Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems
Bi-Heng Liu, Li Li, Yun-Feng Huang, Chuan-Feng Li, Guang-Can Guo, Elsi-Mari Laine, Heinz-Peter Breuer, Jyrki Piilo
EExperimental control of the transition from Markovian to non-Markovian dynamics ofopen quantum systems
Bi-Heng Liu, Li Li, Yun-Feng Huang, Chuan-Feng Li, ∗ Guang-CanGuo, Elsi-Mari Laine, Heinz-Peter Breuer, and Jyrki Piilo † Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, China Turku Centre for Quantum Physics, Department of Physics and Astronomy,University of Turku, FI-20014 Turun yliopisto, Finland Physikalisches Institut, Universit¨at Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany (Dated: May 5, 2011)
Realistic quantum mechanical systems are al-ways exposed to an external environment. Thepresence of the environment often gives rise to aMarkovian process in which the system loses in-formation to its surroundings. However, manyquantum systems exhibit a pronounced non-Markovian behavior in which there is a flow of in-formation from the environment back to the sys-tem, signifying the presence of quantum memoryeffects [1–5]. The environment is usually com-posed of a large number of degrees of freedomwhich are difficult to control, but some sophis-ticated schemes for modifying the environmenthave been developed [6]. The physical realizationand control of dynamical processes in open quan-tum systems plays a decisive role, for example,in recent proposals for the generation of entan-gled states [7–9], for schemes of dissipative quan-tum computation [10], for the design of quan-tum memories [11] and for the enhancement ofthe efficiency in quantum metrology [12]. Herewe report an experiment which allows throughselective preparation of the initial environmen-tal states to drive the open system from theMarkovian to the non-Markovian regime, to con-trol the information flow between the system andthe environment, and to determine the degree ofnon-Markovianity by direct measurements on theopen system.
The standard approach to the dynamics of open quan-tum systems employs the concept of a quantum Markovprocess which is given by a semigroup of completely posi-tive dynamical maps and a corresponding quantum mas-ter equation with a generator in Lindblad form [13, 14].Very recently, a toolbox for the engineering of such quan-tum Markov processes in a multi-qubit system of trappedions has been realized experimentally [15] and technolog-ical developments have also allowed experimental stud-ies of quantum correlations in open systems [16, 17].Within a microscopic approach quantum Markovian mas-ter equations are usually obtained by means of the ∗ Electronic address: cfl[email protected] † Electronic address: jyrki.piilo@utu.fi
Born-Markov approximation which presupposes a weaksystem-environment coupling and several further, mostlyrather drastic approximations. However, in many pro-cesses occurring in nature these approximations are notapplicable, a situation which occurs, in particular, in thecases of strong system-environment couplings, structuredand finite reservoirs, low temperatures, as well as in thepresence of large initial system-environment correlations.In the case of substantial quantitative and qualitative de-viations from the dynamics of a quantum Markov processone often speaks of a non-Markovian process, implyingthat the dynamics is governed by significant memory ef-fects. Quite recently important steps towards the devel-opment of a general consistent theory of non-Markovianquantum dynamics have been made which try to rig-orously define the border between Markovian and non-Markovian quantum evolution and to quantify memoryeffects in the open system dynamics [18–21].The measure for quantum non-Markovianity con-structed in [19] is based on the idea that memory ef-fects in the open system dynamics can be characterizedin terms of the flow of information between the opensystem and its environment. It has been used recently,e.g., to describe this information flow in the energy trans-fer dynamics of photosynthetic complexes [2, 4], and tocharacterize memory effects of the dynamics of qubits inspin baths [5]. Here, we present the results of an experi-ment which enables through a careful preparation of theinitial system-environment states and quantum state to-mography not only to control and to monitor the transi-tion from Markovian to non-Markovian quantum dynam-ics, but also the direct determination of this measure forquantum non-Markovianity.Quantum memory effects are quantified by employingthe trace distance D ( ρ , ρ ) = tr | ρ − ρ | between twoquantum states ρ and ρ . This quantity can be in-terpreted as a measure for the distinguishability of thetwo states [22–24]. Markovian processes tend to contin-uously reduce the distinguishability of physical states,which means that there is a flow of information from theopen system to its environment. In view of this inter-pretation the characteristic feature of a non-Markovianquantum process is the increase of the distinguishability,i.e. a reversed flow of information from the environmentback to the open system. Through this recycling of infor-mation the earlier states of the open system influence its a r X i v : . [ qu a n t - ph ] S e p FIG. 1: The experimental setup. Here, the abbreviations ofthe components are: HWP – half wave plate, QWP – quarterwave plate, IF – interference filter, QP – quartz plate, PBS –polarizing beamsplitter, FP – Fabry-Perot cavity, and SPD –single photon detector. later states, which expresses the emergence of memoryeffect in the open system’s dynamics [19, 20].On the basis of this physical picture one can constructa general measure for the degree of non-Markovianityof a quantum process given by some dynamical map Φ t which maps the initial states ρ (0) of the open system tothe corresponding states ρ ( t ) = Φ t (cid:0) ρ (0) (cid:1) at time t . Thefull time evolution of the open system over some timeinterval from the initial time 0 to the final time T is thengiven by a one-parameter family Φ = { Φ t | ≤ t ≤ T } of dynamical maps. We define the rate of change of thetrace distance by σ ( t, ρ , (0)) = ddt D ( ρ ( t ) , ρ ( t )). Themeasure N (Φ) for the non-Markovianity of the processis then defined by N (Φ) = max ρ , (0) (cid:90) σ> dt σ ( t, ρ , (0)) . (1)Here, the time-integration is extended over all subinter-vals of [0 , T ] in which the rate of change of the tracedistance σ is positive, and the maximum is taken overall pairs of initial states. The quantity (1) thus measuresthe maximal total increase of the distinguishability overthe whole time-evolution, i.e., the maximal total amountof information which flows from the environment back tothe open system.In our experiment the open quantum system is pro-vided by the polarization degree of freedom of photonscoupled to the frequency degree of freedom represent-ing the environment. The experimental setup is shownin Fig. 1. An ultraviolet Argon-Ion laser is used topump two 0 . . FIG. 2: The frequency spectrum of the initial state for variousvalues of the tilting angle θ . tilted in the horizontal plane. A 4nm (full width at halfmaximum) interference filter is placed after the FP cavityto filter out at most two transmission peaks. The corre-sponding interference filter in the other arm is 10nm. Thepolarization and frequency degrees of freedom are cou-pled in a quartz plate in which different evolution timesare realized by varying the thickness of the plate. A po-larizing beam splitter together with a half-wave plate anda quarter-wave plate is used as a photon state analyzer.The half wave plate HWP2 and the tilted FP cavity areused to prepare the initial one-photon states | ψ , (0) (cid:105) = | ϕ , (cid:105) ⊗ | χ (cid:105) , where | ϕ , (cid:105) = 1 √ | H (cid:105) ± | V (cid:105) ) , (2)with | H (cid:105) and | V (cid:105) denoting the horizontal and the ver-tical polarization state, respectively. The environmentalstate | χ (cid:105) = (cid:82) dωf ( ω ) | ω (cid:105) involves the amplitude f ( ω ) forthe photon to be in a mode with frequency ω , which isnormalized as (cid:82) dω | f ( ω ) | = 1. The form of the fre-quency distribution | f ( ω ) | and thus the initial state ofthe environment can be controlled by the tilting angle θ of the FP cavity. Figure 2 shows how θ determinesthe structure of the frequency spectrum and, thus, theenvironmental initial state | χ (cid:105) . By changing the initialstate of the environment in the experiment we modify theopen system dynamics in a way which allows us to ob-serve transitions between Markovian and non-Markovianquantum dynamics.In the first version of the experiment, HWP1 is fixedat zero degree to generate a two-photon state. Pho-ton 2 is directly detected in detector SPD in the endof arm 2 as a trigger of photon 1. Photon 1 first passes FIG. 3: The trace distance (a) and the concurrence betweenthe system and the ancilla (b) as a function of the effectivepath difference for four different values of the tilting angle θ . The solid lines represent the theoretical predictions for σ = 1 . × Hz and ∆ ω = 1 . × Hz. The effective pathdifference is equal to ∆ nL and λ = 702nm. The experimen-tal error bars due to the counting statistics are smaller thanthe symbols. HWP2, preparing it in the state | ϕ (cid:105) or | ϕ (cid:105) . The sub-sequent interaction between polarization and mode de-grees of freedom in the quartz plate is described by aquantum dynamical map Φ t acting on the open system,where the interaction time t is connected to the vari-able length L of the quartz plate by means of t = L/c .Finally, a full state tomography is carried out in detec-tor SPD in the end of the arm 1 to determine the po-larization state ρ , ( t ) = Φ t ( | ϕ , (cid:105)(cid:104) ϕ , | ) of photon 1.This allows the direct experimental determination of thetrace distance D ( ρ ( t ) , ρ ( t )) between the two possibleone-photon states after a certain interaction time t con-trolled by the length L of the quartz plate. Experimentalresults for four different values of the tilting angle θ of theFP cavity are shown in Fig. 3(a). We clearly observe acrossover from a monotonic to a non-monotonic behaviorof the trace distance as a function of time, i.e., a transi-tion from a Markovian to a non-Markovian dynamics ofthe polarization degree of freedom of the photon.The experimental results admit a simple theoreticalanalysis which is based on the fact that the time evolu-tion in the quartz plate may be described by the unitaryoperator U ( t ) which is defined by U ( t ) | λ (cid:105) ⊗ | ω (cid:105) = e in λ ωt | λ (cid:105) ⊗ | ω (cid:105) , where n λ represents the refraction index for light withpolarization λ = H, V . The presence of the quartz platethus leads to the decoherence of superpositions of po-larization states, which is due to a nonzero difference∆ n = n V − n H in the refraction indices of horizontallyand vertically polarized photons. The corresponding dy-namical map Φ t takes the form:Φ t : | H (cid:105)(cid:104) H | (cid:55)→ | H (cid:105)(cid:104) H | , | V (cid:105)(cid:104) V | (cid:55)→ | V (cid:105)(cid:104) V | , | H (cid:105)(cid:104) V | (cid:55)→ κ ∗ ( t ) | H (cid:105)(cid:104) V | , | V (cid:105)(cid:104) H | (cid:55)→ κ ( t ) | V (cid:105)(cid:104) H | , where the decoherence function κ ( t ) is given by theFourier transform of the frequency distribution, κ ( t ) = (cid:90) dω | f ( ω ) | e iω ∆ nt . FIG. 4: The change of the trace distance and of the con-currence as functions of the tilting angle θ . The transitionfrom the non-Markovian to the Markovian regime occurs at θ (cid:39) . ◦ , and from the Markovian to the non-Markovianregime at θ (cid:39) . ◦ . The positive values in the blue re-gions give directly the non-Markovianity measure N (Φ) ofthe process. The negative values in the grey area correspondto N (Φ) = 0, i.e., to Markovian dynamics. The error bars aredue to the uncertainty of the tilting angle and the countingstatistics. With the help of these relations it is easy to show that thetrace distance corresponding to the initial pair of states(2) is equal to the modulus of the decoherence function, D ( ρ ( t ) , ρ ( t )) = | κ ( t ) | . (3)In the experiment the transition from Markovian to non-Markovian dynamics is observed through variation of thetilting angle θ of the FP cavity. As illustrated in Fig. 2,all frequency distributions are very well approximated bya sum of two Gaussians centered at frequencies ω k withamplitudes A k and equal widths σ , which yields | κ ( t ) | = e − σ (∆ nt ) A (cid:112) A + 2 A cos(∆ ω · ∆ nt ) , (4)where A = A , A = A A and ∆ ω = ω − ω . Thetilting angle of the cavity is relatively small and thus thedistance ∆ ω between the peaks remains approximatelyconstant. Therefore, the only relevant parameter control-ling the transition is the relative amplitude A . Equation(4) yields an excellent approximation of the experimentaldata (see the continuous lines in Fig. 3).The experiment also enables a direct determination ofthe measure for non-Markovianity (1). First, we notethat the initial pair (2) is already optimal in the sensethat it yields a maximal increase of the trace distance.The theoretical explanation for this fact is presented inthe Supplementary Information. Second, in our experi-ment the increase of the trace distance is restricted to asingle time interval (see Fig. 3). The non-Markovianitymeasure (1) is thus obtained by determining the differ-ence of the trace distance between the first local mini-mum and the subsequent maximum. Our experimentalresults are shown in Fig. 4. Increasing the tilting angleof the cavity decreases the relative amplitude A betweenthe peaks in the frequency spectrum and thereby reducesthe non-Markovianity of the process until a transitionto Markovian dynamics occurs. Further increasing thetilting angle amplifies the relative amplitude again andbrings the dynamics back to the non-Markovian regime.In Ref. [21] an alternative measure for non-Markovianity has been proposed which is based on theidea that a Markovian dynamics leads to a monotonicdecrease of the entanglement between the open systemand an isomorphic ancilla system, while a non-Markoviandynamics induces a temporary increase of the entan-glement. One can show (see Supplementary Informa-tion) that for the present experiment this measure co-incides with (1) if one uses the concurrence [25, 26] asan entanglement measure. This fact leads to an alter-native and independent method for the measurementof the non-Markovianity by means of our experimen-tal setup. Fixing HWP1 to 22.5 degree, we generate amaximally entangled two-photon state. Photon 1 thenpasses the quartz plate and the composite final state isanalyzed through two-photon state tomography. Exper-imental results are shown in Fig. 3(b) and Fig. 4, clearlydemonstrating the equivalence of both measures for non-Markovianity.Our experiment clearly reveals the measurability of re-cently proposed theoretical measures for quantum non- Markovianity which yield important information aboutthe type of quantum noise and about environmentalproperties, even when the environment is a complexsystem involving an infinite number of degrees of free-dom and is not directly accessible through measurements.Moreover, we have introduced a method for the controlof the information flow between the open system and itsenvironment, which opens the possibility of efficiently ex-ploiting memory effects in future quantum technologies[27]. Acknowledgments
This work was supported by theNational Fundamental Research Program, National Nat-ural Science Foundation of China (Grant Nos. 10874162and 10734060), the Magnus Ehrnrooth Foundation, andthe Graduate School of Modern Optics and Photonics.
Author Contributions
B.-H.L., L.L., Y.-F.H., C.-F.L., and G.-C.G. planned, designed and implementedthe experiments. E.-M.L., H.-P.B., and J.P. carried outthe theoretical analysis and developed the interpretation.B.-H.L., C.-F.L., E.-M.L, H.-P.B., and J.P. wrote thepaper and all authors discussed the contents.
Author Information
The authors declare thatthey have no competing financial interests. Reprintsand permissions information is available online athttp://npg.nature.com/reprintsandpermissions. Corre-spondence and requests for materials should be addressedto C.-F.L. or J.P. [1] Breuer, H.-P & Petruccione, F.
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For an arbitrary initial pair ρ , (0) the trace distanceevolution is given by D ( ρ ( t ) , ρ ( t )) = (cid:112) a + | κ ( t ) b | ,where a = ρ (0) − ρ (0) denotes the difference of theinitial populations, and b = ρ (0) − ρ (0) the differenceof the initial coherences. It follows that any growth of the trace distance is maximal for a = 0 and | b | = 1, inwhich case we obtain the formula (3). Therefore, the pairof states given by Eq. (2) is an optimal initial pair. Moregenerally, all initial pairs of states are optimal which cor-respond to pairs of antipodal points on the equator of theBloch sphere that represents the two-state state system. Equivalence of the two measures
The initial state is the pure, maximally entangledsystem-ancilla state ρ SA (0) = | ψ SA (cid:105)(cid:104) ψ SA | with | ψ SA (cid:105) = √ ( | HH (cid:105) + | V V (cid:105) ) . The dynamical map acts locally onthe system part leading to the state ρ SA ( t ) = (Φ t ⊗ I ) ρ SA (0)= 12 (cid:16) | HH (cid:105)(cid:104) HH | + | V V (cid:105)(cid:104)
V V | + κ ( t ) ∗ | HH (cid:105)(cid:104) V V | + κ ( t ) | V V (cid:105)(cid:104) HH | (cid:17) . This state is diagonal in a basis of maximally entangledstates (Bell state basis), the maximal eigenvalue being p max = (1+ | κ ( t ) | ). Therefore, the concurrence of ρ SA ( t )is given by C ( ρ SA ( t )) = 2 p max − | κ ( t ) | [24]. Thus, wehave the relation D ( ρ ( t ) , ρ ( t )) = C ( ρ SA ( tt