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Erratum to: “On detection of vacuum birefringence using intense laser pulses” [Phys. Lett. A 330 (2004) 429]
Physics Letters A 346 (2005) 385–386 www.elsevier.com/locate/pla
Erratum
Erratum to: “On detection of vacuum birefringence using intense laser pulses” [Phys. Lett. A 330 (2004) 429] Andre N. Luiten ∗ , Jesse C. Petersen 1 School of Physics M013, The University of Western Australia, Nedlands, Western Australia 6009, Australia Received 12 August 2005 Available online 24 August 2005
Since publication of this article it has been brought to our attention that our reliance on the work in Ref. [1] is only valid in particular circumstances. By incorporating the recent work of Rikken and Rizzo [2,3] we find a modification to the result originally predicted. In particular, the sensitivity of the experiment will be improved by a factor of two over the value that was originally presented. Eq. (1) in the former paper was [4] (in SI units): 16 α 2 U 28 α 2 U , n⊥ = 1 + , (1) 45 Ue 45 Ue where α is the fine structure constant, U is the energy density of a counter-propagating optical field and Ue = m4e c5 /h¯ 3 ≈ 1.44 × 1024 J m−3 is the approximate Compton energy density of the electron (me is the electron rest mass). The equivalent expressions given in the more recent work [2] are (where we have converted from the Heaviside–Lorentz units used in the paper to SI units here): n = 1 + 2ξ c2 B02 + E02 + 2cE0 B0 , (2) n⊥ = 1 + 7/2ξ c2 B02 + E02 + 2cE0 B0 , n = 1 +
where ξ = e4 h¯ /(180π0 m4e c7 ), E0 is the electric field amplitude, and B0 is the magnetic field amplitude. This form of the equations is only valid for the particular circumstances in which the two interacting laser beams are exactly counter-propagating (as was also the case with Eq. (1)). In our experimental arrangement, the laser beams are very nearly in this configuration and negligible error is introduced by assuming an exact counter-propagating configuration. By substituting for the average energy of an electromagnetic field, U = 0 E02 into Eq. (1), and using the equivalence of magnetic and electric field energy in an electromagnetic wave, one finds that Eq. (1) is identical to the sum of the first two bracketed terms in Eq. (2) (this is true for both n⊥ and n ). The final cross term in the brackets DOI of original article: 10.1016/j.physleta.2004.08.020. * Corresponding author.
E-mail address: [email protected] (A.N. Luiten). 1 Department of Physics, Simon Fraser University, Burnaby, BC, Canada.
0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.08.038
386
A.N. Luiten, J.C. Petersen / Physics Letters A 346 (2005) 385–386
of Eq. (2) was not included in the expressions obtained in Ref. [1]. That additional term is exactly equal in value to the sum of the first two terms (in this particular situation). This doubles the change in refractive index for a given electromagnetic field with respect to that predicted by Eq. (1). We presented a table (Table 1) in Ref. [4] that summarises the observational time needed to obtain a signal-to-noise ratio of unity in various circumstances. The final column in this table (the observational time) will be improved (reduced) by a factor of 4 from the values given in that paper. We would also like to use this opportunity to correct an impression that we may have inadvertently created in our original article that the project, Biréfringence Magnétique du Vide (BMV) made use of superconducting electromagnets. In fact the project will make use of copper electromagnets that will be cooled using liquid nitrogen [5]. We would like to thank Professor Carlo Rizzo for bringing to our attention the work in Refs. [2,3], and thus the missing term in our original expressions.
References [1] [2] [3] [4] [5]
E.B. Aleksandrov, et al., Sov. Phys. JETP 62 (1985) 680. G.L.J.A. Rikken, C. Rizzo, Phys. Rev. A 63 (2000) 012107. G.L.J.A. Rikken, C. Rizzo, Phys. Rev. A 67 (2003) 015801. A.N. Luiten, J.C. Petersen, Phys. Lett. A 330 (2004) 429. S. Askenazy, et al., in: G. Cantatore (Ed.), Quantum Electrodynamics and the Physics of the Vacuum, American Institute of Physics, 2001, p. 115.
Erratum
Erratum to: “On detection of vacuum birefringence using intense laser pulses” [Phys. Lett. A 330 (2004) 429] Andre N. Luiten ∗ , Jesse C. Petersen 1 School of Physics M013, The University of Western Australia, Nedlands, Western Australia 6009, Australia Received 12 August 2005 Available online 24 August 2005
Since publication of this article it has been brought to our attention that our reliance on the work in Ref. [1] is only valid in particular circumstances. By incorporating the recent work of Rikken and Rizzo [2,3] we find a modification to the result originally predicted. In particular, the sensitivity of the experiment will be improved by a factor of two over the value that was originally presented. Eq. (1) in the former paper was [4] (in SI units): 16 α 2 U 28 α 2 U , n⊥ = 1 + , (1) 45 Ue 45 Ue where α is the fine structure constant, U is the energy density of a counter-propagating optical field and Ue = m4e c5 /h¯ 3 ≈ 1.44 × 1024 J m−3 is the approximate Compton energy density of the electron (me is the electron rest mass). The equivalent expressions given in the more recent work [2] are (where we have converted from the Heaviside–Lorentz units used in the paper to SI units here): n = 1 + 2ξ c2 B02 + E02 + 2cE0 B0 , (2) n⊥ = 1 + 7/2ξ c2 B02 + E02 + 2cE0 B0 , n = 1 +
where ξ = e4 h¯ /(180π0 m4e c7 ), E0 is the electric field amplitude, and B0 is the magnetic field amplitude. This form of the equations is only valid for the particular circumstances in which the two interacting laser beams are exactly counter-propagating (as was also the case with Eq. (1)). In our experimental arrangement, the laser beams are very nearly in this configuration and negligible error is introduced by assuming an exact counter-propagating configuration. By substituting for the average energy of an electromagnetic field, U = 0 E02 into Eq. (1), and using the equivalence of magnetic and electric field energy in an electromagnetic wave, one finds that Eq. (1) is identical to the sum of the first two bracketed terms in Eq. (2) (this is true for both n⊥ and n ). The final cross term in the brackets DOI of original article: 10.1016/j.physleta.2004.08.020. * Corresponding author.
E-mail address: [email protected] (A.N. Luiten). 1 Department of Physics, Simon Fraser University, Burnaby, BC, Canada.
0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.08.038
386
A.N. Luiten, J.C. Petersen / Physics Letters A 346 (2005) 385–386
of Eq. (2) was not included in the expressions obtained in Ref. [1]. That additional term is exactly equal in value to the sum of the first two terms (in this particular situation). This doubles the change in refractive index for a given electromagnetic field with respect to that predicted by Eq. (1). We presented a table (Table 1) in Ref. [4] that summarises the observational time needed to obtain a signal-to-noise ratio of unity in various circumstances. The final column in this table (the observational time) will be improved (reduced) by a factor of 4 from the values given in that paper. We would also like to use this opportunity to correct an impression that we may have inadvertently created in our original article that the project, Biréfringence Magnétique du Vide (BMV) made use of superconducting electromagnets. In fact the project will make use of copper electromagnets that will be cooled using liquid nitrogen [5]. We would like to thank Professor Carlo Rizzo for bringing to our attention the work in Refs. [2,3], and thus the missing term in our original expressions.
References [1] [2] [3] [4] [5]
E.B. Aleksandrov, et al., Sov. Phys. JETP 62 (1985) 680. G.L.J.A. Rikken, C. Rizzo, Phys. Rev. A 63 (2000) 012107. G.L.J.A. Rikken, C. Rizzo, Phys. Rev. A 67 (2003) 015801. A.N. Luiten, J.C. Petersen, Phys. Lett. A 330 (2004) 429. S. Askenazy, et al., in: G. Cantatore (Ed.), Quantum Electrodynamics and the Physics of the Vacuum, American Institute of Physics, 2001, p. 115.