Abstract
We present a new optical scheme enabling the implementation of highly stable and configurable nonMarkovian dynamics. Here one photon qubit can circulate in a multipass bulk geometry consisting of two concatenated Sagnac interferometers to simulate the so called collisional model, where the system interacts at discrete times with a vacuum environment. We show the optical features of our apparatus and three different implementations of it, replicating a pure Markovian scenario and two nonMarkovian ones, where we quantify the information backflow by tracking the evolution of the initial entanglement between the system photon and an ancillary one.
Introduction
Precise control of quantum states is a crucial requirement for future quantum technologies^{1,2}. Their processing protocols should preserve and distribute microscopic correlations in macroscopic scenarios, where countable quantum systems are subjected to environmental noise. It is essential in this context to understand how much robust are the possible quantum dynamical processes and the best way to control the information permeability between the systems and their environment^{3,4,5,6,7,8}.
Quantum dynamical processes do not act merely on the sample system, actually they act in an extended Hilbert space where system and its surrounding environment are in contact^{9,10}. The nonisolated sample system s is called open quantum system (OQS), and is characterized by a state \({\rho }_{s}\in { {\mathcal H} }_{s}\). Similarly, the environment e is characterized by a state \({\rho }_{e}\in { {\mathcal H} }_{e}\). Without loss of generality, one can assume that the extended system s − e that lives in \( {\mathcal H} ={ {\mathcal H} }_{s}\otimes { {\mathcal H} }_{e}\) is closed, then no information can be lost but only distributed inside \( {\mathcal H} \)^{11,12}.
The dynamics of an OQS are called Markovian if each continuous or discrete section of the total evolution is independent of the previous ones, otherwise they are called nonMarkovian^{13}. In the quantum scenario, three different approaches are widely used to quantify the degree of nonMarkovianity of a process^{14,15}. The first method is based on the presence of information backflow towards the system from the environment, that acts in this case as a reservoir of information^{16,17,18}. In the OQS framework the total systemenvironment state \({\rho }_{s,e}\in {\mathcal H} \) evolves according to a quantum process generating a communication link between \({ {\mathcal H} }_{s}\) and \({ {\mathcal H} }_{e}\). Here the strength of the flow of information between system and environment during their interaction can be used to discriminate the level of nonMarkovianity of the process. The second approach studies the divisibility of the process in Completely Positive (CP) maps, defining the evolution as nonMarkovian if this decomposition fails at some time^{14,19}. In cases were this CP divisibility is valid, also master equations can be well defined^{20}. The third method, which has been used in this work, studies the evolution of the entanglement between the system and an isolated ancilla and it is strictly related to the two approaches mentioned above. In order to explain it we refer to the next section and to^{14,21}.
If the environment is represented by an ensemble of spaces \({ {\mathcal H} }_{e}={ {\mathcal H} }_{{e}_{1}}\otimes \ldots \otimes { {\mathcal H} }_{{e}_{k}}\), and the system space \({ {\mathcal H} }_{s}\) interacts sequentially with each of them at discrete times, we obtain the so called collisional model (CM)^{22,23,24,25,26}. It represents a powerful tool to approximate continuoustime quantum dynamics and to analyze nonMarkovian dynamics of OQSs^{27,28,29,30,31,32}. Linear optics platforms have been thoroughly analyzed for the implementation of CMs^{33,34}. A simple and effective implementation has been proposed by some of us^{35}. There the authors consider an initial photon state \({\rho }_{s}({t}_{0})\in { {\mathcal H} }_{s}\), whose spatial mode collides sequentially with the modes of an environment ensemble, which can be considered as a double space environment \({ {\mathcal H} }_{e}={ {\mathcal H} }_{{e}_{1}}\otimes { {\mathcal H} }_{{e}_{2}}\) with one subspace always prepared in a certain generic state \({\rho }_{{e}_{2}}\in { {\mathcal H} }_{{e}_{2}}\) at any stepk of the process. The evolution of \({\rho }_{s}({t}_{k})\) is mainly controlled by the interaction involving the Hilbert spaces \({ {\mathcal H} }_{s}\) and \({ {\mathcal H} }_{{e}_{1}}\), while the memory effects are due to the interenvironment collisions written in \({ {\mathcal H} }_{{e}_{1}}\) and \({ {\mathcal H} }_{{e}_{2}}\), which also produces an effective evolution in \({\rho }_{{e}_{1}}({t}_{k})\in { {\mathcal H} }_{{e}_{1}}\).
The optical implementation described in^{35} is realized by a sequence of MachZehnder interferometers (MZIs) as shown in Fig. 1, where the continuous trajectory is associated to \({\rho }_{s}\), the segmented trajectories to \({\rho }_{{e}_{1}}\) and the dotted trajectory to \({\rho }_{{e}_{2}}\). At the stepk of this process \({\rho }_{s}({t}_{k1})\) interferes with \({\rho }_{{e}_{1}}({t}_{k1})\) in the beam splitter BS_{1}, while the interenvironment collision with \({\rho }_{{e}_{2}}\) occurs in BS_{2}, which posses a variable reflectivity \({R}_{BS}\in [0,1]\) to control the environment memory.
Theoretical Model
In our proposal (shown in Fig. 2a) we consider that all BS_{2} have reflectivity \({r}_{2}=1\), so that they can be substituted with perfectly reflective mirrors (M). Here the continuous trajectories correspond to the system (smode) while the segmented ones correspond to the first environment subspace (e_{1}mode), as in Fig. 1. However, the second environment subspace (e_{2}mode) has no defined path (not present in Fig. 2), since it represents the “absorption environment” after the action of a polarization independent neutral filter F_{k} placed in the e_{1}mode. As seen in Fig. 2b, the superoperator process \(\varepsilon ({t}_{k},{t}_{k1})\) is composed by a quarter wave plate (QWP) in the smode and a half wave plate (HWP) in the e_{1}mode, both at fixed rotation angle \(\varphi =0\). The environment memory is controlled by the transmissivity factor \({T}_{k}\in [0,1]\) of F_{k}, which gives access to the vacuum state stored by \({\rho }_{{e}_{2}}=0\rangle \,\langle 0\), hence effectively mimicking the interaction with the dottedlines of Fig. 1. The phase factor θ_{k} mediates collision mechanism by controlling the optical interference. Accordingly, in this setting a purely Markovian dynamics corresponds to the minimum information backflow from \({ {\mathcal H} }_{{e}_{1}}\) to \({ {\mathcal H} }_{s}\), which is achieved by the maximum loss of information from \({ {\mathcal H} }_{{e}_{1}}\) to \({ {\mathcal H} }_{{e}_{2}}\) (\({T}_{k}=0\)). NonMarkovian dynamics instead can arise whenever using \({T}_{k}\ne 0\). Let us suppose that the smode is initially prepared in a maximally entangled state with an external ancillary system (amode) as \({{\rm{\Psi }}}_{a,s}^{\pm }\rangle =\frac{1}{\sqrt{2}}{(H\rangle }_{a}V{\rangle }_{s}\pm V{\rangle }_{a}H{\rangle }_{s})\), where \(H\rangle \) (\(V\rangle \)) represents the horizontal (vertical) polarization of a photon qubit. Since both emodes are initialized in a vacuum state \(0\rangle \), the actual complete initial state corresponds to \({\rho }_{a,s,{e}_{1},{e}_{2}}({t}_{0})={{\rm{\Psi }}}^{\pm }\rangle \,\langle {{\rm{\Psi }}}^{\pm }\), with
where \({\hat{a}}_{x}^{\dagger }\) are the photon creation operators on each xmode. It is worth stressing that due to the possibility of loosing the sphoton during the propagation after its interaction with the e_{2}mode, our scheme effectively describes the evolution of a qutrit system (with canonical basis given by the states \({1}_{h}{\rangle }_{s}\), \({1}_{v}{\rangle }_{s}\) and \(0{\rangle }_{s}\)), where information is only stored in the bidimensional subspace associated with onephoton sector.
In our prepared scenario the systemenvironment interactions are controlled by a series of operations, such as the BS one,
with \(1\rangle =(\alpha {1}_{h}\rangle +\beta {1}_{v}\rangle )/\sqrt{\alpha {}^{2}+\beta {}^{2}}\) and r as its reflectivity factor. The wave plates act according to
with \({\sigma }^{z}={1}_{h}\rangle \,\langle {1}_{h}{1}_{v}\rangle \,\langle {1}_{v}+0\rangle \,\langle 0\), \({\sigma }^{z/2}={1}_{h}\rangle \,\langle {1}_{h}+i{1}_{v}\rangle \,\langle {1}_{v}+0\rangle \,\langle 0\) and \({\mathbb{I}}={1}_{h}\rangle \,\langle {1}_{h}+{1}_{v}\rangle \,\langle {1}_{v}+0\rangle \,\langle 0\).
The attenuation operation applied by the filter F connects the environment space of the remaining light (\({ {\mathcal H} }_{{e}_{1}}\)) with the space of the absorbed light (\({ {\mathcal H} }_{{e}_{2}}\)) according to
which generates the effective interenvironment collisions that can reset the e_{1}mode to the vacuum state depending on the absorption factor 1 − T. Finally the phase control acts as
Then, the superoperator can be written as follows:
According to our CM represented in Fig. 2, the input state \({\rho }_{a,s,{e}_{1},{e}_{2}}({t}_{0})\) evolves as
at the first step of the evolution. For consecutive steps, the process can be repeated with variations on \(\varepsilon ({t}_{k},{t}_{k1})\) or by using the same operation. Finally, one can extract the ancillasystem state as \({\rho }_{a,s}({t}_{k})=T{r}_{{e}_{1},{e}_{2}}[{\rho }_{a,s,{e}_{1},{e}_{2}}({t}_{k})]\) or the ancillaenvironment state \({\rho }_{a,{e}_{1}}({t}_{k})=T{r}_{s,{e}_{2}}[{\rho }_{a,s,{e}_{1},{e}_{2}}({t}_{k})]\) by tracing out the undesired spaces and measuring bipartite tomographies after the action of the k single step process.
A characterization of the nonMarkovianity of the process can then be obtained by studying the evolution of the concurrence C_{a,s} between the ancilla a and the system s at the various steps of the interferometric propagation. From the results of^{14,36} we know in fact that in the cases where the relation \({C}_{a,s}({t}_{k}) > {C}_{a,s}({t}_{k1})\) holds for some \(k > 1\), a backflow of information from e_{1} to s has occurred, resulting in a clear indication of a nonMarkovian character of the system dynamics. On the contrary a null increase of C_{a,s}(t_{k}) cannot be used as an indication of Markovianity.
The magnitude of all information backflows between two steps of the evolution gauges the degree of nonMarkovianity, which can be estimated by considering the integral of the concurrence variation^{37,38}, over the time intervals in which it increases, i.e. the quantity
As already mentioned our system s is intrinsically 3dimensional. Accordingly the C_{a,s} appearing in Eq. 8 should be the qutrits concurrence^{38} instead of the standard qubit one^{37}. However, for the sake of simplicity, in the experimental implementation which we present in the following sections, we shall restrict the analysis only to the entanglement between the singlephoton sectors of s and a, by property postselecting our data. Accordingly our measurements do not complete capture the full nonMarkovian character implicit in Eq. 8.
Experimental Implementation
The experimental setup is based on two concatenated bulk optics Sagnac interferometers (SIs) as described in Fig. 3a). They are initially prepared in a collinear configuration, that by applying the displacement of a mirror in SI_{1} is transformed in a displaced multipass scheme that replicates the CM of Fig. 2. Here we exploit a geometry endowed with high intrinsic phase stability, where different BSs (present in the scheme of Fig. 1) are substituted by different transversal points on a single BS. In this scheme the odd steps circulate in SI_{1}, while the even ones circulate in SI_{2}. The configuration is equivalent to the model of Fig. 2 since we can choose the smodes and the e_{1}modes as the clockwise and counterclockwise trajectories inside each SI, respectively. For the sake of simplicity, from now on we will use the label “eenvironment” only for the non absorbed space of the environment, because its complementary part cannot be measured in our configuration.
The relative phase factors θ_{k} are implemented by a fixed glass plate intersecting all the emodes inside each SI, while thin glass plates are placed in every smode and tilted independently (see Fig. 3b). The transmissivity factors \({T}_{k}=\frac{{T}_{k}^{e}}{{T}_{k}^{s}}\) are implemented by a single neutral density filter F^{e} with transmissivity T^{e} that intersects all the emodes inside each SI, while another filter F^{s} with transmissivity T^{s} intersects all the smodes for timecompensation between both optical paths. In this configuration both filters introduce only a controlled absorption, that represents an intrinsic degree of Markovianity under any kind of regime. Even so, the se absorptions can be mapped by the relative absorption factor T_{k}. Analogously, a single QWP intersects all smodes of each SI, while a single HWP intersects all the emodes. Since smode and emode contain the same kind of optical elements, we ensure temporally compensated trajectories with an uncertainty of <30 μm per step. The superposition of the 2^{k} trajectories at step k is collected by a singlemode optical fiber (SMF) after the tomography stage of se modes. Analogously, another SMF collects the external amode.
In this work we focus our attention on the case where all the steps are identical, namely by using a unique filtering factor \({T}_{k}=F\) and phase factor \({\theta }_{k}=\theta \). This regime can be described by
and corresponds to the case of a stroboscopic evolution (SE)^{35}.
The entangled state \({\rho }_{a,s}({t}_{0})={{\rm{\Psi }}}_{a,s}^{\pm }\rangle \,\langle {{\rm{\Psi }}}_{a,s}^{\pm }\) is prepared by two indistinguishable processes of TypeII spontaneous parametric down conversion (SPDC) inside a high brilliance, high purity Sagnac source based on a periodicallypoled KTP (PPKTP) nonlinear crystal^{39}. Here a singlemode continuouswave laser at 405 nm is converted into pairs of photons with orthogonal polarizations at 810 nm of wavelength and 0.42 nm of linewidth (measured by techniques described in^{40}). One photon is injected in the smode of the setup, while the other travels through the external amode. Finally we reconstruct the postselected state associated to the singlephoton sectors of the density matrices \({\rho }_{a,s}({t}_{k})\) or \({\rho }_{a,e}({t}_{k})\) by bipartite hypercomplete tomographies between their associated modes (see Fig. 2b).
In Fig. 4 we show a simulation of the possible nonMarkovian dynamics under the SE with maximum environment memory (\(T=1\)) and variable phase factor \(\theta \). These predicted scenarios were obtained by considering an ideal Bell input state \({{\rm{\Psi }}}^{\pm }\rangle \) and ideal optical elements, e.g. symmetric BS and nolosses elements. They are interesting to understand and identify the flows and backflows of information, which can be used in the analysis of engineered se couplings and its permeability or temporally localized communications for noise avoidance. Besides the se collision, the emode also suffers interenvironment collisions with the absorption space of the environment. Thus, there is a complex information exchange where it is difficult to identify particular correlations exclusively between both as and ae concurrence behaviours.
In Fig. 5 we show a comparison between three SEs considering ideal optical elements, the actual experimental input state \({\rm{\Psi }}{\rangle }_{exp}\), a phase factor \(\theta =\pi /2\) and different degrees of memory T. In the case \(T=1\) it results a fast entanglement fluctuation with a nonMarkovian degree of \({\mathscr{N}}=0.475\) up to the sixth step. In the case \(T=1/4\) one obtains a slower entanglement fluctuation that gives \({\mathscr{N}}=0.185\), while in the case \(T=1/16\) it emerges an even slower fluctuation with \({\mathscr{N}}=0.005\) (all values of \({\mathscr{N}}\) reported here are computed on the postselected singlephoton sectors).
Experimental Results
The experimental test was restricted to the case of a SE with \(\theta =0\) as seen in the lightblue lines of Fig. 4a,b, but considering real optical elements. The prepared entangled state \({{\rm{\Omega }}}_{a,s}\) showed a measured Fidelity \(F=\langle {{\rm{\Psi }}}_{a,s}^{\pm }{{\rm{\Omega }}}_{a,s}{{\rm{\Psi }}}_{a,s}^{\pm }\rangle =0.9712\pm 0.0004\), then the simulated data for the imperfect evolution referred to a Werner input mixed state^{41,42,43} \({{\rm{\Omega }}}_{a,s}=\frac{4F1}{3}{{\rm{\Psi }}}_{a,s}^{\pm }\rangle \,\langle {{\rm{\Psi }}}_{a,s}^{\pm }+\frac{1F}{3}{{\mathbb{I}}}_{a}\otimes {{\mathbb{I}}}_{s}\).
In Fig. 6 we present the concurrence fluctuations of the singlephoton sectors expressed in the postselected density matrices \({\rho }_{a,s}({t}_{k})\) and \({\rho }_{a,e}({t}_{k})\) during the SE. These states are reconstructed by normalizing the remaining non absorbed coincident photons, and by consequence the associated concurrence values become invariant under losses. Nevertheless, both dynamics behave according to the simulation for the Wernerlike input state \({{\rm{\Omega }}}_{a,s}\). In the case \(T=1\) of Fig. 6a we obtained the highest possible nonMarkovianity, where the large concurrence fluctuations give us \({\mathscr{N}}=0.3232\). As seen in Fig. 7, in the case \(T=0.209\) we obtained reduced concurrence revivals and nonMarkovianity of \({\mathscr{N}}=0.1442\), while in the case \(T=0\) we confirmed the lowest possible nonmarkovianity by obtaining a near to zero value on \({\mathscr{N}}=0.0044\).
For our particular CM, these results confirm that entanglement revivals are strictly connected to the environment memory. In fact, they show with high precision that decreasing values on T reduce the information backflows to the smode. The slight deviation from the expected theoretical simulations originates from the not perfect superposition of all possible photon trajectories. Even so, this error is strongly minimized by the use of SMFs as final spatial filters.
Conclusions
In this work we presented a linear optics setup that allows to simulate different open quantum systems dynamics. It is based on a novel interferometric structure that guarantees high phase stability and a multipass evolution in a compact setup, that makes possible to study the dynamics up to 6 steps at least. The dynamics studied here represents the first implementation of the socalled collisional model for open quantum systems^{35}, and our results correspond to a particular case of it. The setup is able to simulate a wide variety of stroboscopic evolutions, from strictly Markovian all the way up to strongly nonMarkovian dynamics, where quantum memory effects show their contribution. We can experimentally track the role of systemenvironment and intraenvironment interactions in the arising of nonMarkovian features and characterize the transition between the two regimes. As the field of quantum technologies spreads, more and more attention has being addressed to the study of nonMarkovian dynamics. It can, in principle, be used for efficient information processing^{44,45,46,47,48}, as well as for engineering novel interesting quantum states^{49,50,51}. In this perspective, our scheme can be of great interest, thanks to its stability, modular nature and direct access to the environmental degrees of freedom.
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Acknowledgements
We acknowledge support from the European Commission grants FP7ICT20119600838 (QWAD  Quantum Waveguides Application and Development) and H2020FETPROACT2014 (QUCHIP  Quantum Simulation on a Photonic Chip). We thank partial support from the Chilean agency Comisión Nacional de Investigación Científica y Tecnológica (CONICYT) and its Ph.D. scholarships “Becas Chile”.
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V.G. and P.M. proposed the theoretical frame and the optical scheme presented in Fig. 2, Á.C. proposed and coordinated the experimental multipass implementation, A.G., C.L. and L.D.B. achieved and analysed the experimental measures, A.D.P. and F.S. contributed to the interpretation of results. All authors contributed to the writing of the manuscript.
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Correspondence to Álvaro Cuevas or Vittorio Giovannetti or Paolo Mataloni.
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