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Methods in Theoretical Quantum Optics$

Stephen Barnett and Paul Radmore

Print publication date: 2002

Print ISBN-13: 9780198563617

Published to British Academy Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780198563617.001.0001

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(p.236) Appendix 5 OPERATOR ORDERING THEOREMS

(p.236) Appendix 5 OPERATOR ORDERING THEOREMS

Source:
Methods in Theoretical Quantum Optics
Publisher:
Oxford University Press

In Chapter 3, we derived a number of theorems relating different orderings of the exponential function of operators. Here we summarize these theorems and give some alternative forms and more general expressions.

For two operators Appendix 5 Operator Ordering Theorems and Appendix 5 Operator Ordering Theorems which commute with their commutator Appendix 5 Operator Ordering Theorems we have

(A5.1) Appendix 5 Operator Ordering Theorems
Important examples of this are
(A5.2) Appendix 5 Operator Ordering Theorems Appendix 5 Operator Ordering Theorems
(A5.3) Appendix 5 Operator Ordering Theorems
and the continuum operator analogue of these given by
(A5.4) Appendix 5 Operator Ordering Theorems
and
(A5.5) Appendix 5 Operator Ordering Theorems

The exponential function of the number operator is related to its normal ordered form by

(A5.6) Appendix 5 Operator Ordering Theorems
(p.237) with the continuum operator generalization
(A5.7) Appendix 5 Operator Ordering Theorems
There is an antinormal ordered analogue of (A5.6) given by
(A5.8) Appendix 5 Operator Ordering Theorems
but there is no antinormal ordered continuum generalization.

For two operators Appendix 5 Operator Ordering Theorems and Appendix 5 Operator Ordering Theorems with commutator Appendix 5 Operator Ordering Theorems, we have

(A5.9) Appendix 5 Operator Ordering Theorems
This result may also be applied to superoperators as in Section 5.6.

The angular momentum operators Appendix 5 Operator Ordering Theorems Appendix 5 Operator Ordering Theorems and Appendix 5 Operator Ordering Theorems satisfy the commutation relations Appendix 5 Operator Ordering Theorems and Appendix 5 Operator Ordering Theorems. We have

(A5.10) Appendix 5 Operator Ordering Theorems
which may be generalized to
(A5.11) Appendix 5 Operator Ordering Theorems
where
(A5.12) Appendix 5 Operator Ordering Theorems
(A5.13) Appendix 5 Operator Ordering Theorems
and
(A5.14) Appendix 5 Operator Ordering Theorems

These theorems can be applied to a pair of field modes with annihilation operators Appendix 5 Operator Ordering Theorems and Appendix 5 Operator Ordering Theorems by making the identifications

(A5.15) Appendix 5 Operator Ordering Theorems
(A5.16) Appendix 5 Operator Ordering Theorems

(p.238) The operators Appendix 5 Operator Ordering Theorems and Appendix 5 Operator Ordering Theorems satisfy the commutation relations Appendix 5 Operator Ordering Theorems Appendix 5 Operator Ordering Theorems and Appendix 5 Operator Ordering Theorems We have

(A5.17) Appendix 5 Operator Ordering Theorems
which may be generalized to
(A5.18) Appendix 5 Operator Ordering Theorems
where
(A5.19) Appendix 5 Operator Ordering Theorems
(A5.20) Appendix 5 Operator Ordering Theorems
and
(A5.21) Appendix 5 Operator Ordering Theorems
These theorems can be applied either to a single field mode by making the identifications
(A5.22) Appendix 5 Operator Ordering Theorems
(A5.23) Appendix 5 Operator Ordering Theorems
or to a pair of modes by making the identifications
(A5.24) Appendix 5 Operator Ordering Theorems
(A5.25) Appendix 5 Operator Ordering Theorems

Two operators are equivalent if their matrix elements between any two basis states are equal, for all possible pairs of basis states. This can be used to derive ordering theorems. We illustrate this by obtaining (A5.6) using the number state and coherent state bases. As in Chapter 3, let

(A5.26) Appendix 5 Operator Ordering Theorems
(p.239) The number state matrix elements of each side of (A5.26) are
(A5.27) Appendix 5 Operator Ordering Theorems
and
(A5.28) Appendix 5 Operator Ordering Theorems
using (3.4.1). It follows immediately that Appendix 5 Operator Ordering Theorems. Alternatively we can use the coherent state basis to establish this result. It is sufficient to equate only the diagonal matrix elements in this basis since, apart from a normalization factor Appendix 5 Operator Ordering Theorems depends only on α‎ while Appendix 5 Operator Ordering Theorems depends only on Appendix 5 Operator Ordering Theorems and these are treated as two independent variables. The diagonal coherent state matrix elements of each side of (A5.26) are
(A5.29) Appendix 5 Operator Ordering Theorems
using (3.6.32) and (3.6.24), and
(A5.30) Appendix 5 Operator Ordering Theorems
using Appendix 5 Operator Ordering Theorems. Equating (A5.29) and (A5.30) again gives Appendix 5 Operator Ordering Theorems Appendix 5 Operator Ordering Theorems.