The Hubbard model and the Ising model are the conceptually simplest but standard theoretical frameworks which have been used to explore strongly-correlated condensed matter systems. Due to the lack of exact analytical solutions in higher dimensions and computational limitations in numerical analysis, complete knowledge of the Hubbard model beyond in one dimension is yet to be attained, and its complete link to experimental signatures has been elusive. Quantum simulation of the Hubbard model offers an alternative route to investigate the physics of the model since a quantum simulator outperforms a classical counterpart as it handles intractable quantum many-body systems. In addition, the quantum Ising model can be mapped into any NP-complete problem; hence, the quantum Ising simulator would pave a new way to investigate various problems in physics, computer science, chemistry and engineering fields. We have investigated the theoretical and experimental aspects to implement quantum Hubbard simulator and quantum Ising simulator.

A BEC is generally defined as a macroscopic occupation of a single-particle quantum state, a phenomenon often referred to as off-diagonal long-range order due to non-vanishing off-diagonal components of the one-body density. The wave function of the condensate plays a role as an order parameter, whose phase is essential in characterizing the coherence and superfluid phenomena. The long-range spatial coherence leads to the existence of phase-locked multiple condensates in an array of superfluid helium, superconducting Josephson junctions, or atomic BECs. Under certain circumstances, a quantum phase difference of π is predicted to develop among weakly coupled Josephson junctions. Such a meta-stable π-state was discovered in a weak link of superfluid ^{3}He, which is characterized by a ‘*p*-wave’ order parameter. Possible existence of such a π-state in weakly coupled atomic BECs has also been proposed, but remains undiscovered. We have observed for the first time spontaneous buildup of in-phase (‘zero-state’) and anti-phase (‘π-state’) 'superfluid' states in an exciton-polariton condensate array connected by weak periodic potential barriers in one dimension (1D). These states reflect the band structure of the one-dimensional polariton array and the dynamic characteristics of meta-stable exciton-polariton condensates.

**Figure 1.** Band structure and ‘superfluid’ state in a 1D exciton-polariton condensate array. **(a)** Time-integrated energy versus in-plane of an exciton-polariton array at near zero-detuning. **(b)** Extended-zone scheme of the band structure under a weak periodic potential with a lattice constant *a*. **(c)** Energy versus in-plane momentum at blue detuning above threshold pumping. **(d)** Schematic illustration of the Bloch wavefunctions for states labeled as A, B, C in **(b)**.

We extended the previous 1D work into a two-dimensional condensate-array under a weak periodic square lattice potential profile. We have observed the up to 4 Brillouin zones (BZs) in 2D reciprocal space to exhibit the band structure of a single particle in a 2D square lattice. The three high symmetry points (Γ, X, M) in the 2D BZ exhibit rotational symmetry, where the eigenstates can be classified according to their symmetry properties of the rotation group analogus to atomic orbital state denotation.

Figure 2 displays the observed FF images in momentum (k_{||,x}, k_{||,y}) space at various optical pump powers. These images are taken under the pump direction of ê_{k||,x} - ê_{k||,y}. This particular pump direction was chosen in order to access energy values of s-, p-, d_{xy}- states individually from spectrometer measurements.

**Figure 2.** Lower polariton distribution in reciprocal lattice space. Pump power-dependent far-field images for the pumping direction along ê_{k||,x} - ê_{k||,y}. The relative pump rates (*P*/*P _{th}*) are 0.29, 1, 1.7, 4.3, and 29 from left to right. Narrow peaks in square lattice positions originate from pump laser scatterings by periodic circular aperture, whose distance provides the size of the first BZ.

A many-boson system in an open dissipative environment has the potential to enhance certain computation algorithms, both classical and quantum. Byrnes *et al.* proposed the classical simulated annealing algorithm in such systems, whereby the computation can be accelerated by bosonic final state stimulation (Fig. 3). We are investigating the quantum Zeno effect in a many-boson system, which causes a speed-up in the Landau-Zener transition. Furthermore, the experimental implementations of those algorithms in exciton-polariton condensates using quantum feedback control have been proposed (Fig. 4).

**Figure 3.** The equilibriation time τ for a 4-site Ising model at various error probabilities for the system including **(a)** higher order transition effects and **(b)** no higher order transitions. **(c)** The equilibriation time for various boson numbers *N* per site and error probabilities ε.

**Figure 4.** Schematic of the quantum Ising model simulator including feedback control. **(a)** Each site of the Ising Hamiltonian is encoded as a trapping site, containing *N* bosons. The bosons can occupy one of two states σ = ±1, depicted as either red or blue. **(b)** The interaction between the sites may be externally induced by measuring the average spin on each site *i* via the detectors, which produce a detector current *I _{i}(t)* and apply a local field

C. W. Lai, N. Y. Kim, S. Utsunomiya, G. Roumpos, H. Deng, M. D. Fraser, T. Byrnes, P. Recher, N. Kumada, T. Fujisawa and Y. Yamamoto,”Coherent zero-state and π-state in an exciton-polariton condensate array”, *Nature* **450**, 529 (2007).

T. Byrnes, K. Yan and Y. Yamamoto, “Optimization using Bose-Einstein condensation and measurement-feedback circuits”, arXiv:0909.2530 (2009).

T. Byrnes, P. Recher and Y. Yamamoto, “Mott transitions of exciton-polaritons and indirect excitons in a periodic potential”, *Phys. Rev. B* **81**, 205312 (2010).

Dr. Na Young Kim

Kai Wen

Prof. Yoshihisa Yamamoto

Dr. Tim Byrnes (NII, Japan)

Defense Advanced Research Projects Agency

National Science Foundation

NICT, MEXT, NII FIRST