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
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) . 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. [4–6], 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 [11–14]. An open system driven from the Markovian to the non-Markovian regime was experimentally simulated . A single-qubit thermometry was realized through a simulation of the system–bath interaction . A universal cooling method, which is applicable to any physical system, was demonstrated experimentally . 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 [20–34].
Different from discussing the spatially separated systems, TS problems focus on a single system at different times . 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 [35–38]. In order to precisely quantify TS, a concept of temporal steerable weight was introduced in the literature , 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 . Quite recently, spatiotemporal steering, which generalizes the concepts of TS and EPR steering, has been applied for testing nonclassical correlations in quantum networks .
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 , 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:45] and using the conventional RWA approach may lead to faulty results [18,19].
Assume that the whole system is initially in the state , where is the initial state of the qubit and chosen as a maximally mixed state . The bath is considered to be initially in a vacuum state , with . The system–bath coupling spectrum is assumed as a Lorentz-type
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
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 () selects the horizontally polarized state of the photon, and then a half-wave plate (HWP) and a quarter-wave plate (QWP) can rotate into one of the six eigenstates of the Pauli operators.
Figure 1(b) accomplishes the task of the non-RWA quantum channel. We use the horizontal and vertical polarization modes and to encode the qubit’s basis states. The bath acts by a collective performance of four path modes , with . In order to briefly introduce the implementation of the channel, let us start from the output of , where the and components are spatially separated so that each one can be rotated with the wave plate by angle and the wave plate by angle . After rotating by the wave plate , the mode becomes a superposition as . By transmitting it along a loop, the superposition state undergoes again and couples with the spatial modes and encoded as the path numbers in Fig. 1(b), resulting in the state .
It is worth noting that we make use of two Soleil–Babinet compensators ( and ) in order to append a phase to the passing components (). Here, is the photon frequency and is the traveling time of the photon across the SBC, where () denotes the thickness of , () indicates the indices of refraction (corresponding to the and polarizations), and is the vacuum speed of light. Hence, there are four phases appended to the polarization states of photons passing and . 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 () that deviate the component and transmit the one. Among them, we insert some wave plates to implement operations on the polarization states. The angles of are adjusted as and to transform a single component ( or ) into a superposition form, while the angles of are fixed at to convert into or vice versa. The input–output states of Fig. 1(b) are mapped as follows (see the details in Appendix A):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 , , , and . It is noted that according to the hierarchy equation method [5,9], the qubit state at an arbitrary time can be simulated by adjusting the parameters and 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 , , and adjust according to the theoretical results with RWA. Furthermore, the phases are adjusted in according to the theoretical results, and is set randomly. Then the evolution map in the Schrödinger picture reads as1(b), respectively. If we further set the phases 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.  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 . 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. , is also easy to implement with the setup in Fig. 1. Based on Eq. (7), by setting , , and adjusting , one can have the following evolution map: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 (let in this paper) on the final state captured by Bob at a later time . The TS in the qubit systems can be detected by a concept called as TS parameter , which is defined in terms of a temporal analog of the steering inequality [35,37],10), i.e., , is a sufficient condition for the steerability. Therefore, during an evolution, one can define the steerable durations conditioned by .
A. Experimental and Numerical Results of the TS Parameter
We consider that Alice chooses a pair of the Pauli operators as the observables () measured on the initial state of the qubit. After the measurement, the qubit state collapses to one of the six eigenstates of the Pauli operators with a probability . 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 into one of the six eigenstates of the Pauli operators, and correspondingly multiplies a probability of . This preparation is completed by sequentially using the , 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 and calculates . Theoretically speaking, is a function of the time , the channel parameter , and the parameter . Actually, in our experiment, the dynamics of 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 versus evolution time scaled by . In Figs. 2(a) and 2(b), the measurement bases are () and (), which correspond to the eigenstates of and , respectively. Our experimental and theoretical results are in good agreement and show the oscillation of TS parameter with time. More important is the steering limit, i.e., , 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 . 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, 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 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 () (eigenstates of and , 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.
5. TEMPORAL STEERABLE WEIGHT
Now let us briefly introduce the temporal steerable weight, described by , which is a precise quantifier of TS. Alice measures the observable on the system’s state at an initial time and gets the outcome with a probability of . Assume that there are observables, i.e., with , and each of them is of dimension (the case of is considered in this paper). After the measurement, the system’s state is mapped to . Then, the system is sent to Bob through a quantum channel . At time , Bob receives the system and performs tomography measurements to obtain the state . In order to precisely quantify TS, a concept called TS weight, i.e., , is introduced via a semidefinite program as [31,39]14), where indicates a classical random variable. indicates a set of positive semidefinite matrices held by Bob, and is the deterministic single-party conditional probability according to Alice’s measurement outcome [31,39]. Nonzero 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 comes from a sufficient and necessary characterization of steerability and quantifies the TS precisely.
A. Experimental Results of
We also experimentally test , 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 and are chosen the same as those in Fig. 2. Consequently, we compare the results shown in Fig. 3 with those in Fig. 2. Since is defined according to the sufficient and necessary condition of the existence of TS, the data of precisely tell us when the TS exists and disappears, especially for durations below the TS limit (in Fig. 2), where the criterion is disabled to detect TS.
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 . Moreover, in the non-RWA channel, different choices of Alice’s measurement cause quite different behaviors of , [comparing Fig. 3(a) with Fig. 3(c)]. However, in the RWA case, the behavior of in Figs. 3(b) and 3(d) are similar, which cannot clearly reflect the influence of the different choice of Alice’s measurement.
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)
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.
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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