Abstract
The simulation of open quantum dynamics has recently allowed the direct investigation of the features of systemenvironment interaction and of their consequences on the evolution of a quantum system. Such interaction threatens the quantum properties of the system, spoiling them and causing the phenomenon of decoherence. Sometimes however a coherent exchange of information takes place between system and environment, memory effects arise and the dynamics of the system becomes nonMarkovian. Here we report the experimental realisation of a nonMarkovian process where system and environment are coupled through a simulated transverse Ising model. By engineering the evolution in a photonic quantum simulator, we demonstrate the role played by systemenvironment correlations in the emergence of memory effects.
Introduction
The simulation of quantum processes is a key goal for the grand programme aiming at grounding quantum technologies as the way to explore complex phenomena that are inaccessible through standard, classical calculators^{1,2}. Some interesting steps have been performed in this direction: simple condensed matter and chemical processes have been implemented on controllable quantum simulators^{3,4}. The relativistic motion and scattering of a particle in the presence of a linear potential has been demonstrated in a trappedion quantum simulator^{5,6,7} that opens up the possibility to the study of quantum field theories^{8,9}. The quantum Ising model^{10} has been experimentally analysed under the perspective of universal digital quantum simulators^{11} and very recently scaled up to hundreds of particles^{12}.
This scenario has recently been extended to open quantum evolutions^{13}, marking the possibility to investigate important features of the way a quantum system interacts with its environment, including the socalled sudden death of entanglement induced by a memoryless environment^{14}.
This interaction can destroy the most genuine quantum properties of the system, or involve exchange of coherence between system and environment, giving rise to memory effects and thus making the dynamics “nonMarkovian”^{15}. Characterizing nonMarkovian evolutions is currently at the centre of extensive theoretical and experimental efforts^{16,17,18}. Here we demonstrate experimentally the (non)Markovianity of a process where system and environment are coupled through a simulated transverse Ising model^{10} (see Supplementary Material). By engineering the evolution in a fully controlled photonic quantum simulator, we assess and demonstrate the role that systemenvironment correlations have in the emergence of memory effects.
The paradigmatic description of quantum open dynamics involves a physical system S evolving freely according to the Hamiltonian H^{S} and embedded in an environment E (whose free dynamics is ruled by H^{E}). System and environment interact via the Hamiltonian H^{SE} ^{15}, which we assume to be timeindependent for easiness of description. While the dynamics of the joint state ρ_{SE} is closed and governed by the unitary operator , the state of S, which is typically the only object to be accessible directly, is given by the reduced density matrix ρ_{S}(t) = Tr_{E}{ρ_{SE}(t)}. For factorized initial states ρ_{SE}(0) = ρ_{S}(0) ρ_{E}(0) and moving to an interaction frame defined by the free Hamiltonian of the total system, such reduced evolution can be recast in the operatorsum picture , where {K_{μ}} is the set of tracepreseving, nonunitary Kraus operators of S that are responsible for effects such as the loss of populations and coherence from the state of the system^{19}.
The roots of such decoherence mechanism have long been studied, together with their intimate connection to the socalled measurement problem and the implications of the collapse of the wave function^{20,21,22}. Numerous experimentally oriented techniques have been proposed to counteract decoherence^{19,23,24}. Yet, a complementary viewpoint can be taken, where the possibility to engineering structured environments and tailored SE couplings is seen as a resource to achieve longer coherence times of the system^{25}, prepare entangled states, perform quantum computation and realise quantum memories. This calls for the exploitation of memoryeffects typical of a nonMarkovian dynamics as a useful tool for the processing of a quantum state.
Unfortunately, a satisfactory understanding of nonMarkovianity is yet to be reached, which motivates the recent and intense efforts performed towards the rigorous characterisation of nonMarkovian evolutions, the formulation of criteria for the emergence of nonMarkovian features and the proposition of factual measures for the quantification of the degree of nonMarkovianity of a process^{26,27,28,29}. Some of them have been recently used in order to characterize, both theoretically and experimentally, the character of systemenvironment interactions for few and manybody quantum systems^{16,17,18}. Here we use a photonic setup to simulate a systemenvironment coupling ruled by a transverse Ising spin model (see Supplementary Material). Such interaction gives rise to the nonMarkovian evolution of S, as witnessed by the nonmonotonic behavior of its entanglement with an ancilla A that is shielded from the environmental effects. Our goal is the experimental investigation of the fundamental connection between systemenvironment correlations and nonMarkovianity. The simulator that we propose allows for the implementation of various free evolutions of S and E, as well as the adjustment of their mutual coupling, thus making possible the transition from deeply nonMarkovian dynamics, all the way down to a fully forgetful regime.
Results
Experimental setup and theoretical model
The simulator is experimentally realised using different degrees of freedom of the photonic device shown in Fig. 1, which consists of the concatenation of two MachZehnder interferometers and a single Sagnac loop. The information carriers are two photons (referred hereafter as “high” and “low”): the system S is embodied by the polarisation of the low photon (which can be either horizontal H〉 ≡ 0〉 or vertical V〉 ≡ 1〉), while the environment E is encoded in the longitudinal momentum degree of freedom (the path) of the same photon (which will be right r〉 ≡ 0〉 or left l〉 ≡ 1〉). The ancilla A is embodied by the polarisation of the high photon. High and low photons are emitted by the source of polarizationentangled described in the Methods.
The ancilla is the key tool for our goals. Indeed, to investigate the emergence of nonMarkovianity in the evolution of S due to its interaction with E, we use the method proposed in Ref. 27: we focus on the modifications induced by the SE coupling on a prepared entangled state of S and A. If the SA entanglement decays monotonically in time, the dynamics of S is fully Markovian. Differently, if for certain timewindows there is a kickback from E that makes such entanglement increase, the dynamics is necessarily nonMarkovian. In fact, if the local action of the environment is no longer represented by a continuous family of completely positive maps, the SA entanglement is no longer constrained to decrease monotonically. This is evidence of the flowback of coherence on the system and results in an increment of the SA entanglement.
As in other digital quantum simulators, the dynamics is approximated by a stroboscopic sequence of quantum gates. Conceptually, the simulation consists of the forward evolution of S over discrete time slices^{30} according to a TrotterSuzuki decomposition^{31} of the total time propagator (see Supplementary Material). This approach is known to be effective for quantum simulation^{7} and is implemented here by the sequence of operations shown in Fig. 2a). The free evolution of the environment is accounted for by the Hadamard gate ( is the m = x, y, z Pauli matrix of qubit j = S,E and is the Hadamard gate of qubit j = S,E), while . The SE interaction is engineered by implementing the controlledrotation , which rotates the system according to the general singlequbit operation depending on the state of the environment. This class of conditional operations is obtained from Hamiltonian generators of the twoqubit transverse Ising form (see Supplementary Material), which motivates our choice and is thus the class of systemenvironment interactions that is simulated in this work. Experimentally, we fix ϕ = π/4 so as to implement a conditional() gate. By using the identity , it is straightforward to prove that this gate is locally equivalent (via ) to the composition of a controlled Hadamard and a controlled antiZ (i.e. a gate applying only when E is in 0〉), namely . The extra needed to make the two conditional gates equivalent can thus be absorbed in H^{S}, which thus becomes .
This set of gates is experimentally realised in our photonic simulator as sketched in Fig. 2b). The gate is implemented by means of a beam splitter (BS) in conditions of temporal and spatial indistinguishability of the optical modes. This scheme allows to evolve the input modes r〉 and at each step as
where the phases φ_{i} are varied by rotating thin glass plates intercepting one of the optical modes entering the BS. In order to perform a more general rotation of E it will be necessary to unbalance the output modes (using intensity attenuators) making the probability of occurrence of r〉 and unequal.
The controlled gate is realised by placing a halfwave plate (HWP) on one of the two output modes with optical axis at 22.5° with respect to the vertical direction. The temporal delay introduced by this waveplate is compensated by another HWP on the opposite output mode with optical axis set at 0° with respect to the vertical direction. This implements the gate, as shown in Fig. 2b). In order to assess nonMarkovianity, ancilla and system are prepared in the maximally entangled state . This twoqubit Bell state is engineered using the polarization entanglement source of Ref. 32 (see Methods). The environment is initialised in ( is set by the transmittivity of BS_{1} [cf. Fig. 1] and can be varied by placing an intensity attenuator on one of the two output modes), a state endowed with quantum coherence as it is key due to the conditional nature of the dynamics that we simulate. The stability, modular structure and long coherence time of our interferometric setting allows for the repeated iteration of each block of gates .
As A does not evolve with the environment, the polarisation state of the high photon (depicted in red in Fig. 1) is immediately detected and only the low photon (in yellow in Fig. 1) goes through each sequence of gates at the various steps. A standard optical setup for the performance of quantum state tomography (QST)^{33} is used to reconstruct the state of the SA system after the evolution. The radiation is collected by using an integrated system composed of GRaded INdex (GRIN) lens and singlemode fibre^{34} and is then detected by singlephoton counters. For each step of the simulated dynamics, we measure the state of the environment by projecting the polarisation qubits on the states HH〉, HV〉, VH〉, VV〉. Finally, we trace out the degrees of freedom of the SA system by summing up the corresponding counts measured for every single projection needed for the implementation of singlequbit QST. The Pauli operators for E, generated after the first passage through the BS_{1}, are measured using BS_{2}. The same procedure is followed to perform the QST of the environmental state at each step. The state of the SA system is reconstructed in a similar way, by summing up the counts collected after projecting E onto r〉 and .
NonMarkovian simulation
Figure 3 summarises the results obtained by running through the evolution of the overall system. In order to quantify entanglement we use the entanglement of formation EOF(SA)^{35} between S and A, which is operatively linked to the cost of engineering a given state by means of Bellstate resources (see the Methods section). We are also interested in the correlations shared by the environment with the rest of the system, hence we evaluate the von Neumann entropy of the environment, defined as S(E) = −Tr[ρ_{E} log_{2}(ρ_{E})], which under the assumption of pure total ASE state quantifies the entanglement in the partition AS against E. Figure 3a) shows the experimental entanglement of formation at each time step (blacksquare points) against the results of a theoretical model (red line) that, including all the most relevant sources of imperfections, deviates from the ideal picture sketched above. First, the BSs are not entirely polarisationinsensitive: for BS_{1} and H (V) polarisation the reflectivity over transmittivity ratio R/T = 42/58 (45/55), while for BS_{2} is R/T = 45/55 (55/45). Second, although the desired input state is created with high fidelity (, see Supplementary Material), the entangledstate source generates spurious HV〉 and VH〉 components, accounting for about 5% of the total state, which reduce the initial systemenvironment entanglement to about 0.8. The inclusion of such imperfections makes the agreement between theory and experimental data very good up to the fourth step of our simulation, showing at least one revival of the SA entanglement and thus witnessing the nonMarkovian nature of the evolution^{27}. The fifth experimental point is significantly far from the theoretical behaviour because of the notideal setting of the phases φ_{i} (i = 1, 2, 3). In fact, we have verified that their values are not completely polarization independent. This determines a slight difference for the four contributions HH〉_{SA}, HV〉_{SA},VV〉_{SA},VH〉_{SA} entering the state. This imperfection affects the performance of each Hadamard gate and becomes significant especially for the last step, where the cumulative effect of three gates should be considered. Furthermore, the fifth point is affected by an uncompensated phase shift induced by the multilayer mirrors used in our setting [cf. Figure 1], as explained in the Supplementary Information. Nonetheless, this point still reveals successfully the occurrence of a second entanglement revival, thus strengthening our conclusions.
Figure 3b) compares the experimentally inferred EOF(SA) (black squares) to the von Neumann entropy of the environment S(E) (green circles) quantifying the correlations shared between E and SA. These figures of merit appear to be perfectly anticorrelated, thus giving evidence of a tradeoff between the amount of entanglement that S and A can share at the expenses of SE correlations. This strengthens the idea that correlations with the environment play a fundamental role in this process, a point that has been addressed in Ref. 36 where it is shown that the establishment of systemenvironment correlations is a necessary conditions for the emergence of nonMarkovianity. This result can be bridged with our analysis considering that the mixed systemenvironment state of Ref. 36 can be purified by enlarging the Hilbert space of the system including an appropriate ancilla. The anticorrelation between S(E) and EOF(SA) suggest interesting connections with the monogamous nature of quantum entanglement in such tripartite system^{37}. We have illustrated such connections in the Supplementary Information, where a theoretical simulation for up to 11 steps is reported and a quantitative analysis fof the entanglement sharing features in our system is provided.
To reinforce our point even further, we now explore the Markovian counterpart of our simulation by replacing the unitary evolution of E with an incoherent map that resets the environment into the very same state at each step of the evolution. As the key role in the SE interaction is played by quantum coherence, we chose to reset the environment into a completely mixed state. This is realised by spoiling the temporal indistinguishability of the optical modes entering the BS by mutually delaying the r〉 and components. Intuitively, by making the state of the environment rigid, we wash out any possibility for systemenvironment correlations, thus pushing the dynamics towards Markovianity. This intuition is fully confirmed by the experimental evidences collected via QST: the black squares in Figure 3c) show a monotonic (quasiexponential) decay of EOF(SA) (matching our theoretical predictions, red line), which is in perfect agreement with the absence of SE quantum correlations as signalled by the positivity of the partially transposed SE state.
Discussion
By simulating a nontrivial twoqubit coupling model, we have demonstrated the nonMarkovianity of the evolution induced on S by a dynamical environment and a systemenvironment interaction allowing for kickback of coherence. By adopting a witness that makes use of the effects that nonMarkovianity has on entanglement, we have explored experimentally the link between the emergence of nonMarkovianity and systemenvironment correlations. The next step in this endeavour will be the experimental proof that nonMarkovianity can be used as a resource for the advantageous processing of information^{38,39}, such as the preparation of interesting states^{40}, along the lines of previous studies on state engineering and information manipulation through Markovian processes^{41,42,43}. Our setup will be particularly well suited for this task, in light of the effective control over both system and environment that can be engineered both in space (thanks to the modular nature of the setting) and in time.
Methods
Source of entangled states
The black box shown in Fig. 1 of the main text represents the source used for the generation of the hyperentangled state encoded in the polarization and path degrees of freedom of the photons produced by spontaneous parametric downconversion. Here polarization entanglement, corresponding to the state , was realized by spatial and temporal superposition of photon pair emissions occurring with equal probability, back and forth, from a single barium borate (BBO) typeI crystal under double excitation of a vertically polarized UV CW pump beam. A suitable rotation of the polarization on one of the two possible emission directions of the photons was then applied. The source that we have used is such that the two photons, belonging to the degenerate BBO emission cone, are also path entangled, as explained in Ref. 32. For our pourposes we need to select a pair of correlated directions, belonging to the emission cone surface, along which two photons travel, obtaining in this way the high and low photons shown in red and yellow in Fig. 1.
Entanglement of formation
For two qubits, the entanglement of formation is given by^{35} , with h(x) = −x log_{2} x − (1 − x) log_{2}(1 − x) and C = max{0, λ_{1} − λ_{2} − λ_{3} − λ_{4}}. Here, {λ_{i}} is the set of eigenvalues (arranged in nonincreasing order) of the Hermitian matrix where and ρ* is the conjugate of the density matrix.
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Acknowledgements
This work was supported by EUProject CHISTERAQUASAR, PRIN 2009 and FIRBFuturo in ricerca HYTEQ, the EU under a Marie Curie IEF Fellowship (L.M.) and the UK EPSRC (M.P.) under a Career Acceleration Fellowship and a grant of the “New Directions for Research Leaders” initiative (EP/G004579/1).
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M.P. proposed the original idea, A.C. and P.M. designed the experimental setup. A.C., C.G. and P.M. performed the experiment, L.M. and M.P. analysed and interpreted the data. All authors contributed to the writing of the manuscript.
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Chiuri, A., Greganti, C., Mazzola, L. et al. Linear Optics Simulation of Quantum NonMarkovian Dynamics. Sci Rep 2, 968 (2012). https://doi.org/10.1038/srep00968
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DOI: https://doi.org/10.1038/srep00968
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