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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 19 — Sep. 12, 2011
  • pp: 18091–18108
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Mapping of Ising models onto injection-locked laser systems

Shoko Utsunomiya, Kenta Takata, and Yoshihisa Yamamoto  »View Author Affiliations


Optics Express, Vol. 19, Issue 19, pp. 18091-18108 (2011)
http://dx.doi.org/10.1364/OE.19.018091


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Abstract

We propose a mapping protocol to implement Ising models in injection-locked laser systems. The proposed scheme is based on optical coherent feedback and can be potentially applied for large-scale Ising problems.

© 2011 OSA

1. Introduction

The emergence of optimization problems is ubiquitous in our modern life, for which we have to find the best solution by choosing the particular combination of M variables σ 1, σ 2, ... σM under some constraints. In most cases, such a mission can be formulated as a computational problem of minimizing a cost function E(σ 1, σ 2, ... σM). The mission is to find the specific values for those M variables for which the function E(σ 1, σ 2, ... σM) has the minimum value. If such an optimization problem can be solved in polynomial time, i.e. the computational time varies polynomially as a problem size M, by only nondeterministic machines, such a problem is said to belong to the class NP (NP for nondeterministic polynomial). Among NP problems, there are certain subsets of problems known as NP-complete problems. Any NP problems can be mapped to NP-complete problems using a polynomial step. The Graph Partition Problem (GPP) or MAX-CUT problem is a representative of the NP-complete problem. If one has an efficient machine to solve a certain NP-complete problem of size M, then one can solve any NP problems with an extra cost in time that scales only polynomially with M. However, problems in NP-complete class are considered to be computationally hard, since such a nondeterministic machine cannot be simulated by a deterministic Turing machine without an exponential growth of computational time [1

1. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W. H. Freeman and Company, 1979).

].

Quantum annealing (or quantum adiabatic computation) is proposed to solve Ising models by utilizing quantum uncertainty, more specifically quantum mechanical tunneling across a potential energy landscape (PEL) [6

6. P. Ray, B. K. Chakrabarti, and A. Chakrabarti, “Sherrington–Kirkpatrick model in a transverse field: absence of replica symmetry breaking due to quantum fluctuations,” Phys. Rev. B 39, 11828–11832 (1989). [CrossRef]

11

11. R. D. Somma and C. D. Batista, “Quantum approach to classical statistical mechanics,” Phys. Rev. Lett. 99, 030603 (2007). [CrossRef] [PubMed]

]. Experimental realization of quantum annealing employed either a sample of real magnetic crystal [12

12. J. Brooke, D. Bitko, T. F. Rosenbau, and G. Aeppli, “Quantum annealing of a disordered magnet,” Science 284(5415), 779–781 (1999). [CrossRef] [PubMed]

, 13

13. G. Aeppli and T. F. Rosenbaum, in Quantum Annealing and Related Optimization Methods, A. Das and B. K. Chakrabarti, eds. (Springer Verlag, 2005).

] or molecular NMR technique [14

14. M. Steffen, W. van Dam, T. Hogg, G. Breyta, and I. Chuang, “Experimental implementation of an adiabatic quantum optimization algorithm,” Phys. Rev. Lett. 90, 067903 (2003). [CrossRef] [PubMed]

]. However, in order to map a given mathematical problem onto such a quantum annealing Ising machine, non-local interaction Jij must be implemented artificially irrespective of actual distance between two sites. This is extremely hard to achieve in real crystals or molecules.

An Ising machine that harnesses the effect of Bose-Einstein condensation (BEC) and measurement-feedback control has been recently proposed [15

15. T. Byrnes, K. Yan, and Y. Yamamoto, “Optimization using Bose-Einstein condensation and measurement-feedback circuits,” arXiv:0909.2530v2.

]. The proposed machine utilizes the bosonic final state stimulation property of BEC to attain the speedup in cooling down to the ground state by a factor of bosonic particle number per site [16

16. K. Yan, T. Byrnes, and Y. Yamamoto, “Kinetic Monte Carlo study of accelerated optimization problem search using Bose-Einstein condensates,” Prog. Inform. 8, 1–9 (2011).

]. However, a measurement-feedback circuit which is required to implement nonlocal interactions Jij is an inherently incoherent device and cannot create the quantum coherence between different sites. The cost resulting from this fact is that the computational time scales exponentially as a problem size of M [16

16. K. Yan, T. Byrnes, and Y. Yamamoto, “Kinetic Monte Carlo study of accelerated optimization problem search using Bose-Einstein condensates,” Prog. Inform. 8, 1–9 (2011).

]. After all, the Ising model, itself, is NP-complete [17

17. F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15, 3241–3253 (1982). [CrossRef]

].

In this paper, we propose a new Ising machine based on one master laser and M mutually injection-locked slave lasers. An Ising model is implemented by coherent feedback network using optical interference circuits instead of incoherent electrical measurement-feedback circuits. A spin degree of freedom σiz at each site is represented by right or left circular polarization states of each slave laser. The ground state of an implemented Ising model which is represented with polarization configuration of M slave lasers emerges spontaneously through the natural mode competition induced by cross-gain saturation among all candidate polarization configurations. The cost function E(σ 1, σ 2, ..., σM) mentioned above corresponds to the over-all photon decay rate in our system. The injection-locked laser system oscillates with a specific polarization configuration which minimizes the cost function (photon decay rate).

A semiconductor laser is particularly attractive for this application because of its rapid intra-band spin relaxation, small saturation power, short photon lifetime and compact size. A rapid spin relaxation process realizes equally populated conduction electron spins and valence hole spins, so that the particular polarization configuration with a minimum photon decay rate can oscillate alone by suppressing the oscillation of all the other 2M – 1 polarization configurations. A saturation photon number ns (or inverse fractional spontaneous emission coupling efficiency 1β=ns) of a semiconductor laser is many orders of magnitude smaller than that of the other types of laser systems. Therefore, the above cross-gain saturation can be switched on by an extremely small injection power. A photon lifetime of typical semiconductor lasers is ∼ 1psec, which allows a very large injection-locking bandwidth even with a very small injection power. A stable and fast operation of this computing machine originates from this fact that an injection-locking bandwidth is extremely broad. Finally a large number of arrayed vertical cavity surface emitting semiconductor lasers (VCSEL) can be integrated into one chip. It is even possible to integrate hybridly those VCSELs with a master laser, other linear optical circuits and detectors, which makes a whole computing system very compact even for a large-scale Ising problem M ≫ 1.

2. Proposed system

Our goal is to map the following Ising Hamiltonian into an externally controllable physical system:
=(i<jJijσizσjz+iλiσiz).
(2)
In this Hamiltonian, σiz describes an Ising spin projection on z-axis. Jij is a mutual interaction term between spin i and spin j, and λi is a supplemental Zeeman (external magnetic field) term. The injection-locked laser system we propose in this paper can find the ground state of the Hamiltonian Eq. (2) by the spontaneous formation of a macroscopic population of photons in a specific polarization configuration through a laser phase transition. Every photon in the lasing mode is not localized in any specific slave laser but coherently spreads its wavefunction into all slave lasers as partial waves. The polarization state of each slave laser represents the ground state {σiz, i = 1 ∼ M}.

We use circular polarization degrees of freedom in VCSELs to represent the Ising spin information. An effective spin state σiz at site i is experimentally determined by a majority vote using photodetection signals at the final step of this computational scheme, i.e.,
σiz={1(ifnRi>nLi)1(ifnRi<nLi),
(3)
where nRi and nLi are the number of photons with right and left circular polarizations detected at site i (from i-th slave laser.) We assume that all slave lasers are driven by the same pump power.

A proposed injection-locked laser system is shown in Fig. 1. All slave lasers, which play a role of respective Ising sites, are injection-locked by a single master laser with vertical linear polarization. VCSELs can be used as slave lasers. An external cavity controlled diode laser (ECDL) can be used as a master laser. The injection-locking bandwidth is given by [18

18. S. Kobayashi and T. Kimura, “Injection locking in AIGaAs semiconductor laser,” IEEE J. Quantum Electron. 17(5), 681–689 (1981). [CrossRef]

, 19

19. S. Kobayashi, Y. Yamamoto, and T. Kimura, “Optical FM signal amplification and FM noise reduction in an injection locked AlGaAs semiconductor laser,” Electron. Lett. 17(22), 849–851 (1981). [CrossRef]

]
ΔωL=ωQPinPout,
(4)
where ωQ is the cavity photon decay rate of the slave laser and on the order of ∼1012 ( 1s) for the case of a VCSEL, Pin is the injection signal power from the master laser into the slave laser and Pout is the self-oscillation power of the slave laser. We assume that the linewidth enhancement factor (or anomalous dispersion parameter) is zero for simplicity [20

20. L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990). [CrossRef] [PubMed]

]. The locking bandwidth is on the order of 0.1 ∼ 1GHz in the later discussions on numerical examples.

Fig. 1 A proposed injection-locked laser system for finding the ground state of an Ising model Eq. (2). A master laser output is equally split into M paths and injected into M slave lasers via an optical isolator. At a time t < 0 (initialization), the injection signal from the master laser has a vertical linear polarization so that all slave lasers are initialized in vertical linear polarization states |V1|V2 ... |VM. At a time t = 0, the combined attenuator, HWP and QWP can implement the Zeeman term λi. Also at a time t = 0, each slave laser output is injected to other slave lasers via a horizontal linear polarizer, phase shifter, and attenuator but without an isolator. This mutual injection-locking can implement the Ising interaction term Jij. After a steady state condition is reached, the two polarization components of each slave laser are detected by a polarization beam splitter (PBS) and two photodetectors.

A master laser output is split into M paths and injected into M slave lasers. Linear optical circuits can implement the Zeeman term and Ising interaction term in Eq. (2). To implement the Zeeman term λi, the master laser output is injected into the i-th slave laser with a weak horizontal linear polarization component as well as a strong vertical polarization component. The amplitude and phase of the horizontally polarized injection signal is controlled with half-wave plates (HWP) and quarter-wave plates (QWP) as shown in Fig. 1. The Ising interaction term between site i and site j is implemented with the mutual injection of the slave laser outputs via an attenuator, phase shifter and horizontal linear polarizer. In total, 12M(M1) paths must be optically connected between all slave lasers in the worst case. The details of the above implementation scheme will be presented in Sec. 7.

Two circular polarization modes in all slave lasers have an identical phase due to injection-locking by the master laser with vertical linear polarization. Those 2 × M modes are coherently connected by mutual injection-locking among all slave lasers. Before switching the Hamiltonian Eq. (2) on at t = 0, the entire slave laser systems are prepared in 2M linear superposition states,
|Ψt=0=12(|R+|L)112(|R+|L)212(|R+|L)M=12M(|R1|R2|RM++|L1|L2|LM),
(5)
where |R1 and |L1 represent the orthogonal polarization states (the right and left circular polarizations) which exist in the slave laser 1. Those 2M different polarization configurations compete with each other through the cross gain saturation. When the whole slave laser systems reach a steady state condition after the Hamiltonian Eq. (2) is turned on, it is expected that each photon in the M slave laser systems is in a specific polarization configuration with a high probability, such as |R1|L2 ... |RM as shown in Fig. 2, which has the minimum overall loss among 2M polarization configurations. In order to readout the computational result, we can access each partial wave existing in a specific slave laser, as indicated in Fig. 1. Then the computational result can be determined by a majority vote for each slave laser as in Eq. (3).

Fig. 2 Each single photon occupies M slave lasers simultaneously as its partial waves. Each slave laser starts from the state, |V=12(|R+|L)), and ends in the state |R〉 or |L〉 with a high probability, which is the computational result. This computational process is described by the mode competition or amplitude modulation in the |R〉-|L〉 basis and by the mutual interference or phase modulation in the |D〉-|〉 basis.

3. Theoretical model

We will introduce the coupled rate equations for analyzing our Ising machine, starting with the quantum mechanical Langevin equation for an injection-locked laser [20

20. L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990). [CrossRef] [PubMed]

, 21

21. H. A. Haus and Y. Yamamoto, “Quantum noise of an injection-locked laser oscillator,” Phys. Rev. A 29, 1261–1274 (1984). [CrossRef]

]. The Heisenberg-Langevin equation for a (non-Hermitian) cavity field operator is described as below,
ddtA^(t)=iωrA^(t)12[ωQωμ2(χ˜iiχ˜r)]A^(t)+ωQF0eiωt+f˜G+f^L.
(6)
Here the photon field operators are designated by hats and the electric population operators are denoted by tildes. The electric dipole operator is adiabatically eliminated by assuming the electric dipole moment decay rate is faster than the photon decay rate and the electron population decay rate [22

22. M. Surgent, M. O. Scully, and W. E. Lamb, Laser Physics (Westview Press, 1978), Chap. 20, pp. 331–335.

, 23

23. H. Haug, “Quantum-mechanical rate equations for semiconductor lasers,” Phys. Rev.184, 338–348 (1969). [CrossRef]

]. ωQ describes the cavity photon loss rate through an output coupling mirror (we neglect an internal loss rate) and μ is a non-resonant refractive index. ωr is the empty cavity resonance frequency of the slave laser, while ω is the frequency of the injection signal. The operator χ˜i represents the net stimulated emission gain and χ˜r represents the nonlinear dispersion, both of which include the electron population operator. G is the Langevin noise operator for the electric dipole moment which is associated with random photon emission and absorption by the gain medium. L is the Langevin noise operator for the cavity field, which is associated with the injection signal noise including a vacuum fluctuation. F 0 is a c-number injection signal amplitude.

The net gain operator ωμ2χ˜i=E˜CVE˜VC is composed of the photon emission rate operator CV into a lasing mode and absorption rate operator VC. We assume an absorption loss is negligible, CVVCCV, here after. Using the relation between a photon number operator and field operator as (t) = Â (t)Â(t) in Eq. (6), a quantum mechanical rate equation for photon number operator (t) is derived as follows [20

20. L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990). [CrossRef] [PubMed]

, 23

23. H. Haug, “Quantum-mechanical rate equations for semiconductor lasers,” Phys. Rev.184, 338–348 (1969). [CrossRef]

]:
ddtn^(t)=ωQn^(t)+E˜CVn^(t)+E˜CV+ωQ(F0*A^(t)+A^(t)F0)+F^n(t).
(7)
Here the noise term n(t) for the photon number operator satisfies the following two-time correlation function [23

23. H. Haug, “Quantum-mechanical rate equations for semiconductor lasers,” Phys. Rev.184, 338–348 (1969). [CrossRef]

]:
F^n(t)F^n(s)=δ(ts)[ωQn^+E˜CV(n^+1)]
(8)
where we assume that the injection signal noise is equal to the vacuum fluctuation level. If the injection signal has a squeezed noise, the first term of R. H. S. of Eq. (8) is replaced by ωQn^κ where κ is a squeezing parameter and κ < 1 (or κ > 1) represents the amplitude (or phase) squeezed injection signal [20

20. L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990). [CrossRef] [PubMed]

].

Quantum mechanical rate equation for the total electron number operator Ñ in a slave laser is [20

20. L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990). [CrossRef] [PubMed]

, 23

23. H. Haug, “Quantum-mechanical rate equations for semiconductor lasers,” Phys. Rev.184, 338–348 (1969). [CrossRef]

],
ddtN˜(t)=PN˜(t)τspE˜CVn^(t)E˜CV+F˜c(t),
(9)
where P is an average pump rate, N˜τsp is a total spontaneous emission term, τsp is a spontaneous emission lifetime. The noise term c(t) for the electron number operator is also characterized by [20

20. L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990). [CrossRef] [PubMed]

, 23

23. H. Haug, “Quantum-mechanical rate equations for semiconductor lasers,” Phys. Rev.184, 338–348 (1969). [CrossRef]

]
F˜c(t)F˜c(s)=δ(ts)[P+N˜τsp+E˜CV(n^+1)].
(10)
The two noise operators n and c are negatively correlated, i.e. [20

20. L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990). [CrossRef] [PubMed]

, 23

23. H. Haug, “Quantum-mechanical rate equations for semiconductor lasers,” Phys. Rev.184, 338–348 (1969). [CrossRef]

]
F^n(t)F˜c(t)=δ(ts)[E˜CV(n^+1)].
(11)
If we define β as a fractional coupling efficiency of spontaneous emission into a lasing mode, then a photon emission rate operator into a lasing mode is redefined as E˜CV=βN˜τsp. A typical value of β is in the range of β = 10−5 ∼ 10−4 for VCSELs [24

24. S. F. Yu, Analysis and Design of Vertical Cavity Surface Emitting Laser (Wiley-Interscience, 2003), Chap. 8. [CrossRef]

].

When the noise operator L in Eq. (6) represents a vacuum fluctuation, this means the quantum state of the injection signal is a coherent state. In a such a case, the amplitude (or photon number) noise of the injection-locked slave laser output is squeezed to below the standard quantum limit (SQL) under quiet electrical pumping [25

25. Y. Yamamoto, S. Machida, and O. Neilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986). [CrossRef] [PubMed]

], while the phase noise is above the SQL due to random walk phase diffusion [20

20. L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990). [CrossRef] [PubMed]

]. The product of the photon number noise and the phase noise is close to the Heisenberg minimum uncertainty product [20

20. L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990). [CrossRef] [PubMed]

]. That is, the output field of the injection-locked semiconductor laser is close to the minimum uncertainty wavepacket. In such a case, the amplitude and phase noise of the output-field as well as the electron number noise are small compared to their average values and we can safely neglect the quantum noise in the first-order approximation and replace the operators Â(t), (t) and (t) by the corresponding c-numbers. Due to this approximation, we lose the information about the small quantum noise but can still describe the ensemble averaged measurement results for our Ising machine.

We now expand the c-number photon field amplitude as
A(t)=A0(t)ei[ωt+ϕ0(t)],
(12)
where A 0(t) is the slowly varying amplitude (positive real) and ϕ 0(t) is a slowly varying phase. Substituting Eq. (12) into the classical analog of Eqs. (6) and (7) in which the operators  and are replaced by the corresponding c-numbers and the noise terms are neglected, we obtain the following three equations:
ddtA0(t)=12[(ωQ)ECV]A0(t)+ωQF0cos[ϕ0(t)],
(13)
ddtn(t)=(ωQECV)n(t)+ECV+2ωQF0A0(t)cos[ϕ0(t)],
(14)
ddtϕ0(t)=(ωω0)ωQF0A0(t)sin[ϕ0(t)],
(15)
where ω0=ωr+ω2μ2χr is the self-oscillation frequency of the slave laser, ECV = 〈CV〉 and χr = 〈χ˜r〉. Notice that the third term ECV of R. H. S. in Eq. (14) represents the spontaneous emission rate into the lasing mode induced by the quantum noise [22

22. M. Surgent, M. O. Scully, and W. E. Lamb, Laser Physics (Westview Press, 1978), Chap. 20, pp. 331–335.

, 23

23. H. Haug, “Quantum-mechanical rate equations for semiconductor lasers,” Phys. Rev.184, 338–348 (1969). [CrossRef]

], which we keep in our analysis. The injection-locking bandwidth Eq. (4) is derived from Eq. (15) by requiring |sin[ϕ 0(t)]| ≤ 1 and noting Pin = h̄ω|F 0|2 and Pout=ωQh¯ω|A0(t)|2. When ω 0ω, the slave laser phase ϕ 0(t) is shifted away ( π2<ϕ0(t)<π2) from the master laser phase ϕM = 0. In the following discussions, we assume ω = ω 0 so that the slave laser phase is identically equal to the master laser phase ϕ 0(t) = ϕM = 0.

Since the phase of a slave laser is locked to that of the dominant vertically polarized master laser signal during an entire computational time, the phases of right and left circular polarization modes in all slave lasers are identical to that of the master laser. Then, the last term of R. H. S. of Eq. (14) can be rewritten as 2ωQn(t)F0. If an injection signal is only from the master laser with vertical linear polarization F0=ζωQMnM, where ωQM is the master laser photon decay rate, nM is the average photon number of the master laser and ζ is an amplitude attenuation coefficient including a factor of 12 for the projection factor form the vertical linear polarization to the right (or left) circular polarization. For simplicity, we assume that the cavity quality factor of the master laser QM is equal to that of the slave lasers, QM = Q. Then Eq. (14) can be simplified as,
ddtn(t)=(ωQ)n(t)+ECVn(t)+ECV+2ωQn(t)ζnM.
(16)

We now consider two sets of coupled rate equations for the right and left circularly polarized modes of each slave laser. In a proposed injection-locked laser system before the Hamiltonian Eq. (2) is turned on (t < 0), the rate equations are
ddtnRi=(ωQECVRi)nRi+ECVRi+2ωQnRiζnM,
(17)
ddtnLi=(ωQECVLi)nLi+ECVLi+2ωQnLiζnM,
(18)
where i designates a specific slave laser. The vertically polarized injection signal from the master laser into the slave laser i achieves the proper initialization, i.e., nRi = nLi and ϕRi = ϕLi = 0.

The right and left circularly polarized modes couple to the conduction electrons with opposite spin angular momenta ( Je=±12) and the valence holes with opposite orbital and spin angular momenta ( Jh=±32) due to the selection rule [26

26. C. Weisbuch and B. Vinter, Quantum Semiconductor Structures: Fundamentals and Applications (Academic Press, 1991), Chap. 3, pp. 65–69.

]. We assume the active quantum well layer is doped with acceptor impurities heavily, so that the gain is governed solely by the minority carrier, conduction electrons. If we denote the electron number of two spin components by NRi and NLi, we have
ddtNRi=P2NRi(t)τspECVRinRi(t)ECVRiNRi(t)NLi(t)τspin,
(19)
ddtNLi=P2NLi(t)τspECVLinLi(t)ECVLiNLi(t)NRi(t)τspin.
(20)
Here we assume that the pump current injects equal number of electrons with spin-up and spin-down. τspin is the spin relaxation time in the conduction band. Note that a short τspin contributes to equalize the populations of two spin components. Our proposed Ising machine relies on this very fast spin relaxation process in VCSELs at room temperature. When τspin is much shorter than the spontaneous emission lifetime τsp and the stimulated emission lifetime τst [20

20. L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990). [CrossRef] [PubMed]

], Eqs. (19) and (20) are reduced to the single rate equation for the total electron number Ni = NRi + NLi:
ddtNi(t)=PNi(t)τspECVi[nRi(t)+nLi(t)+2].
(21)
Since NRi = NLi, it follows that ECVRi = ECVLi = ECVi.

At t = 0, we inject a small amount of horizontally polarized signal to the slave laser i from the master laser to implement the Zeeman term λi in Eq. (2). At t = 0, we also inject a small amount of horizontally polarized signal from a slave laser i to a slave laser j and vice versa to implement the Ising interaction term Jij in Eq. (2). Then the rate equations for nRi and nLi after t = 0 are given as follows,
ddtnRi=(ωQECVi)nRi+ECVi+2ωQnRi[(ζηi)nMji12ξij(nRjnLj)]
(22)
ddtnLi=(ωQECVi)nLi+ECVi+2ωQnLi[(ζ+ηi)nM+ji12ξij(nRjnLj)].
(23)
Here an amplitude attenuation coefficient is ηi for the horizontally polarized injection signal from the master laser. The mutual injection term is optically implemented by inserting a horizontal linear polarizer, attenuator and phase shifter in the optical path between two slave lasers i and j. The amplitude attenuation coefficient is 12ξij for the horizontally polarized signal. ηi is positive and real when the phase of the horizontal polarization component is advanced by π2 with respect to the vertical polarization component. In this case, the overall injection signal from the master laser is in a left circular elliptic polarization. ξij is positive and real when the two slave lasers are connected by π phase difference. The detailed explanation for implementation of the Ising model will be presented in Sec. 7.

4. Mapping protocol

When the system reaches a steady state condition after ηi and ξij are switched on at t = 0, ddtnRi=ddtnLi=0 and ddtNi=0 should hold in Eqs. (21)(23). Then, we can solve for ECVi for the sum of Eqs. (22) and (23) as follows,
ECVi=ωQ2ωQζnM(nRi+nLi)nRi+nLi+2ωQnRinLinRi+nLi[ηinMnRi+nLi+ji12ξijnRjnLjnRi+nLi].
(24)
Here we neglect the spontaneous emission term ECVi, the third terms of R.H.S. in Eqs. (22) and (23), which is much smaller than the stimulated emission term ECVinRi (or ECVinLi), the second terms of R. H. S. in Eqs. (22) and (23) where nRi and nLi are of the order of 104 ∼ 105.

We expect that the coherently coupled slave laser systems oscillate with a polarization configuration which minimizes the overall loss by optimizing the polarization configuration (σ 1, σ 2, ... σM). The minimized overall loss is identical to the threshold gain, Σi=1MECVi. In a standard laser oscillator, there are numerous cavity modes with different eigen-frequencies within a gain bandwidth, so that those multi-modes can oscillate simultaneously if a gain medium is inhomogeneously broadened. However, if we differentiate the loss rates among those cavity modes, for instance by placing a mode (or frequency) selective element inside a cavity and a gain medium is homogeneously broadened, a particular single cavity mode with a minimum loss oscillates alone via cross-gain saturation. We expect the same single mode oscillation is realized in the mutually injection-locked laser systems, which is indeed confirmed by the numerical simulations discussed later.

The first and second terms of R.H.S. of Eq. (24) are almost independent of the polarization configurations, so that the spontaneously selected polarization configuration by the single lasing mode in the entire injection-locked laser systems is expected to minimize
inRinLinTi[ηinMnTi+ji12ξijnRjnLjnTi],
(25)
where nTi = nRi + nLi.

If we define an effective Ising spin by σiz=nRinLinTi in Eq. (25), the proposed systems oscillate with the polarization configuration that minimizes the quantity
jηinMnTiσiz+i<jξijσizσjz,
(26)
where −1 ≤ σiz ≤ 1. If we interpret ηinMnTi as a Zeeman term λi and ξij as an Ising interaction term Jij, it is concluded the proposed injection-locked laser systems can find the ground state of an Ising model Eq. (2). When nM = nTi, the two parameters ηi and ξij in the rate equations Eqs. (22) and (23) are determined by the relations:
ηi=αλimax[|Jij|,|λi|],
(27)
ξij=αJijmax[|Jij|,|λi|],
(28)
where α and α′ are extra attenuation parameters that are chosen as small quantities (0 < α ≪ 1, 0 < α′ ≪ 1) to ensure the stable operation of a whole system.

In the numerical simulation below (Figs. 35), we have used these two parameters α and α′ which determine the attenuation coefficients for the two injection signals into the slave laser, one from the master laser and the other from other slave lasers. We numerically found that α′ must be equal to 2α in order to obtain the correct ground state.

Fig. 3 The time evolution of the average photon numbers nRi and nLi. A fractional spontaneous emission coupling efficiency is β = 10−5, an injection current level is I = 32mA ( IIth2) and attenuation parameters are α=1200 and ζ=1500. (a) M = 2, (J 12, λ 1, λ 2)=6, 1, 910, (b) M = 5, (J 12, J 13, J 14, J 15, J 23, J 24, J 25, J 34, J 35, J 45, λ 1, λ 2, λ 3, λ 4, λ 5)=(6, 112, 5, 92, 4, 72, 3, 52, 2, 32, 1, 910, 910, 910, 910), (c) M = 7, (J 12, J 13, J 14, J 15, J 16, J 17, J 23, J 24, J 25, J 26, J 27, J 34, J 35, J 36, J 37, J 45, J 46, J 47, J 56, J 57, J 67, λ 1, λ 2, λ 3, λ 4, λ 5, λ 6, λ 7) = (6, 529, 509, 163, 469, 449, 143, 409, 389, 4, 349, 329, 103, 289, 269, 83, 229, 209, 2, 169, 149, 1, 910, 910, 910, 910, 910, 910) and (d) M = 10, (J 12, J 13, J 14, J 15, J 16, J 17, J 18, J 19, J 110, J 23, J 24, J 25, J 26, J 27, J 28, J 29, J 210, J 34, J 35, J 36, J 37, J 38, J 39, J 310, J 45, J 46, J 47, J 48, J 49, J 410, J 56, J 57, J 58, J 59, J 510, J 67, J 68, J 69, J 610, J 78, J 79, J 710, J 89, J 810, J 910, λ 1, λ 2, λ 3, λ 4, λ 5, λ 6, λ 7, λ 8, λ 9, λ 10) = (6, 539, 529, 173, 509, 499, 163, 479, 469, 5, 449, 439, 143, 419, 409, 133, 389, 379, 4, 359, 349, 113, 329, 319, 103, 299, 289, 3, 269, 259, 83, 239, 229, 73, 209, 199, 2, 179, 169, 53, 149, 139, 43, 119, 109, 1, 910, 910, 910, 910, 910, 910, 910, 910, 910), where ξij=αJijmaxijM[|Jij|,|λi|] and ηi=αλ1maxijM[|Jij|,|λi|] for each M.
Fig. 4 The time evolution of the average electron number Ni for (a) M = 2 to (d) M = 10. The numerical parameters are identical to those in Fig. 3.
Fig. 5 The time evolution of the photon numbers nRi and nLi in a three site problem. Numerical parameters are β = 10−5, Ith = 16mA, IIth2, α=1350 and ζ=1200. In (a) and (c) the Ising Hamiltonian is given with (J 12, J 13, J 23, λ 1, λ 2, λ 3) = (1, 1, 0, 110, 130, 110), for which the ground state is (σ 1, σ 2, σ 3) = (1, −1, −1). In (b) and (d) the Ising Hamiltonian is given with (J 12, J 13, J 23, λ 1, λ 2, λ 3) = (1, 1, 0, 15, 110, 120), for which the ground state is (σ 1, σ 2, σ 3) = (−1, 1, 1). (a)(b) The numerical results without photon number noise. (c)(d) The numerical results with the Poissonian photon number noise. A Gaussian distributed noise term with a zero mean and standard distribution of nRi or nLi is added at every time step Δt = 10−12 sec in the Range-Kutta numerical integration.

5. Numerical simulation results

In order to confirm the validity of the proposed Ising machine based on mutually injection-locked laser systems, we perform a series of numerical simulations directly using Eqs. (21), (22) and (23). For the numerical integration of Eqs. (21), (22) and (23), we used the fourth-order Runge-Kutta (RK4) method. The time step is fixed at Δt = 10−14 sec, which is sufficiently short compared to any time constants appearing in Eqs. (21), (22) and (23). For a certain parameter range {Jij, λi, M}, we obtained the correct ground states. Since we neglect the quantum noise terms in the quantum mechanical rate equations Eqs. (7) and (9), the numerical results shown in this section are for the ensemble averaged quantities. These numerical simulation results are shown in Figs. 3(a)–3(d) and Figs. 4(a)–4(d). The site number M is varied from M = 2 to M = 10, where the Ising model parameters λi and Jij are chosen so that the given Ising model has local minima separated from the ground state by a macroscopic Hamming distance (number of different spins between two spin configurations). We got the right answers for all cases from M = 2 to M = 10.

In Fig. 3, the average photon numbers in two polarization modes are equal, nRi = nLi, at t = 0, due to the initialization induced by the vertically polarized injection-locking signal from the master laser. After a short time delay less than 1nsec, the average photon numbers in two polarization modes depart with each other due to the effect of mutual injection locking among slave lasers. The system reaches a steady state with a delay time of a few nsec. The physical origin for this delay time is the inverse of locking-bandwidth and will be discussed elsewhere [27

27. K. Takata, S. Utsunomiya, and Y. Yamamoto, “Transient time of an Ising machine based on injection-locked lasers: contribution of locking bandwidth and Zeeman component,” (to be submitted).

]. We also notice that the average electron numbers Ni shown in Fig. 4 decrease and reach the steady state values in the same time delay for all slave lasers. This fact confirms our claim that the proposed Ising machine spontaneously selects the specific polarization configuration which minimizes the overall loss and thus achieves the minimum threshold gain Σi ECVi.

We notice that the transient time to reach a steady state does not show a strong dependence on the problem size M. It is nearly constant and on the order of a few nsec for M = 2 to M = 10. However, we do not confirm yet if this encouraging result is valid or not for a larger problem size.

The proposed Ising machine also reports a wrong answer for a specific parameter case. Such an example is shown in Fig. 5(a), where the state (σ 1z, σ 2z, σ 3z) = (−1, 1, 1) (= (L 1, R 2, R 3)) is chosen by our Ising machine, even though the correct answer is (1, −1, −1). The Zeeman term ηi in Eqs. (22) and (23) drives the photon numbers nRi and nLi to certain directions immediately after the term is switched on. On the other hand, the Ising interaction term ξij in Eqs. (22) and (23) does not drive nRi and nLi initially because of the initial condition nRi = nLi at t = 0. The Zeeman term is more powerful than the Ising term in the beginning of computation. If the system without quantum noise sources is trapped at a local minimum rather than the global minimum (ground state), the system cannot escape from the wrong answer. However the real injection-locked laser systems have inherent quantum noise sources, by which the system has a possibility to escape from the local minimum or the system is dragged toward the global minimum from the beginning. Figure 5(c) shows the numerical results for the same parameter as Fig. 5(a) when we introduce the Poissonian photon number fluctuation to nRi and nLi by adding the Gaussian random numbers in the numerical integration of Eqs. (21), (22) and (23). We recover the correct answer with a finite probability as shown in Fig. 5(c). As a comparison we show other examples that the system can find the ground state (−1, 1, 1) for both cases of with and without Poissonian noise in Figs. 5(b) and 5(d), respectively.

6. Alternative picture of the proposed Ising machine

So far we have described the operational principle of the proposed Ising machine in terms of the probability amplitude modulation and cross gain saturation between right and left circular polarization modes in slave lasers. That is, the probability amplitudes of 2M polarization configurations are initially C(σ1,σ2,σM)=12M but at the end of computation,
C(σ1,σ2,σM)={1(forthegroundstate)0(foralltheotherstates).
(29)
The state evolution of each slave laser, shown in Fig. 2, suggests that if we expand the states of the partial waves in terms of two diagonal linear polarization states, |D=12(|V+|H) and |D¯=12(|V|H), instead of two circular polarization states |R=12(|Vı|H) and |L=12(|V+ı|H), the same operational principle should be described in terms of mutual phase modulation and interference between the two diagonal basis states. Indeed, this picture turns out to be true.

We can alternatively expand the partial wave in a slave laser i by the diagonal linear polarization basis, |D〉 and |〉, as shown in Fig. 2:
ADi(t)=ADi0(t)exp{ı[ωt+ϕDi(t)]},
(30)
AD¯i(t)=AD¯i0(t)exp{ı[ωt+ϕD¯i(t)]}
(31)
Substituting Eqs. (30) and (31) into the classical analog of Eqs. (7), we obtain the following equations for the slowly varying amplitude and phase:
ddtADi0(t)=12(ωQECVi)ADi0(t)+ωQnMζ2+ηi2cos[δϕDi(t)]ji12ξijωQ{nDjcos[ϕDj(t)ϕDi(t)]nD¯jcos[ϕD¯j(t)ϕDi(t)]},
(32)
ddtϕDi(t)=(ωω0)+ωQ1QDi0(t){nMζ2+ηi2sin[δϕDi(t)]ji12ξij[nDjsin(ϕDj(t)ϕDi(t))nD¯jsin(ϕD¯j(t)ϕDi(t))]},
(33)
and the corresponding equations of motion for A D̄i0(t) and ϕD̄i(t). Here we assume again Q = QM and δ=tan1(ηiζ). We can numerically integrate those four equations for ADi 0(t), AD̄i 0(t), ϕDi(t) and ϕD̄i(t) together with the corresponding equation of motion for Ni(t),
ddtNi(t)=PNi(t)τspECVi[nDi(t)+nD¯i(t)+2],
(34)
where nDi(t) = A Di0(t)2 and nD̄i(t) = AD̄i 0(t)2.

Figures 6(a) and 6(b) show the average amplitudes ADi 0 and AD̄i 0, and phase ϕDi and ϕD̄i in the two diagonal polarization modes for the three sites (M = 3) problems, calculated by Eqs. (32), (33) and (34). As expected from the picture shown in Fig. 2, A Di0 = A D̄i0 is satisfied at all time (no amplitude modulation between A Di0 and A D̄i0,) but ϕDi and ϕD̄i depart with each other in a time scale shorter than 1nsec and this phase modulation reaches the steady state within a few nsec. The slight and simultaneous increase in the amplitudes ADi 0 and AD̄i 0 results from the overall reduction in the photon loss rate. If we project the complex amplitudes ADi(t) and AD̄i(t) given by Eqs. (30) and (31) onto the photon numbers nRi and nLi in circular polarization basis, the two complex amplitudes interfere with each other and we can recover the amplitude modulation and mode competition behaviour between two circular polarization modes:
nRi(t)=|(1+ı)2ADi0(t)exp[ıϕDi(t)]+(1ı)2AD¯i0(t)exp[ıϕD¯i(t)]|2,
(35)
nLi(t)=|(1ı)2ADi0(t)exp[ıϕDi(t)]+(1+ı)2AD¯i0(t)exp[ıϕD¯i(t)]|2.
(36)
The numerical result is shown in Fig. 6(c). Figure 6(d) shows the numerical simulation results using Eqs. (21), (22) and (23) of the |R〉 and |L〉 basis. We confirm that the two curves shown in Figs. 6(c) and 6(d) are completely identical. The two complementary pictures, the mode competition between |R〉 and |L〉 states and the mutual interference between |D〉 and |〉 states, can explain the same operational principle of the proposed Ising machine equally well.

Fig. 6 (a)The time evolution of the slowly varying amplitudes ADi 0 and AD̄i 0. (b)The time evolution of the slowly varying phases ϕDi and ϕD̄i. The phases of ±π4 correspond to complete circular polarizations. (c)The time evolution of the photon number nRi and nLi calculated in the |D〉 and |〉 basis. (d)The time evolution of the photon number nRi and nLi calculated in the |R〉 and |L〉 basis. The numerical parameters are M = 3, (J 12, J 13, J 23, λ 1, λ 2, λ 3) = (6, 3, 2, 310, 110, 110), α=1250 and η=1200.

7. Implementation

In this section we describe how to implement the Ising model Eq. (2) in injection-locked laser systems. As shown in Fig. 1, the output from the master laser is equally split and injected into all slave lasers via an optical isolator. The master laser output is used for two different purposes. One is to initialize all slave lasers in a linear superposition of right and left circular polarization states or the two diagonal linear polarization states, i.e., in a vertical linear polarization, and another is to implement the Zeeman term in the generalized Ising model Eq. (2). The output from each slave laser is also split and injected into the other slave lasers (without an isolator in an optical path). The final polarization state at each site (slave laser) is read out by using a polarization beam splitter (PBS) and dual-photodetectors. The implementation at each computational step is described below:
  1. Initialization (t<0)

    Before introducing the Zeeman term and Ising interaction term, we prepare the initial states in all slave lasers. The frequency, phase and polarization of all slave lasers are identical to those of the master laser. In Eqs. (22) and (23) or Eqs. (32) and (33), ζ determines the amount of vertical linearly polarized injection signal. This initialization step is implemented by an isolator and simple attenuators.

  2. Zeeman term λi (t≥0)

    At a time t = 0, we implement the amplitude and phase of the Zeeman term λi for each site i. In the rate equations Eqs. (22) and (23) for the photon numbers, the Zeeman term λi is represented by the horizontal polarization component of the injection signal from the master laser, which is expressed by the term proportional to ηi. As shown in Fig. 7(a), the magnitude of λi (|λi|) is implemented by choosing the proper rotation angle θ for the initial vertical linear polarization signal to include the horizontal polarization component. This is easily achieved by inserting a HWP on a path from the master laser to each slave laser. The horizontal polarization component |H=ı2(|R|L) of the master laser signal couples to right and left circular polarization modes of the slave laser with π2 and π2 phases, respectively. Notice that the sign in front of the constant ηi is opposite for nRi and nLi in Eqs. (22) and (23). Here the rotation angle θ is determined by the amount of required horizontal polarization component |H〉. The sign of ηi is minus for negative λi and plus for positive λi in Eq. (27). To implement the polarity of λi, we put a QWP after HWP to introduce ±π2 phase shift. Then the polarization state changes into the right or left ellipsoidal polarization cosθ|V〉 ± ýsinθ|H〉. The QWP rotation angle determines the dominant polarization in a slave laser. When the rotation angle is π2, ηi is positive so that the left circular polarization will be dominant in the slave laser i.

  3. Ising interaction term Jij (t≥0)

    The emitted light from each slave laser is injected to other slave lasers via a horizontal linear polarizer, phase shifter and attenuator as shown in Fig. 7(b). The magnitude of Jij is implemented by the transmission coefficient of the attenuator. The horizontal polarizer output of the slave laser j is proportional to nRjnLj. This mutual injection signal with horizontal linear polarization couples to the right and left circular polarization modes of the slave laser i with equal amplitudes and opposite phases. Notice that the sign in front of the transmission constant ξij is opposite for nRi and nLi in Eqs. (22) and (23). The sign of Jij is implemented by the optical phase shift ϕij between the two slave lasers. ϕij = 2πl is chosen for a negative Jij and ϕij = 2πl + π for a positive Jij, where l is a positive integer.

  4. Readout

    After a steady state condition is reached, the outputs from all slave lasers are input into polarization beam splitters (PBS) to separate the right and left circular polarization components. The photodetectors at each output port of the PBS provides the computational result through a majority vote.

  5. Signal to noise ratio

    Let mRi and mLi denote the right and left circularly polarized and time integrated photon numbers. The electrical signal power is proportional to the squared difference of the average photon numbers 〈mRimLi2. If the two polarization modes are in independent coherent states, the noise power is proportional to the sum of the variances ΔmRi2+ΔmLi2 where mRi=ηD(ωQ)nRiT, mLi=ηD(ωQ)nLiT, ηD is an overall detection quantum efficiency and T is an integration time. Due to the above assumption, the slave laser output photon statistics obey independent Poisson distributions, i.e., ΔmRi2=mRi and ΔmLi2=mLi. Since the Poisson distribution with a large average photon number is well approximated by the Gaussian distribution with the same variance, the bit error rate in this detection scheme is given by the complementary error function of the signal-to-noise ratio [28

    28. S. D. Personick, “Receiver design for digital fiber optic communication systems, I,” Bell Syst. Tech. J. 52(6), 843–874 (1973).

    ]:
    Pe=12erfc(Q2),
    (37)
    where Q=mRimLi/2ΔmRi2+ΔmLi2 is the Personick Q parameter. The required (peak-to-peak) signal to noise ratio is approximately 4Q 2 ∼ 144 (21.6dB) to achieve a prescribed error rate, for instance, Pe = 10−9. Then, we have
    4Q2=mRimLi2ΔmRi2+ΔmLi2=ηD(ωQ)nRiT(1R)21+R144,
    (38)
    where R=nLinRi<1 is the ratio of the average photon numbers of the minor to major polarization modes. If we use the numerical examples of ηD = 0.01, ωQ=1012(1s), 〈nRi〉 = 104 and R = 0.5, the required integration time to satisfy Eq. (38) is T = 8.4psec, which is much shorter than the computational time. If the photon number noise is above the Poisson limit due to the mode partition noise [29

    29. S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2765 (1992). [CrossRef] [PubMed]

    ], we can always satisfy Eq. (38) by employing a longer integration time. Note that the overall error rate is only additive with the site number M, so that the over all error rate is still bounded by Pe = 10−6 even for a large-scale problem M = 1000.

Fig. 7 (a) Implementation of the Zeeman term λi by an attenuator, HWP and QWP. When ηi > 0, the horizontal polarization component of the injection signal couples to the right and left circular polarization modes of the slave laser with π and zero phases, respectively. (b) Implementation of the Ising interaction term Jij by a horizontal linear polarizer, phase shifter and attenuator. The magnitude of the Ising interaction term |Jij| can be implemented by an attenuator. The horizontal linear polarizer output of the slave laser i is proportional to nRinLi. Therefore, the phase of the injection signal from the slave laser i to the slave laser j can implement a sign of the Ising interaction term Jij/|Jij|.

8. Conclusion

We proposed the mapping protocol of an Ising model onto injection-locked laser systems. Each site of an Ising model is represented by an injection-locked slave laser, in which the Ising spin (up or down) corresponds to the right or left circular polarization of the partial wave of each individual photon which coherently spreads in the entire slave laser systems. Before a computation will start, all slave lasers are injection-locked by a vertical linearly polarized master laser output, so that the two polarization modes of all slave lasers have identical average photon numbers and phases. The Ising interaction term Jij can be implemented by mutual injection between the two slave lasers i and j via a horizontal linear polarizer, phase shifter (0 or π phase) and attenuator. The Zeeman term λi can be implemented by injecting a horizontal linearly polarized master laser signal in addition to the vertical linearly polarized master laser output into the slave laser i.

We have confirmed numerically that the proposed Ising machine outputs correct answers for most cases even if the quantum noise source is neglected. For some other cases, the proposed system outputs wrong answers due to the trapping problem in local minimum states. It is found that the Poissonian photon number noise can make the system to escape from the local minimum state and to find the global minimum state. We have analyzed the proposed Ising machine in terms of complementary diagonal linear polarization basis, in which the amplitudes of the two polarization modes |D〉 and |〉 are identical but the phases depart with each other. When such phase modulated diagonal linear polarization modes are projected onto the two circular polarization modes |R〉 and |L〉, we recover the identical results and confirm the two alternative pictures explain the same operational principle of the proposed Ising machine equally well.

A time delay to reach a steady state condition after the Ising and Zeeman energy terms are switched on can be considered as a lower bound of computational time. The transient time does not show a strong dependence on the problem size M according to numerical simulations. It is on the order of a few nsec from M = 2 to M = 10. However, even an efficient physical simulator is often limited by getting trapped in one of many metastable states (local minima) in the case of NP-complete problems when the problem size becomes large [30

30. A. P. Young, S. Knysh, and V. N. Smelyanskiy, “First-order phase transition in the quantum adiabatic algorithm,” Phys. Rev. Lett. 104, 020502 (2010). [CrossRef] [PubMed]

]. The relation of the computational power of the proposed Ising machine to the NP completeness in the limit of large problem size will be a subject of future study.

9. Summary

The proposed Ising machine can be potentially implemented by semiconductor VCSELs, ECDL and linear optical components even for a very large-scale problem M ∼ 1000. Even though a preliminary numerical simulation result is encouraging in terms of the computational time vs. problem size M, its asymptotic behaviour for NP-complete problems with large M should be carefully studied in a future publication. After completing this work, we were informed of the quantum optical simulation of Ising models using atom-cavity QED systems proposed in [31

31. Y. Zhang, Y. Xia, Z. Man, and G. Guo, “Simulation of the Ising model, memory for Bell states and generation of four-atom entangled states in cavity QED,” Sci. China, Ser. G 52, 700–707 (2009). [CrossRef]

].

Acknowledgments

References and links

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M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W. H. Freeman and Company, 1979).

2.

K. Binder and A. A. Young, “Spin glasses: experimental facts, theoretical concepts, and open questions,” Rev. Mod. Phys. 58, 801–976 (1986). [CrossRef]

3.

V. Dotsenko, Introduction to the Replica Theory of Disordered Statistical Systems (Cambridge University Press, 2000). [CrossRef]

4.

H. Nishimori, Statistical Physics of Spin Glasses and Information Processing: An Introduction (Oxford University Press, 2001). [CrossRef]

5.

A. Das and B. K. Chakrabarti, “Colloquium: quantum annealing and analog quantum computation,” Rev. Mod. Phys. 80, 1061–1081 (2008). [CrossRef]

6.

P. Ray, B. K. Chakrabarti, and A. Chakrabarti, “Sherrington–Kirkpatrick model in a transverse field: absence of replica symmetry breaking due to quantum fluctuations,” Phys. Rev. B 39, 11828–11832 (1989). [CrossRef]

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B. Appoloni, C. Carvalho, and D. de Falco, “Quantum stochastic optimization,” Stochastic Proc. Appl. 33, 233–244 (1989). [CrossRef]

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R. Martonak, G. E. Santoro, and E. Tosatti, “Quantum annealing by the path-integral Monte Carlo method: the two-dimensional random Ising model,” Phys. Rev. B 66, 094203 (2002). [CrossRef]

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G. E. Santoro, R. Martonak, E. Tosatti, and R. Car, “Theory of quantum annealing of an Ising spin glass,” Science 295(5564), 2427–2430 (2002). [CrossRef] [PubMed]

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G. E. Santoro and E. Tosatti, “Quantum to classical and back,” Nat. Phys. 3, 593–594 (2007). [CrossRef]

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R. D. Somma and C. D. Batista, “Quantum approach to classical statistical mechanics,” Phys. Rev. Lett. 99, 030603 (2007). [CrossRef] [PubMed]

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J. Brooke, D. Bitko, T. F. Rosenbau, and G. Aeppli, “Quantum annealing of a disordered magnet,” Science 284(5415), 779–781 (1999). [CrossRef] [PubMed]

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G. Aeppli and T. F. Rosenbaum, in Quantum Annealing and Related Optimization Methods, A. Das and B. K. Chakrabarti, eds. (Springer Verlag, 2005).

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M. Steffen, W. van Dam, T. Hogg, G. Breyta, and I. Chuang, “Experimental implementation of an adiabatic quantum optimization algorithm,” Phys. Rev. Lett. 90, 067903 (2003). [CrossRef] [PubMed]

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T. Byrnes, K. Yan, and Y. Yamamoto, “Optimization using Bose-Einstein condensation and measurement-feedback circuits,” arXiv:0909.2530v2.

16.

K. Yan, T. Byrnes, and Y. Yamamoto, “Kinetic Monte Carlo study of accelerated optimization problem search using Bose-Einstein condensates,” Prog. Inform. 8, 1–9 (2011).

17.

F. Barahona, “On the computational complexity of Ising spin glass models,” J. Phys. A: Math. Gen. 15, 3241–3253 (1982). [CrossRef]

18.

S. Kobayashi and T. Kimura, “Injection locking in AIGaAs semiconductor laser,” IEEE J. Quantum Electron. 17(5), 681–689 (1981). [CrossRef]

19.

S. Kobayashi, Y. Yamamoto, and T. Kimura, “Optical FM signal amplification and FM noise reduction in an injection locked AlGaAs semiconductor laser,” Electron. Lett. 17(22), 849–851 (1981). [CrossRef]

20.

L. Gillner, G. Bjork, and Y. Yamamoto, “Quantum noise properties of an injection-locked laser oscillator with pump-noise suppression and squeezed injection,” Phys. Rev A 41(9), 5053–5065 (1990). [CrossRef] [PubMed]

21.

H. A. Haus and Y. Yamamoto, “Quantum noise of an injection-locked laser oscillator,” Phys. Rev. A 29, 1261–1274 (1984). [CrossRef]

22.

M. Surgent, M. O. Scully, and W. E. Lamb, Laser Physics (Westview Press, 1978), Chap. 20, pp. 331–335.

23.

H. Haug, “Quantum-mechanical rate equations for semiconductor lasers,” Phys. Rev.184, 338–348 (1969). [CrossRef]

24.

S. F. Yu, Analysis and Design of Vertical Cavity Surface Emitting Laser (Wiley-Interscience, 2003), Chap. 8. [CrossRef]

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Y. Yamamoto, S. Machida, and O. Neilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986). [CrossRef] [PubMed]

26.

C. Weisbuch and B. Vinter, Quantum Semiconductor Structures: Fundamentals and Applications (Academic Press, 1991), Chap. 3, pp. 65–69.

27.

K. Takata, S. Utsunomiya, and Y. Yamamoto, “Transient time of an Ising machine based on injection-locked lasers: contribution of locking bandwidth and Zeeman component,” (to be submitted).

28.

S. D. Personick, “Receiver design for digital fiber optic communication systems, I,” Bell Syst. Tech. J. 52(6), 843–874 (1973).

29.

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2765 (1992). [CrossRef] [PubMed]

30.

A. P. Young, S. Knysh, and V. N. Smelyanskiy, “First-order phase transition in the quantum adiabatic algorithm,” Phys. Rev. Lett. 104, 020502 (2010). [CrossRef] [PubMed]

31.

Y. Zhang, Y. Xia, Z. Man, and G. Guo, “Simulation of the Ising model, memory for Bell states and generation of four-atom entangled states in cavity QED,” Sci. China, Ser. G 52, 700–707 (2009). [CrossRef]

OCIS Codes
(030.0030) Coherence and statistical optics : Coherence and statistical optics
(140.3520) Lasers and laser optics : Lasers, injection-locked
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: April 21, 2011
Revised Manuscript: July 3, 2011
Manuscript Accepted: August 11, 2011
Published: August 31, 2011

Citation
Shoko Utsunomiya, Kenta Takata, and Yoshihisa Yamamoto, "Mapping of Ising models onto injection-locked laser systems," Opt. Express 19, 18091-18108 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-19-18091


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References

  1. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W. H. Freeman and Company, 1979).
  2. K. Binder and A. A. Young, “Spin glasses: experimental facts, theoretical concepts, and open questions,” Rev. Mod. Phys.58, 801–976 (1986). [CrossRef]
  3. V. Dotsenko, Introduction to the Replica Theory of Disordered Statistical Systems (Cambridge University Press, 2000). [CrossRef]
  4. H. Nishimori, Statistical Physics of Spin Glasses and Information Processing: An Introduction (Oxford University Press, 2001). [CrossRef]
  5. A. Das and B. K. Chakrabarti, “Colloquium: quantum annealing and analog quantum computation,” Rev. Mod. Phys.80, 1061–1081 (2008). [CrossRef]
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