Linear and angular momentum of the quantum vacuum
Bart van Tiggelen and Geert Rikken
• Sébastien Kawka (Ph.D ENS Lyon) •James Babington (postdoc ANR) •Manuel Donaire (Postdoc ANR) Casimir energy
1. Isotropic radiation with power spectrum ω3 is Lorentz-invariant (Einstein, 1917); 2. Van der Waals force 1/r6 (London, 1930) 3. Relation to Cosmological constant (Pauli, 1934, Davies, 1984) 4. Casimir Polder Force 1/r7 (1947) 5. Attraction between metallic plates (Casimir, 1948) comments 6. Lifshitz theory for dielectric media (Lifshitz, 1956, Dzyalonishiniskii 1961) 7. Observation of Casimir effect (Sparnaay, 1958, Lamoureux (5%), 1997), Chan etal, (1%), 2001) 8. Unruh effect & Hawking radiation (Hawking 1974, Unruh 1976) 9. Sonoluminescence (Schwinger, 1993, Eberlein, 1996) comments! 10. Quantum friction and sheering the quantum vacuum (Pendry, 1998), comments! 11. Casimir momentum in magneto-electric media (Feigel, 2004). Comments! 12. Casimir momentum of magneto-chiral media (Donaire, BVT, 2015)
Bart van Tiggelen GDR Aussois 2018 Photons in Homogeneous media Photons interact with a magnetic field (T)
2 2 휔 풌,휎,푩 2 n 2 = 퐤 + 휎푉퐁 Faraday effect (1846) 푐0 Photons are bent by a magnetic field 퐶 푉 Landau & Lifshitz ECM exercice v = 0 풌 + 휎 푩 퐺 푛 푘 Rikken & Van Tiggelen (1997). Photons probe chirality of matter (P) 2 2 휔 풌,휎 2 n 2 = 1 + 휎푔 풌 Rotatory power (1811) 푐0 Photons probe magneto-chirality (PT)
2 Magneto-chiral effect. Pasteur, 1884……. 2 휔 풌,휎,푩 2 Groenewege, Mol. Phys. (1962) n 2 = 퐤 + 푉푔퐁 푐0 Kleindienst & Wagnière, CPL 288 (1998) Rikken & Raupach, Nature (1997)
Bart van Tiggelen GDR Aussois 2018 Photons in Bi-anisotropic Media Dω= εωEω+gωBω Hω= gT ωE()+ μω1Bω v l g ω = g E0B0 B0E0 gij (ω)= iω gδij gij (ω)= (1− ε)εijl ij i j i j c0 Rotatory power Fizeau effect Magneto-electric birefringence 10-2 10-8 10-15
v E0 x B0
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Bart van Tiggelen GDR Aussois 2018 « Momentum from Nothing »
B0 ε,μ,g P=mv E0
Bart van Tiggelen GDR Aussois 2018 Photons in EM Semi-classical quantum vacuum
force on particle dispersive force Abraham force
1 휕 휌풗 + 훁 ∙ 푼 = +푃 휵퐸 + 푀 휵퐵 + 휕 푷 × 푩 푡 푚 푚 푚 푚 푐 푡 1 10 휕푡 푬 × 푩 + 훁 ∙ 푻 = −푃푚훁퐸풎 − 푀푚훁퐵푚 − 휕푡 푷 × 푩 4휋푐0 푐0 - 1 3 2 휕푡 푀풗 + 푑 풓 푬 × 푩 + 푑 푺 ∙ 푻0 = 0 = 0 4휋푐0 force on particle EM momentum radiative force
1 1 2 ω4 d 3k g E B = c g E B k 0 0 3 4 0 0 EB c0 2 3 π c0 0 0 4c 1 1 v 0 d 3k 1 v k casi c0 2 c0
Bart van Tiggelen GDR Aussois 2018 Observation of the Abraham Force
E0=450 V/mm; B0=1 T = − 40 2 ( 3 ) Ex: Helium α(0) 0.22 10 Cm /V 16.6a0 ρ= 0.17 kg/m3 (room T ) (SI units) g= 0.017 10− 22 m/VT
Classical Abraham ε0α( 0)EB F 71032 N vabr = 0.3 nm/sec Force abr 2m 13 p NatFabr 10 N π h v = gEB≈ 0.02 nm /sec Feigel 4 4 Semi-classical QED with cut-off ρλc 0.1 nm (Feigel)
2 vQED vabr Z?α 0.001nm/sec Rigorous QED (Kawka, 2012 )
Bart van Tiggelen GDR Aussois 2018 푑풑 푑푬 = 훼(0) × 푩 푑푡 푑푡 Abraham force Acoustic P(ω)= P0+ α(0)× E× B× ω× cosωt × n× L pressure V= 8 nm/sec+- 0.8 Feigel correction: 2 nm/sec Excluded by errorbars
E=450 V/mm; B=1 T; f= 7.6 kHz
δP/(EB) α(0)
Bart van Tiggelen GDR Aussois 2018 Photons in Heterogeneous, chiral media
I call any geometrical figure, or group of points, chiral, and say it has chirality, if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself
Lord Kelvin, 1904 Chirality is quantified by a rotationally invariant pseudo-scalar
g 풓ퟏ, 풓ퟐ, 풓ퟑ, … = −g −풓ퟏ, −풓ퟐ, −풓ퟑ, … =g 푹풓ퟏ, 푹풓ퟐ, 푹풓ퟑ, … Harris, Kamien & Lubensky, 1999
Twisted H g= 푟2푑 푠푖푛(2훾)
Bart van Tiggelen GDR Aussois 2018 Photons in Heterogeneous, magnetochiral media 푑휎 푑휎 푑휎 풌, 풌′, 푔, 푩 = −풌′, −풌, 푔, −푩 = −풌, −풌′, −푔, 푩 푑훺 푑훺 푑훺 푑휎 = 퐹 풌 · 풌′ +V F 풌 · 풌′ det 푩, 풌, 풌′ +V F 풌 · 풌′ 푩 ·(k+k’) 푑훺 0 퐻 MC Mie scattering Photon Hall Effect : Magneto-chiral super-current: 1908 Rikken BVT 1996 never observed
Bart van Tiggelen GDR Aussois 2018 Photons in semi-classical quantum vacuum
an electromagnetic "London" super current ? B
푱 = 푞휌푨 ⇒ 0 푺 0 ~푔푉ℇ 푩 ?
Bart van Tiggelen GDR Aussois 2018 Photons in semi-classical quantum vacuum
(partially) unproven theorem: FDT ⇒ 0 풅ퟑ풓 푬 × 푯 0 = 0 twisted electric dipoles
1 3 PEM= 0 푑 풓 푬 × 푩 0 = 0 because B=H 4휋푐0 twisted magnetic dipoles finite & nonzero 휒 0 5 negligeable 푷퐸푀 ~ℏ푉푩 14 sin(2훾) 푘0퐿
Bart van Tiggelen GDR Aussois 2018 Casimir momentum in QED
Bart van Tiggelen GDR Aussois 2018 Casimir momentum in QED
Bart van Tiggelen GDR Aussois 2018 Magneto-chiral Casimir momentum in QED A chiral group + chromophoric electron bound anisotropically + magnetic field + quantum vacuum beyond ED approximation
+e
-e
Bart van Tiggelen GDR Aussois 2018 Momentum of Quantum Vacuum
8휋 푒2ℏ 1 1 = 훿푚 3 푐 푘 ℏ휔 + ℏ2푘2/2푚 0 푘 푘
Bart van Tiggelen GDR Aussois 2018 Magneto-chiral Casimir momentum in QED A chiral group + chromophoric electron bound anisotropically + magnetic field + quantum vacuum beyond ED approximation
∗ 푲 = 푷푘푖푛 + 푒푩0 × 풓푁푒 + 푒푨 풓푁 − 푒푨 풓푒 + ℏ풌 푎푘푔푎푘푔 kinetic Longitudinal Casimir 푘푔 momentum momentum
Conserved Pseudo Abraham Transverse Casimir momentum momentum momentum
2훼 푚푁 훽(0) 0 푷푘푖푛 0 = Log + 1 푒푩0 9휋 푚푒 훼(0)
C8H18O: 0.06 nm/sec/Tesla
Bart van Tiggelen GDR Aussois 2018 Angular Momentum of Quantum Vacuum B
n,V
Bart van Tiggelen GDR Aussois 2018 Momentum of Quantum Vacuum Conclusions and outlook ℏ풌
• Quantum vacuum possesses momentum and angular momentum in presence of chirality and magnetic fields • Non-zero but small . • Semi-classical approaches are inadequate - Unable to absorbe UV divergencies - Unable to include « longitudinal » momentum of gauge field - Unable to cope with recoil induces by vacuum photons • Quid cavity QED? Quid excited states?
Bart van Tiggelen GDR Aussois 2018 Angular Momentum of Quantum Vacuum B
e
푒 classically 퐿푧 + 훷 = 푐표푛푠푒푟푣푒푑 conserved 2휋푐0
Bart van Tiggelen GDR Aussois 2018 Angular Momentum of Quantum Vacuum B
e 푒 푒 퐽푧 = 퐿푧 푘푖푛 + 풓 × 푨0 풓 + 풓 × 푨 풓 푐0 푐0 1 + 푎† 푎 + 푎 푎† Φ∗ (풌) ∙ Υ ∙ Φ (풌) 푘2 푘휫 푘휫 푘휫 푘휫 휫 푧 휫 풌휫 훶 = −푖ℏ훿 풌 × 훁 − 푖ℏ휖 푧, 푛푚 푛푚 풌 푧 푧푛푚
Bart van Tiggelen GDR Aussois 2018 Angular Momentum of Quantum Vacuum B
e
ퟎ, 푁 = 0, 푚 퐽푧(푘푖푛) ퟎ, 푁 = 0, 푚 =
2 3 ℏ휔푐 2푚푐0 2 ℏ휔푐 ℏ 2 Log + ℏ 2 휋 푚푐0 ℏ휔푐 5휋 푚푐0
Bart van Tiggelen GDR Aussois 2018