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Methods in Theoretical Quantum Optics$

Stephen Barnett and Paul Radmore

Print publication date: 2002

Print ISBN-13: 9780198563617

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198563617.001.0001

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(p.234) Appendix 4 QUADRATURE EIGENSTATES

(p.234) Appendix 4 QUADRATURE EIGENSTATES

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

In Section 3.2, we introduced the quadrature operator Appendix 4 Quadrature Eigenstates for a single-mode field (see (3.2.20)). Here we describe the properties of this operator and its eigenstates Appendix 4 Quadrature Eigenstates in more detail. We recall that the commutator of Appendix 4 Quadrature Eigenstates and Appendix 4 Quadrature Eigenstates is

(A4.1) Appendix 4 Quadrature Eigenstates
and that
(A4.2) Appendix 4 Quadrature Eigenstates
Further, the quadrature representation Appendix 4 Quadrature Eigenstates of a single-mode state Appendix 4 Quadrature Eigenstates is defined by the relation Appendix 4 Quadrature Eigenstates. Here we show that these relations imply that the eigenstates of Appendix 4 Quadrature Eigenstates cannot satisfy Kronecker-delta orthogonality. For suppose that Appendix 4 Quadrature Eigenstates has a countable set of orthogonal eigenstates Appendix 4 Quadrature Eigenstates with eigenvalues Appendix 4 Quadrature Eigenstates so that
(A4.3) Appendix 4 Quadrature Eigenstates
and
(A4.4) Appendix 4 Quadrature Eigenstates
The matrix element of the commutator on the left-hand side of (A4.1) would then be
(A4.5) Appendix 4 Quadrature Eigenstates
whereas the matrix element of the right-hand side of (A4.1) would be simply Appendix 4 Quadrature Eigenstates. These clearly cannot be equal since the former is zero when m = n, whereas the latter is only non-zero when m = n. Hence the orthonormality in (A4.4) is inconsistent with the commutator (A4.1). The resolution of this difficulty is to employ delta function orthogonality for the eigenstates Appendix 4 Quadrature Eigenstates having a continuum of eigenvalues Appendix 4 Quadrature Eigenstates, so that
(A4.6) Appendix 4 Quadrature Eigenstates
In doing this, we have extended the set of possible states beyond those of square-integrable functions which can be represented in Hilbert space. This is because Appendix 4 Quadrature Eigenstates is not defined. Now taking matrix elements of (A4.1) in the basis of eigenstates Appendix 4 Quadrature Eigenstates we find
(A4.7) Appendix 4 Quadrature Eigenstates
(p.235) Hence
(A4.8) Appendix 4 Quadrature Eigenstates
where we have used the form of the derivative of the delta function given in Appendix 2. This result together with the resolution of the identity in terms of the quadrature eigenstates, given in (3.3.27), leads to a representation for the action of Appendix 4 Quadrature Eigenstates since
(A4.9) Appendix 4 Quadrature Eigenstates
This is just the familiar position representation of the action of the momentum operator on the state Appendix 4 Quadrature Eigenstates. The representation of the action of Appendix 4 Quadrature Eigenstates on Appendix 4 Quadrature Eigenstates is simply Appendix 4 Quadrature Eigenstates since
(A4.10) Appendix 4 Quadrature Eigenstates
We can use (1.3.33), (1.3.36), (3.2.36) and (A3.16) to write the quadrature eigenstate in the form
(A4.11) Appendix 4 Quadrature Eigenstates
From this, we can obtain the quadrature representation of any state. For example, the quadrature representation of the coherent state is
(A4.12) Appendix 4 Quadrature Eigenstates Appendix 4 Quadrature Eigenstates
in agreement with (3.6.45), where k is the final factor in (A4.12).