Non-Markovian Quantum Interference in Multilevel Quantum Systems

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arXiv:1712.06795v1 [quant-ph] 19 Dec 2017 oie ytemmr eoetesse ete down settles state. system steady the its before be to will memory and the time-dependent, by be modified of will patterns system interference quantum in- the the then the den- accessible, If is the [1]. matrix in sity matrix recorded density memory environmental a of by formation described system’s be- be the will will and interactions environment state their its to medium. due and structured entangled come system a infer- open to the quantum couplings Typically to about due information patterns the ence sys- all The open contain [14–21]. multilevel effect tems of memory equations the master as non-Markovian known system, the of be reservoir. cannot Markov that environment quan- a an by the to approximated when coupled lacks is interest system of tum master system are non-Markovian the systems for the multilevel equation quantum because of difficult dynamics stud- technically However, the limit. on long-time the ies the interpolating and scales limit coher- time short-time quantum different of for behaviour dynamics temporal ence be the will study we to in theory, able advance a system recent quantum on the open based With non-Markovian multilevel medium [4]. quantum equation continuum of master a Markov states in steady embedded the systems of for case made the In been have observations [1–13]. important many interference, etc. quantum control, quantum and interference quantum absorption, transport, quantum and dissipation, emission quantum atom such decoherence, interaction, quantum system-environment as the phenom- quantum involving important ena study to recently tensively ∗ ‡ † o-akva unu nefrnei utlvlquantum multilevel in interference quantum Non-Markovian [email protected] [email protected] [email protected] h o-akva vlto eed ntehistory the on depends evolution non-Markovian The ex- used been has approach system quantum Open ASnmes 36.z 36.g 36.d 32.90.+a 03.65.Ud, 03.67.Bg, 03.65.Yz, numbers: the PACS in ful scales. instrumental time not is different has memory in environmental that pattern interference the domain that temporal show a equati we master in non-Markovian wi phenomena that ex or interference show with the results systems Our using four-level by tigated. involving environment examples dissipative Two bosonic paper. a to coupled esuytennMroinqatmitreec phenomeno interference quantum non-Markovian the study We .INTRODUCTION I. 2 etrfrQatmSineadEgneigadDprmn o Department and Engineering and Science Quantum for Center ejn opttoa cec eerhCne,Biig10 Beijing Center, Research Science Computational Beijing tvn nttt fTcnlg,Hbkn e esy07030 Jersey New Hoboken, Technology, of Institute Stevens 1 aoaoyfrQatmOtc n unu Information, Quantum and Optics Quantum for Laboratory uu Chen Yusui Dtd eebr2,2017) 20, December (Dated: qainapproach equation 1 , 2 , ∗ .Q You Q. J. nSc I,w pl h o-akva atrequa- discus- master brief non-Markovian some with the examples apply two investigate we to systems. III, tions multilevel the Sec. for derive In equations to how master show general and the diffu- approach, quantum-state (QSD) of equations ideas sion principle the review briefly temporal the investigate to inference. quantum used we of be case, behaviors can can each and equation derived, In master non-Markovian be exact pulses. the optical that external show in- the con- quantum system by the atomic trolled four-level study a we for where phenomena terference example illustrate presented second to are the results example major in an Our as methods. atom we theoretical our four-level purpose, simple this a For use equation. non-Markovian corresponding diffusion time-local the quantum-state system the from multilevel obtained that a be is may for idea equation tempo- The master the non-Markovian on inference. investigation master our quantum non-Markovian for ral used theoretical exact be a the to equations establish deriving systematically to we approach decoherence First, quantum and [3–13]. transfer, control, coherence quantum quantum processing, ranging information processes quantum physical from important many from systems arising open multilevel in phenomena inference quantum master non-Markovian the of framework equations. the rig- in studied be quantum orously also temporal can the systems multilevel that for show diffusion interference we quantum-state Here, equa- non-Markovian [22]. master equation the non-Markovian exact from success- of tions have set we a dy- Recently, derived quantum fully and [28–31]. simulations analysis namics decoherence for fundamental of importance still general are However, equations [23–27]. master environment non-Markovian bosonic cou- a systems to quantum the pled study open to of tool dynamics powerful non-Markovian a provided has equation fusion 1 h ae sognzda olw.I e.I,w will we II, Sec. In follows. as organized is paper The the study to is purpose primary our paper, this In dif- quantum-state non-Markovian the years, recent In , † n igYu Ting and n r aal fehbtn quantum exhibiting of capable are ons hu xenlcnrlfilsaeinves- are fields control external thout c atreutosdrvdi this in derived equations master act yepoe eoe nparticular, In before. explored ly ne fnnMroinquantum non-Markovian of onset famlisaeaoi system atomic multi-state a of n 1 , 2 04 China 0084, ‡ USA , ytm:Eatmaster Exact systems: Physics, f 2 sions on non-Markovian features. We conclude in Sec. operator in the basis of noise, we expand the O operator IV. and rewrite it in the form of

t ∗ O(t,s)= O0 + ds1zs1 O1(t,s,s1) II. EXACT NON-MARKOVIAN MASTER ˆ0 EQUATIONS t ∗ ∗ + ds1ds2zs1 zs2 O2(t,s,s1,s2)+ ..., (5) ¨0 For a generic multilevel quantum system coupled to a zero-temperature bosonic environment, the total Hamil- where the terms Oj (j = 0, 1, 2, 3...) are the operators tonian is given by [11] (we set ~ = 1) according to the jth order of the noise. By consistency ∂ δ δ ∂ condition ∗ ψt = ∗ ψt, the O operator can be ∂t δzs δzs ∂t Htot = Hs + Hint + Hb, (1) formally determined by its evolution equation, † † ∗ Hint = L gkbk + L gkbk, ¯ ∗ † † δO Xk Xk ∂ O = −iH + Lz − L O,¯ O − L , (6) t s t δz∗ † s Hb = ωkbkbk, Xk ¯ ∗ t ∗ where O(t,z ) = 0 dsα(t,s)O(t,s,z ). Based on the exact QSD equation´ (4), the reduced density matrix ρ where L is the Lindblad operator of the system, describ- t is recovered from ensemble average over all trajectories. ing the coupling between the system and its environment. The formal master equation can be written as For a dissipative coupling, the Lindblad operator L con- tained in the usual form of interaction term is generally a ∗ † ∂tρt = −i [Hs, ρt]+ LM[z Pt] − L M[OP¯ t] ladder operator in the form of L = |nihn +1|, which t n † ¯† typically satisfies the following condition:P + M[ztPt]L −M[PtO ]L, ∗ LN =0, (2) where ρt = M[|ψt(z )ihψt(z)]. We denote the stochastic ∗ reduced density matrix as Pt = |ψt(z )ihψt(z)|. By the ? where the integer N is the number of energy levels. Note Novikov theorem [ ], the ensemble average can be calcu- † ∗ t ∗ δ lated by M[z Pt]= dsM[z zs]M[ Pt]. Since |ψti is that bk(bk) is the creation (annihilation) operator for the t 0 t δzs δ † kth mode in the bosonic environment and gk is the com- independent of noise´ z , we have M[ P ] = M[P O ]. t δzs t t plex coupling constant between the system and the kth Then the above formal master equation can be further environment mode, described by the correlation function written as of the bath at zero temperature, † † ∂tρt = −i [Hs, ρt]+ L, M[PtO¯ ] − L , M[OP¯ t] . (7) 2 −iωk (t−s) α(t,s)= |gk| e . (3) Xk When the environment is Markov and the correspond- ∗ ing correlation function α(t,s)= δ(t,s), then M[zt Pt]= In the QSD approach, the dynamics of quantum sys- M[PtL] = ρtL, and Eq. (7) is reduced to the master tems is described by a stochastic Schrödinger equation. equation of the standard Lindblad form [2]. However, The total state Ψtot for system and environment can O operator generally contains noise in non-Markovian † be expanded in the basis of Bargmann coherent states regime, so M[PtO¯ ] cannot reduce to a simply form. By 1 2 ∗ Ψtot(t) = π d z||zi|ψt(z )i, where ψt is the stochastic repeatedly using Novikov theorem, we can write the nth- wave function´ for the multilevel system. The formal QSD order noise term as equation at zero temperature [23, 24] is M[zs1 zs3 ...zs2n−1 Pt] ∗ ∗ ∗ t ∂tψt(z ) = (−iHs + Lzt ) ψt(z ) δPt = ds α , M zs3 ...zs2 −1 t δ ˆ 2 1 2 n δz∗ † ∗ 0 s2 − L dsα(t,s) ∗ ψt(z ), (4) ˆ δz n n 0 s t δ = ds2, .., ds2n  αj,j+1 M   Pt , ∗ ∗ iωk t ˆ δz∗ where zt = −i k gkzke is the complex Gaussian 0 j=1 j=1 s2j ∗ Y Y stochastic processP with correlation function M[zt zs] =      2 2 (8) d z −|z| α(t,s). The symbol M[..] = π e .. means ensemble average operation over all´ possible stochastic trajec- where αi,j = α(si,sj ). Next, by applying this inference ∗ † tories zt . The QSD equation may be numerically solved of Novikov theorem on M[PtO¯ ], we can have an infinite ∗ by introducing the so-called O operator: O(t,s)ψt(z )= series of increasing power order of O operator and O† δ ∗ ∗ ψt(z ), whose Markov limit is the Lindblad opera- δzs operator. Now let us recall the identity function of O tor L. In most cases, the O operators contain complex operator in Eq. (6), the right side contains up to (N−2)th components. In order to clearly show the structure of O order of noise, while for the left side, the commutator 3

† th term [L O¯j , Ok] generates up to (j + k) order of noise. Next, the ensemble average term R(t) in the formal mas- Naturally, the two sides of Eq. (6) match with each other ter equation (10) is and construct a set of closed differential equations, which ¯ t means that Oj Ok must go to zero if j + k >N − 2. ¯† ¯† For these disappeared terms, we name them “forbidden R(t)= ρtO0 + ds1α1,2M[zs1 Pt]O1 ˆ0 conditions” [22]: t ¯† + ds1α1,2M[zs1 zs2 Pt]O2. (14) Oj Ok =0, (j + k>N − 2). (9) ˆ0 By repeating applying Novikov’s theorem (8), the two Then it’s easy to prove that the M[zs1 zs3 ...zs2n−1 Pt] ¯† must be a sum of finite terms. We will also have de- terms M[zs1 Pt] and M[zs1 zs2 Pt]O2 can be extensively tailed discussion in the next part. If we denote R(t) = written in an infinite long series. However, with the cri- † M[PtO¯ ], the compact form of the exact master equation terion ”forbidden conditions” (9), we have is written as ¯† M[zs1 Pt]O † 1 ∂tρt = −i [Hs, ρt] + [L, R(t)] + [L, R(t)] . (10) t ¯† = ds2α1,2O0(t,s2)ρtO1 ˆ0 t III. TEMPORAL QUANTUM INTERFERENCE † ¯† BASED ON THE EXACT MASTER EQUATIONS + ds2s3s4α1,2α3,4O1(t,s2,s3)ρtO0(t,s4)O1, ˚0

A. Four-level atom model and ¯† M[zs1 zs2 Pt]O2 The first model we consider as an example is the four- t level atom model described by the total Hamiltonian in ¯† = ds3s4α1,3α2,4O0(t,s3)O0(t,s4)ρtO2 the same form as Eq. (1), with ¨0 t 4 + ds s α α O (t,s ,s )ρ O¯†. ¨ 3 4 1,3 2,4 1 3 4 t 2 Hs = ωm|mihm|, 0 mX=1 Substituting these terms into Eq. (14), we obtain that 3 L = κm|mihm +1|, (11) t ¯† ¯† mX=1 R(t)= ρtO0 + ds1ds2α1,2O0(t,s2)ρtO1(t,s1) ¨0 t where the operator |mihm + 1| is the ladder angular- † † 3 ¯ momentum operator. It is obvious that L = 0. Fol- + d¯sα1,2α3,4O1(t,s2,s3)ρtO0(t,s4)O1(t,s1) ˘0 lowing the previous discussion in Sec. II, O operator t contains up to second order of noise and is in the form of ¯† + d¯sα1,3α2,4O0(t,s3)O0(t,s4)ρtO2(t,s1,s2) ˘0 t t ∗ † O(t,s)= O0(t,s)+ ds1zs1 O1(t,s,s1) ¯ ˆ + d¯sα1,3α2,4O1(t,s3,s4)ρtO2(t,s1,s2), (15) 0 ˘0 t + ds ds z∗ z∗ O (t,s,s ,s ). (12) ¨ 1 2 s1 s2 2 1 2 where d¯sdenotes the integral elements ds1ds2ds3ds4. The 0 exact non-Markovian master equation for four-level sys- Substituting this solution into Eq. (6), we have tem is determined as the same form as in Eq. (10). In order to simplify the numerical simulation, † † ∂tO0(t,s) = [−iHs, O0] − L O¯0, O0 − L O¯1(t,s), we choose Ornstein-Uhlenbeck correlation function, α(t,s) = γ e−γ|t−s|, for the advantage of showing ∂ O (t,s,s ) = [−iH , O ] − L†O¯ , O − L†O¯ , O 2 t 1 1 s 1 0 1 1 0 the transference between Markov (γ → ∞) and non- † ¯ − 2L O2(t,s,s1), Markovian (γ is small ) process. However, it needs to † † point out that, in our general derivation, the correlation ∂tO2(t,s,s1,s2) = [−iHs, O2] − L O¯0, O2 − L O¯1, O1 function is arbitrary. † ¯ − L O2, O0 . (13) As shown in Fig. 1, the time evolution of population and coherence is significantly different in Markov and Meanwhile we have the boundary conditions for each non-Markovian regime. Our result clearly reveals that term of O operator: the decay speed of population is much slower in non- Markovian regime than in Markov limit, and the decay [L, O ]= O (t,s,t), 0 1 behavior of interference is similar. As expected, the re- [L, O1]=2O2(t,s,t,s1). vival behavior is discovered in the non-Markovian regime. 4

(a) (b) (a) (b) 0.25 1 1 1 0.2 0.8 44 11 44 33 ρ ρ 0.6 0..5 0..5 0.1 ρ ρ 0.4 0 0 10 20 0 10 20 0 0 ω t ω t 05 10 15 20 0 5 10 15 20 (c) (d) ω t ω t (c) (d) 1 1 0.2 0.2 | | 14 23 | ρ ρ | | 0.1 0.1 43 0..5 22 0..5 ρ |ρ

0 0 0 10 20 0 10 20 0 0 0 510 15 20 0 5 10 15 20 FIG. 1. (Color online) Non-Markovian evolution of four-level ω t ω t atom. The initial state is set as |ψi0 =(|1i+|2i+|3i+|4i)/2. We show the population of highest level (a) and ground level FIG. 3. (Color online) Time evolution of population of |4i, (b), also the coherence of |ρ23| (c) and |ρ14| (d). The solid |3i, |3i and coherence |ρ43| between |4i and |3i. Initial state (red) curve is for γ = 0.2, dash-dotted (blue) is for γ = 0.5 is set as |4i. The constants are set as: ω4 − ω3 = ω, Ω1 = 5ω, and dotted (black) is for γ = 2. Ω2 = 2Ω1, µ = 2ω, κ = 2. Take comparison for two γ values. (1) γ = 0.5, solid (red), (2) γ = 10, dashed (blue).

Hamiltonian can be written as,

Htot = Hs + Hint + Hb, B. Quantum Interference 4 −iµt Hs = ωm|mihm| + (Ω1e |3ih2| + h.c.) mX=1 −iµt + (Ω2e |4ih2| + h.c.),

Hint = gk|jih1|bk + h.c., 4 Xk jX=3,4 H = ω b† b , 3 b k k k Xk

where the Ω1(2) is the Rabi frequency of the driving field for transition between |4(3)i and |2i ( see Fig.2). There Ω1 Ω2 are two decay channels for spontaneous emission. The γ γ decaying constant between the system and kth mode in environment is gk. Thus, the Lindblad operator for this 2 system is written as 1 L = |1ih3| + κ|1ih4|, where κ is the ratio of decaying constants between these FIG. 2. Schematic diagram of quantum interference model, two channels. It is easy to prove that this model satis- which shows the two upper level (|4i, |3i) coupled to the in- fies the “forbidden condition” and L3 = 0. Because the termediate level (|2i) with Rabi frequency Ω1 and Ω2, and decaying to ground state (|1i). dissipative channels are independent of the driving field, we can apply the results in Eq. (3) for O operators and the exact master equation in (10) directly for numerical simulation to study the quantum interference under the external field control. In Figs. 3 and 4, time evolution of the population Quantum interference in multilevel system leads to of level 2, level 3 and level 4 and the coherence of the many interesting effects, such as quantum absorption re- upper two levels are shown. By choosing different de- duction and cancelation [4]. Here in this section, we cay rate γ in the Ornstein-Uhlenbeck correlation func- γ −γ|t−s| consider a four-level system as shown in Fig. 2. The tion 2 e , the short-time evolution in Markov regime 5

(a) (b) 1 1 0.7 population on |4i population on |3i 0.6

44 0.5 33 0.5 ρ ρ

0.5 0 0 05 10 15 20 0 5 10 15 20 0.4 ω t ω t (c) (d) 1 1 0.3 polulation

| 0.2 22

43 0.5 0.5 ρ |ρ 0.1 0 0 05 10 15 20 05 10 15 20 0 ω t ω t 1 1.5 2 2.5 3 3.5 4 Ω /Ω 2 1 FIG. 4. (Color online) Time evolution of population of |1i, |2i, |3i and coherence |ρ43| between |4i and |3i. Initial state is set FIG. 5. (Color online) The population of |4i (red-circle), |3i as |4i. The constants are set as: ω4 − ω3 = ω, Ω1 = Ω2 = 5ω, (blue-square) at long time limit depending on the ratio of µ = 2ω, κ = 1. Take comparison for two γ values. (1) γ = 0.5, Ω2/Ω1. Initial state is set as |4i. The constants are set as: solid (red), (2) γ = 10, dashed (blue). µ = 2ω, κ = 2. and in non-Markovian regime is compared. With our exact master equation, numerical simulation shows that the 0.3 population of the upper two levels are swapping in a high population on |4i frequency, when γ = 0.5 (the red solid line), a typical population on |3i 0.25 non-Markovian regime. At the same time, the average population of the highest level 4 fluctuates between the range from 0.4 to 1. While in the Markov regime, γ = 10 0.2 ( blue dashed line), the population quickly decays to a steady state. Because of the real decay rate we used 0.15 in the simulation, Ornstein-Uhlenbeck noise’s long-time limit is same as its Markov limit. However, as we empha- polulation sized in the beginning, many devices, for instance high-Q 0.1 cavities, do not satisfy the Markov environment assumption. For example, a visible light laser pulse can easily 0.05 lasts in a high-Q cavity longer than 20 femtosecond scale before achieving a steady state. Otherwise, the measure- 0 ment on the system would lead to a large volatility. 1 1.5 2 2.5 3 3.5 4 Ω /Ω The second conclusion is the population inverse only 2 1 happens in the level 4 and level 3 under some particu- lar conditions. In Fig. 4, two parameters are changed, FIG. 6. (Color online) The population of |4i (red-circle), |3i Γ2 =Γ1 and κ = 1. The populations of level 4 and level (blue-square) at long time limit depending on the ratio of 3 of the steady state are close to each other. Although Ω2/Ω1. Initial state is set as |4i. The constants are set as: the conditions of population inverse was mentioned in [4], µ = 2ω, κ = 1 the discussion was based on the spectral density analysis and Markov assumption. Our exact master equation approach supplies a alter way to study the steady state of the system under arbitrary environmental spectrum. Fig. 6, the population inverse only happens when the In order to study the influence of the parameters to ratio Ω2/Ω1 > 1.5 and the steady populations of level 4 the steady state, we simulate the steady state popula- and level 3 are both close to 0.25, once κ is changed from tions of level 4 and level 3 under different κ and the ratio 2 to 1. With our master equation approach, the tempo- Ω2/Ω1. In Fig. 5, the strongest population inverse hap- ral control on multi-level systems can be simulated under pens around the ratio Ω2/Ω1 = 2.5 and the population considering any parameters in the set up and arbitrary of level 4 reaches the maximal value as 0.65. While in environmental spectrum. 6

IV. CONCLUSION scales ranging from short-time limit to the long-time limit which is typically described by a Markov master equation. A future work along this line of research will be the detailed study of quantum interference and quantum In summery, we demonstrate the interesting quantum control of multilevel systems coupled to a super-ohmic or interference patterns induced by the non-Markovian envi- sub-ohimic environment. ronment with the aid of the exact master equation for the multilevel systems. Several interesting examples are used to show how quantum memory aﬀects system’s dynam- ACKNOWLEDGMENTS ics, and in each case, we have shown that the exact master equations can play a very important role in explor- We acknowledge grant support from the NSF PHY- ing non-Markovian open system dynamics and quantum 0925174. J.Q.Y. is supported by the National Key Re- control. One important feature illustrated by the exact search and Development Program of China (Grant No. master equations in each case is that quantum interfer- 2016YFA0301200), the NSFC (Grant No. 11774022), ence can be investigated in all the physically interesting and the NSAF (Grant No. U1530401).

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