arXiv:0705.2045v3 [quant-ph] 29 Apr 2008 Methods for Producing Optical Coherent State Superpositions S. Glancy Mathematical and Computational Sciences Division, National Institute of Standards and Technology, Boulder, Colorado 80305, USA * H. M. Vasconcelos † Departamento de Engenharia Metal´ ugica e de Materiais, Universidade Federal do Cear´a, Campus do Pici - Bloco 714, Fortaleza, Cear´a 60455-760, Brazil (Dated: 2008 April 28) We discuss several methods to produce superpositions of optical coherent states (also known as “cat states”). Cat states have remarkable properties that could allow them to be powerful tools for quantum information processing and metrology. A number of proposals for how one can produce cat states have appeared in the literature in recent years. We describe these proposals and present new simulation and analysis of them incorporating practical issues such as photon loss, detector inefficiency, and limited strength of nonlinear interactions. We also examine how each would perform in a realistic experiment. PACS numbers: 42.50.Dv, 03.67.Lx Keywords: Schr¨ odinger cat state, coherent state superposition, linear optics I. INTRODUCTION For many years a primary goal for researchers in quan- tum optics has been the generation of exotic quantum states of light. These have included squeezed vacuum states, photon number eigenstates (also known as Fock states), and many examples of entangled sets of pho- tons. States such as these have served well in many ba- sic tests of the foundations of quantum theory, and they may eventually prove to be useful for quantum computa- tion, quantum communication, quantum metrology, and lithography. [1] Therefore there is much interest in im- proving methods to produce these states and to generate other types of optical quantum states. The particular quantum state of interest in this pa- per is a superposition of two coherent states with oppo- site phase, which is often referred to as a (Schr¨ odinger) cat state. Cat states may be used as the logical qubit basis in a quantum computer [2, 3]. They may also serve as input states to an interferometer that is able to measure distances with greater accuracy than achievable within the limits usually imposed by the light’s wave- length [4]. Transforming a single coherent state into a cat state through unitary evolution alone would require a strong nonlinearity. Also, cat states are extremely sensitive to decoherence from photon absorption. For these reasons cat states containing more than one photon on average have been produced only in cavity-quantum- electrodynamic experiments in which an atom interacts with the electromagnetic field confined to a high finesse optical cavity [5, 6]. In experiments of this sort the cav- ity confines the optical mode to a small volume so that it * Electronic address: sglancy@boulder.nist.gov † Electronic address: hilma@ufc.br interacts very strongly with an atom passing through the cavity. Unfortunately, because the cat state is confined to a cavity, it can neither be manipulated with tools such as beam splitters or phase shifters, nor be measured with standard optical means such as photon counters or ho- modyne detection. For the uses described in [2, 3, 4] we require cat states that occupy freely propagating optical modes. In recent years researchers have proposed several schemes to produce freely propagating cat states. The purpose of this paper is to provide a comprehensive re- view and critique of these schemes. We examine the performance of these schemes in realistic experimental environments subject to problems such as photon loss, detector inefficiency and noise, and limited strength of nonlinear interactions. We also make recommendations for the design of cat production experiments. In this in- troductory section we review the properties of cat states and optical tools such as beam splitters and photon coun- ters, which are commonly used in the cat production schemes. The introduction also describes our model for photon absorption, how photon absorption affects cat states, and the difficulties of verifying that a cat state has been produced in an experiment. Section 2 exam- ines a method originally proposed by Yurke and Stoler [7] to transform a coherent state into a cat state using the optical Kerr effect. Section 3 briefly discusses the suggestion by Wolinsky and Carmichael [8] that one may make cat states using a degenerate parametric oscillator in the strong coupling regime. Section 4 examines the method proposed by Song, Caves and Yurke [9] to pro- duce cat states based on an optical backaction evasion measurement. Section 5 examines the method of photon subtraction from a squeezed vacuum state proposed by Dakna and co-authors [10]. Section 6 examines a method from Lund, Jeong, Ralph, and Kim [11] by which one may use small amplitude cat states, linear optical devices, and