Complete positivity of a class of maps generated by master equations derived
beyond the secular approximation is discussed. The connection between such
class of evolutions and physical properties of the system is analyzed in depth.
It is also shown that under suitable hypotheses a Zeno dynamics can be induced
because of the high temperature of the bath.Comment: 9 pages, 2 figure

Messina, A.

Militello, B.

Scala, M.

Journal of Physics A Mathematical and Theoretical

English

arXiv

Complete positivity of a class of maps generated by master equations derived beyond the secular approximation is discussed. The connection between such a class of evolutions and the physical properties of the system is analyzed in depth. It is also shown that under suitable hypotheses a Zeno dynamics can be induced because of the high temperature of the bath

MILITELLO, Benedetto

SCALA, Matteo

MESSINA, Antonino

Archivio istituzionale della ricerca - Università di Palermo

IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICALJ. Phys. A: Math. Theor. 46 (2013) 485301 (8pp) doi:10.1088/1751-8113/46/48/485301Zeno dynamics and high-temperature masterequations beyond secular approximationB Militello1, M Scala2 and A Messina11 Dipartimento di Fisica e Chimica, Universita` degli Studi di Palermo, Via Archiraﬁ 36, I-90123,Palermo, Italy2 Department of Physics and Astronomy, University College London, Gower Street,London WC1E 6BT, UKE-mail: benedetto.militello@unipa.itReceived 17 July 2013, in ﬁnal form 13 September 2013Published 12 November 2013Online at stacks.iop.org/JPhysA/46/485301AbstractComplete positivity of a class of maps generated by master equations derivedbeyond the secular approximation is discussed. The connection between sucha class of evolutions and the physical properties of the system is analyzed indepth. It is also shown that under suitable hypotheses a Zeno dynamics can beinduced because of the high temperature of the bath.PACS numbers: 03.65.Xp, 42.50.Lc(Some ﬁgures may appear in colour only in the online journal)1. IntroductionThe quantum Zeno effect (QZE) is an ubiquitous phenomenon in quantum mechanics [1–4],which is relevant in the analysis of fundamental concepts of quantum mechanics [5–8], usefulin applications [9], and has been experimentally demonstrated [10–12].Recently, the connection between the occurrence of Zeno phenomena and the propertiesof the environment of the physical system under scrutiny has been investigated [13–16]. Theconnection between the QZE and high temperature has been studied, inhibition of Landau–Zener transitions induced by temperature has been predicted [17], and the enhancementof QZE phenomena through temperature has been analyzed in depth [18]. In the caseof high temperature, exploitation of the standard approach to derive master equations andconsequent secular approximation could be problematic, due to the fact that in such a casethe secular approximation, generally performed after Born–Markov approximation, can beinappropriate because of an effectively high coupling strength induced by the increasednumber of photons. Nevertheless, in general, avoiding secular approximation would producea generator of a non-completely positive map. Since complete positivity is a necessarycondition for a map describing a physical system [19], this difﬁculty needs to be overcome.Studies of quantum optics models obtained beyond the secular approximation or without therotating wave approximation (two names used in different contexts to address essentially1751-8113/13/485301+08$33.00 © 2013 IOP Publishing Ltd Printed in the UK & the USA 1J. Phys. A: Math. Theor. 46 (2013) 485301 B Militello et althe same approximation) have been reported throughout the years in a number of differentpapers [20].In this paper, we ﬁnd suitable hypotheses under which the Markovian approach, in thelimit of high temperature, can produce a completely positive map even beyond the secularapproximation. Moreover, through the master equation derived along this route, it is possibleto forecast the occurrence of a Zeno phenomenon under suitable hypotheses. This outcomesomehow conﬁrms the results obtained in [17] and [18]. The paper is organized as follows.In the next section we introduce the general approach and assumptions to derive the masterequation in the high-temperature limit beyond the secular approximation. In section 3 weprovide a general statement connected with the appearance of Zeno phenomena, and show anexample in a three-level system where the high-temperature master equation clearly forecaststhe preservation of a subspace, which we address as the thermal QZE. Finally, in the lastsection, we give some conclusive remarks.2. The master equationLet us consider a system interacting with a bosonic environment:HS =∑(), (1)HB =∑kωka†kak, (2)HI = A ⊗ B, B =∑k(g∗kak + gka†k ), (3)where () are projectors on eigenspaces of HS and A = A†.Now, following the standard approach [21, 22] in the weak coupling limit, we can assumethat the total density matrix can be separated in the system and bath degrees of freedom (Bornapproximation): ρT = ρ ⊗ ρB, where ρ is the density operator of the system S and ρB is thethermal state of the bosonic bath—the state of the bath does not signiﬁcantly change, bothbecause of the weak coupling limit and the fact that the environment is much larger than thesystem and therefore negligibly affected by it. Then we formally integrate the von Neumannequation, iterate it and differentiate both members of the relation obtained:dρ(t)dt= −∫ t0ds trB [HI(t), [HI (s), ρ(s) ⊗ ρB]], (4)where, considering the structure of the operator B and the thermal state ρB, one hastrB[ρB,B] = 0.Subsequently, we perform the Markov approximation by substituting the density operatorρ(s) in the integral with ρ(t), and then (by assuming very short correlation time for the bath)we put the second limit of integration to inﬁnity:dρ(t)dt= −∫ ∞0ds trB [HI(t), [HI (t−s), ρ(t) ⊗ ρB]]. (5)Then we introduce the jump operators:A(ω) =∑′−=ω()A(′), (6)where the sum is meant over all  and ′ such that ′ −  = ω. By exploiting such operatorsand before performing the secular approximation, we obtain:dρdt= −i[HS, ρ] +∑ω,ω′(ω)[A(ω)ρ A†(ω′)−A†(ω′)A(ω)ρ] + H.c.,2J. Phys. A: Math. Theor. 46 (2013) 485301 B Militello et alFigure 1. Assumptions on the coupling scheme: the two subspaces 1 and 2 are connectedthrough the operator A, but no transition is induced by A within each subspace.with ω and ω′ spanning over all possible Bohr frequencies of the system, and(ω) =∫ ∞0ds eiωstrB[B†(t)B(t − s)ρB(0)]={|g(ω)|2D(ω)(1 + N(ω)), ω > 0|g(|ω|)|2D(|ω|)N(|ω|), ω < 0, (7)where D(ω) is the bath density of the modes, while g(ω) is the coupling constant gk in thecontinuum limit, and N(ω) = 1/(exp(ω/(kBT ) − 1). In the derivation of equation (7) wehave neglected the Lamb shifts (LS), which, at the other end, if not negligible can be absorbedby the commutator after introducing H˜S = HS + LS and [HS, ρ] → [H˜S, ρ].Generally, equation (7) does not deﬁne the generator of a completely positive map. Whatone usually does is to perform the secular approximation in order to ﬁnd a completely positivegenerator. This implies keeping only the diagonal terms (ω = ω′) in equation (7) and neglectingthe rest.In any case, as we are going to demonstrate, there exist suitable hypotheses under whichthe structure of the Markovian master equation without secular approximation approaches agenerator of a completely positive map.Assume that there are two subspaces, say 1 and 2, corresponding to two projectors 1 and2 generated by projectors on eigenspaces of HS. Assume also that the operator A connectsonly two such subspaces, which means:1 A1 = 0, 2 A2 = 0, (8)1 A2 = 0, 2 A1 = 0. (9)Equations (8) express the fact that A does not induce transitions within each of the twosubspaces (all matrix elements involving any couple of states of the same subspace arevanishing). Transitions from one subspace to the other are allowed instead, according toequations (9). In ﬁgure 1 we provide a graphical representation of this scenario. If ω refers to atransition within 1 or within 2, then the corresponding A(ω) vanishes, due to equation (8).Instead, if ω refers to transitions between the two subspaces then the corresponding A(ω) isnon-vanishing. All this implies that in equation (7) summation can be restricted to ω and ω′lying in the inter-band transition frequencies (frequencies related to transitions from 1 to2 and vice versa). Let us introduce ω0 as the frequency connecting the center of the bandassociated with 1 with the center of the band associated with 2. We will use the notation∑ω∼ω0 to indicate summation over values of ω, whose absolute values are close to ω0.Assume also that the Bohr frequencies related to transitions from any state in 1 to anystate in 2 are much larger than the Bohr frequencies associated either with transitions inside1 or with transitions inside 2.3J. Phys. A: Math. Theor. 46 (2013) 485301 B Militello et alFinally, assume that the spectrum is ﬂat, meaning that |g(|ω|)|2D(|ω|) = γ is independentof ω, at least in a neighborhood of ω0 corresponding to the frequencies of interest for thisphysical problem. Of course, (ω) still contains a dependence on ω due to the term N(ω).Now, in the limit of high temperature (which implies N(ω)  1) and weak coupling(γ → 0), we can consider γ (1 + N(ω)) ≈ γN(ω). Moreover, in the high-temperature limitit turns out that N(ω) ≈ kBT/(ω), and since the frequencies connecting the subspaces 1 and2 are very close—they are all much larger than the Bohr frequencies associated to transitionsinternal to each subspace—then one can deduce that N(ω) does not signiﬁcantly dependon ω, when ω is related to a transition between the two subspaces. In particular, one has(ω = ω0 + δω)N(ω) ≈ N(ω0) + ∂N∂ω∣∣∣∣ω=ω0δω= N(ω0) − [N(ω0)]2 exp(ω0kBT)δωkBT≈ N(ω0) − kBTδωω20, (10)where the last step is legitimized at high temperature, since in this limit N(ω0) ≈ kBT/(ω0)and the exponential exp(ω0/(kBT )) approaches unity.Therefore, equation (7) may be cast in the following form:dρdt≈ −i[HS, ρ] + γ⎧⎨⎩∑ω,ω′∼ω0N(ω)[A(ω)ρ A†(ω′) − A†(ω′)A(ω)ρ] + H.c.⎫⎬⎭≈ −i[HS, ρ] + γN(ω0)⎧⎨⎩∑ω,ω′∼ω0[A(ω)ρ A†(ω′) − A†(ω′)A(ω)ρ] + H.c.⎫⎬⎭+γ O(kBTδω/(ω20)). (11)Since A = ∑ω A(ω) =∑ω∼ω0 A(ω) =∑ω∼ω0 A†(ω) =∑ω A†(ω) = A†, equation (11)can be put in the form:dρdt= −i[HS, ρ] + γN(ω0)[(A ρ A − A2 ρ) + H.c.] + γO(kBT δω/(ω20)), (12)which, in the limit γ kBTδω/(ω20) → 0, becomesdρdt= −i[HS, ρ] + γN(ω0)[(2A ρ A − A2 ρ − ρ A2)]. (13)Since this master equation is in the standard form, the relevant evolution is described bya completely positive map.Two points are noteworthy: the complete positivity of the induced map (even consideringthe derivation beyond the secular approximation), and the independence of the dissipator fromthe speciﬁc form of HS. This second property is related to the fact that at a certain point wesum up over all the jump operators (A(ω)), which ‘reconstructs’ the complete system operatorinvolved in the system–bath interaction (A).It is also interesting to sum up the hypotheses we have exploited. The ﬁrst one is the typicalweak coupling limit (γ → 0) which is important for performing the Born approximationand begins with the derivation of the master equation. The second hypothesis is the high-temperature limit (which implies N(ω)  1) which allows one to consider the rates of thedownward and upward transitions induced by the environment as being essentially equal. The4J. Phys. A: Math. Theor. 46 (2013) 485301 B Militello et alweak coupling limit makes such rates even closer for any given frequency. Finally, the bandstructure for the energy spectrum of the system (also assuming that inter-band transitionsare forbidden) and the narrowness of such bands (δω/ω0 → 0) allow one to consider equalthe rates for different frequencies, making it possible to reconstruct the operator A fromthe relevant jump operators. These three different limits should be taken in such a way thatγ kBTδω/(ω20) → 0 (see equation (11)), which can be realized in many different ways. Inparticular, a weaker system–environment coupling or narrower bands allow one to explore thethermal Zeno phenomenon at higher temperature.3. Thermal Zeno dynamicsGeneral statementThe structure of the master equation in equation (13) suggests that under suitable hypothesesthe dissipator can hinder some dynamical effects of HS, the free Hamiltonian of the system.This statement can be proven through considerations similar to those made in previousworks, in Hamiltonian contexts [2, 4] and in dissipative contexts [17]. Since α ≡ γN(ω0)is very large, we can treat the commutator −i[HS, ρ] as a perturbation with respect to thedissipator. Assuming this point of view, it is easy to convince oneself that the Hamiltoniancannot signiﬁcantly connect subspaces of the dissipator which arewell separated in terms of therelevant eigenvalues. In other words, assuming that ρa(b) is an eigenoperator of the dissipatorassociated with the eigenvalue αλa(b)—this means that 2Aρa(b)A − A2ρa(b) − ρa(b)A2 =λa(b)ρa(b)—then provided α|λa − λb| is much larger than the matrix elements of HS, theaction of the commutator can only weakly connect the operators ρa and ρb. Now, by increasingthe temperature (and then α) the eigenvalues and their differences increase, which makes thefree Hamiltonian HS more and more ineffective at determining transitions between differentsubspaces of the dissipator. The appearance of such constraints in the time evolution is justthe signature of the (thermal) QZE, which gives rise to an inhibition of the time evolution ifthe initial state of the system belongs to a one-dimensional eigenspace of the dissipator, whilegiving rise to a Zeno dynamics (evolution in the restricted subspace) if the initial state belongsto a multiplet of the dissipator.A three-state system in a high-temperature reservoirAs a very special case, we analyze the following example of inhibition of the time evolutioninduced by temperature. Let us consider a three-state system governed by the followingHamiltonian:HS =3∑l=1νl |l〉〈l| + (|1〉〈2| + H.c.), (14)HB =∑kωka†kak, (15)HI = (|2〉〈3| + |3〉〈2|) ⊗∑kgk(ak + a†k). (16)This model, which is represented in ﬁgure 2, has similarities with those analyzed in [17]and [18]. Nevertheless, it differs from the former because in the present case the Hamiltonianis time independent, and it differs from the latter since here counter-rotating terms are includedin the system–bath interaction.5J. Phys. A: Math. Theor. 46 (2013) 485301 B Militello et alFigure 2. A coupling scheme of the considered special case: states 1 and 2 are coupled by someexternal ﬁeld; 2 and 3 are coupled by the environment. The subspace 1 is generated by |1〉 and|2〉, while 2 coincides with |3〉.On the basis of equation (13), at very high temperature we obtaindρdt= Gρ, (17)withGρ ≈ −i[HS, ρ] + α[(2A ρ A − A2 ρ − ρ A2)], (18)A = |2〉〈3| + |3〉〈2|. (19)This master equation has been obtained under the previous assumptions, i.e., hightemperature, band structure and intra-band transitions are forbidden.The matrix representation of the superoperator G is as follows:G =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝0 i∗ 0 −i 0 0 0 0 0i −α − iω12 0 0 −i 0 0 0 00 0 −α − iω13 0 0 0 0 0 0−i∗ 0 0 −α − iω21 i∗ 0 0 0 00 −i∗ 0 i −2α 0 0 0 2α0 0 0 0 0 −2α − iω23 0 2α 00 0 0 0 0 0 −α − iω31 0 00 0 0 0 0 2α 0 −2α − iω32 00 0 0 0 2α 0 0 0 −2α⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(20)where we have assumed the ordering (ρ11, ρ12, ρ13, ρ21, ρ22, ρ23, ρ31, ρ32, ρ33), and exploitedthe notation ωi j = νi − ν j.The structure of this matrix is very similar to that found in [17]. On the basis of the analysisdeveloped therein, one can easily deduce that for very large α a Zeno phenomenon occurs.In fact, in the α → ∞ limit (to better visualize this situation, consider the complementarysituation in which all the other quantities tends toward zero), the state |1〉〈1| turns out to bevery close to an eigenvector of G and therefore its survival probability approaches unity atevery time: the higher α, the closer to unity is the survival probability of |1〉〈1| at every time.6J. Phys. A: Math. Theor. 46 (2013) 485301 B Militello et alThis is in line with the results in [17], where no secular approximation has been made, andgeneralizes the results of [18], where counter-rotating terms are removed from the beginningin the interaction.To reach the limit of very large α, one needs a very high temperature compared toω0, meaning that the thermal energy is supposed to be much larger than all the transitionfrequencies, i.e., α  ω13, ω23 (both ∼ ω0) and, a fortiori, α  ω12. Also, α   must besatisﬁed.4. ConclusionsIn this paper we have presented a derivation of Markovian master equations beyond the secularapproximation which is valid for a class of physical systems satisfying suitable hypotheses.In particular, it is important that the energy levels of the system form two bands and that theallowed transitions induced by the interaction with the bath are only between states of differentbands.The structure of the master equation obtained exhibits some important properties. Firstof all, it generates a completely positive map, in spite of the fact that it is derived beyondthe secular approximation. Second, the structure of the dissipator is independent from theHamiltonian associated with the small system and depends only on the operators involved inthe system–bath interaction. This is a consequence of the ﬁrst property, since summing up overall the jump operators associated with Bohr frequencies of the system restores the completeform of the system operator involved in system–bath coupling. Finally, on the basis of thestructure of the master equation it is easy to forecast the incoming of Zeno phenomena at hightemperature, which supports the results obtained in some previous works.AcknowledgmentsMS acknowledges ﬁnancial support from the EPSRC grant EP/J014664/1.References[1] Misra B and Sudarshan E C G 1997 J. Math. Phys. 18 756[2] Facchi P, Lidar D A and Pascazio S 2004 Phys. Rev. A 69 032314[3] Facchi P and Pascazio S 2008 J. Phys. A: Math. Theor. 41 493001[4] Facchi P, Marmo G and Pascazio S 2009 J. Phys.: Conf. Ser. 196 012017[5] Home D and Whitaker M A B 1997 Ann. Phys. 258 237[6] Schulman L S 1998 Phys. Rev. A 57 1509[7] Panov A D 1999 Phys. Lett. A 260 441[8] Audretsch J, Mensky M B and Panov A D 1999 Phys. Lett. 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Zeno dynamics and high-temperature master equations beyond secular approximation

B Militello

M Scala

A Messina

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Zeno Dynamics and High-Temperature Master Equations Beyond Secular
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Zeno Dynamics and High-Temperature Master Equations Beyond Secular Approximation

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