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) 214020
Institute of Physics and Astronomy, Universität Potsdam, Germany
www.quantum.physik.uni-potsdam.de
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Motivation
Disp ersion Interactions
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i
Casim
e
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o
f
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a
van der W
Planck spectrum
quantum fluct
uations
radiativ
e
heat tra
nsfer
non-equilibrium steady state
nano-scale hea
ting
...
‘field theory’ vs ‘radiation engineering’
Praha, July 2015
Motivation
fields
matter
Maxwell
Schrödinger
ε0 ∂t E = ∇ × H − j
approximations
························
Coulomb
ε0 ∇2 φ = −ρ
························
ρ0
Newton ∂t j ≈ f
m
relaxation time approximation
f (x, t) − feq (x)
∂t f (x, t) = . . . −
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
radiative heating
→ local thermal equilibrium T (r, t)
Praha, July 2015
Motivation
Example
Born-Markov master equation for two-level medium (continuous)
dP
dt
dNe
dt
∂E
ε0
∂t
= −(iωA + Γ)P + iχ(Ng − Ne )E
1
Im(P∗ · E)
h̄
dP
= ∇×H−
dt
= −γNe +
separation of time scales:
‘ultra-fast’
response of field, correlation time
Markovian decay
‘fast’
response of matter, radiative (spontaneous) decay,
‘slow’
re-distribution of populations
flat (white) spectrum
narrow (peaked) spectrum
Praha, July 2015
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
0←
T
F~v
• 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)
Praha, July 2015
Field response
Kubo formula
F~v
T
F~v
0←
Barton’s model: recap potential
Z
Z
φ(r, z, t) = d2 k dω φkω exp(ik · r − kz) akω (t) + h.c.
Z
linear response to charge density δhφ(x)i = dx0 χ(x, x0 )ρ(x0 )
x = (r, z, t)
i
χ(x, x0 ) = − h[φ(x), φ(x0 )]iΘ(t − t0 )
h̄
non-retarded response
correlation
response
χ(z, z 0 ; k, ω) = −
-Ωp
-Ωs
0
Ωs Ωp
−k(z+z 0 )
e
2ε0 k
R(ω)
ε(ω) − 1
R(ω) =
ε(ω) + 1
Drude metal
√
surface plasmon ωs = ωp / 2, damping Γ = 0.3 ωp
Praha, July 2015
Field response
Z
d2 k
Z
dω φkω exp(ik · r − kz) akω (t) + h.c.
Z
linear response to charge density δhφ(x)i = dx0 χ(x, x0 )ρ(x0 )
φ(r, z, t) =
Kubo formula
i
χ(x, x ) = − h[φ(x), φ(x0 )]iΘ(t − t0 )
h̄
0
non-retarded response
correlation
response
0
e−k(z+z )
0
R(ω)
χ(z, z ; k, ω) = −
2ε0 k
-Ωp
-Ωs
x = (r, z, t)
back in time domain (damped oscillator)
0
Ωs Ωp
ωs sin(ωs τ ) e−Γτ /2
χ(~r, ~r , τ ) = −
Θ(τ )
4πε0 |~r − ~r 0im |
0
Drude metal
√
surface plasmon ωs = ωp / 2, damping Γ = 0.3 ωp
image charge at ~
r 0im
Praha, July 2015
Field response & correlations
positive frequency part φ(x) = φ(+) (x) + φ(−) (x) = . . . aκ (t) + . . . a†κ (t)
field vacuum state φ(+) (x)|vaci = 0
Kubo formula
(+)
χ
i (+)
0
(x, x ) = −
φ (x), φ(x ) Θ(t − t0 )
h̄
0
pos freq response
correlation
response
χ(+) (z, z 0 ; k, ω) = −i
-Ωp
-Ωs
e
−k(z+z 0 )
ε0 k
Im R(ω)Θ(ω) ∗
• algebraic ‘fat tail’: τ 1/Γ
0
Ωs Ωp
χ
Drude metal
√
surface plasmon ωs = ωp / 2, damping Γ = 0.3 ωp
(+)
i Im R0 (0) Θ(τ )
(~r, ~r , τ ) ≈
4πε0 |~r − ~r 0im | πτ 2
0
same tail in correlations hφ(x)φ(x0 )ivac
(fluctuation–dissipation / Shiba relation)
Davidson & Kozak, J Math Phys 12 (1971);
Wodkiewicz & Eberly, Ann Phys (NY) 101 (1976)
Praha, July 2015
1
iω
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
Praha, July 2015
Discussing with Bair Budaev
Budaev & Bogy, Ann Phys (Berlin) 523 (2011);
Appl Phys Lett 99 (2011)
T2
d
Thermal radiation with heat current Q̇:
S(ω; T, Q̇) beyond Planck
Ann Phys (Berlin) 2011
• radiation thermalizes poorly
T
signals from astronomy
• matter huge thermal reservoir
the Sun: our black body
• preferred frame: energy current vs crystal lattice
Exit evanescent waves:
reproduce radiative heat transfer without
T+ ! T
T1
CMB spectroscopy
vs condensed matter
(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
Praha, July 2015
Discussing with Bair Budaev
Thermal radiation with heat current Q̇:
S(ω; T, Q̇) beyond Planck
Ann Phys (Berlin) 2011
Exit evanescent waves:
reproduce radiative heat transfer without
T2
d
(cit’n?)
T1
T+ ! T
T
Short-distance limit of heat transfer:
recover homogeneous medium
∆T
→∞
2
d
between dielectrics Q̇ = σ(n)(T14 − T24 )
No! (Kapitza resistance?)
• radiation model does not allow for d → 0
Budaev & Bogy, Appl Phys Lett 2011
• between conductors Q̇ ∼ h(T )
OK! (nonlocality?!)
Ezzahri & Joulain, Phys Rev B 2014
• coupled oscillator model (‘non-LTE’)
e.g. Barton, J Phys Cond Matt 27 (2015)
Praha, July 2015
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 > 0
1–10 K → h̄/kB T ∼ ps
• Locally (near) thermal equilibrium
matter dynamics
phonons couple (Chen group, Nat Commun 2015)
reasonable approximations behind near-field heat transfer
correlation
response
-Ωp
-Ωs
0
Ωs Ωp
www.quantum.physik.uni-potsdam.de
Praha, July 2015