A driven-dissipative spin chain model based on exciton-polariton condensates
H. Sigurdsson, A.J. Ramsay, H. Ohadi, Y.G. Rubo, T.C.H. Liew, J.J. Baumberg, I.A. Shelykh
AA driven-dissipative spin chain model based on exciton-polariton condensates
H. Sigurðsson, ∗ A. J. Ramsay, H. Ohadi, Y. G. Rubo,
4, 5
T.C.H. Liew, J. J. Baumberg, and I. A. Shelykh
1, 7 Science Institute, University of Iceland, Dunhagi-3, IS-107 Reykjavík, Iceland Hitachi Cambridge Laboratory, Hitachi Europe Ltd., Cambridge CB3 0HE, UK Department of Physics, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK Instituto de Energías Renovables, Universidad Nacional Autónoma de México, Temixco, Morelos 62580, Mexico Center for Theoretical Physics of Complex Systems,Institute for Basic Science, Daejeon 34051, Republic of Korea Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,Nanyang Technological University 637371, Singapore ITMO University, St. Petersburg 197101, Russia (Dated: October 8, 2018)An infinite chain of driven-dissipative condensate spins with uniform nearest-neighbor coherentcoupling is solved analytically and investigated numerically. Above a critical occupation thresholdthe condensates undergo spontaneous spin bifurcation (becoming magnetized) forming a binary chainof spin-up or spin-down states. Minimization of the bifurcation threshold determines the magneticorder as a function of the coupling strength. This allows control of multiple magnetic orders viaadiabatic (slow ramping of) pumping. In addition to ferromagnetic and anti-ferromagnetic orderedstates we show the formation of a paired-spin ordered state | . . . ↑↑↓↓ . . . i as a consequence of thephase degree of freedom between condensates. Many-body spin systems, both classical and quantum,have found applications in a number of fields of risingcomplexity. Their Hamiltonians (Ising, XY , Heisenberg,Sherrington-Kirkpatrick, etc.) have been used to studycollective behaviors such as familiarity recognition inneural networks , hysteresis in DNA interactions , com-binatorial optimization problems in logistics, patterning,and economics . Besides their wide application, con-trollable spin lattices also offer insight into physical prob-lems such as frustration , spin-ice , spin-wave dynam-ics , domain wall motion , and spin-glass forma-tion . A driven-dissipative spin lattice, where bothphase and spin of the vertices are free, has yet to beaddressed. Here, in contrast to entropy and minimumenergy principles (as in the Ising model), the station-ary physics of the system is governed by the balanceof gain and decay with remarkably different solutions .Currently, only limited investigation has been devotedto driven-dissipative lattice systems where recent workshave proposed “simulators” based on interacting exciton-polariton condensates and Ising machines with degen-erate optical parametric oscillators .Nonresonantly excited spinor exciton-polariton (orsimply polariton ) condensates have developed intoa popular platform for cutting edge opto-electronic andopto-spintronic technologies . The driven-dissipativecondensates are realized by matching the gain and thedecay of polaritons through continuous external drivingof either optical or electrical nature. These macroscopiccoherent states possess a spin and a phase degree of free-dom, strong nonlinearities and a small effective mass, al-lowing them to interact and synchronize with other spa-tially separate condensates over long distances (hundredsof microns) , making them interesting candidates fordriven-dissipative spin lattices. Recently it was reportedthat a spinor polariton condensate bifurcates at a critical ... ...... ...... ...... ... (a)(b)(c)(d) FMAFMP
Figure 1. (Color online) (a) Schematic showing spinor con-densates coupled together through the same-spin coupling pa-rameter J in an infinite chain. States with equal number ofbond types per condensate can be categorized as (b) FM, (c)AFM, and (d) paired (P). pump intensity into either of two highly circularly polar-ized states using a continuous linearly polarized nonreso-nant excitation . The emission polarization is explicitlyrelated to the polariton condensate pseudospin orienta-tion (from here on spin ) . The system has since thenbeen extended to polariton condensate spin pairs whichcan controllably display alignment of antiferromagnetic(AFM) and ferromagnetic (FM) nature. This spin de-gree of freedom offers a unique way to study orderingamongst coupled spin vertices in various lattices.In this paper, we extend such polariton condensates toan infinite chain model and present methods of control-lably producing different spin-ordered chains. We solveexactly and numerically analyze the stationary statesof the infinite chain of spin-bifurcated condensates withnearest-neighbor same-spin coupling in the tight bind- a r X i v : . [ c ond - m a t . qu a n t - g a s ] A p r ing approach. The stationary solutions correspond toferromagnetic, antiferromagnetic, and paired-spin orderstates of two-up and two-down spins (P) (see Fig. 1).States characterized by FM bonds with zero phase-slipand AFM bonds with π phase-slip are shown to have aminimum bifurcation threshold, and are stable againstlong-wavelength fluctuations. Monte-Carlo trials withadiabatic ramping of the pump intensity on a cyclic sys-tem of 4 condensates give a phase diagram in full agree-ment with the predicted minimum threshold winners asa function of coupling strength. This clear hierarchy forthe probability of formation is an important prerequi-site for a spin-lattice simulator. Non-adiabatic trials onthe other hand result in a complex phase diagram, as aresult of the initial condition progressing to its nearestphase space attractor. In addition to spatially uniformstationary states, we find that frustrated or defect states,with oscillating spinors can appear in this system.Condensation of the bosonic spin ± is regarded as the solid-state analog of cold-atom Bose-Einstein condensates .Its spin structure and strong interactions allow one torealize spinor condensates where macroscopic coherenceand superfluid character give birth to many intriguingphenomena such as the nonlinear optical spin Hall ef-fect , the formation of polariton half-vortices , andspontaneous symmetry breaking . They offer a newpath towards spin manipulation with already promis-ing results on spin-switches , and transistors .Driven-dissipative polariton condensates can be accu-rately modeled using a coherent macroscopic spinor orderparameter Ψ = (Ψ + , Ψ − ) T where Ψ ± are the spin-up andspin-down components respectively. Similarly to the pairof polariton condensates, where the transport of polari-tons from one condensate to another can be regarded asa form of coherent coupling in the tight-binding approxi-mation , a system of many condensates labeled by index n can be described by coupled dynamical equations i ˙Ψ n = − i g ( S n )Ψ n − i γ − i(cid:15) )ˆ σ x Ψ n + 12 (¯ αS n + αS zn ˆ σ z )Ψ n − J X h nm i Ψ m , (1) S n ≡ | Ψ n + | + | Ψ n − | , (2) S zn ≡ | Ψ n + | − | Ψ n − | , (3)where the sum is over nearest neighbors. Here we de-fine g ( S n ) ≡ − W + Γ + ηS n as the pumping-dissipationimbalance, Γ is the (average) dissipation rate, W is theincoherent in-scattering (or pump rate), and η definesthe gain-saturation nonlinearity . The birefringenceof the system corresponds to the splitting of the XY -polarized states in both energy ( (cid:15) ) and decay-rate ( γ ). The interaction parameters are written as ¯ α = α + α , α = α − α , where α and α are the same-spin andopposite-spin polariton-polariton interaction constants,respectively. Finally, < ( J ) > = ( J ) gives the dissipative cou-pling between the condensates . In particular, = ( J ) < W lin = Γ − γ , resulting in a linearlypolarized emission. At higher pump intensities the orderparameter bifurcates into either a spin-up or spin-downstate due to instability in the linearly polarized modesdue to their splitting ( (cid:15) + iγ ) and polariton-polariton in-teractions. For a single condensate, the critical bifurca-tion threshold is W bif = W lin + η (cid:15) + γ α(cid:15) . (4)In the following, we work above this critical pump thresh-old such that each condensate is either in a ‘spin-up’ ora ‘spin-down’ state.Formally, for identical lattice sites, the symmetry-conserving stationary solutions can be found by usingthe following spinor ansatz:Ψ n +1 = e iϕ n +1 Ψ n , (FM bonds) , (5)Ψ n +1 = e iϕ n +1 ˆ σ x Ψ n , (AFM bonds) , (6)where ( ϕ n +1 ) is the phase shift moving from the conden-sate in question to its nearest neighbor n + 1, and is tobe determined. Eq. 1 can now be written as: i ˙Ψ n = − i g + iω J )Ψ n − i γ − i(cid:15) J )ˆ σ x Ψ n + 12 (¯ αS n + αS zn ˆ σ z )Ψ n . (7)This corresponds to a single condensate with complexrenormalized splitting (cid:15) J and energy shift ω J , arisingfrom AFM bonds and FM bonds, respectively. Thestrength of these parameters depends on the relativephases between nearest-neighbors in the system. Themodified parameters of a chain with two nearest neigh-bors can be written as (cid:15) J = (cid:15) + J ( δ i e iϕ i + δ j e iϕ j ) , (8) ω J = − J ((1 − δ i ) e iϕ i + (1 − δ j ) e iϕ j ) , (9)where δ i = 1 , . The requirement for site-independent (cid:15) J and -1-0.50 (a) -1-0.50 (b) -4 -3 -2 -1 0 1 2 3 4-1-0.50 (c) Figure 2. (Color online) Stability analysis. Plot of Lyapunovexponents λ vs k -vector of fluctuations for 3 lowest bifurcationthreshold chain solutions (a-c) at W = 1 . W bif , J/(cid:15) = 0 . k the exponents stay negative corresponding to a stablesolution. ω J , and cyclic boundary condition of integer 2 π for thephase accumulated around the closed chain, restricts thepossible phases of the bonds. For chain systems of eitherFM or AFM ordering it can be shown that the couplingresults in (see Sec. E): (cid:15) AFM J = (cid:15) + 2 J cos (2 πm/N ) , ω AFM J = 0 , (10) (cid:15) FM J = 0 , ω FM J = − J cos (2 πm/N ) . (11)where m = 0 , , , . . . and N is the number of conden-sates in the chain. For AFM chains, N must be an evennumber since the spin unit-cell is |↑↓ i . In addition, thereis a paired-spin (P) state, where each site has one FMand one AFM bond, and the spin unit-cell is |↑↑↓↓ i . Psolutions have ω P J = ± J and (cid:15) P J = (cid:15) ± J , where the signsare independent. Due to the periodicity of the stationarysolutions, the essential physics of the spin-bifurcated con-densate chain system can be captured within a chain of4 condensates characterized by 10 distinct solutions (seeSec. A). Furthermore, we confirm the analogy betweenthe solutions of the tight binding model (Eq. 1) to a 4condensate chain accounting for the ( x, y ) spatial degreesof freedom (see Sec. F).To identify the stable stationary solutions, we performa long wavelength stability analysis (see Sec. B) for theset of coupled equations describing linear fluctuationsalong the periodic 4 condensate chain. Three lowest bi-furcation threshold solutions of FM, AFM, and P spinorder are found to have negative real-part Lyapunov ex-ponents λ within the first Brillouin zone of the chain(see Fig. 2), and hence are completely stable againstfluctuations travelling along the chain. These solutionsare characterized by a 0 phase slip between FM bonded Figure 3. (Color online) Phase diagram of spin order. Proba-bility of a spin state appearing in 10 realizations of numericalexperiment, where the pump is slowly ramped to a final value W . (a-c) AFM, P, and FM solutions with the lowest thresh-old make up 87% of the data. Black dashed lines in panel (c)indicate regimes where the solution becomes unstable. (d)A population of oscillating limit cycle solutions is noticeable(3.5%) for low coupling strengths. condensates and π phase slip between AFM bonded con-densates (see Sec. D for parameter values).To calculate the phase diagram and verify the analy-sis, we perform Monte-Carlo simulations of the periodic 4condensate chain as a function of pump intensity W andcoupling strength J . Fig. 3 shows the result of 10 Monte-Carlo (MC) trials at each site over a 100 ×
100 pixel mapin parameter space. For each realization of the numeri-cal experiment, the pump intensity is linearly increasedfrom W = 0 . W bif to W at a rate 10 − × W bif ps − , sim-ilar to the ramp-times achieved in experiments . Threedistinct phases corresponding to the stable AFM, P, andFM stationary spin patterns are observed, as identifiedby the long-wavelength stability analysis.To explain the regimes of each state (red areas inFig. 3), we plot the spin bifurcation threshold power W bif against J in Fig. 4. The thresholds are calculated usingEq. 4, with (cid:15) → < ( (cid:15) J ), γ → γ + = ( (cid:15) J ), Γ → Γ + = ( ω J ).As the pump power is slowly increased, the state thatreaches the bifurcation threshold first wins, since it hastime to stabilize before competing states can bifurcate.The calculated phase-boundaries of J/(cid:15) = 0 . , .
91 for
AFMPFM
Figure 4. (Color online) Spin-bifurcation threshold vs cou-pling strength J . W (0)bif is the spin-bifurcation threshold ofan uncoupled condensate. The arrows indicate points wherethe AFM solution changes to P ( J/(cid:15) = 0 .
42) and P changesto FM (
J/(cid:15) = 0 . (b)(a) Figure 5. (Color online) (a) Normalized pseudospins of thefour condensates in the limit cycle solution sampled from thedata in Fig. 3(d). Average spin has converged but multipleenergies cause the pseudospins to precess. Plotted trajectoriesare derived from the right panel and have not been scaled. (b)Time evolution of this state over 260 ps showing the steadyoscillation of the circular polarization component of the con-densates. the AFM-P and P-FM boundaries are in close agreementwith the Monte-Carlo simulations of Fig. 3. The asymp-totic behavior in Fig. 4 for the AFM and P states at
J/(cid:15) = 0 . J/(cid:15) = 1 respectively, is a consequenceof (cid:15) J approaching zero, destabilizing the stationary so-lution. We note that the ramp-time of the pump caninfluence the phase diagram. Fast ramp times soften thecompetitive advantage of a low spin-bifurcation thresh-old, resulting in a blurring of the phase-boundaries (seeSec. C). The depression at low W in Fig. 3(c) correspondsto an area of instability outlined by the black dashed linecalculated using linear stability analysis for fluctuationsat k = 0 (see Sec. B). We note that the MC iterations do not always resultin a stationary AFM, P, or FM steady-state describedabove. Nonstationary symmetry-breaking solutions canarise close to stability boundaries due to the finite ramprate of the pump. The analysis of highly nontrivial evo-lutions of the system in this case is beyond the scope ofthis work.In addition to the stationary states, an oscillating limitcycle solution composed of three spins against one op-posite spin is often observed for low J and high W , asshown in Fig. 3(d) and Fig. 5(a). A time-trace of the S z spin component of this state is plotted in Fig. 5(b).Though the average spin on each site has converged, thespin precession indicates a superposition of states thatare phase locked. Interestingly, the energy of the spinor-components of the limit cycle state correspond to thatof P state, ω J = − J and (cid:15) J = (cid:15) − J , except for the mi-nority spin population in the opposing condensate (e.g.,Ψ +4 polaritons from Fig. 5) which also populates a sep-arate peak in energy. Thus the limit cycle solution canbe characterized as a ‘frustrated P-state’, described bymultiple energies ω J and splittings (cid:15) J , and resulting inan oscillating spinor and frequency comb emission, sim-ilar to discussed in Ref. . In larger chains, the limitcycle states can appear as a result of inhomogeneity inthe chain couplings.In conclusion, we have solved analytically and investi-gated numerically solutions in an infinite chain of coupleddriven-dissipative spinor polariton condensates. A mix-ture of intra-spin coupling and nearest-neighbor inter-coupling allows not only controllable formation of an-tiferromagnetic states or ferromagnetic states, but alsoshows solutions with mixed antiferromagnetic and fer-romagnetic bonding. We find that minimum bifurca-tion threshold determines the spin order in the chain.The one-to-one correspondence between the spin and thephase-slips of the lowest threshold states makes this sys-tem binary and opens the possibility of mapping it to bi-nary models such as the 1D Ising Hamiltonian where theminimization of loss (bifurcation threshold) replaces min-imization of energy. Our work is an important step to-wards understanding and controlling spin order in open-dissipative nonlinear spin lattices. Acknowledgements. —This work was supported by theResearch Fund of the University of Iceland, The Ice-landic Research Fund, Grant No. 163082-051, grant EP-SRC EP/L027151/1, the Mexican Conacyt Grant No.251808, and the Singaporean MOE grants 2015-T2-1-055and 2016-T1-1-084. I.A.S. acknowledges support froma mega-grant № 14.Y26.31.0015 and GOSZADANIE №3.2614.2017/ ПЧ of the Ministry of Education and Sci-ence of Russian Federation Appendix A: Complete solutions of 4 condensate chain system
It has been established that stationary chain solutions of either FM or AFM ordering results in a modified single-condensate dynamical equation (Eq. 7) with only shifted parameters according to Eqs. 10-11. Stationary chainsolutions of mixed FM and AFM bonding (P solution) follow the same procedure but with only π or zero phase slipspossible between condensates. Focusing on a chain of 4 condensates (smallest cell to encompass all spin orderingsperiodically) we find 10 distinct solutions which are summarized in Fig. 6. As the number of condensates in the chainincreases, more solutions of FM or AFM ordering become available but the number of P solutions remains fixed. It’sworth mentioning that panel (c) and (f) are special cases where the phases between neighboring condensates resultin a cancellation such that ω J = 0 and (cid:15) J = (cid:15) in Eqs. 8-9. (c)(a) (b) (f)(d) (e) (g) (h)(i) (j) AFMFM P
Figure 6. (Color online) Schematic showing 10 solutions of a 4 condensate chain system. Red dashed lines depict in-phasecondensates ( ϕ i = 0) and blue dotted lines anti-phase condensates ( ϕ = ± π ). Orange and cyan whole lines in panels (c) and(f) correspond to phases causing a cancellation in the coupling, i.e., e iϕ i + e iϕ j = 0. Appendix B: Linear stability analysis
In this section we formulate the linear stability analysis for a periodic solution for an infinite chain of condensates.This solution is constructed by periodic repetition of a particular solution for the closed ring of four condensate, andit has, in general, the period 4 a , where a is the nearest-neighbor distance. The perturbed solution can be written as ψ ( m ) ± + δψ ( m ) ± , where m is the number of the condensate. The unperturbed solution is periodic, ψ ( m ) ± = ψ ( m +4) ± , andthe perturbation is chosen in the form of plane wave ( (cid:126) = 1) δψ ( m ) ± = u ( m ) ± e ikma + λt + v ( m ) ∗± e − ikma + λ ∗ t . (B1)Here the complex amplitudes are also set to be periodic, u ( m ) ± = u ( m +4) ± and v ( m ) ± = v ( m +4) ± . There are 16 linearizedequations for the amplitudes and 16 Lyapunov exponents λ ( k ). The solution is stable when all of them satisfy <{ λ ( k ) } ≤ U = { u (1)+ , v (1)+ , u (1) − , v (1) − , . . . , u (4) − , v (4) − } T can be written in matrix form as iλ U = M · U , where the 16 ×
16 matrix M can be presented in the 4 × M = M (1) M J , M J M J M (2) M J , , M J M (3) M J M J , M J M (4) . (B2)Here, M ( n ) is the matrix describing the fluctuations in the n -th condensate within the elementary cell, n = 1 , , , , is the 4 × M J is the same-spin coupling between nearest neighboring condensates.For the matrices in (B2) we have M ( n ) = M ( n ) E + M ( n ) W + M γ + M (cid:15) + M ( n ) α + M ( n ) α . (B3)The first matrix is defined by the energy of n -th condensate M ( n ) E = − E ( n ) E ( n ) − E ( n )
00 0 0 E ( n ) . (B4)The second matrix arises from the harvest and saturation rates of the condensate from the static reservoir M ( n ) W = − iη | ψ ( n )+ | + | ψ ( n ) − | ( ψ ( n )+ ) ψ ( n )+ ψ ( n ) ∗− ψ ( n )+ ψ ( n ) − ( ψ ( n ) ∗ + ) | ψ ( n )+ | + | ψ ( n ) − | ψ ( n ) ∗ + ψ ( n ) ∗− ψ ( n ) ∗ + ψ ( n ) − ψ ( n ) ∗ + ψ ( n ) − ψ ( n )+ ψ ( n ) − | ψ ( n ) − | + | ψ ( n )+ | ( ψ ( n ) − ) ψ ( n ) ∗ + ψ ( n ) ∗− ψ ( n )+ ψ ( n ) ∗− ( ψ ( n ) ∗− ) | ψ ( n ) − | + | ψ ( n )+ | − i G I , (B5) where G = Γ − W and I is the identity matrix. The third and the fourth matrices arise from coupling between upand down components M γ = iγ − − − − , M (cid:15) = (cid:15) − − . (B6)The fifth and the sixth matrices arise from the interactions: M ( n ) α = α | ψ ( n )+ | ( ψ ( n )+ ) − ( ψ ( n ) ∗ + ) − | ψ ( n )+ | | ψ ( n ) − | ( ψ ( n ) − ) − ( ψ ( n ) ∗− ) − | ψ ( n ) − | , M ( n ) α = α | ψ ( n ) − | ψ ( n )+ ψ ( n ) ∗− ψ ( n )+ ψ ( n ) − −| ψ ( n ) − | − ψ ( n ) ∗ + ψ ( n ) ∗− − ψ ( n ) ∗ + ψ ( n ) − ψ ( n ) ∗ + ψ ( n ) − ψ ( n )+ ψ ( n ) − | ψ ( n )+ | − ψ ( n ) ∗ + ψ ( n ) ∗− − ψ ( n )+ ψ ( n ) ∗− −| ψ ( n )+ | . (B7) The final matrix describes the same-spin coupling between the nearest-neighboring condensates M J = cos ( ka )2 − J J ∗ − J
00 0 0 J ∗ . (B8) Appendix C: Non-adiabatic Monte-Carlo for undamped 4 condensate chain
Here we give results analogous to Fig. 3 but with damping absent ( = ( J ) = 0) and instantaneous switching of thepump intensity at its mark value. Unlike Fig. 3 where states with the lowest bifurcation threshold were dominant,we uncover a more complex probability map in Fig. 7 through 30 MC trials bounded by their k = 0 stability regions(black dashed lines) predicted by Eq. B2.Fig. 7 shows 77% of the data divided between 6 solutions of the 4 condensate chain. The remaining 4 solutionsfrom Fig. 6 are not observed since they are unstable over the entire J − W map. Another 11% of the data (notshown here) ended in oscillating limit cycle states discussed in Fig. 5. The remaining 12% were categorized asnonstationary/chaotic.The complex features of the probability maps in Fig. 7 as opposed to the more simplistic ones in Fig. 3 highlight theimportant role of damping in the system and adiabatic switching of the pump intensity. Intuitively, the complicatedfeatures in Fig. 7 arise from the order parameter overshooting many possible stable minima in the phase space of thesystem. It then becomes a matter of the nearest and strongest attractor to stabilize the solution. Figure 7. (Color online) Colormaps showing the likelihood of a spin state appearing through 30 MC iterations for each pixelin a 100 ×
100 map using Eq. 1. Different from Fig. 3 we get a noticeable population in three more states (FM: ω J = 0. AFM: (cid:15) J = (cid:15) . FM: ω J = +2 J ). Black dashed lines are predicted stability boundaries calculated using Eq. B2 for k = 0. Appendix D: Numerical methods
A QR algorithm is implemented to solve the eigenvalue problem of Eq. B2. Eq. 1 is solved using a variable-orderAdams-Bashforth-Moulton predictor-corrector method. The parameters used for 0D simulations were: η = 0 .
02 ps − ;Γ = 0 . − ; (cid:15) = 0 .
04 ps − ; γ = 0 . (cid:15) ; α = 0 .
01 ps − ; α = − . α .The parameters used for 2D simulations of Eq. F1 were: m ∗ = 5 × − m ; η = 0 .
01 ps − ; Γ = 0 . − ; (cid:15) = 0 . − ; γ = 0 . (cid:15) ; α = 0 .
003 ps − ; α = − . α ; d = 12 µ m; σ = 10 . µ m. Where m is the free electron rest mass. Appendix E: Derivation of the coupling contribution in chain systems
Consider the stationary condensate chain composed entirely of either FM- or AFM bonds. According to Eqs. 8-9,each condensate with two nearest neighbors is presented with a term ω J = − J ( e iϕ i + e iϕ j ) , (E1)for two FM bonds or (cid:15) J = (cid:15) + J ( e iϕ i + e iϕ j ) , (E2)for two AFM bonds. Here ϕ i,j is the phase difference of moving from the condensate in question to its neighbor.This contribution can in general be a complex number appearing equally in each condensate. In this section we showthat this number must stay real for a chain system. The lattice unit cell of the chain system is one condensate andone bond. Assuming that the chain closes on itself, the number of free variables ( ϕ i ) is then equal to the number ofindependent equations. The following can then be generalized to any number of condensates in a chain.Let’s now imagine 4 condensates locked in a chain. We can classify the phase jumps going clockwise as { ϕ , ϕ , ϕ , ϕ } (see Fig. 8). It is obvious that ω J and (cid:15) J must be equal for all condensates in the chain in or-der to have a steady state. Thus the phase contribution e iϕ i + e iϕ j must be the same for all condensates. This meansthat the 4 stationary condensates allow us to write, e iϕ + e − iϕ = e iϕ + e − iϕ = e iϕ + e − iϕ = e iϕ + e − iϕ . (E3) Figure 8. Schematic of the 4 condensate chain. (Left) Phase jumps ϕ i take place moving from one condensate to the next.(Middle) Ψ gets a contribution e iϕ + e − iϕ . (Right) Ψ gets a contribution e iϕ + e − iϕ . Writing e iϕ i = a i + ib i where a i , b i ∈ R it’s then easy to show that, a = a , (E4) a = a , (E5) b = b = b = b . (E6)The real part of the contribution e iϕ i + e iϕ j is thus equal for each condensate but the imaginary part gets canceled.Consequently, from | e iϕ i | = 1 we come to the solution a = ∓ a , which can more clearly be written:cos ( ϕ ) = ∓ cos ( ϕ ) . (E7)The minus sign in Eq. E7 corresponds to a cancellation in the coupling with no shift in ω J or (cid:15) J whereas the plus signmandates the opposite. Applying the constraint e i ( ϕ + ϕ + ϕ + ϕ ) = 1 corresponding to a full cycle in our 4 condensatechain we come to the conclusion that the only possible values of coupling in the latter case are cos ( ϕ ) = cos (2 πm/N )where m = 0 , , , . . . and N is the number of condensates in the chain. As the number of condensates increases inthe chain, more solutions become available.The same procedure can be applied to a state where each condensate has one FM- and one AFM bond (P solutions).Then only e iϕ i = ± Appendix F: 4 condensate chain with spatial degrees of freedom
The tight binding model (Eq. 1) offers a simple solution to the stationary spin patterns in the condensate chain. Wefind that these exact solutions can also be produced with little difficulty accounting for the spatial degree of freedom.The complex Ginzburg-Landau equation can be written then : i ˙Ψ = 12 (cid:20) − i ( g ( S ) + γσ x ) + (1 − i Λ) (cid:18) ¯ αS + αS z σ z − (cid:15)σ x − (cid:126) ∇ m ∗ + g P P ( r ) (cid:19)(cid:21) Ψ . (F1)Here, m ∗ is the effective mass of the polaritons. The exciton reservoir is taken to be completely static and the inducedrepulsive potential is then given by an effective interaction constant g P . The remainder of the parameters serve thesame purpose here as in the tight binding model with W = P ( r ). We note that modeling the system by coupling anexciton reservoir rate equation to the order parameter only requires rescaling of the parameters and does not criticallyaffect the observed solutions in Eq. F1 when the decay rate of the reservoir is taken to be large compared to thepolariton lifetime .The pump P ( r ) is a 3 × d and with a FWHM σ which then form four potential minimum in a 2 × g P , or by increasing the strength ofthe center Gaussian pump spot causing an increased barrier between the condensates which effectively changes thecoupling strength J . Figure 9. (Color online) Density and phase maps of the AFM, P, and FM spin states from Eq. F1. Evaluating the phasedifference between the black crosses in each solution confirms that AFM bonds favor π phase difference and FM bonds 0 phasedifference. ∗ correspondence address: [email protected] J. Hopfield and D. Tank, Science , 625 (1986). N. N. Vtyurina, D. Dulin, M. W. Docter, A. S. Meyer,N. H. Dekker, and E. A. Abbondanzieri, Proceedings ofthe National Academy of Sciences , 4982 (2016). D. L. Stein and C. M. Newman,
Spin Glasses and Com-plexity (Princeton University Press, 2013). J.-P. Bouchaud, Journal of Statistical Physics , 567(2013). K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Ed-wards, J. K. Freericks, G.-D. Lin, L.-M. Duan, andC. Monroe, Nature , 590 (2010). H. Diep,
Frustrated spin systems (World Scientific, 2013). C. Nisoli, R. Moessner, and P. Schiffer, Rev. Mod. Phys. , 1473 (2013). J. Drisko, T. Marsh, and J. Cumings, Nature Communi-cations , 14009 EP (2017), article. J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,and D. C. Ralph, Phys. Rev. Lett. , 3149 (2000). M. M. Glazov and A. V. Kavokin, Phys. Rev. B , 161307(2015). J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Miesner,A. P. Chikkatur, and W. Ketterle, Nature , 345 (1998). T. C. H. Liew, A. V. Kavokin, and I. A. Shelykh, Phys.Rev. Lett. , 016402 (2008). C. Adrados, T. C. H. Liew, A. Amo, M. D. Martín, D. San-vitto, C. Antón, E. Giacobino, A. Kavokin, A. Bramati, and L. Viña, Phys. Rev. Lett. , 146402 (2011). M. Mezard, G. Parisi, and M. A. Virasoro,
Spin GlassTheory and Beyond (World Scientific, Singapore, 1986). A. Le Boité, G. Orso, and C. Ciuti, Phys. Rev. Lett. ,233601 (2013). N. G. Berloff, K. Kalinin, M. Silva, W. Langbein, andP. G. Lagoudakis, ArXiv e-prints (2016), arXiv:1607.06065[cond-mat.mes-hall]. T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto,and H. Takesue, Nat Photon , 415 (2016), article. M. D. Fraser, S. Hofling, and Y. Yamamoto, Nat Mater , 1049 (2016), commentary. D. Sanvitto and S. Kena-Cohen, Nat Mater , 1061(2016), review. T. Liew, I. Shelykh, and G. Malpuech, Physica E: Low-dimensional Systems and Nanostructures , 1543 (2011). D. Sanvitto and S. Kena-Cohen, Nat Mater , 1061(2016), review. B. Nelsen, G. Liu, M. Steger, D. W. Snoke, R. Balili,K. West, and L. Pfeiffer, Phys. Rev. X , 041015 (2013). H. Ohadi, A. Dreismann, Y. G. Rubo, F. Pinsker, Y. delValle-Inclan Redondo, S. I. Tsintzos, Z. Hatzopoulos, P. G.Savvidis, and J. J. Baumberg, Phys. Rev. X , 031002(2015). K. V. Kavokin, I. A. Shelykh, A. V. Kavokin, G. Malpuech,and P. Bigenwald, Phys. Rev. Lett. , 017401 (2004). H. Ohadi, Y. del Valle-Inclan Redondo, A. Dreismann, Y. Rubo, F. Pinsker, S. Tsintzos, Z. Hatzopoulos, P. Sav-vidis, and J. Baumberg, Phys. Rev. Lett. , 106403(2016). J. Kasprzak, M. Richard, S. Kundermann, A. Baas,P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H.Szymańska, R. André, J. L. Staehli, et al. , Nature ,409 (2006). R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, and K. West,Science , 1007 (2007). 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, Nature , 529 (2007). I. Carusotto and C. Ciuti, Rev. Mod. Phys. , 299 (2013). T. Byrnes, N. Y. Kim, and Y. Yamamoto, Nat Phys ,803 (2014), review. I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008). E. Kammann, T. C. H. Liew, H. Ohadi, P. Cilibrizzi,P. Tsotsis, Z. Hatzopoulos, P. G. Savvidis, A. V. Ka-vokin, and P. G. Lagoudakis, Phys. Rev. Lett. , 036404(2012). C. Antón, S. Morina, T. Gao, P. S. Eldridge, T. C. H.Liew, M. D. Martín, Z. Hatzopoulos, P. G. Savvidis, I. A.Shelykh, and L. Viña, Phys. Rev. B , 075305 (2015). K. G. Lagoudakis, T. Ostatnický, A. V. Kavokin, Y. G.Rubo, R. André, and B. Deveaud-Plédran, Science ,974 (2009). H. Ohadi, E. Kammann, T. C. H. Liew, K. G. Lagoudakis, A. V. Kavokin, and P. G. Lagoudakis, Phys. Rev. Lett. , 016404 (2012). I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. , 323 (2004). A. Amo, T. C. H. Liew, C. Adrados, R. Houdré, E. Gia-cobino, A. V. Kavokin, and A. Bramati, Nature Photonics , 361 (2010). A. Dreismann, H. Ohadi, Y. del Valle-Inclan Redondo,R. Balili, Y. G. Rubo, S. I. Tsintzos, G. Deligeorgis, Z. Hat-zopoulos, P. G. Savvidis, and J. J. Baumberg, Nature Ma-terials , 1074 (2016). D. Ballarini, M. De Giorgi, E. Cancellieri, R. Houdré,E. Giacobino, R. Cingolani, A. Bramati, G. Gigli, andD. Sanvitto, Nature Communications , 1778 EP (2013),article. J. Keeling and N. G. Berloff, Phys. Rev. Lett. , 250401(2008). I. L. Aleiner, B. L. Altshuler, and Y. G. Rubo, PhysicalReview B , 121301 (2012). K. Rayanov, B. L. Altshuler, Y. G. Rubo, and S. Flach,Phys. Rev. Lett. , 193901 (2015). M. Wouters and I. Carusotto, Phys. Rev. Lett. , 140402(2007). M. Wouters, T. C. H. Liew, and V. Savona, Phys. Rev. B , 245315 (2010). L. A. Smirnov, D. A. Smirnova, E. A. Ostrovskaya, andY. S. Kivshar, Phys. Rev. B89