Abstract

We develop a dynamical theory of heat transfer between two nanosystems. In particular, we consider the resonant heat transfer between two nanoparticles due to the coupling of localized surface modes having a finite spectral width. We model the coupled nanosystem by two coupled quantum mechanical oscillators, each interacting with its own heat bath, and obtain a master equation for the dynamics of heat transfer. The damping rates in the master equation are related to the lifetimes of localized plasmons in the nanoparticles. We study the dynamics toward the steady state and establish connection with the standard theory of heat transfer in the steady state. For strongly coupled nanoparticles, we predict Rabi oscillations in the mean occupation number of surface plasmons in each nanoparticle.

© 2013 Optical Society of America

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  1. P. Vasa, W. Wang, R. Pomraenke, M. Lammers, M. Maiuri, C. Manzoni, G. Gerullo, and C. Lienau, “Real-time observation of ultrafast Rabi oscillations between excitons and plasmons in J-aggregate/metal hybrid nanostructures,” Nat. Photonics (2013).
    [CrossRef]
  2. A. E. Craig, G. A. Olson, and D. Sarid, “Experimental observation of the long-range surface-plasmon polariton,” Opt. Lett. 8, 380–382 (1983).
    [CrossRef]
  3. S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer, 1989), Vol. 3.
  4. W. Eckhardt, “Macroscopic theory of electromagnetic fluctuations and stationary radiative heat transfer,” Phys. Rev. A 29, 1991–2003 (1984).
    [CrossRef]
  5. G. S. Agarwal, “Quantum electrodynamics in the presence of dielectrics and conductors. I. Electromagnetic-field response functions and black-body fluctuations in finite geometries,” Phys. Rev. A 11, 230–242 (1975).
    [CrossRef]
  6. E. M. Lifshitz, “The theory of molecular attractive forces between solids,” Sov. Phys. JETP 2, 73–83 (1956).
  7. D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303–3314 (1971).
    [CrossRef]
  8. M. Tschikin, S.-A. Biehs, P. Ben-Abdallah, and F. S. S. Rosa, “Radiative cooling of nanoparticles close to a surface,” Eur. Phys. J. B 85, 233–240 (2012).
    [CrossRef]
  9. M. Janowicz, D. Reddig, and M. Holthaus, “Quantum approach to electromagnetic energy transfer between two dielectric bodies,” Phys. Rev. A 68, 043823 (2003).
    [CrossRef]
  10. G. Domingues, S. Volz, K. Joulain, and J.-J. Greffet, “Heat Transfer between two nanoparticles through near field interaction,” Phys. Rev. Lett. 94, 085901 (2005).
    [CrossRef]
  11. P.-O. Chapuis, M. Laroche, S. Volz, and J.-J. Greffet, “Radiative heat transfer between metallic nanoparticles,” Appl. Phys. Lett. 92, 201906 (2008).
    [CrossRef]
  12. P. M. Tomchuk and N. I. Grigorchuk, “Shape and size effects on the energy absorption by small metallic particles,” Phys. Rev. B 73, 155423 (2006).
    [CrossRef]
  13. A. Manjavacas, and F. J. García de Abajo, “Radiative heat transfer between neighboring particles,” Phys. Rev. B 86, 075466 (2012).
    [CrossRef]
  14. A. Pérez-Madrid, J. M. Rubí, and L. C. Lapas, “Heat transfer between nanoparticles: thermal conductance for near-field interactions,” Phys. Rev. B 77, 155417 (2008).
    [CrossRef]
  15. A. Pérez-Madrid, L. C. Lapas, and J. M. Rubí, “Heat exchange between two interacting nanoparticles beyond the fluctuation-dissipation regime,” Phys. Rev. Lett. 103, 048301 (2009).
    [CrossRef]
  16. P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett. 107, 114301 (2011).
    [CrossRef]
  17. A. I. Volokitin and B. N. J. Persson, “Radiative heat transfer between nanostructures,” Phys. Rev. B 63, 205404 (2001).
    [CrossRef]
  18. G. V. Dedkov and A. A. Kyasov, “On the radiative heat exchange between spherical particles at small distances,” Europhys. Lett. 93, 34001 (2011).
    [CrossRef]
  19. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
  20. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley-Interscience, 2007).
  21. S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities and Photons (Oxford University, 2012).
  22. G. S. Agarwal, Quantum Optics (Cambridge University, 2012).
  23. R. Loudon, The Quantum Theory of Light (Oxford University, 2000).
  24. T. K. Hakala, J. J. Toppari, A. Kuzyk, M. Pettersson, H. Tikkanen, H. Kunttu, and P. Törmä, “Vacuum Rabi splitting and strong-coupling dynamics for surface-plasmon polaritons and rhodamine 6G molecules,” Phys. Rev. Lett. 103, 053602 (2009).
    [CrossRef]
  25. A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
    [CrossRef]
  26. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, 1975).
  27. C. Sönnichsen, Plasmons in metal nanostructures, Ph.D. thesis (Ludwig-Maximilians-University of Munich, 2001).
  28. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]

2012 (2)

M. Tschikin, S.-A. Biehs, P. Ben-Abdallah, and F. S. S. Rosa, “Radiative cooling of nanoparticles close to a surface,” Eur. Phys. J. B 85, 233–240 (2012).
[CrossRef]

A. Manjavacas, and F. J. García de Abajo, “Radiative heat transfer between neighboring particles,” Phys. Rev. B 86, 075466 (2012).
[CrossRef]

2011 (3)

P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett. 107, 114301 (2011).
[CrossRef]

G. V. Dedkov and A. A. Kyasov, “On the radiative heat exchange between spherical particles at small distances,” Europhys. Lett. 93, 34001 (2011).
[CrossRef]

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

2009 (2)

T. K. Hakala, J. J. Toppari, A. Kuzyk, M. Pettersson, H. Tikkanen, H. Kunttu, and P. Törmä, “Vacuum Rabi splitting and strong-coupling dynamics for surface-plasmon polaritons and rhodamine 6G molecules,” Phys. Rev. Lett. 103, 053602 (2009).
[CrossRef]

A. Pérez-Madrid, L. C. Lapas, and J. M. Rubí, “Heat exchange between two interacting nanoparticles beyond the fluctuation-dissipation regime,” Phys. Rev. Lett. 103, 048301 (2009).
[CrossRef]

2008 (2)

A. Pérez-Madrid, J. M. Rubí, and L. C. Lapas, “Heat transfer between nanoparticles: thermal conductance for near-field interactions,” Phys. Rev. B 77, 155417 (2008).
[CrossRef]

P.-O. Chapuis, M. Laroche, S. Volz, and J.-J. Greffet, “Radiative heat transfer between metallic nanoparticles,” Appl. Phys. Lett. 92, 201906 (2008).
[CrossRef]

2006 (1)

P. M. Tomchuk and N. I. Grigorchuk, “Shape and size effects on the energy absorption by small metallic particles,” Phys. Rev. B 73, 155423 (2006).
[CrossRef]

2005 (1)

G. Domingues, S. Volz, K. Joulain, and J.-J. Greffet, “Heat Transfer between two nanoparticles through near field interaction,” Phys. Rev. Lett. 94, 085901 (2005).
[CrossRef]

2003 (1)

M. Janowicz, D. Reddig, and M. Holthaus, “Quantum approach to electromagnetic energy transfer between two dielectric bodies,” Phys. Rev. A 68, 043823 (2003).
[CrossRef]

2001 (1)

A. I. Volokitin and B. N. J. Persson, “Radiative heat transfer between nanostructures,” Phys. Rev. B 63, 205404 (2001).
[CrossRef]

1984 (1)

W. Eckhardt, “Macroscopic theory of electromagnetic fluctuations and stationary radiative heat transfer,” Phys. Rev. A 29, 1991–2003 (1984).
[CrossRef]

1983 (1)

1975 (1)

G. S. Agarwal, “Quantum electrodynamics in the presence of dielectrics and conductors. I. Electromagnetic-field response functions and black-body fluctuations in finite geometries,” Phys. Rev. A 11, 230–242 (1975).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1971 (1)

D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303–3314 (1971).
[CrossRef]

1956 (1)

E. M. Lifshitz, “The theory of molecular attractive forces between solids,” Sov. Phys. JETP 2, 73–83 (1956).

Agarwal, G. S.

G. S. Agarwal, “Quantum electrodynamics in the presence of dielectrics and conductors. I. Electromagnetic-field response functions and black-body fluctuations in finite geometries,” Phys. Rev. A 11, 230–242 (1975).
[CrossRef]

G. S. Agarwal, Quantum Optics (Cambridge University, 2012).

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, 1975).

Ben-Abdallah, P.

M. Tschikin, S.-A. Biehs, P. Ben-Abdallah, and F. S. S. Rosa, “Radiative cooling of nanoparticles close to a surface,” Eur. Phys. J. B 85, 233–240 (2012).
[CrossRef]

P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett. 107, 114301 (2011).
[CrossRef]

Biehs, S.-A.

M. Tschikin, S.-A. Biehs, P. Ben-Abdallah, and F. S. S. Rosa, “Radiative cooling of nanoparticles close to a surface,” Eur. Phys. J. B 85, 233–240 (2012).
[CrossRef]

P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett. 107, 114301 (2011).
[CrossRef]

Chapuis, P.-O.

P.-O. Chapuis, M. Laroche, S. Volz, and J.-J. Greffet, “Radiative heat transfer between metallic nanoparticles,” Appl. Phys. Lett. 92, 201906 (2008).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Craig, A. E.

Dedkov, G. V.

G. V. Dedkov and A. A. Kyasov, “On the radiative heat exchange between spherical particles at small distances,” Europhys. Lett. 93, 34001 (2011).
[CrossRef]

Domingues, G.

G. Domingues, S. Volz, K. Joulain, and J.-J. Greffet, “Heat Transfer between two nanoparticles through near field interaction,” Phys. Rev. Lett. 94, 085901 (2005).
[CrossRef]

Eberly, J. H.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, 1975).

Eckhardt, W.

W. Eckhardt, “Macroscopic theory of electromagnetic fluctuations and stationary radiative heat transfer,” Phys. Rev. A 29, 1991–2003 (1984).
[CrossRef]

García de Abajo, F. J.

A. Manjavacas, and F. J. García de Abajo, “Radiative heat transfer between neighboring particles,” Phys. Rev. B 86, 075466 (2012).
[CrossRef]

Garcia-Vidal, F. J.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

Gerullo, G.

P. Vasa, W. Wang, R. Pomraenke, M. Lammers, M. Maiuri, C. Manzoni, G. Gerullo, and C. Lienau, “Real-time observation of ultrafast Rabi oscillations between excitons and plasmons in J-aggregate/metal hybrid nanostructures,” Nat. Photonics (2013).
[CrossRef]

Gonzalez-Tudela, A.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

Greffet, J.-J.

P.-O. Chapuis, M. Laroche, S. Volz, and J.-J. Greffet, “Radiative heat transfer between metallic nanoparticles,” Appl. Phys. Lett. 92, 201906 (2008).
[CrossRef]

G. Domingues, S. Volz, K. Joulain, and J.-J. Greffet, “Heat Transfer between two nanoparticles through near field interaction,” Phys. Rev. Lett. 94, 085901 (2005).
[CrossRef]

Grigorchuk, N. I.

P. M. Tomchuk and N. I. Grigorchuk, “Shape and size effects on the energy absorption by small metallic particles,” Phys. Rev. B 73, 155423 (2006).
[CrossRef]

Hakala, T. K.

T. K. Hakala, J. J. Toppari, A. Kuzyk, M. Pettersson, H. Tikkanen, H. Kunttu, and P. Törmä, “Vacuum Rabi splitting and strong-coupling dynamics for surface-plasmon polaritons and rhodamine 6G molecules,” Phys. Rev. Lett. 103, 053602 (2009).
[CrossRef]

Haroche, S.

S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities and Photons (Oxford University, 2012).

Holthaus, M.

M. Janowicz, D. Reddig, and M. Holthaus, “Quantum approach to electromagnetic energy transfer between two dielectric bodies,” Phys. Rev. A 68, 043823 (2003).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Janowicz, M.

M. Janowicz, D. Reddig, and M. Holthaus, “Quantum approach to electromagnetic energy transfer between two dielectric bodies,” Phys. Rev. A 68, 043823 (2003).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Joulain, K.

P. Ben-Abdallah, S.-A. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett. 107, 114301 (2011).
[CrossRef]

G. Domingues, S. Volz, K. Joulain, and J.-J. Greffet, “Heat Transfer between two nanoparticles through near field interaction,” Phys. Rev. Lett. 94, 085901 (2005).
[CrossRef]

Kravtsov, Y. A.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer, 1989), Vol. 3.

Kunttu, H.

T. K. Hakala, J. J. Toppari, A. Kuzyk, M. Pettersson, H. Tikkanen, H. Kunttu, and P. Törmä, “Vacuum Rabi splitting and strong-coupling dynamics for surface-plasmon polaritons and rhodamine 6G molecules,” Phys. Rev. Lett. 103, 053602 (2009).
[CrossRef]

Kuzyk, A.

T. K. Hakala, J. J. Toppari, A. Kuzyk, M. Pettersson, H. Tikkanen, H. Kunttu, and P. Törmä, “Vacuum Rabi splitting and strong-coupling dynamics for surface-plasmon polaritons and rhodamine 6G molecules,” Phys. Rev. Lett. 103, 053602 (2009).
[CrossRef]

Kyasov, A. A.

G. V. Dedkov and A. A. Kyasov, “On the radiative heat exchange between spherical particles at small distances,” Europhys. Lett. 93, 34001 (2011).
[CrossRef]

Lammers, M.

P. Vasa, W. Wang, R. Pomraenke, M. Lammers, M. Maiuri, C. Manzoni, G. Gerullo, and C. Lienau, “Real-time observation of ultrafast Rabi oscillations between excitons and plasmons in J-aggregate/metal hybrid nanostructures,” Nat. Photonics (2013).
[CrossRef]

Lapas, L. C.

A. Pérez-Madrid, L. C. Lapas, and J. M. Rubí, “Heat exchange between two interacting nanoparticles beyond the fluctuation-dissipation regime,” Phys. Rev. Lett. 103, 048301 (2009).
[CrossRef]

A. Pérez-Madrid, J. M. Rubí, and L. C. Lapas, “Heat transfer between nanoparticles: thermal conductance for near-field interactions,” Phys. Rev. B 77, 155417 (2008).
[CrossRef]

Laroche, M.

P.-O. Chapuis, M. Laroche, S. Volz, and J.-J. Greffet, “Radiative heat transfer between metallic nanoparticles,” Appl. Phys. Lett. 92, 201906 (2008).
[CrossRef]

Lienau, C.

P. Vasa, W. Wang, R. Pomraenke, M. Lammers, M. Maiuri, C. Manzoni, G. Gerullo, and C. Lienau, “Real-time observation of ultrafast Rabi oscillations between excitons and plasmons in J-aggregate/metal hybrid nanostructures,” Nat. Photonics (2013).
[CrossRef]

Lifshitz, E. M.

E. M. Lifshitz, “The theory of molecular attractive forces between solids,” Sov. Phys. JETP 2, 73–83 (1956).

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford University, 2000).

Maiuri, M.

P. Vasa, W. Wang, R. Pomraenke, M. Lammers, M. Maiuri, C. Manzoni, G. Gerullo, and C. Lienau, “Real-time observation of ultrafast Rabi oscillations between excitons and plasmons in J-aggregate/metal hybrid nanostructures,” Nat. Photonics (2013).
[CrossRef]

Manjavacas, A.

A. Manjavacas, and F. J. García de Abajo, “Radiative heat transfer between neighboring particles,” Phys. Rev. B 86, 075466 (2012).
[CrossRef]

Manzoni, C.

P. Vasa, W. Wang, R. Pomraenke, M. Lammers, M. Maiuri, C. Manzoni, G. Gerullo, and C. Lienau, “Real-time observation of ultrafast Rabi oscillations between excitons and plasmons in J-aggregate/metal hybrid nanostructures,” Nat. Photonics (2013).
[CrossRef]

Martin-Cano, D.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

Martin-Moreno, L.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

Moreno, E.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

Olson, G. A.

Pérez-Madrid, A.

A. Pérez-Madrid, L. C. Lapas, and J. M. Rubí, “Heat exchange between two interacting nanoparticles beyond the fluctuation-dissipation regime,” Phys. Rev. Lett. 103, 048301 (2009).
[CrossRef]

A. Pérez-Madrid, J. M. Rubí, and L. C. Lapas, “Heat transfer between nanoparticles: thermal conductance for near-field interactions,” Phys. Rev. B 77, 155417 (2008).
[CrossRef]

Persson, B. N. J.

A. I. Volokitin and B. N. J. Persson, “Radiative heat transfer between nanostructures,” Phys. Rev. B 63, 205404 (2001).
[CrossRef]

Pettersson, M.

T. K. Hakala, J. J. Toppari, A. Kuzyk, M. Pettersson, H. Tikkanen, H. Kunttu, and P. Törmä, “Vacuum Rabi splitting and strong-coupling dynamics for surface-plasmon polaritons and rhodamine 6G molecules,” Phys. Rev. Lett. 103, 053602 (2009).
[CrossRef]

Polder, D.

D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303–3314 (1971).
[CrossRef]

Pomraenke, R.

P. Vasa, W. Wang, R. Pomraenke, M. Lammers, M. Maiuri, C. Manzoni, G. Gerullo, and C. Lienau, “Real-time observation of ultrafast Rabi oscillations between excitons and plasmons in J-aggregate/metal hybrid nanostructures,” Nat. Photonics (2013).
[CrossRef]

Raimond, J.-M.

S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities and Photons (Oxford University, 2012).

Reddig, D.

M. Janowicz, D. Reddig, and M. Holthaus, “Quantum approach to electromagnetic energy transfer between two dielectric bodies,” Phys. Rev. A 68, 043823 (2003).
[CrossRef]

Rosa, F. S. S.

M. Tschikin, S.-A. Biehs, P. Ben-Abdallah, and F. S. S. Rosa, “Radiative cooling of nanoparticles close to a surface,” Eur. Phys. J. B 85, 233–240 (2012).
[CrossRef]

Rubí, J. M.

A. Pérez-Madrid, L. C. Lapas, and J. M. Rubí, “Heat exchange between two interacting nanoparticles beyond the fluctuation-dissipation regime,” Phys. Rev. Lett. 103, 048301 (2009).
[CrossRef]

A. Pérez-Madrid, J. M. Rubí, and L. C. Lapas, “Heat transfer between nanoparticles: thermal conductance for near-field interactions,” Phys. Rev. B 77, 155417 (2008).
[CrossRef]

Rytov, S. M.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer, 1989), Vol. 3.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley-Interscience, 2007).

Sarid, D.

Sönnichsen, C.

C. Sönnichsen, Plasmons in metal nanostructures, Ph.D. thesis (Ludwig-Maximilians-University of Munich, 2001).

Tatarskii, V. I.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer, 1989), Vol. 3.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley-Interscience, 2007).

Tejedor, C.

A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106, 020501 (2011).
[CrossRef]

Tikkanen, H.

T. K. Hakala, J. J. Toppari, A. Kuzyk, M. Pettersson, H. Tikkanen, H. Kunttu, and P. Törmä, “Vacuum Rabi splitting and strong-coupling dynamics for surface-plasmon polaritons and rhodamine 6G molecules,” Phys. Rev. Lett. 103, 053602 (2009).
[CrossRef]

Tomchuk, P. M.

P. M. Tomchuk and N. I. Grigorchuk, “Shape and size effects on the energy absorption by small metallic particles,” Phys. Rev. B 73, 155423 (2006).
[CrossRef]

Toppari, J. J.

T. K. Hakala, J. J. Toppari, A. Kuzyk, M. Pettersson, H. Tikkanen, H. Kunttu, and P. Törmä, “Vacuum Rabi splitting and strong-coupling dynamics for surface-plasmon polaritons and rhodamine 6G molecules,” Phys. Rev. Lett. 103, 053602 (2009).
[CrossRef]

Törmä, P.

T. K. Hakala, J. J. Toppari, A. Kuzyk, M. Pettersson, H. Tikkanen, H. Kunttu, and P. Törmä, “Vacuum Rabi splitting and strong-coupling dynamics for surface-plasmon polaritons and rhodamine 6G molecules,” Phys. Rev. Lett. 103, 053602 (2009).
[CrossRef]

Tschikin, M.

M. Tschikin, S.-A. Biehs, P. Ben-Abdallah, and F. S. S. Rosa, “Radiative cooling of nanoparticles close to a surface,” Eur. Phys. J. B 85, 233–240 (2012).
[CrossRef]

Van Hove, M.

D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303–3314 (1971).
[CrossRef]

Vasa, P.

P. Vasa, W. Wang, R. Pomraenke, M. Lammers, M. Maiuri, C. Manzoni, G. Gerullo, and C. Lienau, “Real-time observation of ultrafast Rabi oscillations between excitons and plasmons in J-aggregate/metal hybrid nanostructures,” Nat. Photonics (2013).
[CrossRef]

Volokitin, A. I.

A. I. Volokitin and B. N. J. Persson, “Radiative heat transfer between nanostructures,” Phys. Rev. B 63, 205404 (2001).
[CrossRef]

Volz, S.

P.-O. Chapuis, M. Laroche, S. Volz, and J.-J. Greffet, “Radiative heat transfer between metallic nanoparticles,” Appl. Phys. Lett. 92, 201906 (2008).
[CrossRef]

G. Domingues, S. Volz, K. Joulain, and J.-J. Greffet, “Heat Transfer between two nanoparticles through near field interaction,” Phys. Rev. Lett. 94, 085901 (2005).
[CrossRef]

Wang, W.

P. Vasa, W. Wang, R. Pomraenke, M. Lammers, M. Maiuri, C. Manzoni, G. Gerullo, and C. Lienau, “Real-time observation of ultrafast Rabi oscillations between excitons and plasmons in J-aggregate/metal hybrid nanostructures,” Nat. Photonics (2013).
[CrossRef]

Appl. Phys. Lett. (1)

P.-O. Chapuis, M. Laroche, S. Volz, and J.-J. Greffet, “Radiative heat transfer between metallic nanoparticles,” Appl. Phys. Lett. 92, 201906 (2008).
[CrossRef]

Eur. Phys. J. B (1)

M. Tschikin, S.-A. Biehs, P. Ben-Abdallah, and F. S. S. Rosa, “Radiative cooling of nanoparticles close to a surface,” Eur. Phys. J. B 85, 233–240 (2012).
[CrossRef]

Europhys. Lett. (1)

G. V. Dedkov and A. A. Kyasov, “On the radiative heat exchange between spherical particles at small distances,” Europhys. Lett. 93, 34001 (2011).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (3)

W. Eckhardt, “Macroscopic theory of electromagnetic fluctuations and stationary radiative heat transfer,” Phys. Rev. A 29, 1991–2003 (1984).
[CrossRef]

G. S. Agarwal, “Quantum electrodynamics in the presence of dielectrics and conductors. I. Electromagnetic-field response functions and black-body fluctuations in finite geometries,” Phys. Rev. A 11, 230–242 (1975).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Sketch of the situation considered. Two nanoparticles separated by a distance d are effectively replaced by two dipoles oscillating with the surface mode resonance frequency ωsp.

Fig. 2.
Fig. 2.

Sketch of our model consisting of two harmonic oscillators A and B interacting through HI. Each oscillator is coupled to a heat bath of independent oscillators in equilibrium at temperatures T1 and T2.

Fig. 3.
Fig. 3.

Plot of (a) aa/n¯1, (b) bb/n¯1, and (c) Im(baab)/n¯1 over time for κ1=κ2κ and different coupling strengths g/κ. For large t the solutions converge to the steady-state solutions found in Eqs. (19) and (20) for Ωab=Ωba=(κ1+κ2) and n¯2=0.

Fig. 4.
Fig. 4.

Plot of Im(baab)/n¯1, aa/n¯1, and bb/n¯1 over time for κ1=κ2=1κ and g/κ=10.

Fig. 5.
Fig. 5.

(a) and (b) Heat transfer rate R/Γ from Eq. (42) between two nanoparticles normalized to the linewidth Γ of the nanoparticles’ surface mode resonances over time in units of Γ1 for (a) d=800nm, g/Γ=0.6 and (b) d=1200nm, g/Γ=0.4. (c) Mean occupation number bb/n¯1 of the surface mode of nanoparticle B from Eq. (28) over time in units of Γ1 for d=100nm; hence g/Γ=5.8. (a)–(c) We use the coupling constant g from Eq. (55) with r0=20nm and d=100, 800, and 1200 nm. The horizontal line is the steady-state expression from Eq. (43).

Equations (55)

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p=ϵ0α(ω)E,
α(ω)=4πr03ϵ(ω)1ϵ(ω)+2,
ϵ(ω)=ϵωp2ω(ω+iγ),
H0=ωspaa+ωspbb.
HI=g(ba+ab).
HB1/B2=jω1/2ja1/2ja1/2j.
ρB1/B2=eβ1/2HB1/B2Tr(eβ1/2HB1/B2),
HAB1=ijg1j(a+a)(a1ja1j),
HBB2=ijg2j(b+b)(a2ja2j),
ρA/B=eβ1/2HA/BTr(eβ1/2HA/B),
H=H0+HI+HB1+HB2+HAB1+HBB2.
ρSt=iωa[aa,ρS]iωb[bb,ρS]ig[ab+ba,ρS]κ1(n¯1+1)(aaρS2aρSa+ρSaa)κ1n¯1(aaρS2aρSa+ρSaa)+(12,ab),
κ1/2=πjg1/2j2δ(ω1/2jωa)
n¯1/2=1eβ1/2ωa/b1
ddtaa=ig(abba)2κ1aa+2κ1n¯1,
ddtbb=ig(baab)2κ2bb+2κ2n¯2,
ddtba=Ωabbaig(bbaa),
ddtab=Ωbaabig(aabb),
aa=A(κ1n¯1+κ2n¯2)+2κ1κ2n¯1A(κ1+κ2)+2κ1κ2,
bb=A(κ1n¯1+κ2n¯2)+2κ1κ2n¯2A(κ1+κ2)+2κ1κ2,
ba=igΩab2κ1κ2(n¯2n¯1)(κ1+κ2)A+2κ1κ2,
ab=ΩabΩbaba,
A=g2Ωab+ΩbaΩabΩba=g22(κ1+κ2)(κ1+κ2)2+(ωaωb)2.
aa=bb=κ1κ1+κ2n¯1+κ2κ1+κ2n¯2.
x˙=Ax+a
A=(2κ10g02κ2g2g2g(κ1+κ2)).
aa|t=0=n¯1,bb|t=0=0,and(baab)|t=0=0,
x=(ixiηi(eλit+2κ1λi[eλit1])iyiηi(eλit+2κ1λi[eλit1])iηi(eλit+2κ1λi[eλit1])),
xi=g2κ1+λiandyi=g2κ2+λi,
bb{n¯1g2t2fortκ11,κ21g2n¯1κ2(κ1+κ2)11+g2κ1κ2fortκ11,κ21,
baab{2n¯1igtfor tκ11,κ21+2ign¯1(κ1+κ2)11+g2κ1κ2fortκ11,κ21.
P=k(xBxA)·x˙B,
xA(a+a)2mωsp,
xB(b+b)2mωsp,
x˙A=pAmi(aa)ωsp2m,
x˙B=pBmi(bb)ωsp2m.
P=ki2m(abba).
HI=k2(xAxB)2
HI=k2mωsp(ab+ba).
P=ωsp(ig)(abba).
P=ωsp(ig)[abab]=ωspR,
R=igiηi[eλit+2κ1λi(eλit1)].
R{2n¯1g2tfor tκ11,κ212g2n¯1(κ1+κ2)11+g2κ1κ2for tκ11,κ21.
RFGR=2π2|f|HI|i|2δ(ωfωi).
RFGR=2πg2[n1(n2+1)n2(n1+1)]δ(ωfωi)=2πg2[n1n2]δ(ωfωi).
RFGR=2πg2n¯1δ(ωaωb).
RFGR=g2n¯12πdω1dω2κ1/π(ωspω1)2+κ12κ2/π(ωspω2)2+κ22δ(ω1ω2)=2g2n¯1κ1+κ2.
P=0dω4π3ωn¯1Im(α)2ω6c6[3(ωcd)6+1(ωcd)4+1(ωcd)2].
Pωspn¯1F(ωsp)0dω4π3Im(α)2,
F(ω)=ω6c6[3(ωcd)6+1(ωcd)4+1(ωcd)2].
Rst-st=n¯1F(ωsp)0dω4π3Im(α)2=n¯1F(ωsp)(4πr03)212+dω4π3(3Im(ϵ)|ϵ+2|2)2.
Rst-st=n¯1F(ωsp)(4πr03)2ωsp6Γ22π3(3ϵ+2)2+dω1(ω2ωsp2+2iωΓ)2(ω2ωsp22iωΓ)2,
Rst-st=n¯1ωsp2Γr062F(ωsp)(3ϵ+2)2.
R=g2Γn¯1.
g=ωspr032F(ωsp)3ϵ+2.

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