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Power laws in the dynamical hysteresis of quantum nonlinear photonic resonators
W. Casteels, F. Storme, A. Le Boité, C. Ciuti
Laboratoire Matériaux et Phénomènes Quantiques Université Paris Diderot-Paris-7
04/06/2016
QlightCrete2016
Chania, Crete, Greece
● Introduction:
● Dynamical hysteresis:
● Conclusions & perspectives
Outline
● Nonlinear photonic resonator● Optical bistability: hysteresis● Semiclassical approach● Quantum description
● Time dependent sweep● Hysteresis area● Characteristic time● Scaling analysis
IntroductionIntroduction
Many-body physics with light
Recent reviews:I. Carusotto and C. Ciuti – Rev. Mod. Phys. 85, 299 (2013)S. Schmidt and J. Koch, Annalen der Physik 525, 395 (2013).M. J. Hartmann – arXiv:1605.00383 (2016)C. Noh and D. G. Angelakis – arXiv:1604.04433 (2016)...
Realizing correlated quantum states driven-dissipative photonic lattices
System Hamiltonian:
Coherent drive:
Total Hamiltonian:
Losses described by Lindblad master equation for density operator:
Kerr model
microcavity polaritons:
Experimental implementations with large U
Superconducting circuit:
F. R. Ong et. al. – Phys. Rev. Lett. 106, 167002 (2011)Daniele Bajoni et. al. – Phys. Rev. Lett. 100, 047401 (2008)
J. Kasprzak et. al. – Nature 443, 409 (2006)
Optical bistability: hysteresis cycleExperimental signature:
H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan Phys. Rev. Lett. 36, 1135 (1976)
More recent experiments:Microcavity polaritons:● D. Bajoni, E. Semenova, A. Lemaître, S. Bouchoule, E. Wertz, P. Senellart, S. Barbay, R.
Kuszelewicz, and J. Bloch, Phys. Rev. Lett. 101, 266402 (2008)● A. Baas, J. Ph. Karr, H. Eleuch, and E. Giacobino – Phys. Rev. A 69, 023809 (2004)● T. K. Paraïso, M. Wouters, Y. Léger, F. Morier-Genoud, and B. Deveaud-Plédran, Nat.
Mater. 9, 655 (2010)● A. Amo, T. C. H. Liew, C. Adrados, R. Houdré, E. Giacobino, A. V. Kavokin and A.
Bramati Nat. Phot. 4, 361 (2010)● H. Abbaspour, S. Trebaol, F. Morier-Genoud, M. T. Portella-Oberli, and B. Deveaud, Phys.
Rev. Lett. 113, 057401 (2014)● T. Boulier, M. Bamba, A. Amo, C. Adrados, A. Lemaitre, E. Galopin, I. Sagnes, J. Bloch,
C. Ciuti, E. Giacobino and A. Bramati – Nat. Comm. 5, 3260 (2014)● S.R.K. Rodriguez, A. Amo, I. Sagnes, L. Le Gratiet, E. Galopin,A. Lemaître, and J. Bloch –
arXiv:1602.07114 (2016)
Superconducting circuits:● I. Siddiqi, R. Vijay, F. Pierre, C. M. Wilson, M. Metcalfe, C. Rigetti, L. Frunzio, and M. H.
Devoret, Phys. Rev. Lett. 93, 207002 (2004)● R. Vijay, M. Devoret, and I. Siddiqi, Rev. Sci. Instrum. 80, 111101 (2009)● F. R. Ong, M. Boissonneault, F. Mallet, A. Palacios-Laloy, A. Dewes, A. C. Doherty, A.
Blais, P. Bertet, D. Vion, and D. Esteve, Phys. Rev. Lett. 106, 167002 (2011)
Opto-mechanical cavities:● A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, Phys. Rev. Lett. 51, 1550
(1983)● P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, JOSA B 2, 1830 (1985);● A. Gozzini, I. Longo, S. Barbarino, F. Maccar- rone, and F. Mango, JOSA B 2, 1841 (1985)● F. Mueller, S. Heugel, and L. J. Wang, Phys. Rev. A 77, 031802 (2008);● C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt,
Phys.Rev. Lett. 101, 133903 (2008)● F. Hao, D. JiangFang, L. Yong, and C. GengYu - Sci. China Phys. Mech. Astron. 58, 1674
(2015)● H. Xu, U. Kemiktarak, J. Fan, S. Ragole, J. Lawall and J. M. Taylor – arXiv:1510.04971
(2015)
Photonic crystals:● M. F. Yanik, S. Fan,and M. Soljačić – Appl.Phys. Lett. 83, 2739 (2003)● M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi and T. Tanabe – Opt. Expr. 13
2678 (2005)
...
Optical bistability: hysteresis cycleExperimental signature:
H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan Phys. Rev. Lett. 36, 1135 (1976)
More recent experiments:Microcavity polaritons:● D. Bajoni, E. Semenova, A. Lemaître, S. Bouchoule, E. Wertz, P. Senellart, S. Barbay, R.
Kuszelewicz, and J. Bloch, Phys. Rev. Lett. 101, 266402 (2008)● A. Baas, J. Ph. Karr, H. Eleuch, and E. Giacobino – Phys. Rev. A 69, 023809 (2004)● T. K. Paraïso, M. Wouters, Y. Léger, F. Morier-Genoud, and B. Deveaud-Plédran, Nat.
Mater. 9, 655 (2010)● A. Amo, T. C. H. Liew, C. Adrados, R. Houdré, E. Giacobino, A. V. Kavokin and A.
Bramati Nat. Phot. 4, 361 (2010)● H. Abbaspour, S. Trebaol, F. Morier-Genoud, M. T. Portella-Oberli, and B. Deveaud, Phys.
Rev. Lett. 113, 057401 (2014)● T. Boulier, M. Bamba, A. Amo, C. Adrados, A. Lemaitre, E. Galopin, I. Sagnes, J. Bloch,
C. Ciuti, E. Giacobino and A. Bramati – Nat. Comm. 5, 3260 (2014)● S.R.K. Rodriguez, A. Amo, I. Sagnes, L. Le Gratiet, E. Galopin,A. Lemaître, and J.
Bloch – arXiv:1602.07114 (2016)
Superconducting circuits:● I. Siddiqi, R. Vijay, F. Pierre, C. M. Wilson, M. Metcalfe, C. Rigetti, L. Frunzio, and M. H.
Devoret, Phys. Rev. Lett. 93, 207002 (2004)● R. Vijay, M. Devoret, and I. Siddiqi, Rev. Sci. Instrum. 80, 111101 (2009)● F. R. Ong, M. Boissonneault, F. Mallet, A. Palacios-Laloy, A. Dewes, A. C. Doherty, A.
Blais, P. Bertet, D. Vion, and D. Esteve, Phys. Rev. Lett. 106, 167002 (2011)
Opto-mechanical cavities:● A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, Phys. Rev. Lett. 51,
1550 (1983)● P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, JOSA B 2, 1830 (1985);● A. Gozzini, I. Longo, S. Barbarino, F. Maccar- rone, and F. Mango, JOSA B 2, 1841 (1985)● F. Mueller, S. Heugel, and L. J. Wang, Phys. Rev. A 77, 031802 (2008);● C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt,
Phys.Rev. Lett. 101, 133903 (2008)● F. Hao, D. JiangFang, L. Yong, and C. GengYu - Sci. China Phys. Mech. Astron. 58, 1674
(2015)● H. Xu, U. Kemiktarak, J. Fan, S. Ragole, J. Lawall and J. M. Taylor – arXiv:1510.04971
(2015)
Photonic crystals:● M. F. Yanik, S. Fan,and M. Soljačić – Appl.Phys. Lett. 83, 2739 (2003)● M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi and T. Tanabe – Opt. Expr.
13 2678 (2005)
...
Optical bistability: semiclassical approach Experimental signature:
H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan Phys. Rev. Lett. 36, 1135 (1976)
Semiclassical analysis:
Steady-state:
Laser-cavity detuning:
→ Bistability for
The quantum solution is unique:
Optical bistability at the quantum level
P. D. Drummond and D. F. Walls – J. Phys. A: Math. Gen. 13, 725 (1980).
Optical bistability at the quantum levelThe quantum solution is unique: Bimodal Wigner distribution:
K. Vogel and H. Risken – Phys. Rev. A 39, 4675 (1989)
P. D. Drummond and D. F. Walls – J. Phys. A: Math. Gen. 13, 725 (1980).
optical bistability at the quantum level
Quantum trajectory:
Semiclassical results
The quantum solution is unique: Bimodal Wigner distribution:
K. Vogel and H. Risken – Phys. Rev. A 39, 4675 (1989)
J. Kerckhoff, M. A. Armen, and H. Mabuchi, Opt. Express 19, 24468 (2011).
P. D. Drummond and D. F. Walls – J. Phys. A: Math. Gen. 13, 725 (1980).
Optical bistability at the quantum level
Optical bistability: hysteresis cycleExperimental signature:
H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan Phys. Rev. Lett. 36, 1135 (1976)
More recent experiments:Microcavity polaritons:● D. Bajoni, E. Semenova, A. Lemaître, S. Bouchoule, E. Wertz, P. Senellart, S. Barbay, R.
Kuszelewicz, and J. Bloch, Phys. Rev. Lett. 101, 266402 (2008)● A. Baas, J. Ph. Karr, H. Eleuch, and E. Giacobino – Phys. Rev. A 69, 023809 (2004)● T. K. Paraïso, M. Wouters, Y. Léger, F. Morier-Genoud, and B. Deveaud-Plédran, Nat.
Mater. 9, 655 (2010)● A. Amo, T. C. H. Liew, C. Adrados, R. Houdré, E. Giacobino, A. V. Kavokin and A.
Bramati Nat. Phot. 4, 361 (2010)● H. Abbaspour, S. Trebaol, F. Morier-Genoud, M. T. Portella-Oberli, and B. Deveaud, Phys.
Rev. Lett. 113, 057401 (2014)● T. Boulier, M. Bamba, A. Amo, C. Adrados, A. Lemaitre, E. Galopin, I. Sagnes, J. Bloch,
C. Ciuti, E. Giacobino and A. Bramati – Nat. Comm. 5, 3260 (2014)● S.R.K. Rodriguez, A. Amo, I. Sagnes, L. Le Gratiet, E. Galopin,A. Lemaître, and J.
Bloch – arXiv:1602.07114 (2016)
Superconducting circuits:● I. Siddiqi, R. Vijay, F. Pierre, C. M. Wilson, M. Metcalfe, C. Rigetti, L. Frunzio, and M. H.
Devoret, Phys. Rev. Lett. 93, 207002 (2004)● R. Vijay, M. Devoret, and I. Siddiqi, Rev. Sci. Instrum. 80, 111101 (2009)● F. R. Ong, M. Boissonneault, F. Mallet, A. Palacios-Laloy, A. Dewes, A. C. Doherty, A.
Blais, P. Bertet, D. Vion, and D. Esteve, Phys. Rev. Lett. 106, 167002 (2011)
Opto-mechanical cavities:● A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, Phys. Rev. Lett. 51,
1550 (1983)● P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, JOSA B 2, 1830 (1985);● A. Gozzini, I. Longo, S. Barbarino, F. Maccar- rone, and F. Mango, JOSA B 2, 1841 (1985)● F. Mueller, S. Heugel, and L. J. Wang, Phys. Rev. A 77, 031802 (2008);● C. Metzger, M. Ludwig, C. Neuenhahn, A. Ortlieb, I. Favero, K. Karrai, and F. Marquardt,
Phys.Rev. Lett. 101, 133903 (2008)● F. Hao, D. JiangFang, L. Yong, and C. GengYu - Sci. China Phys. Mech. Astron. 58, 1674
(2015)● H. Xu, U. Kemiktarak, J. Fan, S. Ragole, J. Lawall and J. M. Taylor – arXiv:1510.04971
(2015)
Photonic crystals:● M. F. Yanik, S. Fan,and M. Soljačić – Appl.Phys. Lett. 83, 2739 (2003)● M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi and T. Tanabe – Opt. Expr.
13 2678 (2005)
...
??
Dynamical HysteresisDynamical Hysteresis
W. Casteels, F. Storme, A. Le Boité, and C. Ciuti – Phys. Rev. A 93, 033824 (2016)
So far: analysis of the steady-state properties
Q: what about time dependence of the experimental sweep?
Time dependent sweep
So far: analysis of the steady-state properties
Q: what about time dependence of the experimental sweep?
Time dependent sweep
Triangular time dependence: → dynamical hysteresis:
So far: analysis of the steady-state properties
Q: what about time dependence of the experimental sweep?
Time dependent sweep
Triangular time dependence: → dynamical hysteresis:
Double power law
Slower sweep
Double power law behavior of the hysteresis area A:
Semiclassical
Quantum
Oscillations with minima at the multi-photonic resonances, i.e. for
Characteristic time
Scaling AnalysisScaling Analysis
Liouvillian gap Liouvillian gap
Liouvillian gap:
Liouvillian gap Liouvillian gap
Slowest system timescale
Liouvillian gap:
Liouvillian gap Liouvillian gap
Sweep timescale
with
Slowest system timescale
Liouvillian gap:
Non-adiabatic region
Liouvillian gapComparison with numerics
Non-adiabatic region
Liouvillian gapComparison with numerics
Tunneling time:
H. Risken, C. Savage, F. Haake and D. F. Walls – Phys. Rev. A 35, 1729 (1987)
Non-adiabatic region
Tunneling time
● Can be very sensitive to system parameters● Quickly becomes astronomically large for weak nonlinearities
Tunneling time
● Can be very sensitive to system parameters● Quickly becomes astronomically large for weak nonlinearities
Conclusions & PerspectivesConclusions & Perspectives
Conclusions & perspectives
● Revealed dynamical hysteresis of optical bistability (↔ semiclassical prediction of static hysteresis).
● Hysteresis area exhibits rich double power law behavior.
● Can be understood qualitatively from the Liouvillian gap.
Conclusions
PerspectivesDynamical hysteresis for bistable states of collective phases?
A. Le Boité, G. Orso and C. Ciuti – Phys. Rev. Lett. 110, 233601(2013)
J. Jin, D. Rossini, M. Leib, M. J. Hartmann and R. Fazio – Phys. Rev. A 90 (2014)
R. M. Wilson, K. W. Mahmud, A. Hu, A. V. Gorshkov, M. Hafezi, and M. Foss-Feig – arXiv:1601.06857 (2016)
→ Also in this case quantum solution is unique
J. J. Mendoza-Arenas, S. R. Clark, S. Felicetti, G. Romero, E. Solano, D. G. Angelakis, and D. Jaksch – Phys. Rev. A 93, 023821 (2016)P. Degenfeld-Schonburg and M. J. Hartmann – Phys. Rev. B 89, 245108 (2014)
Conclusions & perspectives
→ Numerical tools for driven-dissipative lattices:
→ extension for dynamical phenomena (time-dependence, Liouvillian gap, ...)
Perspectives
S Finazzi, A Le Boité, F Storme, A Baksic, C Ciuti – Corner-Space Renormalization Method for Driven-Dissipative Two-Dimensional Correlated Systems – Phys. Rev. Lett. 115, 080604 (2015)
W. Casteels, S. Finazzi, A. Le Boité, F. Storme, C. Ciuti – Truncated correlation hierarchy schemes for driven-dissipative multimode quantum systems – arXiv:1605.00882
Conclusions & perspectives
Nicola Bartolo
Thibaud Lacroix
Jared Lolli
Fabrizio Minganti
Riccardo Rota
Thank you for your attention!
Thanks!
Liouvillian gapArea vs. non-adiabatic region
Qualitative agreements of power laws
Dynamical hysteresis: first exponent