基于超导量子比特的光学双稳性质的研究
|
摘要
近些年来,量子电动力学被广泛的探索。一系列的量子光学中有的效应,在量子电路实验也实现了。例如,兰姆平移、非线性光谱、受激辐射等。最近,超导量子比特与谐振腔耦合的TC模型也被证实了。超导谐振腔在实现量子相干效应和多光子的多体效应中有很重要的作用。
当量子多体系统与谐振腔耦合时,会产生很多新的新的现象是在原子系统中所没有的。在原子系统中,原子能够与谐振腔强耦合在一起,产生的光子会显示一些有趣的效应。在超导电路中,电路中可以有很多种电磁耦合,有很多可控参数,正是由于这些电路的这种多样性,可以设计电路满足一些原子系统所不能实现的多体效应。由于超导量子电路的可控性比较好,可以更容易的用超导量子电路来实现一些量子光学中的现象。
超导量子比特有一些类似于原子的特点,可以称量子比特为人造原子。本文就是在基于量子比特电路的研究,来实现光学双稳态。用矩阵列形式的很多个量子比特与超导传输线腔耦合,通过外加驱动场的作用最后实现光学双稳的现象。本文具体的分为下列几部分来讨论:
第一部分绪论中介绍了约瑟夫逊结的基本知识,几种量子比特的基本模型,光学双稳的基本原理等。
第二部分研究了传输线腔的等效LC电路模型,电路的二次量子化,传输线腔与外部导线耦合以及传输线腔的损耗等问题。
第三部分研究了传输线腔与量子比特耦合作用的系统,对库伯对盒量子化,研究了量子比特对传输线腔的影响,频移。
第四部分研究了外加驱动场与传输线腔-量子比特相互作用的系统,讨论了量子比特的受激辐射过程,最后得到光学双稳特性,并用半经典理论解释光学双稳。
In the past few years, circuit quantum electrodynamics (QED) has been intensively explored. A number of quantum optical effects have been demonstrated in experiments, such as Lamb shift, nonlinear spectrum, and lasing. Most recently, the Tavis-Cummings model has been tested with a number of qubits coupling to a resonator. The superconducting resonator is very important for study of quantum coherence effects and many-body effects of photons.
When a quantum many-body system is coupled to a resonator cavity ,some novel many-body phenomena can be observed which have not been studied previously in the atomic systems. In atomic systems, atoms can be coupled strongly to a resonator that produce photons to demonstrate interesting effects. In the superconducting circuit, as a result of the diversity of the electromagnetic couplings and many controllable parameters, a large variety of many-body effects can be demonstrated which can not be simulated in atomic systems. Since quantum circuit can be easily controlled , it is convenient to engineer quantum circuit for demonstrating quantum optical effects.
As we known that superconducting qubit is similar to atom, they have some analogous features, so qubit can be taken as artificial atom. In this paper, we study qubit in quantum circuit, and get optical bistability. Our work focus on the behavior of the qubits array in this system, the qubits array is coupled strongly to a superconducting transmission line resonator. Through a driving field interact with this system, we can demonstrate the optical bistable phenomenon. The main work of this paper is divided into several sections, as follows:
The first part is the introduction; in this section we introduce the background of our work and some basic concept of Josephson junction. Also, we introduce three type superconducting qubits and fundamental principle of optical bistability
In the second section, we discus LC model of the superconducting transmission line resonator, second quantization, transmission line resonator coupling to external leads and dissipation of the transmission line resonator.
In the third section, we discus qubit-resonator system, quantization of cooper pair box, and the frequency shift induced by the qubit.
At last, we discus microwave driven qubit-resonator system. First, we analyse the lasing action of qubit. Then, we design a quantum circuit and we find that optical bistability can be demonstrated. Semiclassic quantum theory can be used to explain the phenomenon.
当量子多体系统与谐振腔耦合时,会产生很多新的新的现象是在原子系统中所没有的。在原子系统中,原子能够与谐振腔强耦合在一起,产生的光子会显示一些有趣的效应。在超导电路中,电路中可以有很多种电磁耦合,有很多可控参数,正是由于这些电路的这种多样性,可以设计电路满足一些原子系统所不能实现的多体效应。由于超导量子电路的可控性比较好,可以更容易的用超导量子电路来实现一些量子光学中的现象。
超导量子比特有一些类似于原子的特点,可以称量子比特为人造原子。本文就是在基于量子比特电路的研究,来实现光学双稳态。用矩阵列形式的很多个量子比特与超导传输线腔耦合,通过外加驱动场的作用最后实现光学双稳的现象。本文具体的分为下列几部分来讨论:
第一部分绪论中介绍了约瑟夫逊结的基本知识,几种量子比特的基本模型,光学双稳的基本原理等。
第二部分研究了传输线腔的等效LC电路模型,电路的二次量子化,传输线腔与外部导线耦合以及传输线腔的损耗等问题。
第三部分研究了传输线腔与量子比特耦合作用的系统,对库伯对盒量子化,研究了量子比特对传输线腔的影响,频移。
第四部分研究了外加驱动场与传输线腔-量子比特相互作用的系统,讨论了量子比特的受激辐射过程,最后得到光学双稳特性,并用半经典理论解释光学双稳。
In the past few years, circuit quantum electrodynamics (QED) has been intensively explored. A number of quantum optical effects have been demonstrated in experiments, such as Lamb shift, nonlinear spectrum, and lasing. Most recently, the Tavis-Cummings model has been tested with a number of qubits coupling to a resonator. The superconducting resonator is very important for study of quantum coherence effects and many-body effects of photons.
When a quantum many-body system is coupled to a resonator cavity ,some novel many-body phenomena can be observed which have not been studied previously in the atomic systems. In atomic systems, atoms can be coupled strongly to a resonator that produce photons to demonstrate interesting effects. In the superconducting circuit, as a result of the diversity of the electromagnetic couplings and many controllable parameters, a large variety of many-body effects can be demonstrated which can not be simulated in atomic systems. Since quantum circuit can be easily controlled , it is convenient to engineer quantum circuit for demonstrating quantum optical effects.
As we known that superconducting qubit is similar to atom, they have some analogous features, so qubit can be taken as artificial atom. In this paper, we study qubit in quantum circuit, and get optical bistability. Our work focus on the behavior of the qubits array in this system, the qubits array is coupled strongly to a superconducting transmission line resonator. Through a driving field interact with this system, we can demonstrate the optical bistable phenomenon. The main work of this paper is divided into several sections, as follows:
The first part is the introduction; in this section we introduce the background of our work and some basic concept of Josephson junction. Also, we introduce three type superconducting qubits and fundamental principle of optical bistability
In the second section, we discus LC model of the superconducting transmission line resonator, second quantization, transmission line resonator coupling to external leads and dissipation of the transmission line resonator.
In the third section, we discus qubit-resonator system, quantization of cooper pair box, and the frequency shift induced by the qubit.
At last, we discus microwave driven qubit-resonator system. First, we analyse the lasing action of qubit. Then, we design a quantum circuit and we find that optical bistability can be demonstrated. Semiclassic quantum theory can be used to explain the phenomenon.
引文
[1] A. Blais et al. Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation. Phys. Rev. A, 2004, 69:062320
[2] J. Q. You et al. Quantum information processing with superconducting qubits in a microwave field. Phys. Rev. B, 2003, 68:064509
[3] F. Marquardt et al. Superposition of two mesoscopically distinct quantum states: Coupling a Cooper-pair box to a large superconducting island. Phys.Rev.B, 2001, 63:054514
[4] A. Wallraff et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature, 2004, 431:162~167
[5] I. Chiorescu et al. Coherent dynamics of a flux qubit coupled to a harmonic oscillator. Nature, 2004, 431:159~162
[6] J. Johansson et al. Vacuum Rabi Oscillations in a Macroscopic Superconducting Qubit LC Oscillator System. Phys.Rev.Lett, 2006, 96:127006
[7] D. I. Schuster et al. Resolving photon number states in a superconducting circuit. Nature, 2007, 445:515~518
[8] A. Fragner et al. Resolving Vacuum Fluctuations in an Electrical Circuit by Measuring the Lamb Shift. Science, 2008, 322:1357~1360
[9] J. M. Fink et al. Climbing the Jaynes–Cummings ladder and observing its nonlinearity in a cavity QED system. Nature, 2008, 454: 315~318
[10] L. S. Bishop et al. Nonlinear response of the vacuum Rabi resonance. Nat. Phys, 2008, 5:105~109
[11] A.Wallraff, et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics.Nature,2004,431:162-167
[12] Circuit QED. Phys. Rev. Lett, 2009, 103:083601
[13] M. J. Hartmann et al. Strongly interacting polaritons in coupled arrays of cavities. Nat. Phys, 2006, 2:849~855
[14] D.I.Schuster, A.A.Houck, et al. Resolving photon number states in asuperconducting circuit. Nature,2007,445:515-518
[15] Jung-Tsung Shen, M.L.Povinelli, et al. Stopping single photons in one-dimensional circuit quantum electrodynamics systems. Phys. Rev. B, 2007,75:035320
[16] A. A. Houck, et al. Generating single microwave photons in a circuit, arXiv, cond-mat, 2007,0702648
[17] O. Astafiev,et al. Resonance Fluorescence of a Single Artificial Atom. Science, 2010, 327 :840-843
[18] A. D. Greentree et al. Quantum phase transitions of light. Nat. Phys, 2006, 2: 856~861
[19] D. G. Angelakis et al. Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays. Phys. Rev. A, 2007, 76:031805(R)
[20] N. Na et al. Strongly correlated polaritons in a two-dimensional array of photonic crystal microcavities. Phys. Rev. A, 2008, 77:031803(R)
[21] N. Lambert et al. Quantum chaos and critical behavior on a chip. Phys. Rev. B, 2009, 80:165308
[22] P. Nataf et al. Vacuum Degeneracy of a Circuit QED System in the Ultrastrong Coupling Regime. Phys. Rev. Lett, 2010, 104:023601
[23] A.Wallraff, D.Schuster, A.Blais, L.Frunzio, et al., Circuit Quantum Electrodynamics: Coherent Coupling of a Single Photon to a Cooper Pair Box. arXiv:cond-mat,0407325v1
[24] O. Astafiev et al. Single artificial-atom lasing. Nature, 2007, 449:588~590
[25] F. Deppe et al. Two-photon probe of the Jaynes–Cummings model and controlled symmetry breaking in circuit QED. Nat. Phys, 2008, 4:686~691
[26] J. M. Fink et al. Dressed Collective Qubit States and the Tavis-Cummings Model in G. Chen et al. Simulation of the superradiant quantum phase transition in the superconducting charge qubits inside a cavity. Phys. Rev. A, 2007, 76:055803
[27] M. H. Devoret et al. Measurements of Macroscopic Quantum Tunneling out of the Zero-Voltage State of a Current-Biased Josephson Junction. Phys. Rev. Lett, 1985, 55:1908~1911
[28] J. M. Martinis et al. Experimental tests for the quantum behavior of a macroscopicdegree of freedom: The phase difference across a Josephson junction. Phys. Rev. B, 1987, 35:4682~4698
[29] J. M. Martinis et al. Effect of environmental noise on the accuracy of Coulomb-blockade devices. Phys Rev. B, 1993, 48:18316~18319
[30] D. V. Averin and Yu. V. Nazarov . Single-electron charging of a superconducting island. Phys. Rev. Lett. 1992,69:1993–1996
[31] M. H. Devoret, A. Wallraff, J. M. Martinis. Superconducting Qubits: A Short Review.arXiv:cond-mat,2004:0411174
[32] Alexandre Zagoskin, et al. Superconducting qubits. arXiv:Cond-mat,2008, 0805. 0164v1
[33] Alexandre Zagoskin et al. Superconducting qubits. arXiv:Cond-mat, 2008, 0805. 0164v1
[34] Y. Makhlin et al. Quantum-state engineering with Josephson-junction devices. Rev. Mod. Phys, 2001, 73:357~400
[35] G. Wendin et al. Superconducting Quantum Circuits, Qubits and Computing. arXiv :cond-mat, 2005, 0508729v1
[36] R. Friedman et al. Quantum superposition of distinct macroscopic states. Nature, 2000, 406 :43~46
[37] S. Han et al. Observation of Cascaded Two-Photon-Induced Transitions between Fluxoid States of a SQUID. Phys. Rev. Lett, 2000, 84 :1300~1303
[38] M. Steffen et al. Accurate control of Josephson phase qubits. Phys. Rev. B, 2003, 68: 224518
[39] Y. Makhlin,G. Sch?n et al. Quantum-state engineering with Josephson-junction devices.Rev. Mod. Phys. ,2001,73:357–400
[40] G. Wendin and V.S. Shumeiko. Superconducting Quantum Circuits, Qubits and Computing.arXiv,cond-mat,2005,0508729v1
[41] M.R.Geller and A.N.Cleland. Superconducting qubits coupled to nanoelectromechanical resonators: An architecture for solid-state quantum-information processing. Phys. Rev. A, 2005, 71:032311
[42] Liang Bao-Long, Wang Ji-Suo, et al. Cooper-pair number-phase quantizationanalysis in double-Josephson-junction macroscopic circuit coupled by a capacitor. Chinese Physics B,2008,17:697-701
[43] Florian Marquardt and C. Bruder. Superposition of two mesoscopically distinct quantum states: Coupling a Cooper-pair box to a large superconducting island. Phys. Rev. B,2001 63:054514
[44] V. E. Manucharyan, E. Koch, L. Glazman, M. H.Devoret Fluxonium, Single Cooper-Pair Circuit Free of Charge Offsets.Science,2009,326:113-116
[45] Alec Maassen van den Brink. Hamiltonian for coupled flux qubits. Phys. Rev. B, 2005,71:064503
[46] S.M. Girvin, M.H. Devoret, and R..J. Schoelkopf. Circuit QED and engineering charge based superconducting qubits.arXiv:Cond-mat,2009,0912.3902v1
[47] J.M.Fink, et al. Climbing the Jaynes-Cumming ladder and observing its n nonlinearity in a cavity QED system.Nature,2008,454:315-318
[48] J. M. Fink1, R. Bianchetti1 et al. Dressed Collective Qubit States and the Tavis-Cummings Model in Circuit QED .Phys. Rev. Lett.2009,103:083601
[49] S.M.Girvin, Ren-Shou Huang et al. Prospects for Strong Cavity Quantum Electrodynamics with Superconducting Circuits. arXiv:Cond-mat,2003, 0310670v1
[50] Alexandre Blais, Ren-Shou Huang, Andreas Wallraff, et al., Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation. Phys. Rev. A ,2004,69:062320
[51] F. Plastina and G. Falci. Communicating Josephson qubits. Phys. Rev. B , 2003,67:224514
[52] J. Johansson, et al. Vacuum Rabi oscillations in a macroscopic superconducting qubit LC oscillator system. Phys. Rev. Lett.2006,96:127006
[53] D. Vion, et al. Manipulating the Quantum States of an Electrical Circuit. Science,2007,296:886-889
[54] J.M Marinis, S. Nam, J. Aumentado, C. Urbina, Rabi oscillations in a large Josephson-junction qubit .Phys.Rev.Lett,2002,89:117901
[55] MCCALL S L.GIBBS H M.VENKATESAN T C Optical transistor and bistability 1975
[56] R. Bonifacio and L. A. Lugiato,Cooperative effects and bistability for resonance fluorescence, Opt. Commun. 1976,19: 172
[57] Hitoshi Kawaguchi, Multiple Bistability and Multistability in a Fabry-Perot Laser Diode Amplifier, IEEE, 1987, Vol.23, No.9
[58] Demetri Psaltis and Nabil Farhat, Optical information processing based on an associative-memorymodel of neural nets with thresholding and feedback, OPTICS LETTERS, 1985, Vol. 10, No. 2
[2] J. Q. You et al. Quantum information processing with superconducting qubits in a microwave field. Phys. Rev. B, 2003, 68:064509
[3] F. Marquardt et al. Superposition of two mesoscopically distinct quantum states: Coupling a Cooper-pair box to a large superconducting island. Phys.Rev.B, 2001, 63:054514
[4] A. Wallraff et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature, 2004, 431:162~167
[5] I. Chiorescu et al. Coherent dynamics of a flux qubit coupled to a harmonic oscillator. Nature, 2004, 431:159~162
[6] J. Johansson et al. Vacuum Rabi Oscillations in a Macroscopic Superconducting Qubit LC Oscillator System. Phys.Rev.Lett, 2006, 96:127006
[7] D. I. Schuster et al. Resolving photon number states in a superconducting circuit. Nature, 2007, 445:515~518
[8] A. Fragner et al. Resolving Vacuum Fluctuations in an Electrical Circuit by Measuring the Lamb Shift. Science, 2008, 322:1357~1360
[9] J. M. Fink et al. Climbing the Jaynes–Cummings ladder and observing its nonlinearity in a cavity QED system. Nature, 2008, 454: 315~318
[10] L. S. Bishop et al. Nonlinear response of the vacuum Rabi resonance. Nat. Phys, 2008, 5:105~109
[11] A.Wallraff, et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics.Nature,2004,431:162-167
[12] Circuit QED. Phys. Rev. Lett, 2009, 103:083601
[13] M. J. Hartmann et al. Strongly interacting polaritons in coupled arrays of cavities. Nat. Phys, 2006, 2:849~855
[14] D.I.Schuster, A.A.Houck, et al. Resolving photon number states in asuperconducting circuit. Nature,2007,445:515-518
[15] Jung-Tsung Shen, M.L.Povinelli, et al. Stopping single photons in one-dimensional circuit quantum electrodynamics systems. Phys. Rev. B, 2007,75:035320
[16] A. A. Houck, et al. Generating single microwave photons in a circuit, arXiv, cond-mat, 2007,0702648
[17] O. Astafiev,et al. Resonance Fluorescence of a Single Artificial Atom. Science, 2010, 327 :840-843
[18] A. D. Greentree et al. Quantum phase transitions of light. Nat. Phys, 2006, 2: 856~861
[19] D. G. Angelakis et al. Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays. Phys. Rev. A, 2007, 76:031805(R)
[20] N. Na et al. Strongly correlated polaritons in a two-dimensional array of photonic crystal microcavities. Phys. Rev. A, 2008, 77:031803(R)
[21] N. Lambert et al. Quantum chaos and critical behavior on a chip. Phys. Rev. B, 2009, 80:165308
[22] P. Nataf et al. Vacuum Degeneracy of a Circuit QED System in the Ultrastrong Coupling Regime. Phys. Rev. Lett, 2010, 104:023601
[23] A.Wallraff, D.Schuster, A.Blais, L.Frunzio, et al., Circuit Quantum Electrodynamics: Coherent Coupling of a Single Photon to a Cooper Pair Box. arXiv:cond-mat,0407325v1
[24] O. Astafiev et al. Single artificial-atom lasing. Nature, 2007, 449:588~590
[25] F. Deppe et al. Two-photon probe of the Jaynes–Cummings model and controlled symmetry breaking in circuit QED. Nat. Phys, 2008, 4:686~691
[26] J. M. Fink et al. Dressed Collective Qubit States and the Tavis-Cummings Model in G. Chen et al. Simulation of the superradiant quantum phase transition in the superconducting charge qubits inside a cavity. Phys. Rev. A, 2007, 76:055803
[27] M. H. Devoret et al. Measurements of Macroscopic Quantum Tunneling out of the Zero-Voltage State of a Current-Biased Josephson Junction. Phys. Rev. Lett, 1985, 55:1908~1911
[28] J. M. Martinis et al. Experimental tests for the quantum behavior of a macroscopicdegree of freedom: The phase difference across a Josephson junction. Phys. Rev. B, 1987, 35:4682~4698
[29] J. M. Martinis et al. Effect of environmental noise on the accuracy of Coulomb-blockade devices. Phys Rev. B, 1993, 48:18316~18319
[30] D. V. Averin and Yu. V. Nazarov . Single-electron charging of a superconducting island. Phys. Rev. Lett. 1992,69:1993–1996
[31] M. H. Devoret, A. Wallraff, J. M. Martinis. Superconducting Qubits: A Short Review.arXiv:cond-mat,2004:0411174
[32] Alexandre Zagoskin, et al. Superconducting qubits. arXiv:Cond-mat,2008, 0805. 0164v1
[33] Alexandre Zagoskin et al. Superconducting qubits. arXiv:Cond-mat, 2008, 0805. 0164v1
[34] Y. Makhlin et al. Quantum-state engineering with Josephson-junction devices. Rev. Mod. Phys, 2001, 73:357~400
[35] G. Wendin et al. Superconducting Quantum Circuits, Qubits and Computing. arXiv :cond-mat, 2005, 0508729v1
[36] R. Friedman et al. Quantum superposition of distinct macroscopic states. Nature, 2000, 406 :43~46
[37] S. Han et al. Observation of Cascaded Two-Photon-Induced Transitions between Fluxoid States of a SQUID. Phys. Rev. Lett, 2000, 84 :1300~1303
[38] M. Steffen et al. Accurate control of Josephson phase qubits. Phys. Rev. B, 2003, 68: 224518
[39] Y. Makhlin,G. Sch?n et al. Quantum-state engineering with Josephson-junction devices.Rev. Mod. Phys. ,2001,73:357–400
[40] G. Wendin and V.S. Shumeiko. Superconducting Quantum Circuits, Qubits and Computing.arXiv,cond-mat,2005,0508729v1
[41] M.R.Geller and A.N.Cleland. Superconducting qubits coupled to nanoelectromechanical resonators: An architecture for solid-state quantum-information processing. Phys. Rev. A, 2005, 71:032311
[42] Liang Bao-Long, Wang Ji-Suo, et al. Cooper-pair number-phase quantizationanalysis in double-Josephson-junction macroscopic circuit coupled by a capacitor. Chinese Physics B,2008,17:697-701
[43] Florian Marquardt and C. Bruder. Superposition of two mesoscopically distinct quantum states: Coupling a Cooper-pair box to a large superconducting island. Phys. Rev. B,2001 63:054514
[44] V. E. Manucharyan, E. Koch, L. Glazman, M. H.Devoret Fluxonium, Single Cooper-Pair Circuit Free of Charge Offsets.Science,2009,326:113-116
[45] Alec Maassen van den Brink. Hamiltonian for coupled flux qubits. Phys. Rev. B, 2005,71:064503
[46] S.M. Girvin, M.H. Devoret, and R..J. Schoelkopf. Circuit QED and engineering charge based superconducting qubits.arXiv:Cond-mat,2009,0912.3902v1
[47] J.M.Fink, et al. Climbing the Jaynes-Cumming ladder and observing its n nonlinearity in a cavity QED system.Nature,2008,454:315-318
[48] J. M. Fink1, R. Bianchetti1 et al. Dressed Collective Qubit States and the Tavis-Cummings Model in Circuit QED .Phys. Rev. Lett.2009,103:083601
[49] S.M.Girvin, Ren-Shou Huang et al. Prospects for Strong Cavity Quantum Electrodynamics with Superconducting Circuits. arXiv:Cond-mat,2003, 0310670v1
[50] Alexandre Blais, Ren-Shou Huang, Andreas Wallraff, et al., Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation. Phys. Rev. A ,2004,69:062320
[51] F. Plastina and G. Falci. Communicating Josephson qubits. Phys. Rev. B , 2003,67:224514
[52] J. Johansson, et al. Vacuum Rabi oscillations in a macroscopic superconducting qubit LC oscillator system. Phys. Rev. Lett.2006,96:127006
[53] D. Vion, et al. Manipulating the Quantum States of an Electrical Circuit. Science,2007,296:886-889
[54] J.M Marinis, S. Nam, J. Aumentado, C. Urbina, Rabi oscillations in a large Josephson-junction qubit .Phys.Rev.Lett,2002,89:117901
[55] MCCALL S L.GIBBS H M.VENKATESAN T C Optical transistor and bistability 1975
[56] R. Bonifacio and L. A. Lugiato,Cooperative effects and bistability for resonance fluorescence, Opt. Commun. 1976,19: 172
[57] Hitoshi Kawaguchi, Multiple Bistability and Multistability in a Fabry-Perot Laser Diode Amplifier, IEEE, 1987, Vol.23, No.9
[58] Demetri Psaltis and Nabil Farhat, Optical information processing based on an associative-memorymodel of neural nets with thresholding and feedback, OPTICS LETTERS, 1985, Vol. 10, No. 2