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arXiv:0709.2011 · New Journal of Physics

Highly non-Gaussian states created via cross-Kerr nonlinearity

Tomáš Tyc, Natalia Korolkova

Published Feb 26, 2008• 32 citations
arXivHTMLPDF

Abstract

We propose a feasible scheme for generation of strongly non-Gaussian states using the cross-Kerr nonlinearity. The resultant states are highly non-classical states of an electromagnetic field and exhibit negativity of their Wigner function, sub-Poissonian photon statistics and amplitude squeezing. Furthermore, the Wigner function has a distinctly pronounced 'banana' or 'crescent' shape specific for the Kerr-type interactions, which so far has not been demonstrated experimentally. We show that creating and detecting such states should be possible with the present technology using electromagnetically induced transparency in a four-level atomic system in N-configuration.

DOI
10.1088/1367-2630/10/2/023041

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We propose a feasible scheme for generation of strongly non-Gaussian states using the cross-Kerr nonlinearity. The resultant states are highly non-classical states of an electromagnetic field and exhibit negativity of their Wigner function, sub-Poissonian photon statistics and amplitude squeezing. Furthermore, the Wigner function has a distinctly pronounced 'banana' or 'crescent' shape specific for the Kerr-type interactions, which so far has not been demonstrated experimentally. We show that creating and detecting such states should be possible with the present technology using electromagnetically induced transparency in a four-level atomic system in N-configuration.
Non-Gaussian quantum states, described by negative-valued Wigner functions, are important for both fundamental tests of quantum physics and for emerging quantum information technologies. One of the promising ways of generating a non-Gaussian state from a coherent one is the use of cubic (Kerr) optical nonlinearity, which produces the characteristic banana-like shape of the resulting quantum states. However, the Kerr effect is weak in highly transparent optical materials (dimensionless nonlinearity parameter Γ≲10 −6 ). Therefore, a big number of the photons in the optical mode ( n ≳10 6 ) is necessary to generate an observable non-Gaussianity. In this case, the direct approach to calculation of the Wigner function becomes extremely computationally expensive. In this work, we develop quick algorithms for computing the Husimi and Wigner quasi-probability functions of these non-Gaussian states by means of the Kerr nonlinearity. This algorithm can be used for any realistic values of the photon numbers and the nonlinearity.
Non-Gaussian states of light are a critical resource for fault-tolerant quantum computing and enhanced metrology, but are typically faint and often obtained via post-selection. Here, we demonstrate the deterministic generation of a bright non-Gaussian state by introducing a Kerr nonlinearity to a macroscopic state of light called bright squeezed vacuum (BSV). To characterize the resulting state, we use a single-shot f-2f interferometer to sample its Husimi function. We observe a clear transformation from a 2D Gaussian distribution to an 'S'-shaped non-Gaussian profile, which is the direct statistical evidence of the intensity-dependent nonlinear phase. The negativity of the Wigner function, which is an intrinsic property of any pure non-Gaussian state, cannot be observed because BSV is a mixed state even under minute optical loss. However, we show that BSV can be considered as a mixture of pure squeezed coherent states, for some of which Kerr-induced Wigner-function negativity is quite tolerant to loss. This work bridges the gap between quantum optics and ultrafast nonlinear optics, opening a path to quantum applications that require high photon flux.
Strong optical nonlinearities are key to a range of optical technologies, in particular, to the generation of quantum light states. This paper proposes and demonstrates a novel nonlinear mechanism based on an N-level electro-magnetic induced transparency (EIT) in a warm rubidium vapor. Our scheme relies on using an optical frequency comb as a pump and a CW probe signal, which co-propagate and interact in warm rubidium vapor, as shown in Fig. 1(a). Multiple comb lines excite N-level EIT resonances as shown in Fig. 1(b), which modifies the optical response of the medium, creating a giant Kerr nonlinearity [1], which opens the way for different phenomena like slow light propagation or generation of quantum light states. Here, we experimentally create the giant Kerr nonlinearity, with the magnitude ranging from 1.2 x 10–7to 7.7 × 10–7m2V-2.
We study the nonlinear effect in a three-level atomic system in Λ configuration. As two or more fields are confined within the common electromagnetically induced transparency (EIT) window, they directly disturb EIT dark state at the same time, which leads to the self- and cross-Kerr nonlinear interactions. Our scheme is different from the existed schemes based on three-level or multilevel systems only existing single type of nonlinear interactions. In the common EIT windows, the self- and cross-Kerr nonlinearities are established with suppressed linear absorption and our scheme can be generalized polychromatic Kerr nonlinearity. Because the atoms are almost trapped in the ground states and the atom population becomes unchanged, the scheme is robust to the atomic spontaneous emission. Our scheme is a promising avenue for preparation of squeezing and entanglement .
We have introduced the notion of ‘four-photon nonlinear coherent states’ (FPNCSs) as right eigenstates of the fourth power of the generalized annihilation operator A4. It has been shown that there are four possible sets of such states which all can be expressed as the deformed Schrödinger cat type states derived from the superposition of four nonlinear coherent states (NCSs) which are π/2 out of phase. The nonclassical properties of the FPNCSs would be studied for two well-known quantum systems, a Kerr-like medium and a trapped laser-driven ion far from the Lamb-Dicke regime. We have found that the depth or the domain of the nonclassical features of FPNCSs in terms of sub-Poissonian photon statistics, higher order squeezing as well as amplitude-squared squeezing are higher than the corresponding standard states. Specifically, the proposed states in this paper exhibit sub-Poissonian statistics over an extensive range of real amplitudes which is in contrast with the lower level of sub-Poissonian characteristics of four-photon standard coherent states. Also the depth or domain of the nonclassical features of the FPNCSs can be modified by changing the magnitude of the nonlinearity parameters. Analysis of the quasi-probability distributions like Glauber–Sudarshan P-function, Husimi Q-function and Wigner distribution function verifies the nonclassical nature of the presented states. Studying the cavity field evolution in the interaction of a single two-level atom with an even NCS in the framework of f-deformed Jaynes-Cummings model shows that such a system can potentially be considered as a possible source for generating the proposed states.
Non-Gaussian states represent a powerful resource for quantum information protocols in the continuous variables regime. Cat states, in particular, have been produced in the motional degree of freedom of trapped ions by controlled displacements dependent on the ionic internal state. An alternative method harnesses the Kerr nonlinearity naturally present in this kind of system. We perform detailed calculations confirming its feasibility for typical experimental conditions. Additionally, this method permits generation of all other complex non-Gaussian states with negative Wigner functions resulting from Kerr nonlinear interaction. Especially, superpositions of several coherent states are achieved at a fraction of the time necessary to produce the cat state.