Exploration of multiphoton entangled states by using weak nonlinearities

We propose a fruitful scheme for exploring multiphoton entangled states based on linear optics and weak nonlinearities. Compared with the previous schemes the present method is more feasible because there are only small phase shifts instead of a series of related functions of photon numbers in the process of interaction with Kerr nonlinearities. In the absence of decoherence we analyze the error probabilities induced by homodyne measurement and show that the maximal error probability can be made small enough even when the number of photons is large. This implies that the present scheme is quite tractable and it is possible to produce entangled states involving a large number of photons.


Kerr nonlinearities
Before describing the proposed scheme, let us first give a brief introduction of the Kerr nonlinearities. The nonlinear Kerr media can be used to induce a cross phase modulation with Hamiltonian of the form χ = ħ † † H a a a a s s p p , where χ is the coupling constant and a s ( ) a p represents the annihilation operator for photons in the signal (probe) mode. If we assume that the signal mode is initially described by the state ψ = + c c 0 1 s s s 0 1 and the coherent probe beam is α p , then after the Kerr interaction the whole system evolves as where θ χ = t with interaction time t. In order to distinguish different cases, one may perform a homodyne measurement 39 on the probe beam with quadrature operator φ a a e e p p i i , where φ is a real constant. Especially for φ = 0, this operation is conventionally referred to as X homodyne measurement; while for φ π = /2, it is called P homodyne measurement.

Creation of multiphoton entangled states with linear optics and weak nonlinearities
Let , = , , ,  a i n 1 2 i represent input ports with respective spatial modes, namely signal modes, and α is a coherent beam in probe mode. The setup of creating multiphoton entangled states is shown in Fig. 1.
Without loss of generality we may suppose that each input port is supplied with an arbitrary single-photon state. Then, the total input state reads n a a a n a a a a n a a n a a a a n a Each polarizing beam splitter (PBS) is used to transmit H polarization photons and reflect V polarization photons. When the signal photons travel to the PBSs, they will be individually split into two spatial modes and then interact with the nonlinear media so that pairs of phase shifts θ and 2θ are induced on the coherent probe beam, respectively. We here introduce a single phase gate θ θ ( ) = − / R n 3 2 n so as to implement the next X homodyne measurement on the probe beam. We describe our method in details. For n is even, after the interaction between the photons with Kerr media and followed by the action of the phase gate, the combined system evolves as   In order to create the desired multiphoton entangled states, we here perform an X homodyne measurement 38,45 on the probe beam. If the value x of the X homodyne measurement is obtained, then the signal photons become Similarly, for odd n, we have As an example of the applications of interest for the present scheme, we introduce a class of remarkable multipartite entangled states are two orthonormal states, namely Dicke states 53,54 . In view of its "catness", the state Ψ n k can be referred to as cat-like state, and especially for = / k n 2 it can be expressed as the canonical n-partite Greenberger-Horne-Zeilinger (GHZ) state. In the present scheme, obviously, for = / , we can obtain these cat-like states with = , , , /  k n 0 1 2 for even n and = / , / , , /  k n 1 2 3 2 2 for odd n, where the qubits are encoded with the polarization modes ≡ H 0 and ≡ V 1 . Of course, more generally, we may project out a group of multiphoton entangled states involving generalized Dicke states.

Discussion
There are two models commonly employed in the process of Kerr interaction, single-mode model and continuous-time multi-mode model 51 . The former implies that one may ignore the temporal behavior of the optical pulses but the latter is causal, non-instantaneous model involving phase noise. It has been shown that 52 this causality-induced phase noise will preclude the possibility of high-fidelity CPHASE gates created by the cross-Kerr effect in optical fiber. To solve this problem, one may need to find an optimum response function for the available medium, or to exploit more favorable systems, such as cavitylike systems 55 . After all, the ultimate possible performance of Kerr interaction with a larger system is an interesting open issue. More recently, we note that Feizpour et al. 56 showed the first direct measurement of the cross-phase shift due to single photons. It may be possible to open a door for future studies of nonlinear optics in quantum information processing. In the present scheme, we restrict ourselves to ignoring the phase noise and concentrate mainly on showing a method for exploring multiphoton entangled states in the regime of weak cross-Kerr nonlinearities, i.e. θ π  . It is worth noting that, there are only small phase shifts θ and 2θ instead of a series of related functions of the number of photons in the process of interaction with Kerr nonlinearities. This implies that the present scheme is quite tractable especially for creating entangled states with a larger number of photons. In addition, the error probabilities ε k are , which is exactly the result described by Nemoto and Munro in 40 . Obviously, the error probabilities in our scheme are no more than that one even when the number of photons is large. Therefore, by choosing an appropriate coherent probe beam the error probability can be reduced to as low a level as desired and then the present scheme may be realized in a nearly deterministic manner.
In summary, based on linear optics and weak nonlinearities we have shown a fruitful method for exploring a class of multiphoton entangled states, the generalized cat-like states. Evidently, three aspects are noteworthy in the present framework. First, since there are no large phase shifts in the interacting process with weak Kerr nonlinearities, our scheme is more feasible compared with the previous schemes. Second, the system is measured only once with a small error probability and it means that the present scheme might be realized near deterministically. Finally, the fruitful architecture allows us to explore a group of multiphoton entangled states involving a large number of photons, i.e., to produce entangled states approaching the macroscopic domain.