Heating in periodically driven Floquet systems Anushya Chandran - PowerPoint PPT Presentation

QMath 2016 Heating in periodically driven Floquet systems Anushya Chandran Boston University

ννν Floquet system νννννν ννν νννννν ννν νννννν ννν νννννν Periodically driven isolated system Hamiltonian H 0 H ( t ) = H 0 + V cos( ω t ) H 1

Few-body Floquet systems νννννν ννν νννννν ννν νννννν ννν νννννν ννν Rabi oscillations Kapitza pendulum Probability Time Amplitude of drive ➡ Frequency New stable equilibrium

Few-body Floquet systems νννννν ννν νννννν ννν νννννν ννν νννννν ννν Rabi oscillations Kapitza pendulum Probability Time Amplitude of drive ➡ Frequency New stable equilibrium Many-body?

Interest: fundamental & engineering νννννν ννν νννννν ννν νννννν ννν νννννν ννν Simplest non-equilibrium setting: what can happen? Engineer new states out of equilibrium?

Interest: fundamental & engineering νννννν ννν νννννν ννν νννννν ννν νννννν ννν Simplest non-equilibrium setting: what can happen? Engineer new states out of equilibrium? Bi 2 Se 3 Wang et al (Gedik group) Aidelsburger et al (Bloch group) Science (2013) Nature (2014)

Outline νννννν ννν νννννν ννν νννννν ννν νννννν ννν Steady states of Floquet systems Thermal Many-body localized (infinite Time crystal (persistent local temperature) memory) (period doubling in an “integrable” theory)

Thermalization in isolated systems νννννν ννν νννννν ννν νννννν ννν νννννν ννν | ψ ( t ) i Local state A ρ A ( t ) = Tr B | ψ ( t ) ih ψ ( t ) | B

Thermalization in isolated systems νννννν ννν νννννν ννν νννννν ννν νννννν ννν | ψ ( t ) i Local state A ρ A ( t ) = Tr B | ψ ( t ) ih ψ ( t ) | B No driving t →∞ ρ A ( t ) = 1 Z Tr B e − β H lim

Thermalization in isolated systems νννννν ννν νννννν ννν νννννν ννν νννννν ννν | ψ ( t ) i Local state A ρ A ( t ) = Tr B | ψ ( t ) ih ψ ( t ) | B No driving With driving t →∞ ρ A ( t ) = 1 t →∞ ρ A ( t ) = 1 Z Tr B e − β H Z Tr B e − β H lim lim ∝ 1

Eigenstate thermalization hypothesis (ETH) νννννν ννν νννννν ννν νννννν ννν νννννν ννν For all eigenstates E i at inverse temperature β A ρ A = Tr B | E i ih E i | = 1 Z Tr B e − β H B H | E i i = E i | E i i ETH ⇒ thermalization Generically thermalization seems to ⇒ ETH Deutsch (1991) Srednicki (1994)

ννν Driven eigenstates νννννν ννν νννννν ννν νννννν ννν νννννν H ( t ) = H 0 + V cos( ω t ) H 1 Floquet/periodic U ( T ) = Te − i R T 0 H ( t ) dt 0 evolution:

ννν Driven eigenstates νννννν ννν νννννν ννν νννννν ννν νννννν H ( t ) = H 0 + V cos( ω t ) H 1 Floquet/periodic U ( T ) = Te − i R T 0 H ( t ) dt 0 evolution: H 0 Energy

ννν Driven eigenstates νννννν ννν νννννν ννν νννννν ννν νννννν H ( t ) = H 0 + V cos( ω t ) H 1 Floquet/periodic U ( T ) = Te − i R T 0 H ( t ) dt 0 evolution: H 0 U ( T ) Energy Floquet Quasi-energy

ννν Driven eigenstates νννννν ννν νννννν ννν νννννν ννν νννννν H ( t ) = H 0 + V cos( ω t ) H 1 Undriven eigenstates ETH H 0 U ( T ) Hot Cold Energy Floquet Quasi-energy

ννν Driven eigenstates νννννν ννν νννννν ννν νννννν ννν νννννν H ( t ) = H 0 + V cos( ω t ) H 1 Undriven eigenstates ETH Driven eigenstates ? H 0 U ( T ) Hot Cold Energy Floquet Quasi-energy

ννν Driven eigenstates νννννν ννν νννννν ννν νννννν ννν νννννν H ( t ) = H 0 + V cos( ω t ) H 1 Undriven eigenstates ETH Driven eigenstates ? U ( T ) 1 Local drive: p h E β | U ( T ) | E α i ⇠ 2 L α β Floquet Quasi-energy Deutsch (1991) Srednicki (1994) Ponte, AC , Papic, Abanin (2015)

ννν Driven eigenstates νννννν ννν νννννν ννν νννννν ννν νννννν H ( t ) = H 0 + V cos( ω t ) H 1 Undriven eigenstates ETH Driven eigenstates ? U ( T ) 1 Local drive: p h E β | U ( T ) | E α i ⇠ 2 L α ∆ αβ β ∆ αβ ∼ 1 2 L Floquet Quasi-energy Deutsch (1991) Srednicki (1994) Ponte, AC , Papic, Abanin (2015)

ννν Driven eigenstates νννννν ννν νννννν ννν νννννν ννν νννννν H ( t ) = H 0 + V cos( ω t ) H 1 Undriven eigenstates ETH Driven eigenstates ? U ( T ) 1 Local drive: p h E β | U ( T ) | E α i ⇠ 2 L ∆ αβ ∼ 1 2 L Floquet eigenstates mix all temperatures! Floquet Quasi-energy Deutsch (1991) Srednicki (1994) Ponte, AC , Papic, Abanin (2015)

Outline νννννν ννν νννννν ννν νννννν ννν νννννν ννν Steady states of Floquet systems Thermal Many-body localized (infinite Time crystal (persistent local temperature) memory) (period doubling in an “integrable” theory)

Interacting driven bosons νννννν ννν νννννν ννν νννννν ννν νννννν ννν Driven O(N) model H ( t ) = 1 Z d d x ( | Π | 2 + | r Φ | 2 + r ( t ) | Φ | 2 + λ | Φ | 4 ) 2 r ( t ) = r 0 − r 1 cos( γ t ) AC , Sondhi (2015)

Interacting driven bosons νννννν ννν νννννν ννν νννννν ννν νννννν ννν Driven O(N) model H ( t ) = 1 Z d d x ( | Π | 2 + | r Φ | 2 + r ( t ) | Φ | 2 + λ | Φ | 4 ) 2 r ( t ) = r 0 − r 1 cos( γ t ) O(2) version: Near transition from Mott insulator to superfluid r(t): modulating tunneling AC , Sondhi (2015)

Interacting driven bosons νννννν ννν νννννν ννν νννννν ννν νννννν ννν Driven O(N) model H ( t ) = 1 Z d d x ( | Π | 2 + | r Φ | 2 + r ( t ) | Φ | 2 + λ | Φ | 4 ) 2 r ( t ) = r 0 − r 1 cos( γ t ) Equilibrium: canonical model for symmetry-breaking Analytical control in the large-N limit Self-consistent classical equations Z Λ d d k ! d 2 k | 2 + r ( t ) + � dt 2 + | ~ k ( t ) | 2 (2 ⇡ ) d | f ~ f ~ k ( t ) = 0 N λ AC , Sondhi (2015)

Interacting driven bosons νννννν ννν νννννν ννν νννννν ννν νννννν ννν ~ ~ ~ k 3 k 1 k 2 k ( t ) 2 = | k | 2 + r ( t ) + r eff ( t ) ω ~ Feedback term Feedback term prevents parametric resonance Steady state: finite correlations with structure “Integrable”: unknown generalized Gibbs ensemble AC , Sondhi (2015)

Period doubling in the driven ferromagnet νννννν ννν νννννν ννν νννννν ννν νννννν ννν Drive period = π 0 . 6 0 . 4 M ( t ) 0 . 2 0 . 0 − 0 . 2 0 10 20 30 40 50 60 t/ π 0 . 15 0 . 05 M ( t ) − 0 . 05 − 0 . 15 1584 1586 1588 1590 1592 1594 1596 1598 1600 t/ π 0 . 04 F[ M ( t ) ] 0 . 03 0 . 02 0 . 01 0 . 00 − 4 − 3 − 2 − 1 0 1 2 3 4 Frequency AC , Sondhi (2015)

Outline νννννν ννν νννννν ννν νννννν ννν νννννν ννν Steady states of Floquet systems Thermal Many-body localized (infinite Time crystal (persistent local temperature) memory) (period doubling in an “integrable” theory)

ννν Periodic circuits νννννν ννν νννννν ννν νννννν ννν νννννν | ψ ( n + 1) i | ψ ( n ) i = U n | ψ (0) i U | ψ ( n ) i Floquet evolution without H(t)!

ννν Clifford circuits νννννν ννν νννννν ννν νννννν ννν νννννν • Clifford gates: Hadamard, Phase, CNOT • Efficiently simulable (poly(N) time for N qubits) • U † ( X 1 ⊗ Z 2 ⊗ . . . 1 N ) U = Y 1 ⊗ X 2 ⊗ . . . Z N • Can entangle • Infinite temperature locally? Gottesman-Knill (1996) Aaronson & Gotteman (2004)

ννν Thermalization νννννν ννν νννννν ννν νννννν ννν νννννν t X Y Z b a c x U AC , C. R. Laumann (2015)

ννν Thermalization νννννν ννν νννννν ννν νννννν ννν νννννν S A ( t ) = − Tr ρ A log 2 ρ A t X S A ( t ) Y N A = 20 Z b a c x A t U AC , C. R. Laumann (2015)

ννν Thermalization νννννν ννν νννννν ννν νννννν ννν νννννν S A ( t ) = − Tr ρ A log 2 ρ A t X S A ( t ) Y N A = 20 Z b a c x A t U • Operator support grows in time • ρ A = 1 for t > vN A • Simulable system that thermalizes! AC , C. R. Laumann (2015)

Localization νννννν ννν νννννν ννν νννννν ννν νννννν ννν t S A ( t ) A x t • Strictly local integrals of motion: Z i • Block spread of information • Transition to thermalization: percolation of operator support AC , C. R. Laumann (2015)

Outline νννννν ννν νννννν ννν νννννν ννν νννννν ννν Steady states of Floquet systems Thermal Many-body localized (infinite Time crystal (persistent local temperature) memory) (period doubling in an “integrable” theory)