In this work we analyze the dynamics of a quantum particle subject to an asymmetric bistable potential and interacting with a thermal reservoir. We obtain the time evolution of the population distributions in both energy and position eigenstates of the particle, for different values of the coupling strength with the thermal bath. The calculation is carried out using the Feynman-
Vernon functional under the discrete variable representation

A. LA COGNATA

B. SPAGNOLO

Breuer H. P.

D. VALENTI

E. PALADINO

G. FALCI

M. BERRITTA

P. CALDARA

Prokofev N. V.

 In this work we analyze the dynamics of a quantum particle subject to an asymmetric bistable potential and interacting with a thermal reservoir. We obtain the time evolution of the population distributions in both energy and position eigenstates of the particle, for different values of the coupling strength with the thermal bath. The calculation is carried out using the Feynman-Vernon functional under the discrete variable representation. </jats:p

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International Journal of Quantum Information

DYNAMICS OF A QUANTUM PARTICLE IN ASYMMETRIC BISTABLE POTENTIAL WITH ENVIRONMENTAL NOISE

CALDARA, Pasquale

LA COGNATA, Angelo

VALENTI, Davide

SPAGNOLO, Bernardo

Berritta, M

Paladino, E

Falci, G.

Archivio istituzionale della ricerca - Università di Palermo

DYNAMICS OF A QUANTUM PARTICLEIN ASYMMETRIC BISTABLE POTENTIALWITH ENVIRONMENTAL NOISEP. CALDARA*, A. LA COGNATA, D. VALENTIy and B. SPAGNOLODipartimento di Fisica e Tecnologie Relative, Universita di Palermo,Group of Interdisciplinary Physics and CNISM-INFM, Unita di Palermo,Viale delle Scienze, Ed.18, I-90128 Palermo, Italy*pcaldara@gip.dft.unipa.ityvalentid@gip.dft.unipa.itM. BERRITTA, E. PALADINO and G. FALCIDipartimento di Metodologie Fisiche e Chimiche (DMFCI),Universita di Catania, Viale A. Doria 6, I-95125 Catania,Italy CNR-IMM MATIS, Catania, ItalyReceived 24 August 2010In this work we analyze the dynamics of a quantum particle subject to an asymmetric bistablepotential and interacting with a thermal reservoir. We obtain the time evolution of the popu-lation distributions in both energy and position eigenstates of the particle, for di®erent values ofthe coupling strength with the thermal bath. The calculation is carried out using the Feynman-Vernon functional under the discrete variable representation.Keywords: Bistable potential; NES; DVR; Caldeira-Leggett model.1. IntroductionA feature which makes a strong di®erence-between the behaviour of a quantumsystem with respect to a classical one is quantum tunneling. This e®ect often occursin condensed matter physics, such as Josephson junctions and hetero-nanos-tructures.1,2 In a dissipative quantum system interacting with a thermal bath, thequantum tunneling can play an important role on the relaxation time from ametastable state.3 During the last few years, the e®ects of environment on quantumtunneling phenomenon have been intensively studied.47 Commonly, environmentis modelled as a number N (usually N!1) of harmonic oscillators considered atthermal equilibrium, i.e. thermal bath, interacting with the quantum system througha bilinear coupling.812 In this context, symmetric and asymmetric quantum bistablesystems are good enough to analyze superconducting quantum bits and decoherenceInternational Journal of Quantum InformationVol. 9, Suppl. (2011) 119127#.c World Scienti¯c Publishing CompanyDOI: 10.1142/S0219749911007137119phenomena.13,14 Obtaining longer coherence times in such systems, when theyinteract with noisy environment, is one of the major requirements in devising andmanufacturing devices capable of storing quantum bits. In this respect, a main topicis to know the properties of a particle subject to an external potential, in the presenceof random °uctuations. It can also be useful to study the changes occurring in thedynamics of a quantum particle a®ected by noisy perturbations, when di®erentshapes of the potential pro¯le are used. Potentials which model the interaction withlaser beams have most interesting implications for quantum systems such as thecoherent destruction of tunneling,15 the e®ect of quantum stochastic resonance,16 andthe control and reduction of decoherence in open quantum systems.17 In this work, inorder to analyze the evolution of a quantum particle subject to time-independentasymmetric bistable potential and a®ected by environmental noise, we use theCaldeira-Leggett model,4 which allows us to derive a quantum mechanical analogueof the generalized Langevin equation. The study is performed by using the approachof the Feynman-Vernon functional18 in discrete variable representation (DVR).19,202. The ModelOur system consists of a quantum particle with mass M , interacting with a thermalbath which plays the role of environment. The dynamics of the particle is investigatedby using the Caldeira-Leggett model.4 In our analysis, q^ and p^ are one-dimensionaloperators for position and momentum, respectively.The unperturbed Hamiltonian of the system isH^ 0 ¼p^ 22Mþ V^ 0ðq^Þ ð1ÞwhereV^0ðq^Þ ¼M 2!4064Uq^ 4 M!404q^ 2  q^; ð2Þis the asymmetric bistable potential shown in Fig. 1. Here,  and U are the asym-metry parameter and the barrier height, respectively, and !0 is the natural oscillationfrequency. In our study we consider only 8 energy eigenstates. In Fig. 1 these energyeigenvalues are shown on the vertical axis. In the same ¯gure, on the horizontal axis,we indicate the 8 position eigenvalues, obtained by using the DVR-state jqi. Theblack circle marks the initial position of the particle, that is the system at t ¼ 0 is in astate given by a proper linear combination of the 8 eigenstates jqi considered in ouranalysis. The curves shown in the ¯gures are the eigenfunctions corresponding to the 8energy eigenvalues.In order to describe the dynamics of the particle interacting with environment, weconsider the following HamiltonianH^ðtÞ ¼ H^ 0ðtÞ þ H^B; ð3Þ120 P. Caldara et al.whereH^B ¼XNj¼112p^ 2jmjþmj!2j x^j cjmj!2jq^ !2" #ð4Þis the Hamiltonian which describes the thermal reservoir and its interaction with theparticle. As usual, the thermal bath is depicted by an ensemble of N harmonicoscillators with spatial coordinate x^j, momentum p^j, massmj, and frequency !j. Thecoe±cients cj are the coupling constant between system and thermal bath.We note that, as N !1, from Eq. (4) a continuous spectral density is obtained.In our study we use the Ohmic spectral density characterized by an exponentialcut-o® !cJð!Þ ¼ ! exp  !!c : ð5ÞHere,  ¼M with  the strength of the coupling between system and heat bath. Wenote also that !c  !0; !j; .Because of the bilinear coupling between the coordinate q^ of the system and thecoordinate x^ of the thermal bath, this model is the quantum analogue of a classicalsystem a®ected by a constant random force.7 In the next two subsections we brie°ysummarize the mathematical approach used in this study.E0E1E2E3E4E7q0 q1 q2 q3 q4q5q6q7energypositionE5E6initial positionV0(q)Fig. 1. Potential pro¯le V0ðqÞ (see Eq. (2)) for U ¼ 3 and  ¼ 0:5. Energy levels and correspondingeigenstates considered in our analysis are indicated by horizontal lines and curves, respectively. The energyeigenvalues are E0 ¼ 2:01, E1 ¼ 0:92, E2 ¼ 0:11, E3 ¼ 1:08, E4 ¼ 1:97, E5 ¼ 2:69, E6 ¼ 2:76,E7 ¼ 3:27. By using the DVR-state jqi, eigenvalues of the position operator are obtained and shown on thehorizontal axis: q0 ¼ 4:17, q1 ¼ 1:38, q2 ¼ 1:71, q3 ¼ 3:02, q4 ¼ 4:05, q5 ¼ 4:97, q6 ¼ 5:86, q7 ¼ 6:81. Theinitial position is qstart ¼ 0 (black circle).Quantum Relaxation Time in ABP 1212.1. The Feynman-Vernon approachIn order to make our analysis independent on the properties of the heat bath, we traceout the degrees of freedom of the reservoir by using the reduced density operatorðqf ; q 0f ; tÞ ¼Zdq0Zdq00Kðqf ; q 0f ; t; q0; q 00; t0ÞSðq0; q 00; t0Þ; ð6Þwhere the propagator K is given byKðqf ; q 0f ; t; q0; q 00; t0Þ ¼Z qðtÞ¼qfqðt0Þ¼q0DqZ q 0ðtÞ¼q 0fq 0ðt0Þ¼q00Dq 0A½qA½q 0FF V ½q; q 0 ð7ÞandA½q ¼ expðiðSS ½q=}ÞÞ with SS ½q being the classical action functional. In Eq. (7),FF V ½q; q 0 ¼ expððF V ½q; q 0=}ÞÞ is the Feynman-Vernon (FV) in°uence functionalwith the in°uence weight functional F V ½q; q 0 depending on the bath correlationfunction.72.2. Discrete variable representationBy solving the eigenvalue equation connected with the Hamiltonian H^0 (see Eq. (1)),we get the energy eigenstates (see vertical axis in Fig. 2). Within the framework of thediscrete variable representation (DVR)19 it is possible to obtain the basis fjqig ofeigenstates of the position operator q^ (see horizontal axis in Fig. 2).0 0.2 0.4 0.6-2 -1 0 1 2 3populationLog t (units of τ) ρq0 ρq1 ρq2 ρq30 0.2 0.4 0.6-2 -1 0 1 2 3populationLog t (units of τ) ρq4 ρq5 ρq6 ρq70 0.2 0.4 0.6-2 -1 0 1 2 3populationLog t (units of  τ)0 0.2 0.4 0.6-2 -1 0 1 2 3populationLog t (units of  τ)Fig. 2. Time evolution of the diagonal elements, q ( ¼ 0; 1; . . . ; 7), of the density matrix in q-rep-resentation. The matrix elements q are the population distributions in the eight position eigenstatesconsidered. The time evolution is obtained for di®erent values of the coupling strength: (a)  ¼ 0:01, (b) ¼ 0:4, (c)  ¼ 1 and (d)  ¼ 2:8.122 P. Caldara et al.In this representation, using Eq. (7), the continuous real-time path integral givenin Eq. (6) becomes a discrete path with m transitions at times t1; t2; . . . ; tmm mðtÞ ¼X0 0Z ðtÞ¼mðt0Þ¼0DZ 	ðtÞ¼	m	ðt0Þ¼	0D	C½; 	FF V ½; 	0 0 ð8Þwhere C½; 	 ¼ A½qA½q 0 and the in°uence weight functional of the FV functional isFV ½; 	 ¼ Xml¼1Xl1j¼0lSðtl  tjÞj  iXml¼1Xl1j¼0lRðtl  tjÞ	j: ð9ÞHere, the absolute coordinates qj are replaced by the discrete relative coordinatesjðtÞ ¼ qjðtÞ  q 0jðtÞ and center of mass coordinates 	j ¼ qjðtÞ þ q 0jðtÞ.Because we are interested in the evolution of the populations, in Eq. (8) weconsider the diagonal terms mmðtÞ. Applying the non-interacting cluster approxi-mation (NICA),20 we get the following master equation (ME)ðtÞ ¼XN¼1Z tt0dt 0Hðt t 0Þðt 0Þ;  ¼ 1; . . . ;N ; ð10Þwhere N is the number of eigenstates and the kernel H, which indicates the clustermatrix, takes into account of all possible transitions in the DVR paths.3According to the path integral technique based on the Feynman-Vernon theory,using ME corresponds to take into account only the paths connecting diagonalelements of the reduced density matrix of the position operator q^.20Within NICA we neglect all intercluster interactions. We assume furthermorethat the characteristic memory time mem of the matrix elements of H in Eq. (10) isthe smallest time scale of the problem (Markovian limit). By this assumption weobtain the following Markovian approximated master equation:ðtÞ ¼XN¼1ðtÞðtÞ ð11Þwith the time-dependent rate coe±cientsðtÞ ¼Z 10dHðt; t Þ: ð12ÞSince the diagonal elements ðtÞ obey Eq. (11), the long-time dynamics is ruledby a single exponential decay. Thus, Eq. (11) is a set of coupled ordinary ¯rst-orderdi®erential equations, which can be decoupled via a diagonalization procedure. Thediagonalized rate matrix readsXN1;2¼1ðS1Þ112S2 ¼ ; ð13ÞQuantum Relaxation Time in ABP 123where S denotes the element of the transformation matrix and  the eigenvaluesof the rate matrix. The general solution of the Markov approximated ME isðtÞ ¼XN;¼1S ðS1Þeðtt0Þðt0Þ: ð14ÞBecause of the conservation probability, for the diagonal matrix elements holdsðtÞ ¼ X 6¼ðtÞ: ð15ÞThis condition implies that one eigenvalue equals zero, i.e. 1 ¼ 0. Therefore,ðtÞ ¼ 1 þXN¼2XN¼1S ðS1Þeðtt0Þðt0Þ; ð16Þwith 1 ¼PN¼1 S;1ðS1Þ1;ðt0Þ being the asymptotic population of the DVR-state jqi. The rate which determines the dynamics over the largest time-scale is thequantum relaxation rate  minfj<ðÞj;  ¼ 2; . . . ;Ng; ð17Þwhere  are the eigenvalues of the rate matrix and j<ðÞj are the non-zero absolutevalues of the real part of .In the next section we focus our study on the medium-short time behavior of thesystem, using the largest 1 as timescale to analyze the non-equilibrium dynamics ofthe quantum particle in the presence of thermal °uctuations.3. ResultsIn this section, we study the time evolution of our quantum particle taking into accountthe 8 energy levels shown in Fig. 1. We restrict the study to the 8 lowest levels of thesystem, becausewe are interested in the dynamics of a particle that can not reach energylevels higher than the relative maximum of the potential. In particular, we intend toanalyze the time behaviour of the populations for di®erent values of the couplingstrength, focusing on the time behaviour of the state jq0i (left side well of the potential).By using the DVR-state jqi, as initial condition for the particle we choose thenon-equilibrium position qstart ¼ 0. The corresponding state is given byjqstarti ¼ c1jq1i þ c2jq2i ð18Þwith c1 ¼ 0:745 and c2 ¼ 0:667.By integrating Eq. (10) for di®erent values of the parameter , which representsthe intensity of the environmental noise, for each eigenstate jqi we obtain the timebehaviour of the corresponding population q   (see Fig. 2). Moreover, by asimple change of basis, we calculate the time evolution of the populations also in theenergy representation (see Fig. 3). As one can see from Eqs. (11), (13), for each value124 P. Caldara et al.of  there are N relaxation times 1 . Here, we consider the maximum of theserelaxation times, and note that this time increases rapidly for larger values of .Therefore, to describe the time evolution of the system for di®erent values of , wechoose as time scale  the largest of the relaxation times obtained for  ¼ 0:01 andcalculate the evolution of the system for a maximum time t ¼ 600  . This choiceallows to follow the transient dynamics of the system for low and intermediate valuesof the coupling constant (see panels a, b, c in Figs. 2 and 3). For higher values of  thesystem can not reach the regime condition, because of the presence of relaxationtimes longer than the maximum time chosen to calculate the numerical solution (seepanel d in Figs. 2 and 3). This delay in the system dynamics can be explained by thequantum Zeno e®ect, responsible for the suppression of the tunnel e®ect. Moreover,we observe in Fig. 2 a nonmonotonic behaviour of the population q0 as a function ofthe time. Finally, as a consequence of the quantum Zeno e®ect, the eigenstate jq0i canbe maximally populated at di®erent times, varying the coupling strength and,therefore, the value of . This could be useful in view of placing a quantum particle ina given position at a ¯xed time.We note that it would be interesting to compare our results with those obtained inthe case of a harmonic oscillator coupled with a thermal bath without any cuto®, asstudied in previous works.2123 On physical grounds we expect that the time beha-viour of the purity of the system state is strictly connected with the relaxation rates.0 0.2 0.4 0.6-2 -1 0 1 2 3populationLog t (units of  τ)ρE0 ρE1 ρE2 ρE30 0.2 0.4 0.6-2 -1 0 1 2 3populationLog t (units of τ)ρE4 ρE5 ρE6 ρE70 0.2 0.4 0.6-2 -1 0 1 2 3populationLog t (units of τ)0 0.2 0.4 0.6-2 -1 0 1 2 3populationLog t (units of τ)Fig. 3. Time evolution of the diagonal elements, E( ¼ 0; 1; . . . ; 7), of the density matrix in energyrepresentation. The matrix elements Eare the population distributions in the eight energy eigenstatesconsidered. The time evolution is obtained for di®erent values of the coupling strength: (a)  ¼ 0:01, (b) ¼ 0:4, (c)  ¼ 1 and (d)  ¼ 2:8.Quantum Relaxation Time in ABP 125In our analysis the relaxation rates have been used to determine the timescale forobtaining the time evolution of the population distributions. Moreover, we found afreezing phenomenon of the state of the system due to the Zeno e®ect.24 Finally, wenote that the complete description of the dynamics of our initial pure state should beobtained by following the time evolution of all elements of the density matrix asexpressed by Eq. (8). This will be subject of future investigations.4. ConclusionIn this work, we analyze the dynamics of a quantum particle subject to an asym-metric bistable potential and interacting with noisy environment. The study is per-formed exploiting the approach of the Feynman-Vernon functional18 within theframework of the discrete variable representation.19,20 By using the Caldeira-Leggettmodel,4 we describe the transient dynamics of the system for di®erent values of thecoupling strength between the particle and the noisy environment, modelled as athermal bath. Due to the quantum Zeno e®ect, responsible for the suppression of thetunnel e®ect, a delayed dynamics of the system is observed for higher values of thecoupling strength. We ¯nd also that the metastable state inside the left side well ofthe potential can be populated at di®erent times varying the value of the couplingstrength.References1. Y. Nakamura, Yu. A. Pashkin and J. S. Tsai et al., Nature 398 (1999) 786.2. D. Rakoczy, R. Heer, G. Strasser and J. Smoliner, Physica E 16 (2003) 129.3. M. Thorwart, M. Grifoni and P. Hänggi, Ann. Phys. 293 (2001) 15.4. A. O. Caldeira and A. L. Leggett, Phys. Rev. Lett. 46 (1981) 211.5. A. J. 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Viola, E. Knill and S. Lloyd, Phys. Rev. Lett. 82 (1999) 2417.18. R. P. Feynman and F. L. Vernon, Jr., Ann. Phys. 24 (1963) 118.126 P. Caldara et al.19. D. O. Harris, G. G. Engerholm and W. D. Gwinn, J. Chem. Phys. 43 (1965) 1515.20. M. Thorwart, M. Grifoni and P. Hänggi, Ann. Phys. 293 (2001) 15.21. M. G. A. Paris, F. Illuminati, A. Sera¯ni and S. De Siena, Phys. Rev. A 68 (2003) 012314.22. A. Sera¯ni, F. Illuminati, M. G. A. Paris and S. De Siena, Phys. Rev. A 69 (2004) 022318.23. A. Sera¯ni, M. G. A. Paris, F. Illuminati and S. De Siena, J. Opt. B 7 (2005) R19.24. P. Facchi, S. Tasaki, S. Pascazio, H. Nakazato, A. Tokuse and D. A. Lidar, Phys. Rev. A71 (2005) 022302.Quantum Relaxation Time in ABP 127

Dynamics of a Quantum Particle in Asymmetric Bistable Potential with Environmental Noise

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Dynamics of a Quantum Particle in Asymmetric Bistable Potential with Environmental Noise

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Archivio istituzionale della ricerca - Università di Palermo