Journal of Quantum Information Science, 2013, 3, 2733 http://dx.doi.org/10.4236/jqis.2013.31007 Published Online March 2013 (http://www.scirp.org/journal/jqis) NonMarkovian D ynamics of an Open TwoLevel System with AmplitudePhase Damping Ning Tang, Guoyou Wang, Zilong Fan, Haosheng Zeng* Key Laboratory of LowDimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics, Hunan Normal University, Changsha, China Email: *hszeng@hunnu.edu.cn Received February 6, 2013; revised March 10, 2013; accepted March 19, 2013 ABSTRACT By use of the measure, the backflow of information presented recently, we study the nonMarkovianity of the dynamics for a twolevel system interacting with a zerotemperature structured environment via amplitudephase coupling. In the limit of weak coupling between the system and its reservoir, the timelocal nonMarkovian master equation for the re duced state of the system is derived. Under the secular approximation, the exact analytic solution is obtained. Numerical simulations show that the amplitude and phase dampings can produce destructive interference to the backflow of infor mation, leading to the weaker nonMarkovianity of the compound dynamics compared with the dynamics of a single amplitude or phase damping model. We also study the characteristics of the initialstate pairs that maximize the back flow of information. Keywords: NonMarkovianity; AmplitudePhase Dampings; Destructive Interference 1. Introduction The evolution of open quantum systems can be divided into two basic types, i.e., Markovian and nonMarkovian processes. For memoryless Markovian processes, the environment acts as a sink and the information that the system released into the environment during their inter action no longer reflows to the system. However, this is not the case in the nonMarkovian processes, where the lost information will return to the system at a later time, so that the later evolution of the system is affected by its past history, i.e., which appears memory effect. Although almost all early works are devoted to the study of Markovian processes [1], people recently found that Many relevant physical systems, such as the quan tum optical system, quantum dot [2], superconductor sys tem [3], quantum chemistry [4] and biological system [5] etc. can not be described simply by Markovian dynamics. Quantum nonMarkovian processes can lead to distinctly different effects on decoherence and disentanglement [6, 7] of open systems compared with Markovian processes, which are important both for the enriching of the basic theory of quantum mechanics and for some practical ap plications, such as the quantum metrology [8] and quan tum key distribution [9]. Because of these distinct prop erties and extensive applications, more and more atten tions and interest have been devoted to the study of nonMarkovian processes of open systems, including the measures of nonMarkovianity [1017], the positivity [18, 19], and some other dynamical properties [2026] of non Markovian processes. Experimentally, the simulation [2730] of nonMarkovian environment has been real ized. The study of nonMarkovian dynamics of open quan tum systems is typically very involved and often requires some assumptions or approximations. In the previous re searches of nonMarkovian dynamics, only a single cou pling way (amplitudedamping or dephase interaction) is assumed. Further, the rotating wave approximation, that is, neglecting the counterrotating terms in the systemre servoir interaction Hamiltonian, is employed. These as sumptions and approximations limit the serviceable range of the model. In this paper, we consider a complex model which simultaneously consists of amplitude and phase dampings to the environment. We also reserve the counterrotating terms in the systemreservoir interaction Hamiltonian. Our motivation is to compare the non Markovian features of the dynamics for amplitude and phase dampings, and to observe the interference be tween the two nonMarkovian dynamics induced by the two damping ways. The article is organized as follows. In Section 2, we introduce the microscopic Hamiltonian model and derive the nonMarkovian timelocal master equation for a two level system weakly coupled to a vacuum reservoir. In *Corresponding author. C opyright © 2013 SciRes. JQIS
N. TANG ET AL. 28 Section 3, we solve the master equation under the so called secular approximation and present the analytic expression for the calculation of nonMarkovianity based on the measure proposed by Breuer, Laine and Piilo (BLP) [10] recently. In Section 4, we choose the Lor entzian spectra reservoir as an exemplary example and simulate numerically the nonMarkovianity of the system dynamics. The nonMarkovian effects produced by am plitude noise, phase noise and their combination are in vestigated. And finally the conclusion is arranged in Sec tion 5. 2. Microscopic Model Consider a twolevel atom with Bohr frequency 0 in teracting with a zerotemperature bosonic reservoir mod eled by an infinite chain of quantum harmonic oscillators. The total Hamiltonian for this system in the Schrodinger picture is given by. 0 1 2 . kk kkzkk kk kkk k bbgb b gbb (1) where and are the Pauli and inversion opera tors of the atom, , kk b and k b are respectively the frequency, annihilation and creation operators for the kth harmonic oscillator of the reservoir. The atom cou ples to its environment via both amplitude and phase interactions, k is the coupling strength which is as sumed to be real for simplicity. The parameters and describe the relative strengthes of the two couplings which satisfy . Note that we include the counterrotating terms, and 22 1 bk k b , in the interac tion Hamiltonian. The timeconvolutionless (TCL) projection operator technique [1] is most effective in dealing with the dy namics of open quantum systems. In the limit of weak coupling between the system and its environment, by expanding the TCL generator to the second order with respect to coupling strength, the nonMarkovian master equation describing the evolution of the reduced system, in the interaction picture, can be written as d,. dLS tiHtt Dt Dt t (2) where 2, LS Ht StSt (3) is the Lamb shift Hamiltonian which describes a small shift in the energy of the eigenstates of the twolevel atom. In many theoretical researches [20], this term was neglected usually. But in this paper, we will take it into the consideration. The parameters and SS which corresponds to respectively the Lamb shifts of levels 0 and 1 may be written as 0 0dd sin, t St J (4) with 1, 1 and 2 kk k Jg the spectral distribution of the environment. The dissipator Dt that describes the secular motion of the system has the form 2 2 2 0 1, 2 1, 2 , zz Dtt tt tt t tt t (5) where the first term describes the dissipation of the atom to its environment with timedependent decay rate t , and the second term denotes the heating with a rate t . The last term describes the purely dephasing with a rate 0t. These timedependent rates can be written as, 0 0 2d dcos, t tJ (6) with 1,1, 0 . The dissipator Dt represents the contribution of the socalled nonsecular terms, that is, terms oscillat ing rapidly with Bohr frequency 0 , 2 .., z z z z z z Dt ut ivtt tit t tit t ti tt ti tt ttitit t ttitit th c (7) here h.c. denotes the Hermitian conjugation, and the timedependent coefficients are defined as 0 0 2d dcoscos, t utJ tt (8) 0 0 2d dcossin, t vtJ tt (9) 0 0dd cos, t tJt (10) 0 0dd cos, t tJ t (11) Copyright © 2013 SciRes. JQIS
N. TANG ET AL. 29 0 0dd sin, t tJt t (12) 0 0dd sin t tJ . (13) 3. Solution and NonMarkovian Measure We now consider the case where the nonsecular term Dt can be neglected. Just as pointed out by Maniscalco [31], this kind of secular approximation that used after tracing over the bath degrees of freedom is different from the rotating wave approximation before the tracing. It is a more precise approximation that con sists in an average over rapidly oscillating terms, but does not wash out the effect of the counterrotating terms present in the coupling Hamiltonian. Under the secular approximation, the master Equation (2) has the Lindblad like form with timedependant decay rates. It is straight forward and easy to show that the corresponding Bloch equation may be written as 22 22 0 14 2, x bb y SSb(14) 22 22 0 14, 2 yx bbS Sb(15) 22 , zz bb (16) where the three components of the Bloch vector are defined as = j bt trt with and ,,jxyz the Pauli operators. For compactness we omit the argument of all the timedependent coefficients. Employing the method proposed by Hall [32], the corresponding Bloch Equations (14)(16) can be solved exactly which gives e0cos 0sin t xx y btbt bt , , , (17) e0sin 0cos t yx y btbt bt (18) 2 0 e0ede t tts zz btbss s (19) with 22 2 0 0 1d4, 2 t tss ss (20) 2 0d, t ts ss (21) and 2 0d. t tsSsSs (22) In this paper, we employ the (BLP) measure [10] to describe the nonMarkovianity of the considered system. Note that Markovian processes always tend to continu ously reduce the trace distance between any two states of a quantum system, thus an increase of the trace distance during any time interval implies the emergence of non Markovianity. BLP further linked the change of the trace distance to the flow of information between the system and its environment, and concluded that the backflow of information from environment to the system is the key feature of a nonMarkovian dynamics. In quantum in formation science, the trace distance for quantum states 1 and 2 is defined as [33] 121 2 1 , 2 Dtr, (23) with trace norm defined as AA . For a given pair of initial states 1,2 0 of the system, the change of the dynamical tracedistance can be described by its time derivative 1, 212 d ,0 , d tDt t ,t (24) where 1,2 t are the dynamical states of the system with the initial states 1,2 0 . For Markovian processes, the monotonically reduction of the trace distance implies 1,2 ,0t 0 for any initial states 1,2 0 and at any time . If there exists a pair of initial states of the system such that for some evolutional timet, t 1,2 ,0t 0 , then the information takes backflow from environment to the system, and the process is non Markovian. In order to describe the degrees of nonMark ovianity of the whole dynamical process, the quantity, 1,2 1,2 0 0 maxd,0,tt (25) is introduced. Where the time integration is extended over all intervals in which is positive, and the ma ximum is taken over all initialstate pairs of the system. For any Markovian process, . The larger the quantity is, the higher the nonMarkovianity of the process is. 0 For our considered open system and by use of the so lution of Equations (17)(22), Equations (24) becomes 222 2 0 2 22 22 1e4 4 2e , t t xy z Gt bb b (26) where 12 2 22 22 ee tt xy z Gtbbb and 12 0 jj j bb b 0 is the difference between the two Bloch components that correspond to the initial states 1,2 0 Copyright © 2013 SciRes. JQIS
N. TANG ET AL. 30 4. Numerical Simulation In order to demonstrate quantitatively the nonMarkovian characteristics of the system dynamics, we specify our study to a particular reservoir spectra, Lorentzian spectra, 2 0 22 0 , 2π J (27) which describes the interaction of an atom with an im perfect cavity and is widely used in literatures. Where 0 denotes the transition frequency of the atom, 0 is the frequency detuning between the atom and the cavity mode. is the width of Lorentzian dis tribution, which is connected to the reservoir correla tion time 1 R . The parameter 0 can be re garded as the decay rate for the excited atom in the Markovian limit of flat spectrum which is related to the relaxation time 1 0 S . For the Lorentzian spec tra, all the timedependent coefficients can be calcu lated analyticcally, 0 22 esin cos 2 t Stt t , (28) 0 22 esin cos, t tt t (29) with , 0 1,1,1,1, 0, 2 00 00c , and . In Figure 1, we plot the nonMarkovianity as a func tion of the weight factor and the dimensionless de Figure 1. (Color online) NonMarkovianity as a function of the weight factor and the dimensionless detuning 0 for 0 50 and . 0 02 0 . Except for the orange domain in which , all the other pa rameter domains have nonzero nonMarkovianity. tuning 0 , where the parameters are taken as 00 5 and 0 0.2 . It is shown that except for the orange domain in which , all the other parameter domains (i.e., the red and blue domains) have nonzero nonMarkovianity. 0 1 denotes the amplitude dam ping noise in which 0 for a small section of the beginning part of 0 , and then rises with the increase of the detuning, but the change is not monoto nous. The changing regulation is very similar to that of the damped JaynesCummings model case [10]. But here the effect of the counterrotating terms is considered which induces some tiny oscillations of the nonMark ovianity (see the dashed line in Figure 2) compared with the damped JanesCummings model case. 0 de notes the purely dephasing noise in which increases firstly from nonzero and then decreases when the detun ing increases. Interestingly, for the detunings up to 00.79 which indeed covers all the reasonable de tunings of practical experiments, the nonMarkovianity induced by the purely dephasing noise is apparently lager than the one induced by purely amplitudedamping noise (see Figure 2), that is, in the same coupling strength, dephasing is more preferable to induce the backflow of information than dissipation. This is in accordance with the prediction [34] that the main contribution to the nonMarkovianity stems from the evolution of the system coherence, because dephase interaction may be more preferable to induce the time evolution of the offdiago nal elements of the system density matrix than dissipa tion. Further we find that the superimposed interaction of phase and amplitude damplings can sup press the backflow of information, leading to smaller nonMarkovianity for measure . It means that in 10 0 1 2 3 4 0 0.05 0.1 0.15 Δ/γ0 N α=0 α=1 Figure 2. NonMarkovianity as a function of the di mensionless detuning 0 for amplitude and phasing dampings for 0 50 and . 0 02 . For the detunings up to .5. 00 397 0 079 , the nonMarkovianity induced by the phase damping is lager than that in duced by amplitude damping. Copyright © 2013 SciRes. JQIS
N. TANG ET AL. 31 practice we may, not only by the engineering of envi ronmental structure but also by the adjusting of the cou pling ways between the system and its environment, ma nipulate the nonMarkovian dynamics of open quantum systems. In Figure 3, we plot the measure as a function of the weight factor and the dimensionless decay rate 0 , where the parameters is chosen as 025 and 5 . We find that for a given weight factor , the measure increases monotonously with decay rate 0 . This may be understood easily: The parameter is the inverse of the reservoir correlation time. The de crease of indicates an increase of the reservoir cor relation time. The parameter 0 is the inverse of the system relaxation time. Decreasing the system relaxation time is equivalent to increasing the reservoir correlation time. Both the two changes can enhance the nonMark ovianity of the system dynamics [35]. Note that for 1 , the result is similar to the one of damped Jaynescummings model with resonant interaction [36]. Further we also find that the dephase noise 0 can give rise to much more nonMarkovianity than ampli tudedamping noise , and their superposition can reduce the value of the measure . These results once again demonstrate the dominating rules of the quantum coherence to the nonMarko vianity of open quantum system and the effect of de structive interference between amplitude and phase dampings. 1 0 1 We also investigate the problem of what initial state pairs can maximize the measure . Interestingly, in our numerical simulations, we find that there exist only two types of such initial state pairs. One pair is 0 and 1 which in Figures 1 and 3 applies to the red parame Figure 3. (Color online) NonMarkovianity as a func tion of the weight factor and the dimensionless decay rate 0 for 025 and 5 . ter domain, the other pair is 101 2 and 101 2 which applies to the blue parameter domain. Both the initial state pairs have the largest trace distance. For the orange parameter domain in Figure 1, the measure 0 . Note that the initial state pair 0,1 agree with the result [10] for the damped JaynesCummings model, though here we have con sidered the effect of counterrotating terms in the interac tion Hamiltonian. However, the initial state pair , does not coincide with the result ,0 of reference [36]. The former is composed by two or thogonal states with maximal coherence. While the later contains the ground state and a maximal coherence state which have no the largest trace distance. 5. Conclusions In conclusion, we have studied the nonMarkovianity of the dynamics for a twolevel system interacting with a zerotemperature structured environment via amplitude phase damping. In the limit of weak coupling between the system and the reservoir, we have derived the time local nonMarkovian master equation for the reduced state of the system. The exact analytic solution under the secular approximation has been gained and the non Markovian properties for the system dynamics based on the BLP measure have been studied. We found, in the same coupling strength between the system and its envi ronment, that the dephasing interaction is more beneficial to induce the backflow of information from the environ ment to the system compared with amplitudedamping mo del, leading to more stronger nonMarkovianity for the system dynamics. This result further proves the previous viewpoint that the main contribution to the nonMarko vianity stems from the evolution of the system coherence. We also found that the information backflows induced by amplitude and phase damplings can take destructive in terference so as to suppress the nonMarkovianity of the system dynamics. For a given systemreservoir coupling model (a fixed ), the nonMarkovian measure increases monotonously with 0 , but it is not monotonous with dimensionless detuning 0 . The physical explanation for the former case is clear, but for the later case the reason is not clear yet and requires further studying. In the definition of BLP nonMarkovianity measure, a maximization over all possible initial state pairs is in volved. This maximization process usually requires to spend much time or effort on the practical calculations. In our numerical simulations, we found that there are only two pairs of such initial states. One pair was found previously but another pair is new. For various different Copyright © 2013 SciRes. JQIS
N. TANG ET AL. 32 kinds of dynamical models, looking for the characteris tics of such initial state pairs is perhaps a problem worthy of consideration [37,38]. The measure of nonMarkovianity is a fundamental problem in the study of open quantum system dynamics. In the numerical simulations, we have only considered the Lorentzian environment. Actually, our analytic re sults also adapt to other structured environments, such as the Ohmic reservoir, the photonic bandgap material [39], etc. By properly engineering the structure of the envi ronment, one can selectively alter the nonMarkovian dy namical property of the open quantum system, so as to effectively control the evolution of some interesting phy sical quantities, such as the quantum coherence, quan tum entanglement and discord. Therefore, our work will also be helpful for the researches of the related problems. 6. 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