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Abstract JC model is widely applied in the field of quantum information. We propose a way to simulateJC model with linear optical devices. For the case of a two-level atom resonantly interacting witha cavity, the state transformation of the system is unitary when the influence of environment is notconsidered. This unitary evolution exactly corresponds to the evolution of parametric light passingthrough the half wave plate. This proposal mainly simulates the situation that an atom resonantlyinteracts with a cavity, by using linear optical devices. This simulation employs simple devices andis easy to operate, especially with a rigorous map for a JC model. Using several BDs,and HWPs,we also show how to simulate the JC model for the case of the the atom, off-resonant with thecavity.The two simulations may lay the foundation for the future development of linear optics.PACS numbers: 03.67.Lx, 42.50.DvI. INTRODUCTIONMatter, energy and information are the most important elements of nature all the time.As we enter the 20th century, scientists found that the energy and matter can be described asquantization. Then, lots of modern technology, such as the representative, microelectronicsand laser, were risen rapidly. However, when the 21st century comes, information can alsobe quantized, with an immeasurable impact which will change the present communicationtechnology. As an important part of quantum information, linear optics has got a goodresearch in quantum teleportation and simulation.As for simulation, the reachers at University of Science and Technology of China havedone a simulation of Maxwell’s demon by building the corresponding optical path withlinear optical devices.[2]48092
Andrea from Italy has realized quantum random walk simulation ofAnderson localization with linear optics.[3]Recently, the simulation of non-Markov processin quantum open systems has also become the focus of scientific research workers.[4,5,7]As anoperational tool, linear optics has been widely used in the abstract, complex communicationand simulation.JC model has an extremely wide range of applications in quantum state preparation,transmission, and quantum computing. However, how to simulate JC mode in linear opticalsystem has not been proposed. In this paper, we propose a method to simulate JC model bysetting up an optical path with linear optical devices. We simulate the resonance between anatom and a cavity and then show how the states change. Because of the strict correspondenceand the simple devices, this simulation can be widely applied in the program that contains JCmodel and makes further study in the area of linear optics more effectively and conveniently.II. DESCRIPTION OF JC MODELJaynes-Cummings model (JCM) describes a system of a two-level atom interacting witha quantized mode of an optical cavity. The two levels of the atom are the ground state |g⟩and the excited state |e⟩. The JCM is of great interest in atomic physics, quantum optics,and solid-state quantum information circuits, both experimentally and theoretically.The Hamiltonian describing the full system is given by[6] FIG. 1: An atom can jump between the ground state |g⟩ and excited state |e⟩,
induced by thecavity mode.which consists of the atomic free Hamiltonian, the free field Hamiltonian, and the Jaynes-Cummings interaction Hamiltonian:Hatom = 12~ωbσz, (2)Hfeild = ~ωaa†a, (3)Hint = g(σ+a + σ−a†), (4)where the operators a†and a are the bosonic creation and annihilation operators, ωa is theangular frequency of the cavity, and ωb is the angular frequency of the two-level-atom. Inaddition, σ+ and σ− are the raising and lowing operators of the atom, and g is the atom-cavity coupling coefficient. The mathematical expression of σz, σ+,σ− can be written asσz = |e⟩ |e⟩ − |g⟩ |g⟩, σ+ = |e⟩ |g⟩, σ− = |g⟩ |e⟩.For convenience, we assume that ωa = ωb = ω. That is to say there exists a resonancebetween the atom and the cavity. In addittion, we assume ~ = 1 for simplicity. TheHamiltonian can be simplified as :H = ~ωa†a + 12~ωσz + g(σ+a + σ−a†). (5)Sch¨ ordinger equation is given by:i∂∂t|Ψt⟩ = H |Ψt⟩ . (6) |Ψt⟩ = e−iHt|Ψ0⟩ , (7)where |Ψ0⟩ stands for the initial state of the wave function. We use |n⟩ and |n − 1⟩ torepresent an n-photon state and an (n-1) photon state of the cavity. The system evolves ina Hilbert space formed by the two basis states |g⟩|n⟩ and |g⟩|n⟩. The matrix expression ofthe Hamiltonian with this basis vectors can be expressed as:H = ⟨g|⟨n|H |g⟩ |n⟩ ⟨g|⟨n|H |e⟩ |n − 1⟩⟨e|⟨n − 1|H |g⟩ |n⟩ ⟨e|⟨n − 1|H |e⟩ |n − 1⟩, (8)which can be further written as;H =nω − 12ω g√ng√n nω − 12ω, (9)According to the eigenstate equation ˆ H |Ψt⟩ = E |Ψt⟩, and defining the initial state|Ψ0⟩ =[C1 C2]T,we can havenω − 12ω g√ng√n nω − 12ωC1C2 = EC1C2, (10)which can be written asnω − 12ω − E g√ng√n nω − 12ω − EC1C2 = 0. (11)Therefore, we can solve the secular equationdet