TY - JOUR

T1 - Positive P representation

T2 - Application and validity

AU - Gilchrist, A.

AU - Gardiner, C. W.

AU - Drummond, P. D.

PY - 1997/4

Y1 - 1997/4

N2 - The positive P representation is a very successful tool in quantum optics. However, the usual assumption of negligible boundary terms in the time-evolution equations is not always valid. We explore the range of validity of the time-evolution equations both analytically and by numerical investigation of a number of specific examples. We present practical ways of verifying the validity of the use of the positive P representation and find that the standard time-evolution equation can become invalid when nonlinear terms (at unit photon number) are large relative to the damping rate. This is very much larger than is normally the case in nonlinear optics, except possibly near resonances. We are able to show that when the positive P representation is invalid, the boundary terms, normally neglected in an integration by parts, become non-negligible. When numerical simulations are carried out using the positive P representation, specific checks given in this paper should be carried out to verify the compactness of the distribution. In conclusion, we find that (apart from special cases) this technique of quantum time evolution is typically asymptotically valid in the limit of small nonlinearity, rather than exact.

AB - The positive P representation is a very successful tool in quantum optics. However, the usual assumption of negligible boundary terms in the time-evolution equations is not always valid. We explore the range of validity of the time-evolution equations both analytically and by numerical investigation of a number of specific examples. We present practical ways of verifying the validity of the use of the positive P representation and find that the standard time-evolution equation can become invalid when nonlinear terms (at unit photon number) are large relative to the damping rate. This is very much larger than is normally the case in nonlinear optics, except possibly near resonances. We are able to show that when the positive P representation is invalid, the boundary terms, normally neglected in an integration by parts, become non-negligible. When numerical simulations are carried out using the positive P representation, specific checks given in this paper should be carried out to verify the compactness of the distribution. In conclusion, we find that (apart from special cases) this technique of quantum time evolution is typically asymptotically valid in the limit of small nonlinearity, rather than exact.

UR - http://www.scopus.com/inward/record.url?scp=0001312055&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0001312055

VL - 55

SP - 3014

EP - 3032

JO - Physical Review A: covering atomic, molecular, and optical physics and quantum information

JF - Physical Review A: covering atomic, molecular, and optical physics and quantum information

SN - 2469-9926

IS - 4

ER -