Dispersion interactions with long-time tails or beyond local equilibrium

Transcription

1 Dispersion interactions with long-time tails or beyond local equilibrium Carsten Henkel PIERS session Casimir effect and heat transfer (Praha July 2015) merci à : G. Barton (Sussex, UK), B. Budaev (Berkeley, CA) details in: Friction forces on atoms after acceleration F. Intravaia & al, J Phys Cond Matt 27 (2015) Institute of Physics and Astronomy, Universität Potsdam, Germany download slides

2 Motivation Dispersion Interactions Casimir energy quantum fluctuations van der Waals force radiative heat transfer Planck spectrum non-equilibrium steady state nano-scale heating... field theory vs radiation engineering

3 Motivation approximations fields Maxwell ε 0 t E = H j matter Schrödinger Newton t j ρ 0 m f Coulomb relaxation time approximation ε 0 2 φ = ρ t f(x, t) =... f(x, t) f eq(x) τ Ohm j(r; ω) = σ(ω)e(r; ω) why? response of matter system is nonlinear need approximations: popular is Born-Markov central assumption: separation of time scales fast thermalization slow local thermal equilibrium T (r, t) radiative heating

4 Motivation Example Born-Markov master equation for two-level medium (continuous) dp dt dn e dt E ε 0 t = (iω A + Γ)P + iχ(n g N e )E = γn e + 1 h Im(P E) = H dp dt separation of time scales: ultra-fast response of field, correlation time Markovian decay flat (white) spectrum fast response of matter, radiative (spontaneous) decay, slow re-distribution of populations narrow (peaked) spectrum

5 Outline Matter response to radiation: Born-Markov approximation separation of time scales This talk: spectra of (vacuum) field fluctuations ultra-slow, non-markovian correlations F ~ v F ~ v T 0 case study: metallic half-space, two-level system at short distance z λ relevant to quantum friction : G. Barton, Proc Roy Soc (London) A 453 (1997); J Phys Cond Matt 23 (2011) F. Intravaia & al, J Phys Cond Matt 27 (2015)

6 Field response Barton s model: recap potential F ~ v T φ(r, z, t) = d 2 k dω φ kω exp(ik r kz) a kω (t) + h.c. linear response to charge density δ φ(x) = dx χ(x, x )ρ(x ) x = (r, z, t) 0 F ~ v Kubo formula χ(x, x ) = ī h [φ(x), φ(x )] Θ(t t ) correlation response Ω p Ω s 0 Ω s Ω p non-retarded response χ(z, z ; k, ω) = e k(z+z ) 2ε 0 k R(ω) = ε(ω) 1 ε(ω) + 1 R(ω) Drude metal surface plasmon ω s = ω p / 2, damping Γ = 0.3 ω p

7 Field response φ(r, z, t) = d 2 k dω φ kω exp(ik r kz) a kω (t) + h.c. linear response to charge density δ φ(x) = dx χ(x, x )ρ(x ) x = (r, z, t) Kubo formula χ(x, x ) = ī h [φ(x), φ(x )] Θ(t t ) correlation response Ω p Ω s Drude metal 0 Ω s Ω p non-retarded response χ(z, z ; k, ω) = e k(z+z ) 2ε 0 k R(ω) back in time domain (damped oscillator) χ( r, r, τ) = ω s sin(ω s τ) e Γτ/2 4πε 0 r r im Θ(τ) surface plasmon ω s = ω p / 2, damping Γ = 0.3 ω p image charge at r im

8 Field response & correlations positive frequency part φ(x) = φ (+) (x) + φ ( ) (x) =... a κ (t) +... a κ(t) field vacuum state φ (+) (x) vac = 0 Kubo formula χ (+) (x, x ) = ī h [ φ (+) (x), φ(x ) ] Θ(t t ) correlation response Ω p Ω s Drude metal 0 Ω s Ω p surface plasmon ω s = ω p / 2, damping Γ = 0.3 ω p pos freq response χ (+) (z, z ; k, ω) = i e k(z+z ) ε 0 k algebraic fat tail : τ 1/Γ χ (+) ( r, r, τ) i Im R (0) Θ(τ) 4πε 0 r r im πτ 2 same tail in correlations φ(x)φ(x ) vac (fluctuation dissipation / Shiba relation) Im R(ω)Θ(ω) 1 iω Davidson & Kozak, J Math Phys 12 (1971); Wodkiewicz & Eberly, Ann Phys (NY) 101 (1976)

9 Non-Markovian challenges Comment on fluctuation-dissipation relation (kink near zero frequency) Shiba relation, Sassetti & Weiss 1990 (Tauber rule) power law tails... are a problem for Markov approximation ( eternal slip?) Haake & Reibold 1985 Challenge: self-consistent field+atom spectral function near zero frequency, beyond factorising initial conditions Slutskin & al 2011 Q friction power law in velocity v depends on shape of spectrum near ω = 0 Intravaia & al 2014/15

10 Discussing with Bair Budaev Budaev & Bogy, Ann Phys (Berlin) 523 (2011); Appl Phys Lett 99 (2011) T 2 d Thermal radiation with heat current Q: S(ω; T, Q) beyond Planck Ann Phys (Berlin) 2011 radiation thermalizes poorly matter huge thermal reservoir preferred frame: energy current vs crystal lattice T 1 signals from astronomy the Sun: our black body T +! T T CMB spectroscopy vs condensed matter Exit evanescent waves: reproduce radiative heat transfer without (cit n?) required by boundary conditions (charges, impurities, interfaces, nano-scale objects) Berry, J mod Opt 48 (2001) virtual photons tunnelling current large k ω/c non-radiative transport channels attached to matter near field

11 Discussing with Bair Budaev Thermal radiation with heat current Q: S(ω; T, Q) beyond Planck Ann Phys (Berlin) 2011 T 2 Exit evanescent waves: reproduce radiative heat transfer without (cit n?) T d 1 T +! T T Short-distance limit of heat transfer: recover homogeneous medium between conductors Q h(t ) T OK! (nonlocality?!) d2 between dielectrics Q = σ(n)(t 4 1 T 4 2 ) No! (Kapitza resistance?) radiation model does not allow for d 0 Budaev & Bogy, Appl Phys Lett 2011 Ezzahri & Joulain, Phys Rev B 2014 coupled oscillator model ( non-lte ) e.g. Barton, J Phys Cond Matt 27 (2015)

12 Summary & Perspectives Fluctuation-dissipation (Shiba) relation for vacuum field kink spectrum near zero frequency fat correlations Challenge: self-consistent field+atom spectral function near zero frequency Sassetti & Weiss, Phys Rev Lett 65 (1990); beyond factorising initial conditions Slutskin & al, Europhys Lett 96 (2011) Easy way out : restore T > K h/k B T ps Locally (near) thermal equilibrium matter dynamics phonons couple (Chen group, Nat Commun 2015) reasonable approximations behind near-field heat transfer correlation response Ω p Ω s 0 Ω s Ω p

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