Out-of-equilibrium incommensurate lattice models Manas Kulkarni International Centre for Theoretical Sciences ICTS- TIFR, Bangalore Collaboration: A. Purkayastha (ICTS-TIFR) S. Sanyal (ICTS-TIFR à Columbia, USA) A. Dhar (ICTS-TIFR) Phys. Rev. B [Editors choice] (2018, in press) Phys. Rev. B 96, 180204 (Rapid, 2017) Also: Phys. Rev. A 94, 052134, (2016) Phys. Rev. A 93, 062114, (2016) Acknowledgement: H. Spohn (Munich) I. Bloch (Munich)
Mesoscopic systems + Photons/Phonons Quantum dots coupled to photons (QD-cQED) Quantum devices Analogy to Molecular Junctions Heavy Fermions with light (Kondo) Jiang, Kulkarni, Segal, Imry (PRB 2015) Hartle, Kulkarni (PRB 2015) Kulkarni, Cotlet, Tureci (PRB 2014) Agarwalla, Kulkarni, Mukamel, Segal (PRB 2016) Cold Atoms + Quantum Optics Open quantum phase transitions Non-hermitian matrices and Studying random matrices Kulkarni, Makris, Tureci (Unpublished) Kulkarni, Oztop, Tureci (PRL 2013) Driven Quantum Systems Quantum Hamiltonian + Bath Engineering Preparation of entangled states Open analogs of condensed matter systems (spin chains) Time dynamics, Transport, Nonequilibrium Steady States Kulkarni, Hein, Kapit, PRB (2018) Aron, Kulkarni, Tureci (PRX, 2016) Aron, Kulkarni, Tureci (PRA 2014) Schwartz, Martin, Flurin, Aron, Kulkarni, Tureci, Siddiqi (PRL 2016) Purkayastha, Dhar, Kulkarni (PRA 2016, PRB 2018)
Classical Integrable Models Finite Dimensional Systems Calogero Family Solions for finite dimensional systems Duality Nonlinear Dynamics, Quenches, Universality M. K. Polychronakos (2018, ongoing) M. K, Polychronakos, J. Phys. A: Math. Theor. (2017) Franchini, Gromov, M. K, Trombettoni, J. Phys. A: Math (2015) Abanov, Gromov, M. K, J. Phys. A: Math. Theor. (2011) Continuous Systems KdV in cold atoms / nonlinear optics, Benjamin-Ono equation (Non-local Integrable) Multi-component NLS and KdV systems (integrable) Quenches, Solitons and Nonlienar Dynamics Universality Swarup, Vasan, M. K (2018, ongoing) Franchini, M. K, Trombettoni, New J. Phys (2015) M. K., Abanov (PRA, 2012)
Contents Purkayastha, Dhar, Kulkarni (PRB Rapid 2017) Motivation and Experiments Non-equilibrium setup and methods Probes of the phase diagram Generalized Aubry-Andre-Harper Model Couple it to two baths Ganeshan et al, PRL (2015) Closed System Properties (for e.g., Anomalous yet Brownian) Results of the traditional AAH Conclusions and Outlook arxiv:1702.05228, (2017) Purkayastha, Sanyal, Dhar, Kulkarni (almost Mathieu operator) (also arxiv:1703.05844, Verma, Mulatier, Znidaric)
Questions What happens when you connect a system (which has ballistic-critical-delocalised phase) to baths? Do exponents obtained from open system properties (Non-eq Steady State current) match with exponents of closed system? New Diagnostics for phase diagram (NESS current, NESS particle density profile) Universality in Open System Properties? Left Bath T L,µ L Aubtry-Andre or Generalized Aubtry Andre System Right Bath T R,µ R
Nature Letters 2008 Motivation and Experiments
Motivation and Experiments Non-interacting AAH Science 349, 842 (2015) and other experiments from Bloch group (including controlled open system)
Deformations of the Aubry-Andre-Harper Model Purkayastha, Dhar, Kulkarni (PRB Rapid 2017) GAAH E = 2sgn( )(a ) mobility edge Quantities Computed Imbalance, Step Profiles NESS Current and scaling with system size P(x, t), moments
Examples of Typical Eigenfunctions (traditional AAH) =0 0.01 =0.5 Extended 0.00 0 50 100 150 200 250 10(r) 2 0.2 0.1 0.0 =1 0 50 100 150 200 250 Critical 1.0 0.5 =2 Localized 0.0 0 50 100 150 200 250 r Energy Spectrum is a Cantor Set, Avila, S. Jitomirskaya, Ann. of. Math. 170 303 (2009) The Ten Martini Problem
Our non-equilibrium setup and methods Hamiltonian Bath Ĥ SB = (â 1ˆb (1) 1 +â Nˆb (N) 1 + h.c.) System-Bath Coupling Each bath (left and right) has its own temperature and chemical potential Our method valid for arbitrary system-bath coupling (unlike Lindblad) Our method valid for arbitrary inter-system coupling (unlike Local-Lindblad)
I = hˆn r i = Z 2tB 2t B d! Z 2tB 2 J 1(!)J N (!) G 1N (!) 2 [ n (1) F (!) n(n) F (!)], 2t B d! 2 Quantities of Interest h G r1 (!) 2 n (1) F i (!)+ G rn(!) 2 n (N) F (!) where G(!) = h i 1!I H S (1) (!) (N) (!) (1) 2 11 (!) = (N) NN (!) =! 2t 2 B + i 2 J(!) s J(!) = 2 2 1 t B! 2t B 2
Generalized Aubry-Andre-Harper Fraction of localized eigenstates Purkayastha, Dhar, Kulkarni (arxiv:1707.03749) Imbalance in steady state Imbalance Dynamics Imbalance is insensitive to few extended states Current is insensitive to few localized states Step Profile Evolution (Phase coexistence)
Exploring the Single-Particle Mobility Edge in a One-Dimensional Quasiperiodic Optical Lattice Recent experiment from group of Immanuel Bloch (arxiv:1709.03478) Imbalance and Expansion are complementary Imbalance is insensitive to few extended states Expansion is insensitive to few localized states
Wave function matching between GAAH and AAH Pth state of GAAH (near self-dual point) matches with the pth state of critical AAH
GAAH Current Scaling with system size Purkayastha, Dhar, Kulkarni (arxiv:1707.03749) arxiv:1702.05228, (2017) Purkayastha, Sanyal, Dhar, Kulkarni (also arxiv:1703.05844, Verma, Mulatier, Znidaric) AAH showed 1.4 GAAH shows 2.0 Scaling exponent on the line is remarkably different than the AAH critical point
Closed System Properties of GAAH IPR 10 0 10 1 10 2 10 3 10 4 (a) =1.5, =0.39 1.14x 0.31 =1.5, =0.19 0.67x 0.02 =1.5, =0.78 2.91x 1.0 10 1 10 2 10 3 N m2 10 7 10 5 10 3 10 1 (b) =1.5,a =0.388 0.528x 1 =1.5,a =0.194 2.027x 0 =1.5,a =0.776 0.027x 2 10 0 10 1 10 2 10 3 10 4 t Experiments in a different context: PNAS 2009, Anamolous yet Brownian Nature Materials 2012, When Brownian diffusion is not Gaussian p tp (x, t) 10 5 10 3 10 11 10 19 =1.5 =0.388 (c) t=256.0 t=512.0 t=1024.0 10 5 0 5 10 x/ p t tp (x, t) 10 4 10 1 10 2 10 5 =1.5 =0.776 (d) t=256.0 t=512.0 t=1024.0 0.4 0.2 0.0 0.2 0.4 x/t Phase Co-existence: Experiments on a similar Model (I. Bloch)
Closed System Properties GAAH Model is Anomalous yet Brownian for all irrational numbers Experiment, PNAS, 2009 Experiment, Nature Materials, 2012 We will now discuss traditional AAH model, i.e, =0
AAH arxiv:1702.05228, (2017) Current Scaling with system Size at critical point Sub-diffusive for all irrational number and experimental incommensurate lattice Experimental numbers shows eventual flattening Different scaling at Fibonacci
AAH De-localized/Ballistic Regime Localized Regime 10 1 golden mean =0.25 10 2 golden mean =1.1 10 3 I N- independent I 10 4 10 5 e N 10 6 10 7 10 2 10 0 10 1 10 2 10 3 10 4 N 10 8 10 20 30 40 50 60 70 80 90 100 N
AAH arxiv:1702.05228, (2017) Purkayastha, Sanyal, Dhar, Kulkarni NESS particle density profile for the three regimes Remarkably different behavior in the regimes NESS particle spatial density profile provides a novel real-space experimentally measurable probe of the localized, critical and de-localized phases.
AAH arxiv:1702.05228, (2017) Purkayastha, Sanyal, Dhar, Kulkarni All Mostly follow: We find that Kurtosis K = m 4 /(m 2 ) 2 > 3 K >3 means non-gaussian Kurtosis is time-dependent signifying multi-scaling
AAH: Closed System Properties
Conclusions Phys. Rev. B [Editors choice] (2018, in press) Phys. Rev. B 96, 180204 (Rapid, 2017) Interesting phase diagram GAAH model. Open quantum system set-up of the GAAH model captures rich and interesting physics which are missed by the standard closed system description. NESS particle spatial density profile provides a novel real-space experimentally measurable probe of the localized, critical and de-localized phases. Sub-diffusive nature of critical line cannot be obtained from closed system calculations. Phys. Rev. A 93, 062114, (2016), Purkayastha, Dhar, Kulkarni (extensive analysis of Local Lindblad, Eigenbasis lindblad, Redfied, brute-force numercs, NEGF for ordered systems, both Steady State and Time Dynamics)
Other Research/Outlook Quantum Gases placed in cavities M. Kulkarni, B. Oztop, H. E. Tureci, PRL 2013 M. Kulkarni, K. G. Makris, H. E. Tureci (Unpublished) Esslinger Lab, ETH Vuletic Lab, MIT Cold Gases, Integrable Models, Universalities Non-local driven dissipative GPE Polaritons, Caldeira-Leggett Physics Non-Hermitian Matrices Open Quantum Phase Transistions and criticality Thomas Lab, Duke/NCSU Schmiedmayer Lab, Vienna Zwierlein Lab, MIT J. Joseph, J. E. Thomas, M. Kulkarni, A. G. Abanov, PRL 2011 M. Kulkarni, A. Lamacraft, PRA Rapid, 2013 M. Kulkarni, H. Spohn, D. A. Huse, PRA 2015 Dynamical critical pheomenon of coupled stochastic equations Quenching a nonequilibrium profile (soliton-quench) Field theory of integrable models Kardar-Parisi-Zhang Universality
Other Research/Outlook Hybrid Systems of quantum dots and photons Petta Lab, Princeton Agarwalla, Kulkarni, Mukamel, Segal (PRB 2016) M. Kulkarni, O. Cotlet, H. E. Tureci (PRB, 2014) Hartle, Kulkarni (PRB 2015) Interplay between electrons, phonons photons and the quantum dots Kondo Physics, heavy fermions with light Single photon sources, novel microwave amplifiers Tunable solid-state lasers Photon Statistics Many body phases in large scale driven dissipative systems g Houck Lab, Princeton Siddiqi Lab, Berkeley C. Aron, M. Kulkarni, H. E. Tureci (PRA, 2014) C. Aron, M. Kulkarni, H. E. Tureci (PRX 2016) Schwartz, Martin, Flurin, Aron, Kulkarni, Tureci, Siddiqi (PRL 2016) Entanglement in scalable open systems, Realization of XY spin chain, magnetic systems Large scale quantum computation and simulation g g Kulkarni, Hein, Kapit, Aron, arxiv:1707.03749 (2017)