Recurrence formula for generalized optical state truncation by projection synthesis

Abstract

Several interesting states of the quantized electromagnetic field are studied nowadays. Among them one can cite the number and ~its complementary! phase states @1#, the coherent @2# and squeezed states @3#, superposition states having the previous basic states as components @4#, the interpolating states @5#, which vary between two limiting states, going from one to the other, etc. So, in view of the existence of an abundance of states in quantum optics, it is of interest to know how to prepare these states experimentally, this procedure being known in the literature as ‘‘quantum state engineering’’ @6#. The engineering schemes may consider either the case of stationary waves prepared inside a ~high-Q) cavity @6,7# or the case of traveling waves @8,9#. In the realm of cavity QED phenomena, Vogel et al. @6# employed a resonant atom-field interaction to build up a trapped field in an initially empty cavity, while the proposal in Ref. @10# considers both resonant and dispersive atom-field interactions for the preparation of a general superposition in the empty cavity. An alternative proposal has been presented @11#, named the sculpture of quantum states, where a coherent state is previously injected into the cavity and the Wigner distribution function of the desired state is sculptured, through atom-field interaction, from that representing the initial coherent state, the atoms playing the role of quantum chisels. In recent work by Pegg, Phillips, and Barnett @8#, preparation of an arbitrary running-wave superposition of the vacuum and one-photon states, C0u0&1C1u1&, without using cavities was demonstrated. In this way, a traveling field would be available for further applications, including performing measurements on other field states @12,13#. The scheme in @8# obtains the above mentioned superposition by physical truncation of the photon number superposition making up a coherent state. The proposal requires no additional extension of current experiments and is reasonably insensitive to photodetection efficiency for the fields most likely to be used in practice. Since the scheme works via a truncation of the Hilbert space, it has been called a ‘‘quantum scissors’’ device @8#. As mentioned in @8#, states including superpositions of higher photon numbers might be fabricated by superposing fields prepared as superpositions of zeroand onephoton number states @14#. In this connection, we will present here an alternative way to prepare arbitrary truncated states, i.e., C0u0&1C1u1&1C2u2&1•••1CNuN&, N 51,2,3, . . . . . The method is a direct extension of that in @8#, called the optical truncation of a state by projection synthesis.