Optimized preparation of quantum states by conditional measurements.

State preparation of quantum systems is a prerequisite for studying fundamental aspects of quantum measurement theory @1#, as well as for encoding quantum information @2# and its processing ~computing! @3#. Problems of state preparation have been dealt with most extensively in the realm of cavity quantum electrodynamics @4–8#. One proposal for state preparation of a cavity field mode @8# relies entirely on its unitary evolution, via coupling with a rather complex system, an atom having several Zeeman sublevels. This coupling results ~under perfectly adiabatic conditions! in a oneto-one mapping of the initial sublevel superposition to a superposition of Fock ~photon-number! states. Can one alternatively use a simple field-atom interaction, e.g., the resonant Jaynes-Cummings ~JC! model @4# or the off-resonant Kerrlike interaction @5#, followed by a measurement on the atom leaving the cavity, and repeat the process over and over again until the desired state of the field is attained? In general, measurements of atomic observables after the interaction would yield random results for the prepared field state @4,5#. In order to prepare predetermined field states, the conditional measurement ~CM! approach has been suggested @6#. In this approach, only those sequences of atoms in which each atom is found after the interaction to be in a chosen state are used to guide the field evolution to the desired state, whereas all other measurement sequences are discarded, at the price of atomic post selection probability, which is less than unity at each step of the sequence. The CM approach has been significantly enriched by a recipe for constructing an arbitrary superposition of Fock states @7#. It is based on a recurrence relation, which allows one to retract the desired superposition back to the starting vacuum state, by determining the possible initial atomic states and interaction times ~in the JC model! at each step of the CM sequence. The practical restriction on this recipe is that the probability of the resulting CM sequences falls off rapidly with the maximal photon number in the superposition. Our aim here is to address the basic questions of state preparation via quantum measurements: ~a! Given a simple field-atom interaction, as in the JC model, and a choice of experimentally realizable initial field states ~e.g., coherent states!, can the field converge to any desired ‘‘target’’ state to within the required accuracy via a finite number of measurements? Hilbert space topology arguments supported by numerical calculations are given to show that such convergence is in general attained, provided that the number of control parameters per CM is comparable to the dimensionality of the target-state subspace. ~b! How can one choose a CM sequence connecting the initial and ‘‘target’’ states, so as to maximize its success probability and minimize its length ~the required number of CMs!? Although in principle it should be possible to choose the CM sequence with the highest ratio of probability to length, in practice such optimization amounts to the formidable task of a global search over a huge parameter space ~whose dimensionality is the number of parameters per CM times the maximal admissible number of CMs!. We demonstrate that there is a simple and computationally fast alternative, namely, stepwise optimization by search over the parameter space of one CM at a time, allowing one to choose a high-probability CM sequence from among those that converge to the target state monotonically. Let us first formulate our strategy in general terms, suitable for any dynamical model. Suppose that we have started from the field state uc0&5SnCn ,0un&. After K21 CMs the field state is ucK21&5SnCn ,K21un& and we are trying to obtain the ‘‘target’’ state uc t&n5nmin n max Cn ,tun& via an optimal route in Hilbert space. Choosing the next atom to be in initial state ufK &, we unitarily evolve the initial field-atom product state ucK21& ^ ufK & over time tK by the operator U(tK) and perform a CM by projecting the resulting entangled state onto a postselected final atomic state ufK ( f &. This choice of initial and final atomic states and of tK corresponds to choosing a field-state CM transformation