Abstract

We experimentally simulate a quantum channel in a linear optical setup, which is modeled by a two-level system (i.e., qubit) interacting with a bosonic bath. Unlike the traditional works, we treat the system–bath interaction without applying the Born approximation, the Markov approximation, or the rotating-wave approximation (RWA). To the best of our knowledge, this is the first experimental simulation of a quantum channel without any of the approximations mentioned above by using linear optical devices. This non-RWA channel provides a more accurate picture of the quantum open-system dynamics. It not only reveals the effect of the counterrotating terms but also enables us to consider arbitrarily strong coupling regimes. With the proposed channel, we further experimentally investigate the dynamics of the quantum temporal steering (TS), i.e., a temporal analog of Einstein–Podolsky–Rosen steering. The experimental and theoretical results are in good agreement and show that the counterrotating terms significantly influence the TS dynamics. The TS in non-RWA and RWA channels presents different dynamics. However, we emphasize that the results without RWA are closer to realistic situations and thus more reliable. Due to the close relationship between TS and the security of the quantum cryptographic protocols, our findings are expected to have useful applications in secure quantum communications. This work also inspires future interest in studying other quantum coherence properties in the non-RWA channels.

© 2017 Optical Society of America

1. INTRODUCTION

Real quantum systems always interact with the surrounding bath. An accurate description of the quantum open-system dynamics is one of the most challenging issues in quantum mechanics. For simplicity, in most theoretical and experimental studies, the treatment of the system–bath interaction usually starts with a perturbation theory and involves various approximations, such as the Born approximation, the Markov approximation, and the rotating-wave approximation (RWA) [1]. However, it is widely believed that the approximation approaches introduce more or less the incompleteness of the description of the bath. Therefore, several theoretical methods were developed to exactly characterize the bath influence, such as the path integral methods [2,3]. Another kind of powerful mean, the so-called hierarchy equation method, was introduced by Tanimura et al. [46], who established a set of hierarchical equations that includes all orders of system–bath interactions and avoids using the aforementioned approximations. This method has been successfully employed to describe the quantum dynamics of various physical and chemical systems [7,8] as well as some quantum devices [9,10].

Unfortunately, in experimental studies on open-system dynamics, there has been a lack of an accurate simulation of the quantum channel. With linear optical setups, many experimental works on simulating dynamical quantum correlations have been reported [1114]. An open system driven from the Markovian to the non-Markovian regime was experimentally simulated [15]. A single-qubit thermometry was realized through a simulation of the system–bath interaction [16]. A universal cooling method, which is applicable to any physical system, was demonstrated experimentally [17]. However, all of these studies are based on the approximated simulations of the quantum channel, such as the amplitude decay channel and the phase damping channel, which adopt the RWA treatment of system–bath interactions and the weak coupling approximation. It has been found that the RWA may lead to faulty results in describing the quantum concepts, which depend heavily on the characterization of the system–bath interaction [18,19]. Hence, how to experimentally simulate a quantum channel without applying approximations is of interest and importance.

Another focus of this work is on quantum temporal steering (TS), which has attracted much attention recently. TS is a temporal analog of the Einstein–Podolsky–Rosen (EPR) steering, which is one of the most essential features in quantum mechanics. For a bipartite system in an entangled state, EPR steering problems refer to the quantum nonlocal correlations, which allow one of the subsystems to remotely prepare or steer the other one via local measurements. EPR steering is usually treated as an intermediate scenario, lying in between the entanglement and the Bell nonlocality. Recently, quantum steering problems have attracted considerable interest [2034].

Different from discussing the spatially separated systems, TS problems focus on a single system at different times [35]. In this frame, a system is sent to a distant receiver (say Bob) through a quantum channel. Then a detector or manipulator (say Alice) performs some operations (including measurements) before Bob receives the system and performs his measurement. The nonzero TS accounts for how strongly Alice’s choice of measurements at an initial time can influence the final state captured by Bob. In addition, TS also reveals a unique link between a quantum system’s past and future features. It should be pointed that quantum channel plays a particular role on TS rather than on EPR steering because the formation of the quantum correlation between the system’s initial and final state lies on the quantum channel.

Several kinds of inequalities were developed to detect TS, which have become useful tools in verifying the suitability of a quantum channel for a certain quantum key distribution process [3538]. In order to precisely quantify TS, a concept of temporal steerable weight was introduced in the literature [39], where the authors found that the TS characterized by the weight can be used to define a sufficient and practical measure of strong non-Markovianity. Moreover, it was found that TS is intrinsically associated with realism and joint measurability [40,41]. TS has also found an application in magnetoreception [42]. Quite recently, spatiotemporal steering, which generalizes the concepts of TS and EPR steering, has been applied for testing nonclassical correlations in quantum networks [43].

Motivated by the above, in this work we will first propose a linear optical setup to experimentally simulate a quantum channel, modeled by a two-level quantum system (i.e., qubit) interacting with a bosonic bath, without applying Born, Markov, rotating-wave, and perturbation approximations. The experimental parameters accounting for the dynamics of the qubit are set by means of the hierarchy equation method (see some details of the method in Supplement 1). To the best of our knowledge, this is the first experimental simulation of this kind of quantum channel in a linear optical setup. The proposed channel can present a more accurate picture of the open-system dynamics. It allows us not only to reflect the special roles of the counterrotating terms but also to consider the system–bath couplings in arbitrarily strong regimes. Therefore, it can play an important role in the experimental study of the quantum correlation whose dynamics are sensitive to the counterrotating terms.

With the proposed non-RWA channel, we then experimentally investigate the TS problems. We note that in the first experimental observation of TS [44], which enabled a determination of the TS weight, only a phenomenologically designed channel was considered. For such a channel, it is impossible to highlight the important role of the system–bath interaction. In contrast, the channel under our study, governed by a system–bath Hamiltonian without RWA, shows that the counterrotating terms significantly influence the TS dynamics. The TS in non-RWA and RWA channels presents different dynamics. However, the results without RWA are closer to realistic situations and thus more reliable.

2. SYSTEM–BATH MODEL

We consider a qubit system interacting with a bosonic bath, described by a full Hamiltonian:

H=HS+HB+HInt,
where HS=ω02σz is the free Hamiltonian of the qubit (assuming =1), with σz being the Pauli operator of the qubit and ω0 standing for the transition frequency between the two levels of the qubit; HB=kωkbkbk is the free Hamiltonian of the bosonic bath with bk and bk being the bosonic creation and annihilation operators of the kth mode of frequency ωk, respectively; and
HInt=kσx(gkbk+gk*bk)
is the interaction Hamiltonian between the qubit and the bath with gk being the coupling strength between the qubit and the kth mode of the bath. One essential aspect of our study is that the interaction Hamiltonian HInt is in a non-RWA form. Because of the difficulty in studying this kind of non-RWA interaction, previous studies have used a RWA treatment for simplicity by assuming the interaction Hamiltonian as
HIntRWA=k(gkσ+bk+gk*σbk),
where the effect of the counterrotating terms were omitted. Recently, it has been found that the counterrotating terms are necessary in order to accurately describe the spin–boson interaction [45] and using the conventional RWA approach may lead to faulty results [18,19].

Assume that the whole system is initially in the state ρTot(0)=ρS(0)ρB, where ρS(0) is the initial state of the qubit and chosen as a maximally mixed state ρS(0)=I/2. The bath is considered to be initially in a vacuum state ρB=|vacBBvac|, with |vacBk|0k. The system–bath coupling spectrum is assumed as a Lorentz-type

J(ω)=12πγλ2(ωω0)2+λ2,
where λ is the broadening width of the bosonic mode of frequency ω, which is connected to the bath correlation time τB=λ1. The relaxation time scale τS, on which the state of the system changes, is related to γ by τS=γ1. The γ partly reflects the system–bath coupling strength because one will obtain the effective coupling strength as geff2=12γλ after integrating the spectrum J(ω) over the entire region of ω.

The evolution under the total Hamiltonian in Eq. (1) can be translated into the language of quantum channel. Thus, the evolution map of the basis vectors can be described as

|gS|vacBp|gS|evenB,g+1p|eS|oddB,g,|eS|vacBq|eS|evenB,e+1q|gS|oddB,e,
where |gS (|eS) represents the ground (excited) state of the qubit, while the vector |evenB,i (|oddB,i) describes the evolved vector of the bath, which is actually a superposition of all the number states (e.g., k|nkk) with an even (odd) excitation number (i.e., knk is even or odd). Note that the subscript i(=e,g) of |evenB,i (|oddB,i) corresponds to the initial vector |iS|vacB. The vectors satisfy the orthogonality even|oddB,jB,i=0 (including i=j and ij). They also satisfy the normalizing condition even|evenB,iB,i=1 and odd|oddB,iB,i=1. The even|evenB,jB,i or odd|oddB,jB,i (ij) outputs a complex number. The probabilities p and q are time dependent, with p(t)[0,1] and q(t)[0,1]. Then the reduced density matrix of the qubit becomes
ρS(t)=[ρ11(t)ρ12(t)ρ12*(t)ρ22(t)],
where ρ11(t)=qρ11(0)+(1p)ρ22(0), ρ22(t)=(1q)ρ11(0)+pρ22(0), and ρ12(t)=ρ12(0)pqZ1(t)+ρ21(0)(1q)(1p)Z2(t). Here, ρi,j(0) (i,j=1,2) stands for the matrix elements of the qubit’s initial state, while Z1(t)B,geven|evenB,e and Z2(t)B,eodd|oddB,g are the time-dependent complex numbers.

3. EXPERIMENTAL SETUP AND SIMULATION OF THE NON-RWA CHANNEL

Our experimental setup is sketched in Fig. 1. In Fig. 1(a), pairs of photons with an 810 nm wavelength are produced by pumping a type-I beta-barium borate (BBO) crystal with ultraviolet pulses at a 405 nm centered wavelength. Then one photon is led into a state-preparation process. That is, the first polarized beam splitter (PBS1) selects the horizontally polarized state |H of the photon, and then a half-wave plate (HWP) and a quarter-wave plate (QWP) can rotate |H into one of the six eigenstates of the Pauli operators.

 figure: Fig. 1.

Fig. 1. Experimental setup and the stages of the experiment. (a) Photon pairs with an 810 nm wavelength are produced via spontaneous parametric downconversion. One of the two photons is used as the trigger for the coincident counts. The other photon is led to the preparation unit, consisting of a polarized beam splitter (PBS1), a half-wave plate (HWP), and a quarter-wave plate (QWP). In the TS problem, this photon is prepared into one of the six eigenstates of the Pauli operators σx,y,z. (b) Simulation of the quantum channel without RWA in Eq. (5). The angles of HWP1,2,3,4 are adjusted in [0,π/4], while the angles of HWP5,6,7,8 are set at π/4. Two Soleil–Babinet compensators (SBC1 and SBC2) add relative phases to the passing components H and V, respectively. The birefringent calcite beam displacers (BD1,2,3,4) couple the polarization states |H and |V with the spacial modes |ip (i=0,1,2,3). (c) Quantum state tomography is implemented by two QWPs, four HWPs, and two PBSs. Finally, two single-photon detectors equipped with two 10 nm interference filters (IFs) are used for the photon counting.

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Figure 1(b) accomplishes the task of the non-RWA quantum channel. We use the horizontal and vertical polarization modes |H and |V to encode the qubit’s basis states. The bath acts by a collective performance of four path modes |ip, with i{0,1,2,3}. In order to briefly introduce the implementation of the channel, let us start from the output of PBS2, where the H and V components are spatially separated so that each one can be rotated with the wave plate HWP1 by angle θ1[0,π/4] and the wave plate HWP2 by angle θ2[0,π/4]. After rotating by the wave plate HWP1, the |H mode becomes a superposition as |Hcos2θ1|H+sin2θ1|V. By transmitting it along a loop, the superposition state undergoes PBS2 again and couples with the spatial modes |0p and |1p encoded as the path numbers in Fig. 1(b), resulting in the state cos2θ1|H|0p+sin2θ1|V|1p.

It is worth noting that we make use of two Soleil–Babinet compensators (SBC1 and SBC2) in order to append a phase ϕi=ντi to the passing components H (V). Here, ν is the photon frequency and τi=Lknl/c is the traveling time of the photon across the SBC, where Lk (k=1,2) denotes the thickness of SBC1,2, nl (l=H,V) indicates the indices of refraction (corresponding to the H and V polarizations), and c is the vacuum speed of light. Hence, there are four phases ϕ1,2,3,4[0,2π] appended to the polarization states of photons passing SBC1 and SBC2. This innovative design enables us to realistically describe the varying phase of the off-diagonal elements of the system’s density matrix.

In Fig. 1(b), there are several birefringent calcite beam displacers (BD1,2,3,4) that deviate the H component and transmit the V one. Among them, we insert some wave plates to implement operations on the polarization states. The angles of HWP3,4 are adjusted as θ3[0,π/4] and θ4[0,π/4] to transform a single component (H or V) into a superposition form, while the angles of HWP5,6,7,8 are fixed at θ5,6,7,8=π/4 to convert H into V or vice versa. The input–output states of Fig. 1(b) are mapped as follows (see the details in Appendix A):

|H|0pcos2θ1eiϕ1|H|0p+sin2θ1eiϕ2|V|Ψ1,3,|V|0pcos2θ2eiϕ3|V|Ψ2,0sin2θ2eiϕ4|H|3p,
where |Ψ1,3cos2θ3|1psin2θ3|3p and |Ψ2,0cos2θ4|2psin2θ4|0p. By comparing Eq. (7) with Eq. (5), it can be seen that the reduced density operator of the system in Eq. (6) is successfully reproduced by setting the parameters q=cos22θ1, p=cos22θ2, Z1=sin(2θ4)ei(ϕ3ϕ1), and Z2=sin(2θ3)ei(ϕ2ϕ4). It is noted that according to the hierarchy equation method [5,9], the qubit state at an arbitrary time t can be simulated by adjusting the parameters θi and ϕi according to the theoretical results obtained by the hierarchy equation method.

In Fig. 1(c), the density matrix of the output state is reconstructed by the quantum tomography process where ten different coincidence measurement bases are set by QWPs, HWPs, and PBSs. Eight of the bases are set along paths 0 and 3, while the rest are set along paths 1 and 2. Finally, the photons are detected by single-photon detectors equipped with 10 nm interference filters. The extinction rate of our interferometer can reach over 300:1, and the average fidelity between the ideal states and actual states can reach higher than 99%.

Our proposed experimental setup (Fig. 1) can also be used to implement the channel governed by the RWA treatment of the interaction Hamiltonian in Eq. (3). Based on the evolution map in Eq. (7), one can set the HWP angles θ1=0, θ4=π/4, and adjust θ2[0,π/4] according to the theoretical results with RWA. Furthermore, the phases ϕ1,3 are adjusted in [0,2π] according to the theoretical results, and ϕ2,4 is set randomly. Then the evolution map in the Schrödinger picture reads as

|H|0peiϕ1|H|0p,|V|0pcos(2θ2)eiϕ3|V|0p+sin(2θ2)eiϕ4|H|3p,
where the spacial modes |0p and |3p correspond to the paths 0 and 3 in Fig. 1(b), respectively. If we further set the phases ϕ1,3,4=0 according to the interaction picture, Eq. (8) describes the so-called amplitude decay channel, which is just the channel implemented in Refs. [46,47]. The authors in Ref. [46] tested strong coupling cases in order to explore abundant non-Markovianity. However, it was theoretically predicated that the channels with or without RWA can cause quite different non-Markovian behavior, especially in strong-coupling cases [48]. From this point of view, considering the quantum channels without RWA is of importance and an interesting question.

Another well-known channel, the so-called phase damping channel considered in Ref. [47], is also easy to implement with the setup in Fig. 1. Based on Eq. (7), by setting θ1,2=0, ϕ1,2,3,4=0, and adjusting θ4[0,π/4], one can have the following evolution map:

|H|0p|H|0p,|V|0psin(2θ4)|V|0pcos(2θ4)|V|2p,
where the spacial modes |0p and |2p correspond to the paths 0 and 2 in Fig. 1(b), respectively.

4. TS INEQUALITY AND TS PARAMETER

In the TS problem, a system is sent to a distant receiver (say Bob) through a quantum channel. Before Bob receives the system, a detector or manipulator (say Alice) performs some measurements on the system. Then the TS problem refers to the characterization of the influence of Alice’s measurement at an initial time tA (let tA=0 in this paper) on the final state captured by Bob at a later time tB. The TS in the qubit systems can be detected by a concept called as TS parameter SN, which is defined in terms of a temporal analog of the steering inequality [35,37],

SNi=1NE(Bi,tBAi,tA2)1,
where Ai,tA (Bi,tB) stands for the ith observable measured by Alice (Bob) at tA (tB) and the number of observables is N=2 or 3. We have
E(Bi,tBAi,tA2)a=±1P(a|Ai,tA)Bi,tBAi,tA=a2,
with P(a|Ai,tA) being the probability of Alice’s measurement outcome a=+1 or 1. Bob’s expectation value, conditioned on Alice’s measurement outcome, is defined as
Bi,tBAi,tA=ab=±1bP(Bi,tB=b|Ai,tA=a),
where P(Bi,tB=b|Ai,tA=a) denotes the condition probability of Bob’s measurement outcome b (at tB) on the evolved state starting from the collapsed version after Alice’s measurement with the outcome a (at tA). In this paper, we take the case of N=2 into account. The violation of the inequality in Eq. (10), i.e., S2>1, is a sufficient condition for the steerability. Therefore, during an evolution, one can define the steerable durations conditioned by S2>1.

A. Experimental and Numerical Results of the TS Parameter

We consider that Alice chooses a pair of the Pauli operators {σi,σj}(i,j=x,y,z) as the observables (Ai) measured on the initial state ρS=I/2 of the qubit. After the measurement, the qubit state collapses to one of the six eigenstates of the Pauli operators with a probability P(a|Ai,tA)=1/2. This process is usually difficult to implement in experiments since it requires a set of nondestructive measurements. An equivalent way, adopted in this experiment, is to assume that Alice prepares qubit states by rotating the polarization mode |H into one of the six eigenstates of the Pauli operators, and correspondingly multiplies a probability of P(a|Ai,tA). This preparation is completed by sequentially using the PBS1, a HWP, and a QWP [see Fig. 1(a)]. Then the qubit in the prepared state is sent through the quantum channel [simulated in Fig. 1(b)] to Bob, who performs tomography measurements [Fig. 1(c)]. Therefore, Bob obtains the condition probabilities P(Bi,tB=b|Ai,tA=a) and calculates S2. Theoretically speaking, S2 is a function of the time t, the channel parameter γ, and the parameter λ. Actually, in our experiment, the dynamics of S2 are simulated by adjusting the angles of the HWPs and the thicknesses of the SBCs. The experimental errors are estimated from the statistical variation of photon counts, which satisfy the Poisson distribution.

Figure 2 shows S2 versus evolution time scaled by ω0. In Figs. 2(a) and 2(b), the measurement bases are |+ (|) and |0 (|1), which correspond to the eigenstates of σx and σz, respectively. Our experimental and theoretical results are in good agreement and show the oscillation of TS parameter S2 with time. More important is the steering limit, i.e., S2=1, which is marked by a red-dashed horizontal line in Fig. 2. Above this limit, steerability is valid. The vertical dashed lines highlight the steerable durations corresponding to S2>1. For the sake of comparison, we study two kinds of channels, i.e., the non-RWA channel and the RWA channel. The former is modeled by the Hamiltonian in Eq. (2) and experimentally simulated according to the evolution map in Eq. (7), while the latter is modeled by the Hamiltonian in Eq. (3) and experimentally simulated based on the evolution map in Eq. (8). In both of the cases of RWA and non-RWA channels, S2 is suppressed below the steering limit in most of the evolution periods due to the quantum decoherence effects. However, the difference between the two cases is obvious. There are more peaks of S2 over the steering limit in the RWA case [Fig. 2(b)] than the non-RWA case [Fig. 2(a)], which implies that the RWA channel appears to provide longer steerable durations. Similar conclusions can be made by comparing the results in Figs. 2(c) and 2(d), where another set of measurement bases are chosen, i.e., |+ (|) and |R (|L) (eigenstates of σx and σy, respectively). However, we should point out that the extra steerable durations in RWA cases are inauthentic, due to the essential defects in characterizing the system–bath interaction by using the RWA.

 figure: Fig. 2.

Fig. 2. TS parameter S2 versus scaled time ω0t in the non-RWA and RWA channels. Parts (a) and (b) correspond to the measuring bases |+(|) and |0(|1), which are the eigenstates of σx and σz, respectively. Parts (c) and (d) correspond to the measuring bases |+(|) and |R(|L), which are the eigenstates of σx and σy, respectively. The channel parameters, i.e., the system–bath coupling parameters γ=2.5ω0 and the broadening width of the bath mode λ=0.05ω0, which result in an effective strength of the system–bath coupling, i.e., geff=0.25ω0. Horizontal red dashed lines indicate the steering limit. Vertical dashed lines point out the steerable durations corresponding to S2>1. Inset: enlarged drawing with more data for the peaks of S2 close to or beyond the steering limit.

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5. TEMPORAL STEERABLE WEIGHT

Now let us briefly introduce the temporal steerable weight, described by WTS, which is a precise quantifier of TS. Alice measures the observable Ai on the system’s state at an initial time tA and gets the outcome a with a probability of P(a|Ai,tA). Assume that there are N observables, i.e., Ai with i=1,,N, and each of them is of m dimension (the case of m=2 is considered in this paper). After the measurement, the system’s state is mapped to ρa|Ai. Then, the system is sent to Bob through a quantum channel Λ. At time tB, Bob receives the system and performs tomography measurements to obtain the state ςa|Ai=Λ(ρa|Ai). In order to precisely quantify TS, a concept called TS weight, i.e., WTS, is introduced via a semidefinite program as [31,39]

WTS1maxTrλϱλ,
subject to
ς˜a|AiλDλ(a|Ai)ϱλ0,a,Ai,
and
ϱλ0,
where ς˜a|AiP(a|Ai,tA)ςa|Ai stands for the unnormalized states received by Bob. The task of Bob is to check whether the states he receives can be written in a hidden-state form, i.e., λDλ(a|Ai)ϱλ in Eq. (14), where λ(=1,,mN) indicates a classical random variable. ϱλ indicates a set of positive semidefinite matrices held by Bob, and Dλ(a|Ai) is the deterministic single-party conditional probability according to Alice’s measurement outcome [31,39]. Nonzero WTS implies that Bob cannot classically fabricate Alice’s measurement results, and thus the quantum temporal correlation (i.e., the TS) exists between Alice and Bob. Note that WTS comes from a sufficient and necessary characterization of steerability and quantifies the TS precisely.

A. Experimental Results of WTS

We also experimentally test WTS, as illustrated in Fig. 3. The experimental implementations in the input-state preparation (at Alice’s side) and the tomography measurement on the output states (at Bob’s side) are the same as those in Fig. 2. The non-RWA case [Figs. 3(a) and 3(c)] and the RWA case [Figs. 3(b) and 3(d)] are investigated. The parameters γ and λ are chosen the same as those in Fig. 2. Consequently, we compare the results shown in Fig. 3 with those in Fig. 2. Since WTS is defined according to the sufficient and necessary condition of the existence of TS, the data of WTS precisely tell us when the TS exists and disappears, especially for durations below the TS limit S2=1 (in Fig. 2), where the criterion S2 is disabled to detect TS.

 figure: Fig. 3.

Fig. 3. TS weight versus scaled time ω0t in the non-RWA and RWA channels. The measuring bases in (a) and (b) are |+(|) and |0(|1), which are the eigenstates of σx and σz, respectively. Parts (c) and (d) correspond to the measuring bases |+(|) and |R(|L), which are the eigenstates of σx and σy, respectively. The values of parameters γ and λ are chosen the same as those in Fig. 2.

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By comparing Fig. 3(a) with Fig. 3(b) [also comparing Fig. 3(c) with Fig. 3(d)], more interesting phenomena are found, i.e., “sudden death” and “revival” of TS in the non-RWA channel, whereas they never appear in the RWA channel where TS tends to zero asymptotically. We shall emphasize that the quantum correlation like TS inevitably undergoes a sudden change to zero rather than a gradual decrease, especially when the characterization of the system–bath interaction becomes close to the actual situation. This also reminds us of the previous famous report on the sudden death of entanglement [49]. Moreover, in the non-RWA channel, different choices of Alice’s measurement cause quite different behaviors of WTS, [comparing Fig. 3(a) with Fig. 3(c)]. However, in the RWA case, the behavior of WTS in Figs. 3(b) and 3(d) are similar, which cannot clearly reflect the influence of the different choice of Alice’s measurement.

6. CONCLUSION

With the proposed setup, we have experimentally implemented a non-RWA quantum channel and simulated the dynamics of a qubit system interacting with a bosonic bath, without applying Born, Markov and rotating-wave approximations. This kind of quantum channel provides a more realistic description of the environmental impact and reveals the special effects of the counterrotating terms.

Based on this channel, we have studied the TS problem experimentally. Our investigation shows that although the RWA channel seems to provide longer steerable durations (detected by the TS inequality) than the non-RWA case, the latter is closer to realistic situations and thus more reliable. The data of the TS weight show us some new interesting phenomena in the non-RWA channel, i.e., the “sudden death” and “revival” of the TS, which, however, do not appear in the RWA channel. Moreover, the influence of the measurement choice on the TS weight can only be clearly presented in the non-RWA channel.

Our experimental simulation exemplifies a test bed for the precise studies of open quantum systems. Moreover, the non-RWA single channel can be extended to a multichannel (i.e., multiqubit case). Our study will stimulate future reexamination of many previous studies on the quantum correlation dynamics and will motivate broad investigations on many interesting physical problems or phenomena associated with the non-RWA quantum channels.

APPENDIX A: IMPLEMENTATION OF THE EVOLUTION MAP IN EQ. (7)

|HPBS2,HWP1cos2θ1|H|2+sin2θ1|V|1SBC1,2cos2θ1eiϕ1|H|2+sin2θ1eiϕ2|V|1BD1,2cos2θ1eiϕ1|H|0+sin2θ1eiϕ2|V|1HWP5,3cos2θ1eiϕ1|V|0+sin2θ1eiϕ2(cos2θ3|Vsin2θ3|H)|1BD3,4cos2θ1eiϕ1|V|0+sin2θ1eiϕ2(cos2θ3|V|1sin2θ3|H|3)HWP7,8cos2θ1eiϕ1|H|0+sin2θ1eiϕ2(cos2θ3|V|1sin2θ3|V|3),
|VPBS2,HWP2cos2θ2|V|2sin2θ2|H|1SBC1,2cos2θ2eiϕ3|V|2sin2θ2eiϕ4|H|1BD1,2cos2θ2eiϕ3|V|2sin2θ2eiϕ4|H|3HWP6,4cos2θ2eiϕ3(cos2θ4|Vsin2θ4|H)|2sin2θ2eiϕ4|V|3BD3,4cos2θ2eiϕ3(cos2θ4|V|2sin2θ4|H|0)sin2θ2eiϕ4|V|3HWP7,8sin2θ2eiϕ4|H|3+cos2θ2eiϕ3(cos2θ4|V|2sin2θ4|V|0),

Funding

NKRDP of China (2016YFA0301802); National Natural Science Foundation of China (NSFC) (11375003, 11174081, 61472114, 11775065, 11774076, 11374083); Zhejiang Natural Science Foundation (LY17A050003, LZ13A040002); Natural Science Foundation of Shanghai (16ZR1448300); Program for HNUEYT (2011-01-011); Hangzhou City for the Hangzhou-City Quantum Information and Quantum Optics Innovation Research Team.

Acknowledgment

The authors thank Prof. Chuan-Feng Li for valuable suggestions and Dr. Kai Sun, Dr. Xiaoming Hu, and Dr. Zhiyuan Zhou for helpful discussions on the experimental implementation.

 

See Supplement 1 for supporting content.

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3. D. Kast and J. Ankerhold, “Persistence of coherent quantum dynamics at strong dissipation,” Phys. Rev. Lett. 110, 010402 (2013). [CrossRef]  

4. Y. Tanimura, “Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath,” Phys. Rev. A 41, 6676–6687 (1990). [CrossRef]  

5. M. Tanaka and Y. Tanimura, “Multistate electron transfer dynamics in the condensed phase: exact calculations from the reduced hierarchy equations of motion approach,” J. Chem. Phys. 132, 214502 (2010). [CrossRef]  

6. A. G. Dijkstra and Y. Tanimura, “Non-Markovian entanglement dynamics in the presence of system–bath coherence,” Phys. Rev. Lett. 104, 250401 (2010). [CrossRef]  

7. A. Ishizaki and Y. Tanimura, “Dynamics of a multimode system coupled to multiple heat baths probed by two-dimensional infrared spectroscopy,” J. Phys. Chem. A 111, 9269–9276 (2007). [CrossRef]  

8. J. S. Jin, X. Zheng, and Y. J. Yan, “Exact dynamics of dissipative electronic systems and quantum transport: hierarchical equations of motion approach,” J. Chem. Phys. 128, 234703 (2008). [CrossRef]  

9. J. Ma, Z. Sun, X. Wang, and F. Nori, “Entanglement dynamics of two qubits in a common bath,” Phys. Rev. A 85, 062323 (2012). [CrossRef]  

10. Z. Sun, L. W. Zhou, G. Y. Xiao, D. Poletti, and J. B. Gong, “Finite-time Landau–Zener processes and counterdiabatic driving in open systems: beyond Born, Markov, and rotating-wave approximations,” Phys. Rev. A 93, 012121 (2016). [CrossRef]  

11. M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316, 579–582 (2007). [CrossRef]  

12. J. S. Xu, C. F. Li, X. Y. Xu, C. H. Shi, X. B. Zou, and G. C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. 103, 240502 (2009). [CrossRef]  

13. J. S. Xu, X. Y. Xu, C. F. Li, C. J. Zhang, X. B. Zou, and G. C. Guo, “Experimental investigation of classical and quantum correlations under decoherence,” Nat. Commun. 1, 7 (2010). [CrossRef]  

14. J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. Lo Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013). [CrossRef]  

15. B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011). [CrossRef]  

16. L. Mancino, M. Sbroscia, I. Gianani, E. Roccia, and M. Barbieri, “Quantum simulation of single-qubit thermometry using linear optics,” Phys. Rev. Lett. 118, 130502 (2017). [CrossRef]  

17. J. S. Xu, M. H. Yung, X. Y. Xu, S. Boixo, Z. W. Zhou, C. F. Li, A. Aspuru-Guzik, and G. C. Guo, “Demon-like algorithmic quantum cooling and its realization with quantum optics,” Nat. Photonics 8, 113–118 (2014). [CrossRef]  

18. J. Larson, “Absence of vacuum induced berry phases without the rotating wave approximation in cavity QED,” Phys. Rev. Lett. 108, 033601 (2012). [CrossRef]  

19. Z. Sun, J. Ma, X. Wang, and F. Nori, “Photon-assisted Landau–Zener transition: role of coherent superposition states,” Phys. Rev. A 86, 012107 (2012). [CrossRef]  

20. H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007). [CrossRef]  

21. S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein–Podolsky–Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007). [CrossRef]  

22. M. Marciniak, A. Rutkowski, Z. Yin, M. Horodecki, and R. Horodecki, “Unbounded violation of quantum steering inequalities,” Phys. Rev. Lett. 115, 170401 (2015). [CrossRef]  

23. H. Zhu, M. Hayashi, and L. Chen, “Universal steering inequalities,” Phys. Rev. Lett. 116, 070403 (2016). [CrossRef]  

24. V. Handchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 598–601 (2012). [CrossRef]  

25. K. Sun, X. J. Ye, J. S. Xu, X. Y. Xu, J. S. Tang, Y. C. Wu, J. L. Chen, F. C. Li, and C. G. Guo, “Experimental quantification of asymmetric Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016). [CrossRef]  

26. S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015). [CrossRef]  

27. J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014). [CrossRef]  

28. Q. Y. He, L. Rosales-Zarate, G. Adesso, and M. D. Reid, “Secure continuous variable teleportation and Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 115, 180502 (2015). [CrossRef]  

29. A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007). [CrossRef]  

30. M. F. Pusey, “Negativity and steering: a stronger Peres conjecture,” Phys. Rev. A 88, 032313 (2013). [CrossRef]  

31. P. Skrzypczyk, M. Navascués, and D. Cavalcanti, “Quantifying Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 180404 (2014). [CrossRef]  

32. M. T. Quintino, T. Vértesi, and N. Brunner, “Joint measurability, Einstein–Podolsky–Rosen steering, and Bell nonlocality,” Phys. Rev. Lett. 113, 160402 (2014). [CrossRef]  

33. R. Uola, T. Moroder, and O. Gühne, “Joint measurability of generalized measurements implies classicality,” Phys. Rev. Lett. 113, 160403 (2014). [CrossRef]  

34. M. Piani and J. Watrous, “Necessary and sufficient quantum information characterization of Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 114, 060404 (2015). [CrossRef]  

35. Y. N. Chen, C. M. Li, N. Lambert, S. L. Chen, Y. Ota, G. Y. Chen, and F. Nori, “Temporal steering inequality,” Phys. Rev. A 89, 032112 (2014). [CrossRef]  

36. C. Emary, N. Lambert, and F. Nori, “Leggett–Garg inequalities,” Rep. Prog. Phys. 77, 016001 (2014). [CrossRef]  

37. K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Temporal steering and security of quantum key distribution with mutually unbiased bases against individual attacks,” Phys. Rev. A 93, 062345 (2016). [CrossRef]  

38. M. Tomamichel and R. Renner, “Uncertainty relation for smooth entropies,” Phys. Rev. Lett. 106, 110506 (2011). [CrossRef]  

39. S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-Markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016). [CrossRef]  

40. C. M. Li, Y. N. Chen, N. Lambert, C. Y. Chiu, and F. Nori, “Certifying single-system steering for quantum-information processing,” Phys. Rev. A 92, 062310 (2015). [CrossRef]  

41. H. S. Karthik, J. Prabhu Tej, A. R. Usha Devi, and A. K. Rajagopal, “Joint measurability and temporal steering,” J. Opt. Soc. Am. B 32, A34–A39 (2015). [CrossRef]  

42. H. Y. Ku, S. L. Chen, H. B. Chen, N. Lambert, Y. N. Chen, and F. Nori, “Temporal steering in four dimensions with applications to coupled qubits and magnetoreception,” Phys. Rev. A 94, 062126 (2016). [CrossRef]  

43. S. L. Chen, N. Lambert, C. M. Li, G. Y. Chen, Y. N. Chen, A. Miranowicz, and F. Nori, “Spatio-temporal steering for testing nonclassical correlations in quantum networks,” Sci. Rep. 7, 3728 (2017). [CrossRef]  

44. K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Experimental temporal quantum steering,” Sci. Rep. 6, 38076 (2016). [CrossRef]  

45. V. V. Albert, “Quantum Rabi model for n-state atoms,” Phys. Rev. Lett. 108, 180401 (2012). [CrossRef]  

46. F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014). [CrossRef]  

47. O. J. Farí, G. H. Aguilar, A. Valdés-Hernández, P. H. Ribeiro, L. Davidovich, and S. P. Walborn, “Observation of the emergence of multipartite entanglement between a bipartite system and its environment,” Phys. Rev. Lett. 109, 150403 (2012). [CrossRef]  

48. Z. Sun, J. Liu, J. Ma, and X. Wang, “Quantum speed limits in open systems: non-Markovian dynamics without rotating-wave approximation,” Sci. Rep. 5, 8444 (2015). [CrossRef]  

49. T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004). [CrossRef]  

References

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  1. I. de Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems,” Rev. Mod. Phys. 89, 015001 (2017).
    [Crossref]
  2. W. M. Zhang, P. Y. Lo, H. N. Xiong, M. W.-Y. Tu, and F. Nori, “General non-Markovian dynamics of open quantum systems,” Phys. Rev. Lett. 109, 170402 (2012).
    [Crossref]
  3. D. Kast and J. Ankerhold, “Persistence of coherent quantum dynamics at strong dissipation,” Phys. Rev. Lett. 110, 010402 (2013).
    [Crossref]
  4. Y. Tanimura, “Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath,” Phys. Rev. A 41, 6676–6687 (1990).
    [Crossref]
  5. M. Tanaka and Y. Tanimura, “Multistate electron transfer dynamics in the condensed phase: exact calculations from the reduced hierarchy equations of motion approach,” J. Chem. Phys. 132, 214502 (2010).
    [Crossref]
  6. A. G. Dijkstra and Y. Tanimura, “Non-Markovian entanglement dynamics in the presence of system–bath coherence,” Phys. Rev. Lett. 104, 250401 (2010).
    [Crossref]
  7. A. Ishizaki and Y. Tanimura, “Dynamics of a multimode system coupled to multiple heat baths probed by two-dimensional infrared spectroscopy,” J. Phys. Chem. A 111, 9269–9276 (2007).
    [Crossref]
  8. J. S. Jin, X. Zheng, and Y. J. Yan, “Exact dynamics of dissipative electronic systems and quantum transport: hierarchical equations of motion approach,” J. Chem. Phys. 128, 234703 (2008).
    [Crossref]
  9. J. Ma, Z. Sun, X. Wang, and F. Nori, “Entanglement dynamics of two qubits in a common bath,” Phys. Rev. A 85, 062323 (2012).
    [Crossref]
  10. Z. Sun, L. W. Zhou, G. Y. Xiao, D. Poletti, and J. B. Gong, “Finite-time Landau–Zener processes and counterdiabatic driving in open systems: beyond Born, Markov, and rotating-wave approximations,” Phys. Rev. A 93, 012121 (2016).
    [Crossref]
  11. M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316, 579–582 (2007).
    [Crossref]
  12. J. S. Xu, C. F. Li, X. Y. Xu, C. H. Shi, X. B. Zou, and G. C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. 103, 240502 (2009).
    [Crossref]
  13. J. S. Xu, X. Y. Xu, C. F. Li, C. J. Zhang, X. B. Zou, and G. C. Guo, “Experimental investigation of classical and quantum correlations under decoherence,” Nat. Commun. 1, 7 (2010).
    [Crossref]
  14. J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. Lo Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
    [Crossref]
  15. B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
    [Crossref]
  16. L. Mancino, M. Sbroscia, I. Gianani, E. Roccia, and M. Barbieri, “Quantum simulation of single-qubit thermometry using linear optics,” Phys. Rev. Lett. 118, 130502 (2017).
    [Crossref]
  17. J. S. Xu, M. H. Yung, X. Y. Xu, S. Boixo, Z. W. Zhou, C. F. Li, A. Aspuru-Guzik, and G. C. Guo, “Demon-like algorithmic quantum cooling and its realization with quantum optics,” Nat. Photonics 8, 113–118 (2014).
    [Crossref]
  18. J. Larson, “Absence of vacuum induced berry phases without the rotating wave approximation in cavity QED,” Phys. Rev. Lett. 108, 033601 (2012).
    [Crossref]
  19. Z. Sun, J. Ma, X. Wang, and F. Nori, “Photon-assisted Landau–Zener transition: role of coherent superposition states,” Phys. Rev. A 86, 012107 (2012).
    [Crossref]
  20. H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
    [Crossref]
  21. S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein–Podolsky–Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007).
    [Crossref]
  22. M. Marciniak, A. Rutkowski, Z. Yin, M. Horodecki, and R. Horodecki, “Unbounded violation of quantum steering inequalities,” Phys. Rev. Lett. 115, 170401 (2015).
    [Crossref]
  23. H. Zhu, M. Hayashi, and L. Chen, “Universal steering inequalities,” Phys. Rev. Lett. 116, 070403 (2016).
    [Crossref]
  24. V. Handchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 598–601 (2012).
    [Crossref]
  25. K. Sun, X. J. Ye, J. S. Xu, X. Y. Xu, J. S. Tang, Y. C. Wu, J. L. Chen, F. C. Li, and C. G. Guo, “Experimental quantification of asymmetric Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
    [Crossref]
  26. S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015).
    [Crossref]
  27. J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
    [Crossref]
  28. Q. Y. He, L. Rosales-Zarate, G. Adesso, and M. D. Reid, “Secure continuous variable teleportation and Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 115, 180502 (2015).
    [Crossref]
  29. A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
    [Crossref]
  30. M. F. Pusey, “Negativity and steering: a stronger Peres conjecture,” Phys. Rev. A 88, 032313 (2013).
    [Crossref]
  31. P. Skrzypczyk, M. Navascués, and D. Cavalcanti, “Quantifying Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 180404 (2014).
    [Crossref]
  32. M. T. Quintino, T. Vértesi, and N. Brunner, “Joint measurability, Einstein–Podolsky–Rosen steering, and Bell nonlocality,” Phys. Rev. Lett. 113, 160402 (2014).
    [Crossref]
  33. R. Uola, T. Moroder, and O. Gühne, “Joint measurability of generalized measurements implies classicality,” Phys. Rev. Lett. 113, 160403 (2014).
    [Crossref]
  34. M. Piani and J. Watrous, “Necessary and sufficient quantum information characterization of Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 114, 060404 (2015).
    [Crossref]
  35. Y. N. Chen, C. M. Li, N. Lambert, S. L. Chen, Y. Ota, G. Y. Chen, and F. Nori, “Temporal steering inequality,” Phys. Rev. A 89, 032112 (2014).
    [Crossref]
  36. C. Emary, N. Lambert, and F. Nori, “Leggett–Garg inequalities,” Rep. Prog. Phys. 77, 016001 (2014).
    [Crossref]
  37. K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Temporal steering and security of quantum key distribution with mutually unbiased bases against individual attacks,” Phys. Rev. A 93, 062345 (2016).
    [Crossref]
  38. M. Tomamichel and R. Renner, “Uncertainty relation for smooth entropies,” Phys. Rev. Lett. 106, 110506 (2011).
    [Crossref]
  39. S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-Markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016).
    [Crossref]
  40. C. M. Li, Y. N. Chen, N. Lambert, C. Y. Chiu, and F. Nori, “Certifying single-system steering for quantum-information processing,” Phys. Rev. A 92, 062310 (2015).
    [Crossref]
  41. H. S. Karthik, J. Prabhu Tej, A. R. Usha Devi, and A. K. Rajagopal, “Joint measurability and temporal steering,” J. Opt. Soc. Am. B 32, A34–A39 (2015).
    [Crossref]
  42. H. Y. Ku, S. L. Chen, H. B. Chen, N. Lambert, Y. N. Chen, and F. Nori, “Temporal steering in four dimensions with applications to coupled qubits and magnetoreception,” Phys. Rev. A 94, 062126 (2016).
    [Crossref]
  43. S. L. Chen, N. Lambert, C. M. Li, G. Y. Chen, Y. N. Chen, A. Miranowicz, and F. Nori, “Spatio-temporal steering for testing nonclassical correlations in quantum networks,” Sci. Rep. 7, 3728 (2017).
    [Crossref]
  44. K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Experimental temporal quantum steering,” Sci. Rep. 6, 38076 (2016).
    [Crossref]
  45. V. V. Albert, “Quantum Rabi model for n-state atoms,” Phys. Rev. Lett. 108, 180401 (2012).
    [Crossref]
  46. F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014).
    [Crossref]
  47. O. J. Farí, G. H. Aguilar, A. Valdés-Hernández, P. H. Ribeiro, L. Davidovich, and S. P. Walborn, “Observation of the emergence of multipartite entanglement between a bipartite system and its environment,” Phys. Rev. Lett. 109, 150403 (2012).
    [Crossref]
  48. Z. Sun, J. Liu, J. Ma, and X. Wang, “Quantum speed limits in open systems: non-Markovian dynamics without rotating-wave approximation,” Sci. Rep. 5, 8444 (2015).
    [Crossref]
  49. T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004).
    [Crossref]

2017 (3)

I. de Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems,” Rev. Mod. Phys. 89, 015001 (2017).
[Crossref]

L. Mancino, M. Sbroscia, I. Gianani, E. Roccia, and M. Barbieri, “Quantum simulation of single-qubit thermometry using linear optics,” Phys. Rev. Lett. 118, 130502 (2017).
[Crossref]

S. L. Chen, N. Lambert, C. M. Li, G. Y. Chen, Y. N. Chen, A. Miranowicz, and F. Nori, “Spatio-temporal steering for testing nonclassical correlations in quantum networks,” Sci. Rep. 7, 3728 (2017).
[Crossref]

2016 (7)

K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Experimental temporal quantum steering,” Sci. Rep. 6, 38076 (2016).
[Crossref]

S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-Markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016).
[Crossref]

H. Y. Ku, S. L. Chen, H. B. Chen, N. Lambert, Y. N. Chen, and F. Nori, “Temporal steering in four dimensions with applications to coupled qubits and magnetoreception,” Phys. Rev. A 94, 062126 (2016).
[Crossref]

Z. Sun, L. W. Zhou, G. Y. Xiao, D. Poletti, and J. B. Gong, “Finite-time Landau–Zener processes and counterdiabatic driving in open systems: beyond Born, Markov, and rotating-wave approximations,” Phys. Rev. A 93, 012121 (2016).
[Crossref]

H. Zhu, M. Hayashi, and L. Chen, “Universal steering inequalities,” Phys. Rev. Lett. 116, 070403 (2016).
[Crossref]

K. Sun, X. J. Ye, J. S. Xu, X. Y. Xu, J. S. Tang, Y. C. Wu, J. L. Chen, F. C. Li, and C. G. Guo, “Experimental quantification of asymmetric Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref]

K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Temporal steering and security of quantum key distribution with mutually unbiased bases against individual attacks,” Phys. Rev. A 93, 062345 (2016).
[Crossref]

2015 (7)

M. Piani and J. Watrous, “Necessary and sufficient quantum information characterization of Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 114, 060404 (2015).
[Crossref]

M. Marciniak, A. Rutkowski, Z. Yin, M. Horodecki, and R. Horodecki, “Unbounded violation of quantum steering inequalities,” Phys. Rev. Lett. 115, 170401 (2015).
[Crossref]

S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015).
[Crossref]

Q. Y. He, L. Rosales-Zarate, G. Adesso, and M. D. Reid, “Secure continuous variable teleportation and Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 115, 180502 (2015).
[Crossref]

C. M. Li, Y. N. Chen, N. Lambert, C. Y. Chiu, and F. Nori, “Certifying single-system steering for quantum-information processing,” Phys. Rev. A 92, 062310 (2015).
[Crossref]

H. S. Karthik, J. Prabhu Tej, A. R. Usha Devi, and A. K. Rajagopal, “Joint measurability and temporal steering,” J. Opt. Soc. Am. B 32, A34–A39 (2015).
[Crossref]

Z. Sun, J. Liu, J. Ma, and X. Wang, “Quantum speed limits in open systems: non-Markovian dynamics without rotating-wave approximation,” Sci. Rep. 5, 8444 (2015).
[Crossref]

2014 (8)

F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014).
[Crossref]

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

P. Skrzypczyk, M. Navascués, and D. Cavalcanti, “Quantifying Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 180404 (2014).
[Crossref]

M. T. Quintino, T. Vértesi, and N. Brunner, “Joint measurability, Einstein–Podolsky–Rosen steering, and Bell nonlocality,” Phys. Rev. Lett. 113, 160402 (2014).
[Crossref]

R. Uola, T. Moroder, and O. Gühne, “Joint measurability of generalized measurements implies classicality,” Phys. Rev. Lett. 113, 160403 (2014).
[Crossref]

Y. N. Chen, C. M. Li, N. Lambert, S. L. Chen, Y. Ota, G. Y. Chen, and F. Nori, “Temporal steering inequality,” Phys. Rev. A 89, 032112 (2014).
[Crossref]

C. Emary, N. Lambert, and F. Nori, “Leggett–Garg inequalities,” Rep. Prog. Phys. 77, 016001 (2014).
[Crossref]

J. S. Xu, M. H. Yung, X. Y. Xu, S. Boixo, Z. W. Zhou, C. F. Li, A. Aspuru-Guzik, and G. C. Guo, “Demon-like algorithmic quantum cooling and its realization with quantum optics,” Nat. Photonics 8, 113–118 (2014).
[Crossref]

2013 (3)

D. Kast and J. Ankerhold, “Persistence of coherent quantum dynamics at strong dissipation,” Phys. Rev. Lett. 110, 010402 (2013).
[Crossref]

M. F. Pusey, “Negativity and steering: a stronger Peres conjecture,” Phys. Rev. A 88, 032313 (2013).
[Crossref]

J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. Lo Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
[Crossref]

2012 (7)

V. Handchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 598–601 (2012).
[Crossref]

W. M. Zhang, P. Y. Lo, H. N. Xiong, M. W.-Y. Tu, and F. Nori, “General non-Markovian dynamics of open quantum systems,” Phys. Rev. Lett. 109, 170402 (2012).
[Crossref]

J. Ma, Z. Sun, X. Wang, and F. Nori, “Entanglement dynamics of two qubits in a common bath,” Phys. Rev. A 85, 062323 (2012).
[Crossref]

J. Larson, “Absence of vacuum induced berry phases without the rotating wave approximation in cavity QED,” Phys. Rev. Lett. 108, 033601 (2012).
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Z. Sun, J. Ma, X. Wang, and F. Nori, “Photon-assisted Landau–Zener transition: role of coherent superposition states,” Phys. Rev. A 86, 012107 (2012).
[Crossref]

O. J. Farí, G. H. Aguilar, A. Valdés-Hernández, P. H. Ribeiro, L. Davidovich, and S. P. Walborn, “Observation of the emergence of multipartite entanglement between a bipartite system and its environment,” Phys. Rev. Lett. 109, 150403 (2012).
[Crossref]

V. V. Albert, “Quantum Rabi model for n-state atoms,” Phys. Rev. Lett. 108, 180401 (2012).
[Crossref]

2011 (2)

B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
[Crossref]

M. Tomamichel and R. Renner, “Uncertainty relation for smooth entropies,” Phys. Rev. Lett. 106, 110506 (2011).
[Crossref]

2010 (3)

J. S. Xu, X. Y. Xu, C. F. Li, C. J. Zhang, X. B. Zou, and G. C. Guo, “Experimental investigation of classical and quantum correlations under decoherence,” Nat. Commun. 1, 7 (2010).
[Crossref]

M. Tanaka and Y. Tanimura, “Multistate electron transfer dynamics in the condensed phase: exact calculations from the reduced hierarchy equations of motion approach,” J. Chem. Phys. 132, 214502 (2010).
[Crossref]

A. G. Dijkstra and Y. Tanimura, “Non-Markovian entanglement dynamics in the presence of system–bath coherence,” Phys. Rev. Lett. 104, 250401 (2010).
[Crossref]

2009 (1)

J. S. Xu, C. F. Li, X. Y. Xu, C. H. Shi, X. B. Zou, and G. C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. 103, 240502 (2009).
[Crossref]

2008 (1)

J. S. Jin, X. Zheng, and Y. J. Yan, “Exact dynamics of dissipative electronic systems and quantum transport: hierarchical equations of motion approach,” J. Chem. Phys. 128, 234703 (2008).
[Crossref]

2007 (5)

A. Ishizaki and Y. Tanimura, “Dynamics of a multimode system coupled to multiple heat baths probed by two-dimensional infrared spectroscopy,” J. Phys. Chem. A 111, 9269–9276 (2007).
[Crossref]

M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316, 579–582 (2007).
[Crossref]

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref]

S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein–Podolsky–Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007).
[Crossref]

A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

2004 (1)

T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004).
[Crossref]

1990 (1)

Y. Tanimura, “Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath,” Phys. Rev. A 41, 6676–6687 (1990).
[Crossref]

Acin, A.

A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

Adesso, G.

Q. Y. He, L. Rosales-Zarate, G. Adesso, and M. D. Reid, “Secure continuous variable teleportation and Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 115, 180502 (2015).
[Crossref]

Aguilar, G. G.

F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014).
[Crossref]

Aguilar, G. H.

O. J. Farí, G. H. Aguilar, A. Valdés-Hernández, P. H. Ribeiro, L. Davidovich, and S. P. Walborn, “Observation of the emergence of multipartite entanglement between a bipartite system and its environment,” Phys. Rev. Lett. 109, 150403 (2012).
[Crossref]

Albert, V. V.

V. V. Albert, “Quantum Rabi model for n-state atoms,” Phys. Rev. Lett. 108, 180401 (2012).
[Crossref]

Almeida, M. P.

M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316, 579–582 (2007).
[Crossref]

Alonso, D.

I. de Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems,” Rev. Mod. Phys. 89, 015001 (2017).
[Crossref]

Andersson, E.

J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. Lo Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
[Crossref]

Ankerhold, J.

D. Kast and J. Ankerhold, “Persistence of coherent quantum dynamics at strong dissipation,” Phys. Rev. Lett. 110, 010402 (2013).
[Crossref]

Armstrong, S.

S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015).
[Crossref]

Aspuru-Guzik, A.

J. S. Xu, M. H. Yung, X. Y. Xu, S. Boixo, Z. W. Zhou, C. F. Li, A. Aspuru-Guzik, and G. C. Guo, “Demon-like algorithmic quantum cooling and its realization with quantum optics,” Nat. Photonics 8, 113–118 (2014).
[Crossref]

Bachor, H. A.

S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015).
[Crossref]

Barbieri, M.

L. Mancino, M. Sbroscia, I. Gianani, E. Roccia, and M. Barbieri, “Quantum simulation of single-qubit thermometry using linear optics,” Phys. Rev. Lett. 118, 130502 (2017).
[Crossref]

Bartkiewicz, K.

K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Experimental temporal quantum steering,” Sci. Rep. 6, 38076 (2016).
[Crossref]

K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Temporal steering and security of quantum key distribution with mutually unbiased bases against individual attacks,” Phys. Rev. A 93, 062345 (2016).
[Crossref]

Boixo, S.

J. S. Xu, M. H. Yung, X. Y. Xu, S. Boixo, Z. W. Zhou, C. F. Li, A. Aspuru-Guzik, and G. C. Guo, “Demon-like algorithmic quantum cooling and its realization with quantum optics,” Nat. Photonics 8, 113–118 (2014).
[Crossref]

Bowles, J.

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

Breuer, H. P.

B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
[Crossref]

Brunner, N.

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

M. T. Quintino, T. Vértesi, and N. Brunner, “Joint measurability, Einstein–Podolsky–Rosen steering, and Bell nonlocality,” Phys. Rev. Lett. 113, 160402 (2014).
[Crossref]

A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

Cakmak, B.

F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014).
[Crossref]

Castelano, L. K.

F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014).
[Crossref]

Cavalcanti, D.

P. Skrzypczyk, M. Navascués, and D. Cavalcanti, “Quantifying Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 180404 (2014).
[Crossref]

Cernoch, A.

K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Experimental temporal quantum steering,” Sci. Rep. 6, 38076 (2016).
[Crossref]

K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Temporal steering and security of quantum key distribution with mutually unbiased bases against individual attacks,” Phys. Rev. A 93, 062345 (2016).
[Crossref]

Chen, G. Y.

S. L. Chen, N. Lambert, C. M. Li, G. Y. Chen, Y. N. Chen, A. Miranowicz, and F. Nori, “Spatio-temporal steering for testing nonclassical correlations in quantum networks,” Sci. Rep. 7, 3728 (2017).
[Crossref]

Y. N. Chen, C. M. Li, N. Lambert, S. L. Chen, Y. Ota, G. Y. Chen, and F. Nori, “Temporal steering inequality,” Phys. Rev. A 89, 032112 (2014).
[Crossref]

Chen, H. B.

H. Y. Ku, S. L. Chen, H. B. Chen, N. Lambert, Y. N. Chen, and F. Nori, “Temporal steering in four dimensions with applications to coupled qubits and magnetoreception,” Phys. Rev. A 94, 062126 (2016).
[Crossref]

Chen, J. L.

K. Sun, X. J. Ye, J. S. Xu, X. Y. Xu, J. S. Tang, Y. C. Wu, J. L. Chen, F. C. Li, and C. G. Guo, “Experimental quantification of asymmetric Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref]

Chen, L.

H. Zhu, M. Hayashi, and L. Chen, “Universal steering inequalities,” Phys. Rev. Lett. 116, 070403 (2016).
[Crossref]

Chen, S. L.

S. L. Chen, N. Lambert, C. M. Li, G. Y. Chen, Y. N. Chen, A. Miranowicz, and F. Nori, “Spatio-temporal steering for testing nonclassical correlations in quantum networks,” Sci. Rep. 7, 3728 (2017).
[Crossref]

H. Y. Ku, S. L. Chen, H. B. Chen, N. Lambert, Y. N. Chen, and F. Nori, “Temporal steering in four dimensions with applications to coupled qubits and magnetoreception,” Phys. Rev. A 94, 062126 (2016).
[Crossref]

S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-Markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016).
[Crossref]

Y. N. Chen, C. M. Li, N. Lambert, S. L. Chen, Y. Ota, G. Y. Chen, and F. Nori, “Temporal steering inequality,” Phys. Rev. A 89, 032112 (2014).
[Crossref]

Chen, Y. N.

S. L. Chen, N. Lambert, C. M. Li, G. Y. Chen, Y. N. Chen, A. Miranowicz, and F. Nori, “Spatio-temporal steering for testing nonclassical correlations in quantum networks,” Sci. Rep. 7, 3728 (2017).
[Crossref]

H. Y. Ku, S. L. Chen, H. B. Chen, N. Lambert, Y. N. Chen, and F. Nori, “Temporal steering in four dimensions with applications to coupled qubits and magnetoreception,” Phys. Rev. A 94, 062126 (2016).
[Crossref]

S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-Markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016).
[Crossref]

C. M. Li, Y. N. Chen, N. Lambert, C. Y. Chiu, and F. Nori, “Certifying single-system steering for quantum-information processing,” Phys. Rev. A 92, 062310 (2015).
[Crossref]

Y. N. Chen, C. M. Li, N. Lambert, S. L. Chen, Y. Ota, G. Y. Chen, and F. Nori, “Temporal steering inequality,” Phys. Rev. A 89, 032112 (2014).
[Crossref]

Chiu, C. Y.

C. M. Li, Y. N. Chen, N. Lambert, C. Y. Chiu, and F. Nori, “Certifying single-system steering for quantum-information processing,” Phys. Rev. A 92, 062310 (2015).
[Crossref]

Compagno, G.

J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. Lo Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
[Crossref]

Davidovich, L.

O. J. Farí, G. H. Aguilar, A. Valdés-Hernández, P. H. Ribeiro, L. Davidovich, and S. P. Walborn, “Observation of the emergence of multipartite entanglement between a bipartite system and its environment,” Phys. Rev. Lett. 109, 150403 (2012).
[Crossref]

M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316, 579–582 (2007).
[Crossref]

de Melo, F.

M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316, 579–582 (2007).
[Crossref]

de Oliveira, M. C.

F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014).
[Crossref]

de Vega, I.

I. de Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems,” Rev. Mod. Phys. 89, 015001 (2017).
[Crossref]

Dijkstra, A. G.

A. G. Dijkstra and Y. Tanimura, “Non-Markovian entanglement dynamics in the presence of system–bath coherence,” Phys. Rev. Lett. 104, 250401 (2010).
[Crossref]

Doherty, A. C.

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref]

S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein–Podolsky–Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007).
[Crossref]

Eberle, T.

V. Handchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 598–601 (2012).
[Crossref]

Eberly, J. H.

T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004).
[Crossref]

Emary, C.

C. Emary, N. Lambert, and F. Nori, “Leggett–Garg inequalities,” Rep. Prog. Phys. 77, 016001 (2014).
[Crossref]

Fanchini, F. F.

F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014).
[Crossref]

Farí, O. J.

O. J. Farí, G. H. Aguilar, A. Valdés-Hernández, P. H. Ribeiro, L. Davidovich, and S. P. Walborn, “Observation of the emergence of multipartite entanglement between a bipartite system and its environment,” Phys. Rev. Lett. 109, 150403 (2012).
[Crossref]

Franz, T.

V. Handchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 598–601 (2012).
[Crossref]

Gianani, I.

L. Mancino, M. Sbroscia, I. Gianani, E. Roccia, and M. Barbieri, “Quantum simulation of single-qubit thermometry using linear optics,” Phys. Rev. Lett. 118, 130502 (2017).
[Crossref]

Gisin, N.

A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

Gong, J. B.

Z. Sun, L. W. Zhou, G. Y. Xiao, D. Poletti, and J. B. Gong, “Finite-time Landau–Zener processes and counterdiabatic driving in open systems: beyond Born, Markov, and rotating-wave approximations,” Phys. Rev. A 93, 012121 (2016).
[Crossref]

Gong, Q. H.

S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015).
[Crossref]

Gühne, O.

R. Uola, T. Moroder, and O. Gühne, “Joint measurability of generalized measurements implies classicality,” Phys. Rev. Lett. 113, 160403 (2014).
[Crossref]

Guo, C. G.

K. Sun, X. J. Ye, J. S. Xu, X. Y. Xu, J. S. Tang, Y. C. Wu, J. L. Chen, F. C. Li, and C. G. Guo, “Experimental quantification of asymmetric Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref]

Guo, G. C.

J. S. Xu, M. H. Yung, X. Y. Xu, S. Boixo, Z. W. Zhou, C. F. Li, A. Aspuru-Guzik, and G. C. Guo, “Demon-like algorithmic quantum cooling and its realization with quantum optics,” Nat. Photonics 8, 113–118 (2014).
[Crossref]

J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. Lo Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
[Crossref]

B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
[Crossref]

J. S. Xu, X. Y. Xu, C. F. Li, C. J. Zhang, X. B. Zou, and G. C. Guo, “Experimental investigation of classical and quantum correlations under decoherence,” Nat. Commun. 1, 7 (2010).
[Crossref]

J. S. Xu, C. F. Li, X. Y. Xu, C. H. Shi, X. B. Zou, and G. C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. 103, 240502 (2009).
[Crossref]

Handchen, V.

V. Handchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 598–601 (2012).
[Crossref]

Hayashi, M.

H. Zhu, M. Hayashi, and L. Chen, “Universal steering inequalities,” Phys. Rev. Lett. 116, 070403 (2016).
[Crossref]

He, Q. Y.

S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015).
[Crossref]

Q. Y. He, L. Rosales-Zarate, G. Adesso, and M. D. Reid, “Secure continuous variable teleportation and Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 115, 180502 (2015).
[Crossref]

Hor-Meyll, M.

M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316, 579–582 (2007).
[Crossref]

Horodecki, M.

M. Marciniak, A. Rutkowski, Z. Yin, M. Horodecki, and R. Horodecki, “Unbounded violation of quantum steering inequalities,” Phys. Rev. Lett. 115, 170401 (2015).
[Crossref]

Horodecki, R.

M. Marciniak, A. Rutkowski, Z. Yin, M. Horodecki, and R. Horodecki, “Unbounded violation of quantum steering inequalities,” Phys. Rev. Lett. 115, 170401 (2015).
[Crossref]

Huang, Y. F.

B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
[Crossref]

Ishizaki, A.

A. Ishizaki and Y. Tanimura, “Dynamics of a multimode system coupled to multiple heat baths probed by two-dimensional infrared spectroscopy,” J. Phys. Chem. A 111, 9269–9276 (2007).
[Crossref]

Janousek, J.

S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015).
[Crossref]

Jiménez Farías, O.

F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014).
[Crossref]

Jin, J. S.

J. S. Jin, X. Zheng, and Y. J. Yan, “Exact dynamics of dissipative electronic systems and quantum transport: hierarchical equations of motion approach,” J. Chem. Phys. 128, 234703 (2008).
[Crossref]

Jones, S. J.

S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein–Podolsky–Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007).
[Crossref]

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref]

Karpat, G.

F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014).
[Crossref]

Karthik, H. S.

Kast, D.

D. Kast and J. Ankerhold, “Persistence of coherent quantum dynamics at strong dissipation,” Phys. Rev. Lett. 110, 010402 (2013).
[Crossref]

Ku, H. Y.

H. Y. Ku, S. L. Chen, H. B. Chen, N. Lambert, Y. N. Chen, and F. Nori, “Temporal steering in four dimensions with applications to coupled qubits and magnetoreception,” Phys. Rev. A 94, 062126 (2016).
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Laine, E. M.

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S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-Markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016).
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[Crossref]

S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-Markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016).
[Crossref]

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Supplementary Material (1)

NameDescription
» Supplement 1       Brief introduction of the hierarchy equation method

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Figures (3)

Fig. 1.
Fig. 1. Experimental setup and the stages of the experiment. (a) Photon pairs with an 810 nm wavelength are produced via spontaneous parametric downconversion. One of the two photons is used as the trigger for the coincident counts. The other photon is led to the preparation unit, consisting of a polarized beam splitter (PBS1), a half-wave plate (HWP), and a quarter-wave plate (QWP). In the TS problem, this photon is prepared into one of the six eigenstates of the Pauli operators σx,y,z. (b) Simulation of the quantum channel without RWA in Eq. (5). The angles of HWP1,2,3,4 are adjusted in [0,π/4], while the angles of HWP5,6,7,8 are set at π/4. Two Soleil–Babinet compensators (SBC1 and SBC2) add relative phases to the passing components H and V, respectively. The birefringent calcite beam displacers (BD1,2,3,4) couple the polarization states |H and |V with the spacial modes |ip (i=0,1,2,3). (c) Quantum state tomography is implemented by two QWPs, four HWPs, and two PBSs. Finally, two single-photon detectors equipped with two 10 nm interference filters (IFs) are used for the photon counting.
Fig. 2.
Fig. 2. TS parameter S2 versus scaled time ω0t in the non-RWA and RWA channels. Parts (a) and (b) correspond to the measuring bases |+(|) and |0(|1), which are the eigenstates of σx and σz, respectively. Parts (c) and (d) correspond to the measuring bases |+(|) and |R(|L), which are the eigenstates of σx and σy, respectively. The channel parameters, i.e., the system–bath coupling parameters γ=2.5ω0 and the broadening width of the bath mode λ=0.05ω0, which result in an effective strength of the system–bath coupling, i.e., geff=0.25ω0. Horizontal red dashed lines indicate the steering limit. Vertical dashed lines point out the steerable durations corresponding to S2>1. Inset: enlarged drawing with more data for the peaks of S2 close to or beyond the steering limit.
Fig. 3.
Fig. 3. TS weight versus scaled time ω0t in the non-RWA and RWA channels. The measuring bases in (a) and (b) are |+(|) and |0(|1), which are the eigenstates of σx and σz, respectively. Parts (c) and (d) correspond to the measuring bases |+(|) and |R(|L), which are the eigenstates of σx and σy, respectively. The values of parameters γ and λ are chosen the same as those in Fig. 2.

Equations (17)

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H=HS+HB+HInt,
HInt=kσx(gkbk+gk*bk)
HIntRWA=k(gkσ+bk+gk*σbk),
J(ω)=12πγλ2(ωω0)2+λ2,
|gS|vacBp|gS|evenB,g+1p|eS|oddB,g,|eS|vacBq|eS|evenB,e+1q|gS|oddB,e,
ρS(t)=[ρ11(t)ρ12(t)ρ12*(t)ρ22(t)],
|H|0pcos2θ1eiϕ1|H|0p+sin2θ1eiϕ2|V|Ψ1,3,|V|0pcos2θ2eiϕ3|V|Ψ2,0sin2θ2eiϕ4|H|3p,
|H|0peiϕ1|H|0p,|V|0pcos(2θ2)eiϕ3|V|0p+sin(2θ2)eiϕ4|H|3p,
|H|0p|H|0p,|V|0psin(2θ4)|V|0pcos(2θ4)|V|2p,
SNi=1NE(Bi,tBAi,tA2)1,
E(Bi,tBAi,tA2)a=±1P(a|Ai,tA)Bi,tBAi,tA=a2,
Bi,tBAi,tA=ab=±1bP(Bi,tB=b|Ai,tA=a),
WTS1maxTrλϱλ,
ς˜a|AiλDλ(a|Ai)ϱλ0,a,Ai,
ϱλ0,
|HPBS2,HWP1cos2θ1|H|2+sin2θ1|V|1SBC1,2cos2θ1eiϕ1|H|2+sin2θ1eiϕ2|V|1BD1,2cos2θ1eiϕ1|H|0+sin2θ1eiϕ2|V|1HWP5,3cos2θ1eiϕ1|V|0+sin2θ1eiϕ2(cos2θ3|Vsin2θ3|H)|1BD3,4cos2θ1eiϕ1|V|0+sin2θ1eiϕ2(cos2θ3|V|1sin2θ3|H|3)HWP7,8cos2θ1eiϕ1|H|0+sin2θ1eiϕ2(cos2θ3|V|1sin2θ3|V|3),
|VPBS2,HWP2cos2θ2|V|2sin2θ2|H|1SBC1,2cos2θ2eiϕ3|V|2sin2θ2eiϕ4|H|1BD1,2cos2θ2eiϕ3|V|2sin2θ2eiϕ4|H|3HWP6,4cos2θ2eiϕ3(cos2θ4|Vsin2θ4|H)|2sin2θ2eiϕ4|V|3BD3,4cos2θ2eiϕ3(cos2θ4|V|2sin2θ4|H|0)sin2θ2eiϕ4|V|3HWP7,8sin2θ2eiϕ4|H|3+cos2θ2eiϕ3(cos2θ4|V|2sin2θ4|V|0),

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