Linear Response in Fluctuational Electrodynamics

Max Planck Institute Stuttgart Linear Response in Fluctuational Electrodynamics Matthias Krüger Group Members: Artem Aerov Roberta Incardone Moritz Förster MIT - Student: Vladyslav Golyk Collaborators: Joseph Brader, Fribourg Mehran Kardar, MIT Thorsten Emig, Paris Giuseppe Bimonte, Naples T 2 T 1

Table of Contents 1 Introduction Fluctuation induced Interactions QED: Scattering Theory 2 Non-equilibrium Formalism Radiation Force Transfer 3 Linear response Perturbing temperatures Perturbing velocities Onsager Relation

Fluctuation induced interactions Medium Temperature Objects Examples Quantum Casimir-effect in vacuum Critical Casimir-effect Fluid membranes/interfaces Radiative heat transfer

Fluctuation induced interactions Medium Temperature Objects Examples Quantum Casimir-effect in vacuum Critical Casimir-effect Classical Fluid membranes/interfaces Classical Radiative heat transfer

Fluctuation induced interactions Medium Temperature Objects Examples Quantum Casimir-effect in vacuum QED Critical Casimir-effect Classical Fluid membranes/interfaces Classical Radiative heat transfer QED

QED: Scattering Theory μm ε1 ε T-operator: E tot = E in + T E in 2 reflection coefficient U: Translation Lambrecht, Maia Neto and Reynaud (2006); Kenneth and Klich (2006); Milton, Parashar, and Wagner (2008); Reid, Rodriguez, White and Johnson (2009)

QED: Scattering Theory μm ε1 ε T-operator: E tot = E in + T E in 2 reflection coefficient U: Translation Equilibrium Rahi, Emig, Graham, Jaffe, Kardar (2009) F = k B T logz = c 2π dκ } 0 {{ } Quantum log det(i T 1 U 12 T 2 U 21 ) }{{} Classical Lambrecht, Maia Neto and Reynaud (2006); Kenneth and Klich (2006); Milton, Parashar, and Wagner (2008); Reid, Rodriguez, White and Johnson (2009)

Non-equilibrium Objects at different temperatures, in motion, excited states... T env T 2 T1 Eq. Non-eq. Equilibrium Non-equilibrium Free energy defined not defined Forces restricted (no stable points) less restricted Transfer heat transfer Literature on temperature differences: Antezza & Pitaevskii et. al. (2005,2008): Parallel plates, atom-plate: Repulsion Bimonte (2009): Corrugated surfaces Messina and Antezza (2011): General shapes Rytov 1960 Polder, van Hove (1971): Parallel plates Volokitin, Persson (2001): Dipole-plate Narayanaswamy, Chen, Sasihithlu (2008,2011): Two spheres Otey and Fan (2011): Sphere-plate McCauley et. al. (2011): Cone-plate

QED in non-equilibrium (Rytov 1960) Phenomenology Single object: Heat radiation n T p T s 1 T 1 Krüger, Bimonte, Emig, Kardar, Phys. Rev. B 86, 115423 (2012) Krüger, Emig, Kardar, Phys. Rev. Lett. 106, 210404 (2011) Messina, Antezza (2011) Rodriguez, Reid, Johnson (2012) Narayanaswamy, Zheng (2013)

QED in non-equilibrium (Rytov 1960) Phenomenology Single object: Heat radiation Two objects: Coupled fluctuations T p 2 n T s 1 T 2 T 1 Krüger, Bimonte, Emig, Kardar, Phys. Rev. B 86, 115423 (2012) Krüger, Emig, Kardar, Phys. Rev. Lett. 106, 210404 (2011) Messina, Antezza (2011) Rodriguez, Reid, Johnson (2012) Narayanaswamy, Zheng (2013)

QED in non-equilibrium (Rytov 1960) Phenomenology Single object: Heat radiation Two objects: Coupled fluctuations Force (2012) Formula for N arbitrary objects Generalizes equilibrium result T p 2 n T s 1 T 2 T 1 F 1 (T 1 ) = 2 π Z 8 9 1 < dω Im Tr ω : T 2 P [1 + T 1 h 2] U Re[T 1 ] + T 1 T iu 1 = 1 k e B T 1 UT 1 UT 2 1 T 1 1 2 U T 1 U ;. Krüger, Bimonte, Emig, Kardar, Phys. Rev. B 86, 115423 (2012) Krüger, Emig, Kardar, Phys. Rev. Lett. 106, 210404 (2011) Messina, Antezza (2011) Rodriguez, Reid, Johnson (2012) Narayanaswamy, Zheng (2013)

QED in non-equilibrium (Rytov 1960) Phenomenology Single object: Heat radiation Two objects: Coupled fluctuations Force (2012) Formula for N arbitrary objects Generalizes equilibrium result Heat transfer Formula (2012) Symmetric 1 2 H 0 T p 2 T 2 n T s 1 T 1 H = 2 π Z 8 9 ω < h dω Tr Re[T ω : 2 ] + T i 2 T 1 h 2 U Re[T 1 ] + T 1 T iu 1 = 1 k e B T 1 UT 1 UT 2 1 T 1 1 2 U T 1 U ;. Krüger, Bimonte, Emig, Kardar, Phys. Rev. B 86, 115423 (2012) Krüger, Emig, Kardar, Phys. Rev. Lett. 106, 210404 (2011) Messina, Antezza (2011) Rodriguez, Reid, Johnson (2012) Narayanaswamy, Zheng (2013)

QED in non-equilibrium (Rytov 1960) Phenomenology Single object: Heat radiation Two objects: Coupled fluctuations Force (2012) Formula for N arbitrary objects Generalizes equilibrium result Heat transfer Formula (2012) Symmetric 1 2 H 0 T p 2 T 2 n T s 1 T 1 Classical scattering (T 1, T 2 ) Forces and transfer Krüger, Bimonte, Emig, Kardar, Phys. Rev. B 86, 115423 (2012) Krüger, Emig, Kardar, Phys. Rev. Lett. 106, 210404 (2011) Messina, Antezza (2011) Rodriguez, Reid, Johnson (2012) Narayanaswamy, Zheng (2013)

Heat radiation: Sphere and Cylinder H = 2 π 0 dω ω e ω k BT 1 } {{ } Quantum Tr [Re T + T 2] } {{ } Classical Stefan Boltzmann law H = AσT 4 ǫ λ T = c k B T 7.6µm R λ T : Surface R λ T : Volume Cylinder: Polarization Radiation Rate [σ T 4 A] 1 10-1 10-2 R Class. δ 2 R sphere cylinder cylind. cylind. 10-1 1 10 10 2 R [µm] M. Krüger, T. Emig and M. Kardar, Phys. Rev. Lett. 106, 210404 (2011) λ T 2 R Öhman 1961, Kattawar and Eisner (1970), Bimonte et. al. 2009 V. A. Golyk, M. Krüger and M. Kardar, Phys. Rev. E 85, 046603 (2012) Maghrebi, Jaffe and Kardar (2012)

Heat radiation: Sphere and Cylinder H = 2 π Stefan Boltzmann law 0 dω ω e ω k BT 1 } {{ } Quantum H = AσT 4 ǫ Recent experiment (nano fiber) λ T = c k B T 7.6µm R λ T : Surface R λ T : Volume Cylinder: Polarization Radiation Rate [σ T 4 A] 1 10-1 10-2 Tr [Re T + T 2] } {{ } Classical R Class. δ 2 R sphere cylinder cylind. cylind. 10-1 1 10 10 2 R [µm] M. Krüger, T. Emig and M. Kardar, Phys. Rev. Lett. 106, 210404 (2011) λ T 2 R Öhman 1961, Kattawar and Eisner (1970), Bimonte et. al. 2009 V. A. Golyk, M. KrügerWuttke, and M. Rauschenbeutel, Kardar, Phys. Rev. Phys. E 85, Rev. 046603 Lett. (2012) 111, 024301 (2013) Maghrebi, Jaffe and Kardar (2012)

Non-equilibrium Casimir force Equilibrium F eq (T = 0) = 161 c α 4π d 8 1 α 2 F eq (T > 0) = 18 c α d 7 λ 1 α 2 T 2R 2 Tenv T env d T 2R 1 1 Non-equilibrium 1 F (T 1, T 2, T env) 1 d 2 2 Repulsive 3 Stable points 4 Self-propelled F [10-18 N] 1 10-1 10-2 4 2 T 1 =0 K,T 2 =0 K T 1 =300 K, T 2 =300 K T 1 =0 K, T 2 =300 K T 1 =300 K, T 2 =0 K d -2 1 T env = 0, R i = 1µm solid: attraction dashed: repulsion 3 10-3 4 5 6 7 8 9 10 15 20 d [µm] M. Krüger, T. Emig, G. Bimonte and M. Kardar, Europhys. Lett. 95 21002 (2011)

Non-equilibrium Casimir force Equilibrium F eq (T = 0) = 161 c α 4π d 8 1 α 2 F eq (T > 0) = 18 c α d 7 λ 1 α 2 T 2R 2 Tenv T env d T 2R 1 1 Non-equilibrium 1 Levitating a hot nano-sphere 1 F (T 1, T 2, T env) 1 d 2 0.9 2 Repulsive 10-1 0.8 3 Stable points 4 g 0.7 4 Self-propelled d T env = 0, R i = 1µm solid: attraction dashed: repulsion F [10-18 N] d [µm] 10-2 0.6 T 1 =0 K,T 2 =0 K T 1 =300 K, T 2 =300 K T 1 =0 K, T 2 =300 K 0.5 10-3 T s =300K 3 incl. transfer 0.4 4 0 5 0.2 6 7 0.48 90.6 10 0.8 151 20 time d [ms] [µm] Krüger, Emig, Bimonte and Kardar, Phys. Rev. B 86, 115423 (2012) M. Krüger, T. Emig, G. Bimonte and M. Kardar, Europhys. Lett. 95 21002 (2011) F / F G 2 2 T 1 =300 K, T 2 =0 K 1 0-1 -2 0.5 1 1.5 2 d [µm] no transfer d -2 1

Heat Transfer Transfer Rate H s [σ T 4 2πR 2 ] 0.7 0.65 0.6 0.55 0.5 PTA Ratio 1.8 1.6 1.4 1.2 1 0.8 10-2 10-1 10 0 d / R full solution one reflection H s (d = ) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 d / R 10 0 10-1 10-2 Class. R 10-1 10 0 10 1 10 2 R [µm] d d = 2 R R. Incardone, T. Emig and M. Krüger, Arxiv: 1402.5369 (2014)

Heat Transfer Transfer Rate H s [σ T 4 2πR 2 ] 0.7 0.65 0.6 0.55 0.5 PTA Ratio 1.8 1.6 1.4 1.2 1 0.8 10-2 10-1 10 0 d / R full solution one reflection H s (d = ) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 d / R 10 0 10-1 10-2 Class. R 10-1 10 0 10 1 10 2 R [µm] d d = 2 R Interplay: geometry material 20 10 Re ε(ω) Im ε(ω) ε(iω) 10 0.4 0.5 0.6 0.7 ω/c [rad/µm] R. Incardone, T. Emig and M. Krüger, Arxiv: 1402.5369 (2014)

Linear response for objects in vacuum H = H 0 + Ah(t) Fluctuation Dissipation Theorem (classical) B(t) h = 1 t eq Ḃ(t t )A h(t ) k B T Perturbations Temperatures T Velocities v Observables Energy exchange H Force F

Perturbing temperatures Green Kubo relations Equilibrium d H (β) dt α = 1 eq k B T 2 Z 0 dt D E eq H (α) (t)h (β) (0) Near Equilibrium T + δt Golyk, Krüger, Kardar, Phys. Rev. B, 88, 155117 (2013)

Perturbing temperatures Green Kubo relations Equilibrium d H (β) dt α = 1 eq k B T 2 Z 0 dt D E eq H (α) (t)h (β) (0) d F (β) dt α = 1 k eq B T 2 Z 0 dt D E eq H (α) (t)f (β) (0) Near Equilibrium T + δt Golyk, Krüger, Kardar, Phys. Rev. B, 88, 155117 (2013)

Perturbing temperatures Green Kubo relations Equilibrium d H (β) dt α = 1 eq k B T 2 Z 0 dt D E eq H (α) (t)h (β) (0) d F (β) dt α = 1 k eq B T 2 Z 0 dt D E eq H (α) (t)f (β) (0) Near Equilibrium T + δt Measure e.g. transfer in equilibrium? Extend deeper into non-equilibrium? Golyk, Krüger, Kardar, Phys. Rev. B, 88, 155117 (2013)

Perturbing velocities Linear response relation d F (β) dv α = 1 eq k B T Z 0 dt D E eq F (β) (t)f (α) (0) Golyk, Krüger, Kardar, Phys. Rev. B, 88, 155117 (2013); Volokitin, Persson (2007); Jaekel, Reynaud (1992) Maia Neto, Reynaud (1993); Pendry (1997); Mkrtchian, Parsegian, Podgornik, Saslow (2003) ; Démery and Dean (2011) Lach, DeKieviet and Jentschura (2012), Maghrebi, Golestanian and Kardar (2013), Pieplow and Henkel (2013)

Perturbing velocities Linear response relation d F (β) dv α = 1 eq k B T Z 0 dt D E eq F (β) (t)f (α) (0) (Thermal) Friction for arbitrary objects d F (α) = 2 Z e ω/k B T dω dv α πk eq B T 0 e ω/k B T 2 1 8 < Im Tr : 1 i(1 + UT β ) U[i( j T α 1 UT Tα j) 2T α j Im[U]T α ] 1 αut β 9 = 1 U T β U T ; α Golyk, Krüger, Kardar, Phys. Rev. B, 88, 155117 (2013); Volokitin, Persson (2007); Jaekel, Reynaud (1992) Maia Neto, Reynaud (1993); Pendry (1997); Mkrtchian, Parsegian, Podgornik, Saslow (2003) ; Démery and Dean (2011) Lach, DeKieviet and Jentschura (2012), Maghrebi, Golestanian and Kardar (2013), Pieplow and Henkel (2013)

Perturbing velocities Linear response relation d F (β) dv α = 1 eq k B T Z 0 dt D E eq F (β) (t)f (α) (0) (Thermal) Friction for arbitrary objects d F (α) = 2 Z e ω/k B T dω dv α πk eq B T 0 e ω/k B T 2 1 8 < Im Tr : 1 i(1 + UT β ) U[i( j T α 1 UT Tα j) 2T α j Im[U]T α ] 1 αut β Single spherical mirror d F dv = 896π7 R 6. eq 135 λ 8 T 9 = 1 U T β U T ; α Golyk, Krüger, Kardar, Phys. Rev. B, 88, 155117 (2013); Volokitin, Persson (2007); Jaekel, Reynaud (1992) Maia Neto, Reynaud (1993); Pendry (1997); Mkrtchian, Parsegian, Podgornik, Saslow (2003) ; Démery and Dean (2011) Lach, DeKieviet and Jentschura (2012), Maghrebi, Golestanian and Kardar (2013), Pieplow and Henkel (2013)

Onsager Relation d H (α) dv β vβ =0 = T d F(β) dt α {Tα}=T env=t v Golyk, Krüger, Kardar, Phys. Rev. B, 88, 155117 (2013)

Summary QED in (thermal) non-equilibrium Non-equilibrium force Spheres: stable points and self propelled pairs Interplay of shape and materials: Levitation Heat radiation and transfer Radiation / Polarization depending on shape Transfer can depend on orientation d g Properties of fluctuations in non-equilibrium: Linear response relations Near Equilibrium