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6
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Phys. Rev. AJ. Gea-Banacloche, N. Lu, L. M. Pedrotti, S. Prasad, M. O. Scully, and K. Wódkiewicz, 41, 369 (1990)
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Phys. Rev. A
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Gea-Banacloche, J.1
Lu, N.2
Pedrotti, L.M.3
Prasad, S.4
Scully, M.O.5
Wódkiewicz, K.6
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10
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0038515030
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Chap. 5; PRPLCM
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H. Dekker, Phys. Rep. 80, 1 (1981), Chap. 5
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(1981)
Phys. Rep.
, vol.80
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Dekker, H.1
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15
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0001309782
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Phys. Rev. Lett.D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld, 58, 1320 (1987)
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Phys. Rev. Lett.
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Heinzen, D.J.1
Childs, J.J.2
Thomas, J.E.3
Feld, M.S.4
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19
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85036319199
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See Ref. c12 for the relation between the two classes of dissipative systems: one in which the coupling is mediated through a term in the total Hamiltonian, and the other in which the coupling is mediated by a boundary condition—the continuity of φ across [Formula Presented] in the present case
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See Ref. 12 for the relation between the two classes of dissipative systems: one in which the coupling is mediated through a term in the total Hamiltonian, and the other in which the coupling is mediated by a boundary condition—the continuity of φ across x=a in the present case.
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21
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85036357540
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Springer-Verlag, Singapore, S. C. Lim, R. Abd-Shukor, K. H. Kwek
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P. T. Leung, A. Maassen van den Brink, and K. Young, in Frontiers in Quantum Physics, Proceedings of the International Conference, edited by S. C. Lim, R. Abd-Shukor, and K. H. Kwek (Springer-Verlag, Singapore, 1998).
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(1998)
Frontiers in Quantum Physics, Proceedings of the International Conference
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Leung, P.T.1
Maassen van den Brink, A.2
Young, K.3
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23
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4244188552
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E. S. C. Ching, P. T. Leung, W. M. Suen, and K. Young, Phys. Rev. Lett. 74, 2414 (1995)
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(1995)
Phys. Rev. Lett.
, vol.74
, pp. 2414
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Ching, E.S.C.1
Leung, P.T.2
Suen, W.M.3
Young, K.4
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27
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85036392276
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Since the positivity of ρ limits its singularity at [Formula Presented] to at most a δ function, φ itself is continuous. Stronger singularities in ρ would also leave equations such as Eq. (2.4) undefined distributionally
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Since the positivity of ρ limits its singularity at x=a to at most a δ function, φ itself is continuous. Stronger singularities in ρ would also leave equations such as Eq. (2.4) undefined distributionally.
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30
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0028405633
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Phys. Rev. AP. T. Leung, S. Y. Liu, S. S. Tong, and K. Young, 49, 3068 (1994).
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(1994)
Phys. Rev. A
, vol.49
, pp. 3068
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Leung, P.T.1
Liu, S.Y.2
Tong, S.S.3
Young, K.4
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33
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85036171158
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The technical condition that φ and [Formula Presented] be differentiable, as well as the speed of convergence of the sum in Eq. (2.8), are outside the present scope; see Ref. c12
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The technical condition that φ and φ^/ρ be differentiable, as well as the speed of convergence of the sum in Eq. (2.8), are outside the present scope; see Ref. 12.
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34
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85036184275
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Notice that, while novel for open classical systems as in Eq. (2.8), the concept of two-component expansion—where the same [Formula Presented] are used to expand both φ and [Formula Presented] as in Eq. (5.8)—enjoys a wide textbook popularity for closed quantum systems. See, e.g., Ref. c16
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Notice that, while novel for open classical systems as in Eq. (2.8), the concept of two-component expansion—where the same αj(†) are used to expand both φ and φ^ as in Eq. (5.8)—enjoys a wide textbook popularity for closed quantum systems. See, e.g., Ref. 16.
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35
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85036327328
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A. Messiah, Quantum Mechanics, Vol. II (North-Holland, Amsterdam, 1961)
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A. Messiah, Quantum Mechanics, Vol. II (North-Holland, Amsterdam, 1961).
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36
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0001350093
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P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, Phys. Rev. E 57, 6101 (1998).PLEEE8
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(1998)
Phys. Rev. E
, vol.57
, pp. 6101
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Leung, P.T.1
Suen, W.M.2
Sun, C.P.3
Young, K.4
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40
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0003851731
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Dover, New York, M. Abramowitz, I. A. Stegun
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Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (Dover, New York, 1970).
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(1970)
Handbook of Mathematical Functions
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41
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85036348750
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The fact that the sum in Eq. (2.8) converges more slowly if [Formula Presented] c12 is irrelevant in the present formal development. In any case, for operator fields the large-wave-number components never get small as they do for smooth classical fields, irrespective of the expansion method. For numerical applications, it is best to optimize the final formulas for observable physical quantities for convergence along the lines of Sec. III of Ref. c13. Notice, however, that comparison of methods [Formula Presented] [Formula Presented] in Ref. c12 shows the surface term in Eq. (4.3) to improve convergence also for function pairs outside Γ, in particular, to be essential for convergence to the second component
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The fact that the sum in Eq. (2.8) converges more slowly if φ∉Γ 12 is irrelevant in the present formal development. In any case, for operator fields the large-wave-number components never get small as they do for smooth classical fields, irrespective of the expansion method. For numerical applications, it is best to optimize the final formulas for observable physical quantities for convergence along the lines of Sec. III of Ref. 13. Notice, however, that comparison of methods B and C in Ref. 12 shows the surface term in Eq. (4.3) to improve convergence also for function pairs outside Γ, in particular, to be essential for convergence to the second component.
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42
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85036405070
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However, we shall not notationally distinguish between corresponding classical and quantized fields, since the meaning will be clear from the context
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However, we shall not notationally distinguish between corresponding classical and quantized fields, since the meaning will be clear from the context.
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43
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85036431662
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Notice that in general [Formula Presented] provided that one chooses [Formula Presented] a relation that is imposed by normalization only up to a sign
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Notice that in general a-j=aj† provided that one chooses f-j=fj*, a relation that is imposed by normalization only up to a sign.
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44
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85036241689
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For simplicity of notation we discuss Eq. (6.5) for real ω only, but its proof can be extended to complex ω by continuing both sides analytically
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For simplicity of notation we discuss Eq. (6.5) for real ω only, but its proof can be extended to complex ω by continuing both sides analytically.
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45
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85036181266
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This expression is actually symmetric under [Formula Presented] as will be shown in Appendix D
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This expression is actually symmetric under x↔y, as will be shown in Appendix D.
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48
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85036396163
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Equations (7.3), (7.4), (7.9), and (7.10) apply for real ω, and the usual care c20 is needed when performing an analytic continuation from either positive or negative frequencies
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Equations (7.3), (7.4), (7.9), and (7.10) apply for real ω, and the usual care 20 is needed when performing an analytic continuation from either positive or negative frequencies.
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49
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85036199227
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Per term; after summation over [Formula Presented] the factor multiplying [Formula Presented] is seen to vanish near [Formula Presented] as required by consistency
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Per term; after summation over n the factor multiplying ω -1 is seen to vanish near ω=0, as required by consistency.
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51
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85036196268
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the case [Formula Presented] our numbering violates the convention [Formula Presented] Likewise Eq. (8.4) violates the sign convention given earlier. However, these cause no problems here
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In the case n>n0 our numbering violates the convention ω-j=-ωj*. Likewise Eq. (8.4) violates the sign convention given earlier. However, these cause no problems here.
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57
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85036258946
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This corresponds to a model of an optical cavity with a thin slab of material of high dielectric constant, forming a mirror with some output coupling; see, e.g., Ref. c1
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This corresponds to a model of an optical cavity with a thin slab of material of high dielectric constant, forming a mirror with some output coupling; see, e.g., Ref. 1.
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58
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85036393699
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Of course the projection formula for the latter, as opposed to the one for a QNM expansion, involves complex conjugation
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Of course the projection formula for the latter, as opposed to the one for a QNM expansion, involves complex conjugation.
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62
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85036326720
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Up to normalization [Formula Presented] the MU wave function ψ of Appendix D being defined on a continuum of frequencies in the limit [Formula Presented]
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Up to normalization f(x,ω)=ψ(x,ω), the MU wave function ψ of Appendix D being defined on a continuum of frequencies in the limit Λ→∞.
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63
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85036268502
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Eq. (5.8), the normalization (2.11) forces the [Formula Presented] to be real in the conservative limit, in contrast to the standard formalism where their phase is arbitrary. For uniformity we choose the ψ in Eq. (D1) to be real as well; this shows that the result in Eq. (D2) is well defined
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In Eq. (5.8), the normalization (2.11) forces the fj to be real in the conservative limit, in contrast to the standard formalism where their phase is arbitrary. For uniformity we choose the ψ in Eq. (D1) to be real as well; this shows that the result in Eq. (D2) is well defined.
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