Chapter 2: Quantum Master Equations

Transcription

1 Chapter 2: Quantum Master Equations I. THE LINDBLAD FORM A. Superoperators an ynamical maps The Liouville von Neumann equation is given by t ρ = i [H, ρ]. (1) We can efine a superoperator L such that Lρ = i/ [H, ρ]. It is calle a superoperator because it is an object that acts on an operator an results in a new operator. If the Hamiltonian is time-inepenent, we may formally integrate the Liouville von Neumann equation an obtain ρ(t) = exp (Lt) ρ() V(t)ρ(), (2) where V is another superoperator that maps the ensity matrix from its initial form to its form at time t an therefore is calle a ynamical map. It is relate to the unitary evolution operator U(t) = exp( iht/ ) accoring to V(t)ρ() = U(t)ρ()U(t). (3) B. Quantum ynamical semigroups Now let us inclue also interaction between the system of interest an its environment. In the following, we will use ρ S for the reuce ensity operator for the system an Tr E {ρ} for the partial trace over the environment. As the ynamics of the combination of system an environment is fully coherent, we have ρ(t) = U(t)ρ()U(t), (4) which after taking the trace over the environment on both sies results in ρ S (t) = Tr E {U(t)ρ()U(t) }. (5) In many typical situations, the initial state between the system an the environment is a prouct state of the form ρ() = ρ S () ρ E (). (6)

2 2 Then, we may again think of the right han sie of the previous equation to efine a superoperator representing a ynamical map V(t), but now for S alone! Furthermore, we may use the following ecompositions [1] ρ E () = α V(t)ρ S () = αβ λ α ψ α ψ α (7) W αβ (t)ρ S ()W αβ (t), (8) where the operators W αβ act only on the Hilbert space of the system an are given by W αβ (t) = λ β ψ α U(t) ψ β. (9) From the completeness of the states ψ α on the Hilbert space of the environment, we may ientify the relation from which follows W αβ (t)w αβ (t) = 1 S, (1) αβ Tr{V(t)ρ S ()} = Tr{ρ S } = 1, (11) i.e., the ynamical map is trace-preserving. Moreover, it is completely positive, mapping a positive ensity matrix onto another positive ensity matrix [1]. In many important cases, we can make one further assumption on the ynamical map V(t). If correlations in the environment ecay much faster than the timescale of the evolution in the system of interest, we may neglect memory effects escribing how the system has previously interacte with the environment. This is also known as the Markov approximation. For example, consier a thermal state of the environment of the form ρ E = n 1 Z exp( βe n) n n, (12) where Z = n exp( βe n) is the partition function, β is the inverse temperature an E n is the energy of the state n. Then, if the environment is large an its ynamics is fast enough, any energy exchange with the system will quickly issipate away to form a new thermal state with almost exactly the same temperature. Then, from the viewpoint of the system, the state of the environment will appear to be almost constant all the time. Formally, we can express the consequences of the Markov approximation on the ynamical map as [1] V(t 1 )V(t 2 ) = V(t 1 + t 2 ) t 1, t 2. (13)

3 3 Note that the constraint on the times being positive means that we can only piecewise propagate the system forwar in time, i.e, the inverse of the ynamical map oes usually not exist. This is in contrast to coherent ynamics, where there is an inverse operation corresponing to negative time arguments in the unitary evolution operator. Hence, while the ynamical maps of coherent systems form a group, the ynamical maps for open quantum systems only form a semigroup. The generator of the semigroup is the Liouvillian L, which is a generalization of the superoperator appearing on the right han sie of the Liouville von Neumann equation. One important consequence of this generalization is that the von Neumann entropy is no longer a conserve quantity. However, the Liouvillian has to fulfill the property of being the generator of a completely positive an trace-preserving ynamical map. In the following, we will see how the most general form of the Markovian master equation will look like. C. Most general form of the ynamics Similar to the case of a close quantum system, we can write the ynamical map of an open quantum system as an exponential of the generator of the semigroup, V(t) = exp (Lt). (14) The superoperator L reuces to the one of the Liouville von Neumann equation in the case of purely coherent ynamics, but in general will have aitional incoherent terms. Expaning the ynamical maps for short times τ, we obtain ρ(t + τ) = V(τ)ρ(t) = (1 + Lt) ρ(t) + O(τ 2 ), (15) which in the limit τ yiels a first-orer ifferential equation known as a quantum master equation, ρ(t) = Lρ(t). (16) t Let us now erive an explicit form for the master equation. For this, we nee to efine an operator basis {F i }. The inner prouct for operators is efine as { } F i, F j Tr F i F j. (17) A complete orthonormal set consists of N 2 operators, where N is the Hilbert space imension. It is convenient to choose one of the operators as proportional to the ientity, i.e.,

4 4 F 2 N = 1/ N [1]. Then, all other operators are traceless. For example, in a two-level system, the remaining operators are proportional to the Pauli matrices. We can now express the action of the ynamical map using this operator basis as V(t)ρ = where the coefficients c ij (t) is given by c ij (t) = αβ N 2 i,j=1 c ij (t)f i ρf j, (18) F i, W αβ (t) F j, W αβ (t) (19) with the operators W αβ efine as above. The coefficients c ij (t) form a postive matrix C, as for any N 2 -imensional vector v, we have [1] v Cv = v i F i, W αβ (t) αβ i Inserting this expansion into the quantum master equation, we obtain V(t)ρ ρ Lρ = lim τ τ [ 1 c N = lim 2 N 2(τ) N ρ + 1 N 2 1 ( cin 2(τ) τ N τ N τ i=1 ] N i,j=1 c ij (τ) F i ρf j τ 2. (2) F i ρ + c ) N 2 i(τ) ρf i τ (21) (22), (23) where we separate off all terms containing F N 2 N 2 = 1 N. As a next step, we efine the following quantities c ij (τ) a ij = lim τ τ F = 1 N 2 1 lim N i=1 G = 1 2N lim τ τ i, j = 1,..., N 2 1 (24) c in 2(τ) F i (25) τ c N 2 N 2(τ) N τ (F + F ) (26) H = 1 2i (F F ). (27) Note that the operator H is Hermitian, although F is not because the coefficients c in 2 complex. Using these efinitions, we fin for the generator are Lρ = i[h, ρ] + {G, ρ} + N 2 1 ij=1 a ij F i ρf j. (28)

5 5 Since the ynamical map is trace-preserving, the trace over quantum master equation has to vanish, i.e., Tr {Lρ} = Tr {( 2G + from which we can rea off that G has to be G = 1 2 N 2 1 ij=1 N 2 1 ij=1 a ij F i F j ) ρ } =, (29) a ij F i F j. (3) Substituting this result back into the quantum master equation, we obtain Lρ = i[h, ρ] + N 2 1 i,j=1 a i j ( F i ρf j 1 { F j 2 F i, ρ} ). (31) Finally, the matrix forme by the coefficients a ij is again Hermitian an positive, so we can iagonalize it to obtain positive eigenvalues γ i. Then, we fin the most general form of a Markovian quantum master equation to be given by Lρ = i[h, ρ] + N 2 1 i=1 ( γ i A i ρa i 1 { A i 2 A i, ρ} ), (32) where the operators A i are appropriate linear combinations of the operators F i obtaine from the iagonalization proceure. This form of the quantum master equation is known as the Linbla form, as Linbla first showe that the generator of a Markovian master equation has to be of that form [2]. The system Hamiltonian is containe in the Hermitian operator H, but the latter can also inclue aitional terms coming from the interaction with the environment. Furthermore, the eigenvalues γ i correspon to relaxation rates escribing incoherent ecay processes in the system. Typically, these ecay processes will result in the system eventually reaching a stationary state characterize by t ρ =. However, such a stationary state oes not necessarily mean the absence of any ynamics: for example, a single realization of two-level system in a maximally mixe state might still violently jump between both levels! Only when the ensemble average is taken, the ynamics will vanish.

6 6 II. THE QUANTUM OPTICAL MASTER EQUATION A. Weak-coupling approximation In the following, we will consier a system S that is weakly couple to a bath B. The Hamiltonian of the combine system is given by H = H S + H B + H I, (33) where H S enotes the part of Hamiltonian only acting on S, H B only acts on B, an H I accounts for the interaction between the two. We first perform a unitary transformation ρ = U(t)ρ U(t) into the interaction picture, i.e. (assuming = 1), U(t) = exp[ i(h S + H B )t]. (34) Inserting the transforme ensity matrix into the Liouville von Neumann equation, we obtain which can be brought into the convenient form ( U(t)ρ U(t) ) = i[h, U(t)ρ U(t) ], (35) t t ρ = i[h I(t), ρ]. (36) The interaction Hamiltonian is now time-epenent accoring to H I (t) = U (t)h I U(t) = exp[i(h S + H B )t]h I exp[ i(h S + H B )t]. (37) We can formally integrate the Liouville von Neumann equation an obtain t ρ(t) = ρ() i s[h I (s), ρ(s)]. (38) This expression can be inserte back into the Liouville von Neumann equation an after taking the trace over the bath, we arrive at [1] t t ρ S(t) = str B {[H I (t), [H I (s), ρ(s)]]}, (39) where we have assume that the initial state is such that the interaction oes not generate any (first-orer) ynamics in the bath, i.e., Tr B {[H I (t), ρ()]} =. (4)

7 7 So far, we have mae little progress, as the right-han sie of the equation of motion still contains the ensity operator of the full system. To obtain a close equation of motion for ρ S only, we assume that the interaction is weak such that the influence on the bath is small. This is also known as the Born approximation. Then, we may treat the bath as approximately constant an write for the total ensity operator ρ(t) = ρ S (t) ρ B. (41) Then, we obtain a close integro-ifferential equation for the ensity operator ρ S, t t ρ S(t) = str B {[H I (t), [H I (s), ρ S (s) ρ B ]]}. (42) Such an integro-ifferential equation is very ifficult to hanle as the ynamics at time t epens on the state of the system in all previous times. The equation of motion can be brought into a time-local form by replacing ρ S (s) by ρ S (t). As we will see later on, this is not really an approximation, but alreay implicit in the weak-coupling assumption. This step brings us to an equation known as the Refiel master equation [1], t t ρ S(t) = str B {[H I (t), [H I (s), ρ S (t) ρ B ]]}. (43) This equation is still a non-markovian master equation an oes not guarantee to conserve positivity of the ensity matrix ue to approximations we have mae. B. Markov approximation To obtain a Markovian master equation, we first substitute s by t s in the integran, which oes not change the bouns of the integration. Then, we can unerstan the parameter s as inicating how far we go backwars in time to account for memory effects, which can be characteristic timescale τ B, over which correlations in the bath ecay. Uner the Markov approximation, these memory effects are short-live an therefore the integran ecays very quickly for s τ B. Then, we may replace the upper boun of the integration by infinity, an obtain a Markovian master equation, t ρ S(t) = str B {[H I (t), [H I (t s), ρ S (t) ρ B ]]}. (44)

8 8 The typical timescale of the ynamics generate by this master equation is characterize by τ R, which is the relaxation time of the system ue to the interaction with the bath. For the Markov approximation to be vali, this relaxation time has to be long compare to the bath correlation time, i.e, τ R τ B. For quantum optical systems, τ B is the inverse of the optical frequency, i.e., several inverse THz, while the lifetime of an optical excitation is in the inverse MHz range. Therefore, the Markov approximation is well justifie in quantum optical systems. The two approximation we have mae so far are often groupe together as the Born- Markov approximation. However, they still o not guarantee that the resulting master equation generates a quantum ynamical semigroup an hence cannot be cast into a Linbla form. For this, another approximation is necessary. C. Secular approximation The secular approximation involves iscaring fast oscillating terms in the Markovian master equation. It is therefore similar to the rotating-wave approximation (RWA) use in NMR an quantum optics. However, applying the RWA irectly on the level on the interaction Hamiltonian can cause problems such as an incorrect renormalization of the system Hamiltonian [3]. The secular approximation (which is sometimes also calle RWA), however, is carrie out on the level of the quantum master equation. To be explicit, let us write the interaction Hamiltonian in the form H I = α A α B α, (45) where the Hermitian operators A α an B α only act on system an bath, respectively. Assume, we have alreay iagonalize the system Hamiltonian H S, so we know its eigenvalues ε an its projectors onto eigenstates Π(ε). Then, we can project the operators A α on subspaces with a fixe energy ifference ω [1], A α (ω) = Π(ε)A α Π(ε ). (46) ε ε=ω As the eigenvectors of H S form a complete set, we can recover A α by summing over all frequencies, A α (ω) = A ω = A α. (47) ω ω

9 9 Then, we can write the interaction Hamiltonian as H I = α,ω A α (ω) B α = α,ω A α (ω) B α. (48) Using the relation exp(ih s t)a α (ω) exp( ih s t) = exp( iωt)a α (ω) (49) an its Hermitian conjugate, we can write the interaction Hamiltonian in the interaction picture as H I (t) = α,ω exp( iωt)a α (ω) B α (t) = α,ω exp(iωt)a α (ω) B α (t), (5) where we have introuce the interaction picture operators of the bath, B α (t) = exp(ih B t)b α exp( ih B t). (51) Inserting this expression into the Markovian master equation leas us to [1] t ρ = str B {H I (t s)ρ S (t)ρ B H I (t) H I (t)h I (t s)ρ S (t)ρ B } + H.c. (52) = e i(ω ω)t Γ αβ (ω) [ A β (ω)ρ S (t)a α (ω ) A α (ω ) A β (ω)ρ S (t) ] + H.c., (53) ω,ω αβ where we have use the one-sie Fourier transform of the bath correlation functions, Γ αβ = s exp(iωs)tr B { Bα (t) B β (t s)ρ B }. (54) In the case where the state of the bath ρ B is an eigenstate of the bath Hamiltonian H B, the bath correlations o not epen on time. The secular approximation is then referre to as the omission of all terms with ω ω, as these terms oscillate fast an average out. Again, in quantum optical systems, this comes from the fact that optical transition frequencies are much larger than the ecay rates of excite states, i.e., τ R τ S. Hence, we obtain t ρ = ω Γ αβ (ω) [ A β (ω)ρ S (t)a α (ω) A α (ω) A β (ω)ρ S (t) ] + H.c. (55) αβ

10 1 This master equation can now be cast into a Linbla form. For this, we split real an imaginary parts of the coefficients Γ αβ accoring to Γ αβ (ω) = 1 2 γ αβ(ω) + is αβ (ω), (56) where the real part can be written as γ αβ (ω) = Γ αβ (ω) + Γ αβ (ω) = s exp(iωs) B α (s) B β () (57) an forms a positive matrix [1]. Then, iagonlizing the coefficient matrix yiels the Linbla form for the yanmics t ρ = i [H LS, ρ S (t)] + ω,k ( γ k (ω) A k (ω)ρ S A k (ω) 1 { } ) Ak (ω) A k (ω), ρ S, (58) 2 where the Lamb shift Hamiltonian H LS commutes with the system Hamiltonian an hence results in a renormalization of the energy levels. This master equation is still expresse in the interaction picture, it can be transforme back to the Schringer picture by aing the system Hamiltonian H S to the coherent part of the ynamics. D. Interaction with the raiation fiel In the case of an atom interacting with the quantize raiation fiel, the Hamiltonian for the latter is given by a sum of harmonic oscillators, H B = kλ ω k a kλ a kλ, (59) where λ is the polarization inex [4]. If the wavelength of the raiation fiel is much larger than the spatial extent of the atomic wavefunction, the ominant interaction term is given by the electric ipole term, H I = E, (6) where is the ipole operator of the atom an E is the quantize electric fiel, E = i hωk V e ( ) λ a kλ a kλ. (61) kλ

11 11 We now assume the reference state of the bath, ρ B is the vacuum without any photons. Then, we can use the following relations for the reservoir correlations [1] a kλ a k λ = a kλ a k λ = a kλ a k λ = (62) a kλ a k λ = δ kk δ λλ (63) e i kλ e j kλ = δ ij k ik j k. (64) 2 Using this, we fin for the spectral correlation tensor λ Γ ij = k hω k V Going to the continuum limit, we can use 1 V k Ω ( δ ij k ) ik j s exp[ i(ω k 2 k ω)s]. (65) ( δ ij k ik j k 2 (2π) 3 c 3 ω k ωk 2 Ω (66) ) = 8π 3 δ ij. (67) k (2π) = 1 3 Fortunately, this means that the spectral correlation tensor is alreay in a iagonal form an we can irectly obtain a Linbla form for the master equation. For the s-integration, we make use of the relation s exp[ i(ω k ω)s] = πδ(ω k ω) + PV 1 ω k ω, (68) where PV enotes the Cauchy principal value. We can thus split the spectral correlation tensor into its real an imaginary part to obtain the ecay rate an the Lamb shift, respectively. Note. however, that the expression for the Lamb shift is ivergent as we have neglecte relativistic effects that become relevant for large values of ω k. Since it only gives a slight renormalization of the energy levels of the atom, we neglect it in the following. For simplicity, let us consier the case where we are intereste in the transition between only two atomic levels iffering by the frequency ω. Then, we finally arrive at the master equation in Linbla form, where we have use the spontaneous emission rate ( t ρ = γ σ ρσ + 1 ) 2 {σ +σ, ρ}, (69) γ = 4ω2 3 c 3 (7)

12 12 an the spin flip operators σ ± = 1 2 (σ x ± iσ y ). (71) The master equation can be solve by expaning it into Pauli matrices. The vector σ is known as the Bloch vector an the equations of motion for its component ecouple, t σ x = γ 2 σ x (72) t σ y = γ 2 σ y (73) t σ z = γ( σ z + 1). (74) From this, we see that the off-iagonal elements of the ensity matrix ecay exponentially with the rate γ/2, a phenomenon which is calle ecoherence. The z-component of the Bloch vector ecays exponentially with the rate γ to a steay state of σ z = 1, i.e., the atom ens up in the state with lower energy. E. Resonance fluorescence Let us assume now that in aition to the raiation fiel being in the vacuum state, there is a single moe that is being riven by an external laser. As the laser only affects a single moe, we can ignore its effect on the issipative ynamics, i.e., the only consequence results from the Hamiltonian of the form H L = E cos(ωt), (75) where we have assume that the laser is on resonance with the atom. Denoting the two levels of the atom by g, an e, respectively, we can go into the rotating frame of the laser riving by making the unitary transformation e = exp(iωt) e. (76) Inserting back into the master equation, we see that the rotation exactly compensates the energy ifference between e an g, an the laser term becomes H L = E [1 + exp(2iωt)] (77) 2

13 13 In the rotating wave approximation we neglect the fast oscillating term. Then, we can write the laser Hamiltonian as H L = Ω 2 (σ + + σ ), (78) where we have introuce the Rabi frequency Ω = E. The total quantum master equation then reas ( t ρ = i [H L, ρ] + γ σ ρσ + 1 ) 2 {σ +σ, ρ}. (79) The interesting aspect about this master equation is that it exhibits a competition between the coherent ynamics generate by the laser an the issipative ynamics arising from the ecay into the vacuum of the raiation fiel. As before, we can write the master equation as an equation of motion for the Bloch vector [1], t σ = G σ + b, (8) using the matrix G an the vector b given by γ/2 G = γ/2 Ω, b =, (81) Ω γ γ respectively. This equation of motion is calle the optical Bloch equation. Its stationary state σ s can be foun from the conition /t σ = an is given by Note that the population of the excite state, σ z s = γ2 γ 2 + 2Ω 2 (82) σ + s = σ s = Ωγ γ 2 + 2Ω. 2 (83) p e = 1 2 (1 + σ z s ) = Ω 2 γ 2 + 2Ω 2 (84) is always less than 1/2 even in the limit of strong riving, i.e., Ω γ. Thus, it is not possible to create population inversion in a two level system in the stationary state by coherent riving. The population will merely saturate at p e = 1/2. It is also interesting to look at the relaxation ynamics towars the stationary state. For this, we introuce another vector expressing the ifference between the Bloch vector an the stationary solution [1], i.e., δ σ = σ σ s. (85)

14 14 The ynamics of this vector is escribe by a homogeneous ifferential equation, We fin the eigenvalues of G to be δ σ = Gδ σ. (86) t λ 1 = γ 2 (87) λ 2/3 = 3 γ ± iµ (88) 4 where µ is given by ( γ ) 2. µ = Ω 2 (89) 4 Since all eigenvalues have negative real parts, all coefficients of the δ σ vector will eventually ecay an the stationary state σ s is reache. If the atom is initially in the state g, the resulting ynamics is given by [ Ω 2 p e = 1 exp ( 34 ) ( γ 2 + 2Ω γt cos µt + 3 γ 2 4 µ σ ± = Ωγ [ 1 exp ( 34 ) ( γ 2 + 2Ω γt cos µt + 2 )] sin µt { γ 4µ Ω2 γµ (9) } )] sin µt. (91) III. PROJECTION OPERATOR TECHNIQUES A. Relevant an irrelvant part While Markovian master equations have a broa range of applications, in some situations the assumption of rapily ecaying correlations in the environment is not vali. In particular, this is the case for isolate homogeneous quantum many-boy systems, where the equilibration of local observables can only occur ue to interaction with its surrounings. The complete ynamics is governe by the Liouville-von Neumann equation of the full system accoring to t ρ = i [H, ρ] = L(t)ρ. (92) Here, we aim at eriving a close reuce ynamical equation for the subunit uner consieration. This is one by introucing a projection superoperator P that projects onto the relevant part of the full ensity operator ρ [1]. The ynamics of the reuce system is no longer unitary, but escribe by P ρ = PL(t)ρ. (93) t

15 15 In many cases P efines a projection onto a tensor prouct state of one subsystem an a reference state ρ E of the environment acooring to Pρ = Tr E {ρ} ρ E. (94) However, it is also possible to generelize this approach to inclue correlations between system an environment [5]. For example, consier the case of a two-level system resonantly couple to an environment consisting of two energy bans at the eigenenergies of the system. Then, we can introuce a projection superator given by Pρ = Tr {Π µ ρ} 1 Π µ, (95) n µ µ=1,2 where n is the number of levels in the energy ban an the projectors Π µ are given by Π µ = n µ µ µ n µ n µ. (96) As these projectors act on both the system an the environment, the relevant part inclues correlations between system an environment. In orer to obtain a close equation for the relevant part, we efine another projection superoperator Q projecting on the irrelevant part of the full ensity matrix ρ, i.e., leaing to the ynamics escribe by Qρ = ρ Pρ, (97) Q ρ = QL(t)ρ. (98) t Being projection operators onto ifferent parts of the system, P an Q have to satisfy the relations P + Q = I (99) P 2 = P (1) Q 2 = Q (11) PQ = QP =, (12) where I is the ientity operation. Then, we can erive a set of ifferential equations escribing the ynamics, P ρ = PL(t)Pρ + PL(t)Qρ (13) t Q ρ = QL(t)Pρ + QL(t)Qρ. (14) t

16 16 B. The Nakajima-Zwanzig master equation One possibility to tackle these equations is to formally solve the secon equation for Qρ, resulting in Qρ(t) = t t s G(t, s)ql(s)pρ(s), (15) with an appropriate propagator G(t, s), while assuming factorizing initial conitions, i.e., Qρ(t ) =. Inserting this solution for the irrelevant part into the ifferential equation for the relevant part leas to a close integro-ifferential equation known as the Nakajima-Zwanzig equation [6, 7]. Although it allows for a systematic perturbation expansion its structure is usually very complicate because every orer requires the integration over superoperators involving the complete history of Pρ. Therefore, its applicability to many-boy problems is rather limite. C. The time-convolutionless master equation A ifferent approach tries to explicitly avoi the integral over the complete history by looking at the inverse of the time evolution. We replace the ρ(s) in the above equation by ρ(s) = G(t, s)(p + Q)ρ(t), (16) where G(t, s) is the backwar propagator of the full system, i.e., the inverse of its unitary evolution. We then can write the solution of the irrelevant part as Qρ(t) = Σ(t)(P + Q)ρ(t), (17) where we have introuce the superoperator t Σ(t) = s G(t, s)ql(s)pg(t, s). (18) t We then move all occurrences of Qρ(t) to the left-han sie. The superoperator 1 Σ(t) may be inverte for small times or weak interactions with the irrelevant part [1], leaing to Qρ(t) = [1 Σ(t)] 1 Pρ(t). (19)

17 17 Inserting this result into the ifferential equation for the relevant part yiels P t ρ(t) = PL(t)[1 Σ(t)] 1 Pρ(t). (11) Since this ifferential equation for the relevant part oes not involve a convolution integral as in the Nakajima-Zwanzig equation, it is calle the time-convolutionless (TCL) master equation [8]. The TCL generatior is efine as K(t) = PL(t)[1 Σ(t)] 1 P. (111) In orer to perform a perturbation expansion of K(t) we rewrite [1 Σ(t)] 1 as a geometric series, allowing to write the TCL generator as K(t) = n PL(t)Σ(t) n P n λ n K n (t), (112) where λ is the coupling constant in which the series expansion is performe. One may then use the efinition of Σ(t) an the series expansions of G(t, s) an G(t, s) to compute the K n. In many situations, the o terms of the series expansion vanish [1], an the leaing term given by the secon orer expansion is K 2 = leaing to the secon-orer TCL master equation P t ρ = In the interaction picture, it may be explicitely written as P t ρ = t t spl(t)l(s)p, (113) t t t spl(t)l(s)pρ. (114) t sp[h I (t), [H I (s), Pρ]]. (115) The secon-orer TCL master equation iffers from the Nakajima-Zwanzig equation only by the ensity operator in the integral epening on t instea of s. If the projection operator escribes a projection onto a prouct state between the system an a fixe state in the environment, it is ientical to the Refiel equation that we have alreay encountere earlier.

18 18 IV. DECAY INTO TWO BANDS We consier a two-level systems couple to an environment consisting of two istinct energy bans. The system is escribe by the Hamiltonian H = H S + H E + H I. (116) The system Hamiltonian H S is an orinary two-level system with an energy ifference of E,accoring to H S = E e e = Eσ + σ, (117) where we have set the energy of the groun state g to zero. For the energy bans of the environment, we assume a linear spectrum, i.e, H E = N 1 n 1 =1 δε N 1 n 1 n 1 n 1 + N 2 n 2 =1 ( E + δε ) n 2 n 2 n 2, (118) N 2 where N 1 an N 2 are the total number of states in each ban. The interaction Hamiltonian is given by H I = N 1 N 2 n 1 =1 n 2 =1 λ n1,n 2 σ + n 1 n 2 + H.c., (119) i.e., whenever the two-level system is e-excite, an excitation is create in the environment, an vice versa. The matrix elements λ n1,n2 are ranom Gaussianly istribute variables with a variance λ 2. In the following, we assume a separation of energy scales accoring to E δe λ. This moel has some very interesting properties. First of all, the size of the environment is clearly finite, meaning that excitations in the environment may not ecay fast enough an the Markov approximation may not be applicable. This is even further strengthene when taking into account that the interaction will lea to very strong correlations between system an environment. At the same time, the weak interaction naturally gives rise to a peturbative expansion. We procee by efining a correlate projection operator for the relevant part of the ynamics [5], accoring to Pρ = Tr {Π 1 ρ} 1 N 1 Π 1 + Tr {Π 2 ρ} 1 N 2 Π 2 = P e 1 N 1 Π 1 + P g 1 N 2 Π 2, (12)

19 19 where we have introucte the projectors Π 1 = Π 2 = N 1 n 1 =1 N 2 n 2 =1 1 n 1 n 1 (121) 1 n 2 n 2 (122) an P e an P g enote the probability to fin the two-level system in its excite or groun state, respectively. Because of conservation of probability, we have P g = 1 P e, meaning that the relevant part of the ynamics consists of a single variable! In the following, we will erive its equation of motion. The secon orer TCL master equation is given by P t ρ = t spl(t)l(s)pρ, (123) which in our case can be written as P 1 e Π 1 + (1 P N e ) 1 t Π 2 = 1 N 2 str {Π 1 L(t)L(s)Pρ} 1 N 1 Π 1 + Tr {Π 2 L(t)L(s)Pρ} 1 N 2 Π 2. (124) since the projection an the time erivative commute. Multiplying with Π 1 an taking the trace yiels t P e = = str {Π 1 L(t)L(s)Pρ} (125) t [ [ 1 str {Π 1 H I (t), H I (s), P e Π 1 + (1 P e ) 1 ]]} Π 2. (126) N 1 N 2 Expaning the commutator an making use of the relation Π i Π j = δ ij Π i results in t P e = { 1 str P e Π 1 H I (s)h I (t) (1 P e ) 1 } Π 1 H I (t)h I (s) + H.c.. (127) N 1 N 2 We can now introuce two relaxation rates γ 1,2 escribing the transition rates between the two bans. Specifically, we have γ i = 1 N i t s n1 n 1 H I (s)h I (t) + H.c n 1. (128)

20 2 As the interaction Hamiltonian couples oes not contain terms that leave the ban inex unchange, we may write the matrix elements as n 1 H I (s)h I (t) + H.c n 1 = n 2 n 1 H I (s) n 2 n 2 H I (t) n 1 + c.c.. (129) As we work in the interaction picture, we have n 1 H I (s) n 2 n 2 H I (t) n 1 = exp(iω 12 s) exp( iω 12 t) λ n1,n2 2, (13) where we have introuce the transitions frequencies ω 12 = δε(n 1 /N 1 n 2 /N 2 ). Using the statistical properties of the interaction Hamiltonian an performing the integration over s finally yiels [9] γ i = 2λ2 N i n1,n2 sin(ω 12 t) ω 12. (131) For the next step, we note that the sinc function appearing in the relaxation rates is a representation of the Dirac δ istribution, i.e., πδ t (ω 12 ) = lim t sin(ω 12 t) ω 12. (132) Then, at large enough times an small enough couplings [1], we may approximate the rates by γ 1 2πλ2 N 1 k,l δ(e n1 E n2 ) = 2πλ 2 N 2 δε (133) γ 2 2πλ 2 N 1 δε. (134) The equation of motion for the excite state probability then reas P e = (γ 2 + γ 1 )P e + γ 2. (135) Assuming the two-level system is initially in its excite state, the stationary state given by P s e = γ 2 /(γ 1 + γ 2 ) is approache in an exponential ecay accoring to P e (t) = P s e + γ 1 γ 1 + γ 2 exp[ 2(γ 1 + γ 2 )t]. (136) This solution is in excellent agreement with numerical simulations of the full Schringer equation [11]. It is important to stress that this efficient escription of such a complex system is only possible because our projection operator correctly captures the essential features of the

21 21 ynamics. If one choses a ifferent projection operator, say, one without correlations between system an environments, accoring to Pρ = Tr E {ρ} ρ E, (137) then the secon orer TCL master equation no longer correctly escribes the ynamics, as higher orers are ivergent [11]. [1] H.-P. Breuer an F. Petruccione, The Theory of Open Quantum Systems (Oxfor University Press, Oxfor, 22). [2] G. Linbla, Commun. Math. Phys. 48, 119 (1976). [3] C. Fleming, N. I. Cummings, C. Anastopoulos, an B. L. Hu, J. Phys. A 43, 4534 (21). [4] D. F. Walls an G. J. Milburn, Quantum Optics (Springer, Berlin, 1994). [5] H.-P. Breuer, Phys. Rev. A 75, 2213 (27). [6] S. Nakajima, Progr. Theo. Phys. 2, 948 (1958). [7] R. Zwanzig, J. Chem. Phys. 33, 1338 (196). [8] F. Shibata, Y. Takahashi, an N. Hashitsume, J. Stat. Phys. 17, 171 (1977). [9] H. Weimer, M. Michel, J. Gemmer, an G. Mahler, Phys. Rev. E 77, (28). [1] F. Schwabl, Quantum mechanics (Springer, Berlin, Germany, 21). [11] H.-P. Breuer, J. Gemmer, an M. Michel, Phys. Rev. E 73, (26).