Population transfer via a finite temperature state
Wei Huang, Baohua Zhu, Wei Wu, Shan Yin, Wentao Zhang, Chu Guo
PPopulation transfer via a finite temperature state
Wei Huang, Baohua Zhu, Wei Wu, Shan Yin, ∗ Wentao Zhang, † and Chu Guo ‡ Guangxi Key Laboratory of Optoelectronic Information Processing,Guilin University of Electronic Technology, Guilin 541004, China School of material science and engineering, Guilin University of Electronic Technology, Guilin 541004, China Department of Physics, College of Liberal Arts and Sciences,National University of Defense Technology, Changsha 410073, China Henan Key Laboratory of Quantum Information and Cryptography, Zhengzhou, Henan 450000, China (Dated: June 5, 2020)We study quantum population transfer via a common intermediate state initially in thermal equi-librium with a finite temperature T , exhibiting a multi-level Stimulated Raman adiabatic passagestructure. We consider two situations for the common intermediate state, namely a discrete two-level spin and a bosonic continuum. In both cases we show that the finite temperature stronglyaffects the efficiency of the population transfer. We also show in the discrete case that strong cou-pling with the intermediate state, or a longer duration of the controlled pulse would suppress theeffect of finite temperature. In the continuous case, we adapt the thermofield-based chain-mappingmatrix product states algorithm to study the time evolution of the system plus the continuum undertime-dependent controlled pulses, which shows a great potential to be used to solve open quantumsystem problems in quantum optics. I. INTRODUCTION
Stimulated Raman adiabatic passage (STIRAP) is oneof the most important technologies to implement com-plete population transfer from an initial state to a targetstate via a common intermediate state [1–3]. In the stan-dard implementation of STIRAP, two controlled laserpulses in Gaussian shapes, namely the P pulse and S pulse, are used to couple the initial state and the tar-get state to the intermediate state respectively. Whenthe two pulses are applied in a counter-intuitive order,that is, the S pulse occurs before (but overlapping) the P pulse, complete population transfer could be achievedwith negligible excitation of the intermediate state. As aresult this technique is very robust against the noises inthe pulses as well as the dissipation in the intermediatestate.Due to robustness of STIRAP, there are many applica-tions in different quantum systems to achieve completedpopulation transfer from one quantum state to another,such as quantum optics [4], ion-trap system [5], super-conducting qubits [6, 7], cavity system [8] and quantumdots system [9]. Interestingly, STIRAP technique canbe employed not only in quantum systems but also insome specific classical systems, since the equations ofmotions governing these systems are analogous to theSchrodinger equation. For example, we can employ STI-RAP to waveguide coupler to achieve complete trans-fer of intensity of light from input waveguide to outputwaveguide [10]. STIRAP can also be used in surface plas-mon polaritons (SPPs) coupler excited by light on curved ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] graphene sheets [11], the integrated terahertz device [12],and wireless energy transfer [13].Since its initial proposition in a standard three-levelconfiguration, the setup of STIRAP has been general-ized in various directions. For example, Fractional STI-RAP [14], Bright-State STIRAP [15], Straddle STIRAP[16, 17], Two-State STIRAP [2] and Composite-PulseSTIRAP [18]. These developments mainly focus on en-hancing the robustness of STIRAP, or applying STIRAPin more general scenarios of multiple energy levels.In this paper, we study the setup of straddle STI-RAP where population transfers from one energy levelto another via multiple intermediate energy levels. It hasbeen shown that complete population transfer could beachieved as long as the couplings between the two energylevels and the intermediate energy levels satisfy certainconditions [19, 20]. In [21], it is further shown that near-perfect population transfer could also be achieved for afinite-width continuum of intermediate states, and it isrobust under moderate dissipation. However, to our bestknowledge, most of the STIRAP-related works have as-sumed that the intermediate energy levels are initially inunoccupied (vacuum) states. In real applications, a fre-quently met situation is that the intermediate levels areinitial in the thermal equilibrium state. For example, twospins coupled via an optical fiber, or an optical cavity[22](or chain of cavities) initially in thermal equilibrium. Insuch cases, the excitations in the intermediate levels mayparticipate in and intertwine the process, thus destroy-ing the previous physical picture for STIRAP. Here wefill this gap by directly studying STIRAP-like populationtransfer via an intermediate thermal state. We mainlyfocus on two different setups: 1) Population transfer viaa discrete two-level system initially in a thermal statewith a temperature T and 2) Population transfer via abosonic thermal continuum. We study the effect of afinite-temperature intermediate state on the population a r X i v : . [ qu a n t - ph ] J un transfer efficiency by numerically solving the quantumLiouville equation in those setups.Our paper is organized as follows. In the Sec.II, we in-troduce the discrete version of the model which considerspopulation transfer via a two-level system initially in athermal state , and show the effect of the finite temper-ature on the efficiency of population transfer. In Sec.III,we introduce the continuous version of the model whichconsiders population transfer via a thermal bosonic con-tinuum and show the effect of the finite temperature inthis case. We conclude in Sec.IV. ⌦ P ( t )
First we consider population transfer via a discretethermal state. For simplicity, we consider two spinswhich are coupled to two bosonic modes which act asthe ‘flying qubits’. The two bosonic modes are then cou-pled to a common intermediate spin. The Hamiltonianof the whole system can be written asˆ H ( t ) = ω q, σ z + ω q, σ z + ω a, ˆ a † ˆ a + ω a, ˆ a † ˆ a +Ω P ( t )(ˆ a † ˆ σ − + ˆ a ˆ σ +1 ) + Ω S ( t )(ˆ a † ˆ σ − + ˆ a ˆ σ +2 )+ ω m σ zm + g (cid:16) ˆ a † ˆ σ − m + ˆ a ˆ σ + m (cid:17) + g (cid:16) ˆ a † ˆ σ − m + ˆ a ˆ σ + m (cid:17) , (1)where ω q, and ω q, are the energy differences of thetwo qubits, ω a, and ω a, are the frequencies of the twobosonic modes, and the time-dependent couplings Ω P ( t )and Ω S ( t ) between the two qubits and the bosonic mode are induced by two controlled pulses, which are definedas Ω P ( t ) = Ω exp (cid:32) − ( t − τ / τ (cid:33) ; (2)Ω S ( t ) = Ω exp (cid:32) − ( t + τ / τ (cid:33) , (3)with τ the standard deviation of Gaussian pulses, τ thetime delay between the two pulses, and Ω the maximumstrength of the pulses. ω m is the energy difference of theintermediate spin, and g is the coupling strength betweenthe intermediate spin and the two bosonic modes. Wehave set ¯ h = 1. The dynamics of this system is describedby the quantum Liouville equation d ˆ ρ ( t ) dt = − i[ ˆ H ( t ) , ˆ ρ ] . (4)In the rest of this work we will always use the resonantcondition such that ω q, = ω q, = ω a, = ω a, = ω m .The bosonic modes are not occupied initially while theintermediate spin is assumed to be in a thermal statewith a temperature T , that isˆ ρ m = 11 + e − βω m | m (cid:105)(cid:104) m | + e − βω m e − βω m | m (cid:105)(cid:104) m | . (5)Here β is the inverse temperature β = 1 /T and we haveset the Boltzmann constant k B = 1. Thus the initialstate of the whole system can be written asˆ ρ i = | q (cid:105)(cid:104) q | ⊗ | a (cid:105)(cid:104) a | ⊗ ˆ ρ m ⊗ | a (cid:105)(cid:104) a | ⊗ | q (cid:105)(cid:104) q | , (6)where | q i (cid:105) ( | q i (cid:105) ) means the ground (excited) state of thespin q i , and | a i (cid:105) means the vacuum state for the bosonicmode a i , with i = 1 ,
2. The final state after the timeevolution is denoted as ˆ ρ f , namely ˆ ρ f = ˆ ρ ( ∞ ). Moreover,we denote F i ( t ) as the occupation on the excited state ofthe spin q i , that is, F ( t ) = (cid:104) q | ˆ ρ q ( t ) | q (cid:105) , (7) F ( t ) = (cid:104) q | ˆ ρ q ( t ) | q (cid:105) , (8)where ˆ ρ q i ( t ) means the reduced density operator of thespin q i . In our setup, we have F ( −∞ ) = 1 and F ( −∞ ) = 0, and perfect population is achieved if F ( −∞ ) = 0 and F ( −∞ ) = 1. In the following wewill use F = F ( ∞ ) to denote the final fidelity.To show the effect of the temperature T and the inter-play between T and the other parameters, we simulatethe dynamics of Eq. 4 in a wide parameter range, andthe results are shown in Fig. 2. In Fig.2(a), we show thedependence of the final fidelity F on the temperature T and the coupling strength g between the bosonic modesand the intermediate spin. We can see that F is greatlysuppressed when increasing T , showing that a highly oc-cupied excited state would strongly affect the efficiency of FIG. 2: Dependence of F on temperature T in the discretecase, where T ranges from 0 to 20 in all panels. (a) F as afunction of g and T , where g ranges from 1 to 10. (b) F asa function of Ω and T , where Ω ranges from 1 to 4. (c) F as a function of τ and T , where τ ranges from 0 . τ to 4 τ .(d) F as a function of τ and T , where τ ranges from 1 to5. The other parameters used in all panels, unless otherwisespecified, are g = 10, Ω = 2, τ = 1 and τ = 2. population transfer. We can also see that F slightly goesup with g , especially at higher temperature. This is ex-pected since a standard requirement for perfect STIRAPis the strong coupling between the initial (final) stateswith the intermediate states. This result is interesting inthat it show that although STIRAP is known to be robustagainst the dissipation of the intermediate state, howeverit will be strongly affected if the intermediate spin is in ahighly mixed state. In Fig. 2(b), we show the dependenceof F on T and the maximum amplitude of the laser pulseΩ, with g = 10 , τ = 1 , τ = 2. We see a similar effect toFig. 2(a) that F increases with larger Ω and decreaseswith larger T . This is because Ω play a similar role as g which determines the coupling strength between the ini-tial (final) state and the intermediate spin. In Fig. 2(c),we show the dependence of F on T and the time delay τ .We can see that there is a pick around τ = τ , and sincethe coupling strength g = 10 and Ω = 2 are large enough,relatively high population transfer efficiency could still beachieved at high temperature. In Fig. 2(d), we show thedependence of F on T and the period of driving τ . Wecan see that F is larger with larger τ . This is expectedsince larger τ means the time evolution is slower, thusmore adiabatic, which is another standard requirementof STIRAP. For large T , population transfer efficiencyis slightly suppressed but much less significant than in cases of Fig.2(a, b). III. CONTINUOUS INTERMEDIATE STATE
Now we further consider the case that the twoqubits are coupled via an intermediate finite-temperaturebosonic continuum. The Hamiltonian of the whole sys-tem can be written asˆ H ( t ) = ω q, σ z + ω q, σ z + (cid:90) dωω ˆ b † ω ˆ b ω +Ω P ( t ) (cid:90) dω (cid:112) J ( ω ) (cid:16) ˆ σ +1 ˆ b ω + ˆ σ − ˆ b † ω (cid:17) +Ω S ( t ) (cid:90) dω (cid:112) J ( ω ) (cid:16) ˆ σ +2 ˆ b ω + ˆ σ − ˆ b † ω (cid:17) , (9)where J ( ω ) is the spectrum function. We choose a simplesub-ohmic spectrum as J ( ω ) = √ ω, (10)and we also choose a sharp cut-off ω c such that J ( ω ) = 0for ω > ω c , as a signature of a finite-width continuum. Incomparison with the discrete case considered in Sec.II, wehave removed the two intermediate ‘flying qubits’ a and a , which will allow an easier numeric treatment while theresulting physics is still similar. In case the continuumis initially in the zero temperature state, the dynamicsof Eq. 9 can be easily solved based on a discretizationof the continuum and an exact diagonalization approachsince only the single excitation sector needs to be con-sidered [21]. However, for a finite temperature T , thecontinuum is a mixture of different bosonic particles andthe Hilbert space size is in general exponentially large.As a result exact diagonalization would be impossible inthis case. Moreover, in this case a Markovian quantummaster equation, such as the Lindblad equation [23, 24],would likely be problematic since here we consider strongsystem-continuum coupling.In recent years, there is a growing activity to use thesystem-bath approach in combination with matrix prod-uct states method to study the dynamics of open quan-tum systems. The system and the bath are evolved to-gether as a whole, and the dynamics of the system isobtained by tracing out the bath degrees of freedoms.Here we use a thermofield-based chain-mapping matrixproduct states algorithm (TCMPS) [25–31] to study thedynamics of the system plus bath which the the contin-uum in our case. The main advantage of this method isthat the finite temperature bath is mapped into anotherenlarged bath which is initially at zero temperature, thusin favoring of a MPS simulation. TCMPS include threemajor steps: 1) Discretization of the bath [32] , for whichwe use a simple linear discretization scheme with a fre-quency step size δ , the discretized Hamiltonian after thisstep would beˆ H dis ( t ) = ω q, σ z + ω q, σ z + N (cid:88) j =1 ω j ˆ b † j ˆ b j + Ω P ( t ) N (cid:88) j =1 J j (cid:16) ˆ a ˆ b † j + ˆ a † ˆ b j (cid:17) + Ω S ( t ) N (cid:88) j =1 J j (cid:16) ˆ a ˆ b † j + ˆ a † ˆ b j (cid:17) , (11)where we have used N = ω c /δ , ω j = jδ , ˆ b j = ˆ b ( ω j ),ˆ b † j = ˆ b † ( ω j ), J j = (cid:112) J ( ω j ) δ . The time-dependentcouplings J ,j ( t ) and J ,j ( t ) in Fig. 1(b) correspond toΩ P ( t ) J j and Ω S ( t ) J j respectively. In the limit N → ∞ ,ˆ H dis ( t ) is equivalent to ˆ H ( t ) [32, 33]; 2) Thermofieldtransformation maps the bath of N bosonic modes intoan enlarged but equivalent bath with 2 N bosonic modes,and at the same time the thermal state corresponding tothe original bath is mapped into the vacuum state of theenlarged bath. The Hamiltonian after this step isˆ H T ( t ) = ω q, σ z + ω q, σ z + N (cid:88) j =1 ω j (cid:16) ˆ c † ,j ˆ c ,j − ˆ c † ,j ˆ c ,j (cid:17) + Ω P ( t ) N (cid:88) j =1 g ,j (cid:16) ˆ a ˆ c † ,j + ˆ a † ˆ c ,j (cid:17) + Ω P ( t ) N (cid:88) j =1 g ,j (cid:16) ˆ a ˆ c ,j + ˆ a † ˆ c † ,j (cid:17) + Ω S ( t ) N (cid:88) j =1 g ,j (cid:16) ˆ a ˆ c † ,j + ˆ a † ˆ c ,j (cid:17) + Ω S ( t ) N (cid:88) j =1 g ,j (cid:16) ˆ a ˆ c ,j + ˆ a † ˆ c † ,j (cid:17) , (12)where g ,j = J j cosh( θ j ) and g ,j = J j sinh( θ j ), withcosh( θ j ) = (cid:112) n ( ω j ), sinh( θ j ) = (cid:112) n ( ω j ) and n ( ω ) =1 / (cid:0) e βω − (cid:1) to be the Bose-Einstein distribution. 3) Starto chain mapping, which maps the system-bath from thestar configuration into a chain configuration. The finalHamiltonian after those three steps would beˆ H TC ( t ) = ω q, σ z + ω q, σ z + (cid:88) ν =1 N chain (cid:88) j =1 α ν,j ˆ d † ν,j ˆ d ν,j + Ω P ( t ) N chain (cid:88) j =1 (cid:16) β , ˆ a ˆ d † ,j + β , ˆ a ˆ d ,j + H . c . (cid:17) + Ω S ( t ) N chain (cid:88) j =1 (cid:16) β , ˆ a ˆ d † ,j + β , ˆ a ˆ d ,j + H . c . (cid:17) + (cid:88) ν =1 N chain − (cid:88) j =1 β ν,j +1 (cid:16) ˆ d † ν,j ˆ d ν,j +1 + H . c . (cid:17) , (13) t . . . a ) 0 . . . T ( b ) FIG. 3: Population transfer via a thermal bosonic contin-uum. (a) The solid lines from top down correspond to T = 0 , . , . , . F as a functionof time t . The dotted lines from down to top correspond to T = 0 , . , . , . F as a functionof time t . (b) The blue dashed line with circle correspondsto F ( ∞ ) as a function of temperature T , while the yellowdashed line with square corresponds to F ( ∞ ) as a functionof temperature T . where α ,j and β ,j are the diagonal termsand off-diagonal terms resulting from the Lanc-zos tri-diagonalization of the diagonal matrixdiag([ ω , ω , . . . , ω N ]) with the initial vector[ g , , g , , . . . , g ,N ], while α ,j and β ,j are the di-agonal terms and off-diagonal terms resulting fromthe Lanczos tri-diagonalization of the diagonal ma-trix diag([ − ω , − ω , . . . , − ω N ]) with the initial vector[ g , , g , , . . . , g ,N ] [27]. The size of the vectors α ,j and β ,j , denoted as N chain , is usually chosen to be lessthan N . To the best of our knowledge, this work is thefirst time to apply TCMPS to study an open quantumsystem with time-dependent driving.We then evolve ˆ H TC ( t ) with the same initial state forthe two spins as for the discrete case, and vacuum statefor the enlarged continuum corresponding to the set ofmodes ˆ d ν,j . In our simulations we have chosen ω c = 2, δ = 0 . N chain = 50, a time step size dt = 0 .
01 andwe have kept 400 auxiliary states. The largest singularvalue truncation error observed during the time evolutionis of the order 10 − . The simulation results are shownin Fig. 3. In Fig. 3(a), we plot F ( t ) and F ( t ) as afunction of time t , we can see that in case of T = 0, al-most perfect population transfer can be achieved, whichis also shown in [21]. As T increases, the efficiency ofpopulation transfer goes down significantly. In Fig. 3(b),we plot F ( ∞ ) and F ( ∞ ) as a function of the temper-ature T , from which we can see more clearly that theefficiently of population transfer goes down significantlywhen T increases. At T = 1, only about half of the pop-ulation are successfully transferred from q to q . Theseresults show that for STIRAP via an infinite number ofintermediate states, the non-zero temperature stronglyaffects the population transfer efficiency. IV. CONCLUSION
We propose two models to study quantum popula-tion transfer between two spins via an intermediate statewhich is initially in thermal equilibrium. In the first case,we consider a discrete model where the two spins arecoupled to two bosonic modes by two controlled pulsesΩ P ( t ) and Ω S ( t ) which act as ‘flying qubits’, which arethen coupled to a common intermediate spin initially ina thermal state. In the second case, we consider a con-tinuous model where the two spins are directly coupledto a thermal bosonic continuum by the two controlledpulses. In both cases, we show that the efficiency of thepopulation transfer is strongly dependent on the finitetemperature of the intermediate state, in contrast withprevious results that the population transfer efficiency isrobust against the details of the intermediate states aslong as certain control parameters are well tuned.Moreover, in this work we have adapted the TCMPSmethod, which is a recently developed numeric tech-nique used to solve open quantum many-body systems, tostudy quantum population transfer via a thermal bosonic continuum. Our results show that TCMPS could bea perfect numerical tool to study open quantum opticsproblems in presence of a finite temperature environmentand time-dependent driving. V. ACKNOWLEDGEMENT
This work is acknowledged for funding NationalScience and Technology Major Project (grant no.2017ZX02101007-003); National Natural Science Foun-dation of China (grant no. 61565004; 6166500;61965005); the Natural Science Foundation of GuangxiProvince (Nos. 2017GXNSFBA198116 and 2018GXNS-FAA281163); the Science and Technology Program ofGuangxi Province (No. 2018AD19058). W.H. is ac-knowledged for funding from Guangxi oversea 100 tal-ent project and W.Z. is acknowledged for funding fromGuangxi distinguished expert project. C. G acknowl-edges support from National Natural Science Foundationof China under Grants No. 11805279. [1] N. Vitanov, M. Fleischhauer, B. Shore, andK. Bergmann, Advances in Atomic Molecular and Op-tical Physics , 55 (2001).[2] N. V. Vitanov, A. A. Rangelov, B. W. Shore, andK. Bergmann, Reviews of Modern Physics , 015006(2017).[3] B. W. Shore, Advances in Optics and Photonics , 563(2017).[4] W. Huang, B. W. Shore, A. Rangelov, and E. Kyoseva,Optics Communications , 196 (2017).[5] D. Møller, J. L. Sørensen, J. B. Thomsen, andM. Drewsen, Physical Review A , 062321 (2007).[6] K. Kumar, A. Veps¨al¨ainen, S. Danilin, and G. Paraoanu,Nature communications , 10628 (2016).[7] J. Siewert, T. Brandes, and G. Falci, Optics communi-cations , 435 (2006).[8] C. Ye, V. Sautenkov, Y. V. Rostovtsev, and M. Scully,Optics letters , 2213 (2003).[9] U. Hohenester, F. Troiani, E. Molinari, G. Panzarini,and C. Macchiavello, Applied Physics Letters , 1864(2000).[10] S. Longhi, Physical Review E , 026607 (2006).[11] W. Huang, S.-J. Liang, E. Kyoseva, and L. K. Ang,Carbon , 187 (2018).[12] W. Huang, S. Yin, W. Zhang, K. Wang, Y. Zhang, andJ. Han, New Journal of Physics , 113004 (2019).[13] A. A. Rangelov and N. V. Vitanov, Annals of Physics , 2245 (2012).[14] N. Sangouard, S. Gu´erin, L. Yatsenko, and T. Halfmann,Physical Review A , 013415 (2004).[15] B. W. Shore, Acta Physica Slovaca , 361 (2013).[16] N. Vitanov, Physical Review A , 2295 (1998).[17] N. Vitanov, B. W. Shore, and K. Bergmann, The Euro-pean Physical Journal D-Atomic, Molecular, Optical andPlasma Physics , 15 (1998). [18] B. T. Torosov and N. V. Vitanov, Physical Review A ,043418 (2013).[19] N. Vitanov, B. W. Shore, and K. Bergmann, The Euro-pean Physical Journal D-Atomic, Molecular, Optical andPlasma Physics , 15 (1998).[20] N. Vitanov and S. Stenholm, Physical Review A , 3820(1999).[21] W. Huang, S. Yin, B. Zhu, W. Zhang, and C. Guo,Physical Review A , 063430 (2019).[22] L. Contreras-Pulido and R. Aguado, Physical Review B , 155420 (2008).[23] G. Lindblad, Communications in Mathematical Physics , 119 (1976).[24] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan,Journal of Mathematical Physics , 821 (1976).[25] I. de Vega and M.-C. Banuls, Physical Review A ,052116 (2015).[26] E. Mascarenhas and I. De Vega, Physical Review A ,062117 (2017).[27] C. Guo, I. de Vega, U. Schollw¨ock, and D. Poletti, Phys-ical Review A , 053610 (2018).[28] D. M. Fugger, A. Dorda, F. Schwarz, J. von Delft, andE. Arrigoni, New journal of physics , 013030 (2018).[29] F. Schwarz, I. Weymann, J. von Delft, and A. Weichsel-baum, Physical review letters , 137702 (2018).[30] X. Xu, J. Thingna, C. Guo, and D. Poletti, PhysicalReview A , 012106 (2019).[31] T. Chen, V. Balachandran, C. Guo, and D. Poletti, arXivpreprint arXiv:2004.05017 (2020).[32] I. de Vega, U. Schollw¨ock, and F. A. Wolf, PhysicalReview B , 155126 (2015).[33] R. Bulla, T. A. Costi, and T. Pruschke, Reviews of Mod-ern Physics80