Power laws in the dynamical hysteresis of quantum … laws in the dynamical hysteresis of quantum nonlinear photonic resonators W. Casteels, F. Storme, A. Le Boité, C. Ciuti [email protected]

Power laws in the dynamical hysteresis of quantum nonlinear photonic resonators

W. Casteels, F. Storme, A. Le Boité, C. Ciuti

[email protected]

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)

...








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)




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,



(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:


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)


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).


Optical bistability at the quantum level








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)




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,



(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




Triangular time dependence: → dynamical hysteresis:




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:


Slowest system timescale

Liouvillian gap:


Sweep timescale

with

Slowest system timescale

Liouvillian gap:

Non-adiabatic region

Liouvillian gapComparison with numerics


Liouvillian gapComparison with numerics

Tunneling time:

H. Risken, C. Savage, F. Haake and D. F. Walls – Phys. Rev. A 35, 1729 (1987)


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)


→ 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


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

Documents

Power laws in the dynamical hysteresis of quantum … laws in the dynamical hysteresis of quantum nonlinear photonic resonators W. Casteels, F. Storme, A. Le Boité, C. Ciuti [email protected]