Abstract
Photonic quantum simulators are promising candidates for providing insight into other small to mediumsized quantum systems. Recent experiments have shown that photonic quantum systems have the advantage to exploit quantum interference for the quantum simulation of the ground state of Heisenberg spin systems. Here we experimentally characterize this quantum interference at a tuneable beam splitter and further investigate the measurementinduced interactions of a simulated fourspin system by comparing the entanglement dynamics using pairwise concurrence. We also study theoretically a foursite square lattice with nextnearest neighbor interactions and a sixsite checkerboard lattice, which might be in reach of current technology.
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Introduction
More than a quarter of a century ago, Richard Feynman^{1,2} envisioned that a wellcontrolled quantum mechanical system can be used for the efficient simulation of other quantum systems and thus is capable of calculating properties that are unfeasible for classical computers. Quantum simulation promises potential returns in understanding detailed quantum phenomenon of inaccessible quantum systems^{3}, from molecular structure to the behavior of hightemperature superconductors. Moreover, quantum simulations are conjectured to be less demanding than quantum computations by being less stringent on explicit gate operations or error correction^{4,5}. Due to these reasons, quantum simulation has led to many theoretical proposals^{6,7,8,9}. Recently, various quantum simulators based on different physical platform are being constructed, such as atoms in optical lattice^{10,11,12,13,14,15}, trapped ions^{16,17,18,19,20,21}, nuclear magnetic resonance^{22,23}, superconducting circuits^{24}, as well as single photons^{25,26,27,28,29,30,31,32,33}. For recent reviews, see ref. 34,35,36,37.
Motivated by the seminal work of Knill, Laflamme and Milburn^{38} photons are proven to be a suitable system for efficient quantum computing and quantum simulation. Precise singlephoton manipulations and tunable measurementinduced twophoton interactions are the essential ingredients for photonic analog quantum simulation and have been demonstrated. In addition to the highlevel quantum control, such photonic quantum simulators can produce exotic entangled states which are important for understanding the manybody dynamics in quantum chemistry and solidstate physics^{8,9,31}. Note that the recently proposed universal quantum computation based on coherent photon conversion provides an alternative avenue for achieving photonphoton interaction^{39}.
In ref. 31, a spin tetramer, the simplest example of the Heisenberginteracting spin system, has been investigated. In this work we will not only present the detailed study on the tunable measurementinduced photonphoton interaction, but also further investigate more complicated Heisenberg models for the foursite square lattice with nearest and nextnearest couplings and the sixsite checkerboard lattice.
Analog quantum simulations with photons and linear optics
In the case of the simulation of spin1/2 particles the photon's polarization is ideally suited as horizontallypolarized states H〉 and verticallypolarized states V〉 represent for example spinup and spindown states. Moreover, the ability to prepare symmetric polarizationentangled states , and the antisymmetric state enables the establishment of states with bosonic and fermionic character^{30,31,40}. The latter shares the same quantum correlations as Heisenberginteracting spins or socalled valence bond states^{41}.
The theoretical investigation of stronglycorrelated spin systems has led to few exact theorems which in some cases are of importance for the quantum simulation of chemical and physical models. In the particular case of a nearestneighbor antiferromagnetic Heisenberginteracting spin system it was shown by Marshall^{42} that the ground state for N spins on a bipartite lattice has total spin zero. This theorem and its extension^{43} lead to the fact that the ground state must be built as a linear superposition of singlet spin states or valence bonds. This constraint, that forces the ground state's total spin to be zero, gives rise to various valencebond configurations that are either localized or fluctuating as superposition of different singlet partitionings. Localized configurations are typically referred to valence bond solids and delocalized valencebond states correspond to frustrated quantum spin liquids or resonating valencebond states^{44,45}. Recently, the photonic quantum simulation of a four spin1/2 square lattice as an archetype system^{31} showed that quantum monogamy^{46,47} plays an important role in frustrated Heisenberg spin systems.
The experimental setup for studying variable measurementinduced interactions is shown in Fig. 1. Our pump source is a modelocked Ti:sapphire femtosecond laser with a pulse duration of 140 fs and a repetition rate of 80 MHz. The central wavelength of the pump is at 808 nm. Then we use a βbarium borate crystal (BBO0) to upconvert the pump pulses to ultraviolet (UV) pulses via second harmonic generation. The upconverted UV pulses' central wavelength is 404 nm with a pump power of 700 mW. Then we clean the UV pulses with several dichroic mirrors (DM). Photons 1 and 2 are generated from a BBO crystal (BBO1) via spontaneous parametric down conversion (SPDC) in a noncollinear typeII phase matching configuration and in an polarizationentangled state after walkoff compensation^{48}. Photons 3 and 4 are generated from BBO2 in a collinear typeII phase matching configuration and are separated by a polarizing beam splitter (PBS). We guide photons 1 and 3 to a tunable directional coupler (TDC) and then detect them by avalanche photodiodes (APD), which together enable the tunable measurementinduced photonphoton interactions to happen. The relative temporal delay between the photons is adjusted with a motorized translating stage mounted on the fiber coupler of photon 1. Fiber polarization controllers (PC) are employed to eliminate the polarization distinguishability of the two interfering photons.
Bunching due to the twophoton HongOuMandel (HOM) interference^{49} and the corresponding antibunching effect^{40} are crucial for many quantum information processing protocols, especially for photonic quantum computation experiments (Cphase gate^{50,51}, entanglement swapping^{52,53}, etc.), as well as for our photonic quantum simulation^{31}. The bosonic nature of the photons shows up when the input states are superimposed at the TDC such that a detection, even in principle, cannot distinguish either of them. This leads to a superposition of double occupations on both outgoing modes and thus suppression of the coincidence detections, where one photon is detected in each output mode. The visibility of this HOM dip is one when the TDC is set to have equal splitting of transmitted and reflected photons similar to a 50/50 beam splitter. As soon as the two input photons can be partially distinguished by unbalancing the splitting ratio the visibility decreases. The dependence of the ideal visibility (V_{ideal}) upon the reflectivity of TDC (η) is the following:
In Fig. 2 this reflectivity dependent ideal visibility is plotted in the black solid curve. Experimental imperfections due to highorder emissions from SPDC and group velocity mismatch reduce V_{ideal}. The measured visibilities (black squares) are also shown as well as the corresponding fit (dashed red curve).
Pairwise entanglement dynamics
The main advantage of the precise quantum control of individual particles is that interparticle entanglement dynamics can be investigated. By using a similar experimental configuration as in Ref. 31, we study the entanglement distributions among different particles with respect to the effective interaction strength that was tuned by the TDC. For the quantification of the bipartite entanglement in our system, we use the measure of concurrence^{54}, which, for a given state ρ, is , where λ_{i} are the eigenvalues of the matrix ρΣρ^{T}Σ in nonincreasing order by magnitude with , where σ_{Y} = −i0〉 〈1 + i1〉 〈0. While the previous quantum simulation characterized the pairwise energy dynamics of Heisenberg interactions that were directly extractable from measured coincidence counts, this experiment requires the reconstruction of the density matrices to obtain the concurrence values. In the experiment we tune the reflectivity of the TDC and hence vary the photonphoton interaction strength. Various fourphoton quantum states are tomographically measured and the density matrices of them are reconstructed^{55,56}. The concurrence of the twophoton subsystems is calculated from the fourphoton density matrices by tracing out the other two photons.
Due to the quantum monogamy relations^{31,46,47} the total amount of pairwise entanglement stays constant while the change of interaction strength affects the distribution and thus the ground state configurations. When tracking the change in entanglement by using concurrence, the expected “sudden death” and “sudden birth” of entanglement^{58} can be seen (Fig. 3). While this concept is typically used for studying environmentinduced decoherence^{59}, a similar behavior can be observed here too. In fact our tunable interactions allow to mimic a controlled interaction with the environment of two additional particles, which opens the possibility to obtain insights into complex decoherence mechanisms. We explicitly show the concurrence of different photonpair configuations with respect to the TDC angle θ. The relation of the TDC angle and its reflectivity is given by θ = arctan . One can see that the concurrence for one photon pair (e.g. C_{14}) decreases rapidly as we increase θ. At θ = 0.274, all of a sudden the entanglement between another spin pair (e.g. C_{13}) is born at the cost of the reduced concurrence C_{14}, which vanishes as θ is further increased. The observation of similar disappearance or emergence of entanglement among the other photons demonstrates the capability of our quantum simulator for manipulating the quantum correlations^{57,58,59}.
Generalized heisenberg spin model on a foursite square lattice and a sixsite checkerboard lattice
For the foursite square lattice we extend the model in Ref. 31 by adding a nextnearest neighbor interaction term (Fig. 4a). The Hamiltonian for this system is:
where S_{i} is the Pauli spin operator for spin i and J_{1}, J_{2} and J_{3} are the coupling strength parameters between different spins, respectively (see Fig. 4a).
We consider the antiferromagnetic case with the couplings J_{1}, J_{2}, J_{3} ≥ 0. For each of the three terms in H, the ground state is a pair of singlets, , and (up to normalization). In Ref. 31 we showed that by tuning J_{2}/J_{1} from 0 to ∞, the ground state, Φ_{g}, gradually changes from Φ_{=} 〉 to Φ_{}〉.
With the introduction of J_{3}, the ground state of the system is still a superposition Φ_{g}〉 = αΦ_{=} 〉 + βΦ_{}〉 with normalization condition 2(α^{2} + β^{2} + α + β^{2}) = 1.
Remarkably, tuning J_{3}/J_{1} can induce sharp phase transitions with a sudden change of the ground state configuration due to the competing of the valence bond configurations. In Fig. 5, the ground state configurations for different coupling regimes are shown. There are three particular interesting phase transitions:

For J_{1} = J_{3}, Φ_{g}〉 suddenly changes from Φ_{=} 〉 + Φ_{×}〉 to Φ_{}〉 when J_{2}/J_{1} is tuned across 1

For J_{2} = J_{3}, Φ_{g}〉 suddenly changes from Φ_{=} 〉 to Φ_{}〉 − Φ_{×}〉 when J_{2}/J_{1} is tuned across 1

For J_{1} = J_{2}, Φ_{g}〉 suddenly changes from Φ_{=} 〉 + Φ_{}〉 to Φ_{×}〉 when J_{3}/J_{1} is tuned across 1
The case with J_{1} = J_{2} ≠ J_{3} is widely studied for square lattice systems due to its relevance to cuprates, Febased superconductors and other materials^{60,61,62,63}. Previous studies have shown that in the thermodynamic limit, when J_{3} > J_{1}, the system is in a diagonal Neel ordered state^{64}, which is consistent with the ground state Φ_{×}〉 for a minimum of four sites, as discussed above. In the regime where J_{2} < J_{1} the configuration for the ground state is still under debate due to regions that appear to be non magnetic. Numerical calculation have recently shown that this region is highly likely to be a quantum spin liquid with Z_{2} topological order^{64}. However, for only four spins, the region with J_{2} < J_{1} has only a single ground state configuration Φ_{=} 〉 + Φ_{}〉.
We also investigate theoretically the ground states of a J_{1} − J_{2} Heisenberg model on a sixsite checkerboard lattice. The geometry of this system is shown in Fig. 4b, where J_{1} is the coupling strength between the nearest neighbor sites and J_{2} is the coupling strength on the cross bonds. The coupling ratio J_{2}/J_{1} is the only parameter of this system. The introduction of the next nearest neighbor coupling J_{2} makes this system a simple frustrated magnetic model. The study of this model on thermodynamics limits is motivated by the threedimensional pyrochlore materials^{65,66}. The twodimensional model has been studied by several groups^{67,68,69,70,71,72,73}. It is known that for the regime where the coupling ratio the system has a colinear Neel order. At J_{2}/J_{1} ≈ 1 numerical calculation^{68} suggests a plaquette valence bond solid ground state while the ground state at remains under debate^{67,69,70}. For this reason future photonic quantum simulations might provide answers to these open questions.
Numerical studies of the sixsite checkerboard system were done by diagonalizing the Hamiltonian with open boundary conditions. In Fig. 6 the energy spectrum of the six low lying states as a function of the ratio J_{2}/J_{1} in the subspace is presented. The energy spectrum shows an avoid level crossing around J_{2}/J_{1} ≈ 1, which indicates a dramatic change of the ground state properties there. In analogy to the J_{1} − J_{2} Heisenberg model on a plaquette, where the ground states can be expressed as linear superposition of two different dimer coverings whose coefficients depend on the ratio J_{2}/J_{1}, the ground states for the sixsite checkerboard lattice can also be expressed as superpositions of various dimer configurations. For the discussed sixsite lattice system, fifteen different dimer coverings exist. However, only six are independent. By taking the symmetry of our system into consideration only four out of the six coverings are allowed (Fig. 7).
Therefore, the ground states can be described as superpositions of four dimer coverings, ψ_{1}〉, ψ_{2}〉, ψ_{3}〉, ψ_{4}〉 in all region of J_{2}/J_{1}. In Fig. 8 we show the contribution of each dimer configuration with respect to the coupling ratio J_{2}/J_{1}. At J_{2}/J_{1} = 1 the coefficients for ψ_{1}〉 and ψ_{3}〉 are equal and coefficients for ψ_{2}〉 and ψ_{4}〉 are exact zero. In our current convention of the dimer covering wave function this particular superposition gives us a plaquette state on the right four sites as it is shown in the inset. This coincides with the plaquette valence bond solid state in an infinite system. Remarkably, the sixsite checkerboard lattice thus provides already a valuable hint for the true ground state in the thermodynamic limit. The ground states at the ratio J_{2} > J_{1} have close to equal contribution from ψ_{1}〉 and ψ_{2}〉, which suggests a significant contribution from a crossdimer state on the left plaquette. We would like to mention that a crossdimer ground state has also been suggested as a potential ground state for large J_{2} couplings^{69}. It is interesting that only a small contribution from the dimer configuration ψ_{4}〉 can be found in all the possible cases.
Summary and outlook
In conclusion, today's available photonic quantum technology is reaching the stage where significant advantages arise for the simulation of particular interesting questions in solidstate physics and quantum chemistry. Therefore photonic quantum simulations provide exciting opportunities to cover, for example, the direct construction of customtailored manybody wave functions. Impressively, the usage of optical elements such as tunable nonpolarizing or polarizing beam splitters enables entangled fewphoton states to construct manybody valence bond wave functions for molecular and solidstate systems due to nonclassical interferences. As we have shown above, this is of particular interest in condensed matter physics as it provides insight into the frustration of stronglycorrelated spin systems and the onset of quantum phase transitions. On the other hand, in quantum chemistry it effectively allows studying delocalized bonds in chemical structures and chemical reactions^{6}. Thus being able to monitor the full dynamics of individual particles and bonds provide some fascinating perspective for the quantum simulation of small molecules or reactive centers.
The main future challenge will be to increase the number of photons or degrees of freedom to realize a sufficient amount of qubits such that quantum computers can outperform their classical counter parts. In general, up to the level of approximately twenty qubits it presently appears possible to conceive a system based on bare physical qubits. However, given the current experimental limitations, operational fidelities and noise sources, it seems that useful system consisting of more than twenty qubits could not be realized without some level of error correction. But in contrast to the implementations of wellknown quantum algorithms, such as Shor's algorithm for a computationally relevant keylength, the requirements for faulttolerance are much less demanding. As such an open problem relates to how much error correction is needed to achieve a useful quantum simulation.
Recent work^{6,27} has shown that quantum systems with less than a dozen physical qubits are capable of simulating chemical systems with a precision that cannot be achieved by conventional computers, when processed via almost a thousand discrete gate operations. Although such small quantum systems are feasible by using present quantum technology, the requirements in terms of gate operations is tremendous. Thus, analog quantum gate operations look promising in reducing the technical complexity of performing such quantum simulation experiments by requiring fewer number of physical gates.
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Acknowledgements
X.S.M., B.D., S.K., W.N., Y.H.C., A.Z. and P.W. thank F. Verstraete and Č. Brukner for helpful discussions. We further acknowledge support from the European Commission, QESSENCE (No. 248095), ERC Advanced Senior Grant (QIT4QAD), QUILMI (No. 295293), EQUAM (No. 323714), PICQUE (No. 608062), GRASP (No. 613024) and the ERANet CHISTERA project QUASAR, the John Templeton Foundation, the Vienna Center for Quantum Science and Technology (VCQ), the Austrian Nanoinitiative NAP Platon, the Austrian Science Fund (FWF) through the SFB FoQuS (No. F4006N16), START (No. Y585N20) and the doctoral programme CoQuS, the Vienna Science and Technology Fund (WWTF) under grant ICT12041 and the Air Force Office of Scientific Research, Air Force Material Command, United States Air Force, under grant number FA86551113004. XSM was supported by a Marie Curie International Outgoing Fellowship within the 7^{th} European Community Framework Programme. Y.H.C., Z.X.G. and L.M.D. thank H.C. Jiang for helpful discussion and acknowledge support from the NBRPC (973 Program) 2011CBA00300 (2011CBA00302) and the DARPA OLE program.
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X.S.M., S.K. and W.N. designed and performed experiments, analyzed data and wrote the manuscript. B.D. provided the theoretical analysis, analyzed data and wrote the manuscript. Y.H.C., Z.X.G. and L.M.D. developed the theoretical analysis of the general Heisenberg spin model on 4site and 6site lattices. A.Z. supervised the project and edited the manuscript. P.W. designed experiments, analyzed data, wrote the manuscript and supervised the project.
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Ma, Xs., Dakić, B., Kropatschek, S. et al. Towards photonic quantum simulation of ground states of frustrated Heisenberg spin systems. Sci Rep 4, 3583 (2014). https://doi.org/10.1038/srep03583
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DOI: https://doi.org/10.1038/srep03583
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