Energy constraints for quantum logic via nonlinear optical processes

The question of the constraints that the quantum nature of electromagnetic fields places on the realization of quantum logical operations by nonlinear optical processes is addressed by considering the effect of spontaneous emission, in free space, on phase gates, for both atomic and photonic qubits. In all cases, we find an error probability that scales as 1/n¯, where n¯ is the number of photons in the field, just as in previous studies. For the photonic qubits, the possibility of single-photon cross-phase modulation is studied in some detail, first in a simple three-level model, and then in a four-level scheme in which electromagnetically-induced transparency (EIT) is used to enhance the nonlinearity and reduce the linear absorption. It is nonetheless shown that, even in the latter case, an error probability of the order of 1/n¯c is obtained, where n¯c is the number of photons in the EIT “coupling” field.