t) is almost the same for all the values of γ/Ω differing from the case studied earlier that showed different stationary values (see figure 6.1), but remember that this case Ω/! = 0.1 its outside the range of validity of. 44.

(50) !Σz"F 0.2. 0.1. !t 1. 2. 3. 4. 5. "0.1. "0.2. Figure 7.2: Population Inversion for the Driven 2-level system in the Floquet Basis when Ω/! = 0.1. The 3 different curves correspond to the parameters γΩ0 = 1/10 (Blue), γΩ0 = 1/2 (Purple) and γ0 Ω = 1 (Yellow).. our approximation so we stick with the results of section 6.3.. For the other two cases: Figure 7.2 Ω/! = 1 (exact) and 7.4 Ω/! = 0.5 (valid), we see that a very strong driving γ/Ω → 1 is needed for the system to oscillate. Also the stationary value for the system’s population inversion shows a clear dependence on the strength of the driving field as it gets stronger, tending to decrease and approaching 1/2 so that the probabilities of finding the system in any of the two Floquet states become identical.. All three graphs show, in the stationary limit, a tendency of the probability of being in state |φ+ (t)$ bigger than the one in being in state |φ− (t)$. Actually this is cause by the initial conditions of the system. If we choose for example ρ = −1/2 the graphs would be upside down showing a bigger probability of being in state |φ− (t)$. This is actually pretty interesting because although we made a Markovian treatment of the problem we still see that the initial conditions determine the stationary value showing some memory effects. These effects do not appear when the intensity of. 45.

(51) !Σz"F 1.0. 0.5. !t 2. 4. 6. 8. 10. 12. 14. "0.5. "1.0. Figure 7.3: Population Inversion for the Driven 2-level system in the Floquet Basis when Ω/! = 1. The 3 different curves correspond to the parameters γΩ0 = 1/10 (Blue), γΩ0 = 1/2 (Purple) and γ0 Ω = 1 (Yellow).. the light Ω is much smaller than the difference between the atom levels !.. To get a better understanding of what’s happening in the system consider first the states |φ±,n (t)$ Ω with Floquet energies ε̃± = ε± ± . See equations (4.10) and (7.10). 2 In terms of these states the operators A(ω, t) (7.16) look like. A(!, t) = A(! + Ω, t) = A(! − Ω, t) =. d0 {|φ+,0 (t)$ &φ+,1 (t)| − |φ−,0 (t)$ &φ−,1 (t)|} 2 d0 |φ−,0 (t)$ &φ+,1 (t)| 2 d0 |φ+,0 (t)$ &φ−,1 (t)| 2. 46. (7.30).

(52) !Σz"F 1.0. 0.5. !t 2. 4. 6. 8. 10. 12. 14. "0.5. "1.0. Figure 7.4: Population Inversion for the Driven 2-level system in the Floquet Basis when Ω/! = 0.5. The 3 different curves correspond to the parameters γΩ0 = 1/10 (Blue), γΩ0 = 1/2 (Purple) and γ0 Ω = 1 (Yellow).. and they act over these new Floquet states like. A(! + Ω, t) |φ+,n (t)$ =. d0 d0 |φ−,0 (t)$ &φ+,1 (t)| |φ+,n (t)$ = |φ−,n−1 (t)$ 2 2. (7.31). A(! + Ω, t) |φ−,n (t)$ = 0 d0 A(! − Ω, t) |φ−,n (t)$ = |φ+,n−1 (t)$ 2 A(! − Ω, t) |φ+,n (t)$ = 0 d0 A(!, t) |φ+,n (t)$ = |φ+,n−1 (t)$ 2 d0 A(!, t) |φ−,n (t)$ = |φ−,n−1 (t)$ 2 We now make a correspondence with these states and the dressed states of an atom [10]. Consider the Hamiltonian & ' # $ 1 ! † H = σz + ! a a + + g σ + a + σ − a† 2 2. 47. (7.32).

(53) This Hamiltonian describes a 2-level system interacting with a single mode of light that is in resonance with the energy gap of the 2-level system. This is a particular case of the JaynesCummings Hamiltonian. It can be diagonalized with eigenergies and eigenvalues 1 Ω ; |1, n$ = √ {|e, n − 1$ − |g, n$} 2 2 Ω 1 = n! + ; |2, n$ = √ {|e, n − 1$ + |g, n$} 2 2. e1,n = n! −. (7.33). e2,n. (7.34). Figure 7.5: Energy level Diagram for dressed states where |g(e), n$ indicates that the atom is in the ground (excited) state and there are n excitations in the system. Figure 7.5 illustrates an energy level diagram for the dressed states. It is clear that states that have the same number of excitation |g, n$ or |e, n$ have the same energy, but this is not true for the dressed states |2, n$ and |1, n$. We make the equivalence |φ−,n (t)$ ⇐⇒ |1, n$. (7.35). |φ+,n (t)$ ⇐⇒ |2, n$. (7.36). 48.

(54) The Floquet states of the driven 2-level system correspond to the dressed states of an atom. Hence we can now have a complete interpretation of the action of the operators A(ω, t). They induce transitions between the dressed states of the atom and the transitions between these states correspond to a frequency spectrum {!, ! + Ω, ! − Ω}. Figure 7.6 shows these allowed transitions in an energy level diagram. Note that for example transition |2, n$ (−→ |1, n$ is not allowed.. Figure 7.6: Energy level Spectrum and Allowed Transitions for the Driven 2-Level Atom. So the oscillations described in the results presented earlier of our master equation can be viewed as being oscillations between the dressed states of the atom with transitions happening at frequencies that correspond to Mollow‘s triplet.. 7.3. Harmonic oscillator. We consider a one dimensional quantum harmonic oscillator driven by a time periodic force. The Hamiltonian of this system is HS (t) = ω0 a† a + C(a† + a) sin(ωP t) 49. (7.37).

(55) Where C is a constant. This oscillator is coupled to a thermal bath at temperature T and its interaction is given in the dipole approximation by HSB = −D· E. We are interested in describing the evolution of the reduced system so we use master equation (7.11), that is in the Schrödinger picture. In this case, we take the dipole operator as X = a† + a and look for the decomposition. X(t, t0 ) = US† (t, t0 )XUS (t, t0 ) =. !. A(ω, t0 )e−iωt. (7.38). ω. We have seen that these operators are obtained from the Floquet states of the time periodic Hamiltonian of the system. The solution for the Floquet problem for this Hamiltonian is very complicated and it is solved in [10].. We follow a different approach this time and try to find operators A(ω, t) without solving the Floquet problem. Notice that equation (7.38) is just the Heisenberg representation of operator X, d so it satisfies X(t, t0 ) = −i [X(t, t0 ), HS (t)]. dt We solve the equation only for a(t, t0 ), with initial condition that at t = t0 , a(t, t0 ) = a. The equation takes the form d a(t, t0 ) = −iω0 a(t, t0 ) + iC sin(ωP t) dt. (7.39). This equation is easy to solve multiplying by eiω0 t at both sides and integrating. Its solution is. 3 iωP t 4( % e e−iωP t iC eiωP t iC e−iωP t iC − + − a− 2 ω0 + ωP ω0 − ωP 2 ω0 + ωP 2 ω0 − ωP. (7.40). % 3 4( iC e−iωP t eiωP t iC e−iωP t iC eiωP t † a + − − + 2 ω0 + ωP ω0 − ωP 2 ω0 + ωP 2 ω0 − ωP. (7.41). −iω0 (t−t0 ). a(t, t0 ) = e. And for a† (t, t0 ) †. iω0 (t−t0 ). a (t, t0 ) = e. Summing this two we have 50.