Quantum-Classical Correspondence

of Shortcuts To Adiabaticity

Kazutaka TakahashiTokyo Tech

M Okuyama and KT, Quantum-Classical Correspondence of Shortcuts to Adiabaticity, JPSJ 86, 043002 (2017) Editors’ Choice

M Okuyama and KT, From Classical Nonlinear Integrable Systems to Quantum Shortcuts to Adiabaticity, PRL 117, 070401 (2016)

OIST, March 22, 2017

Summary

STA extended from Q to C systems

Solvable systems in Q/C STA

Q-C correspondence of STA

► talk for non-general audience:

somewhat formal, technical, and too many equations!

► your ideas on interesting applications welcome

Principle of STA: Dynamical Invariant

State given by an instantaneous eigenstate of F

Hamiltonian divided into two parts:

► CD Driving: CD term obtained for a given H0

► Inverse Engineering: H designed for a given F

Implementations:

CD term

Lewis-Riesenfeld (1969)

► state is an instantaneous eigenstate of H0

► CD term prevents nonadiabatic transitions

Brief History of STA

1969 Lewis-Riesenfeld Dynamical Invariant

2000 Emmanouilidou Zhao Ao Niu Early application to 2-level system

03, 05 Demirplak-Rice CD driving

08 Masuda-Nakamura Fast-Forward

09 Berry Formula of CD Driving

10 Bilbao G Inverse Engineering, “STA” named

13 Jarzynski C-STA, Scale-Invariant Driving

13 del Campo Scale-Invariant Driving

13 KT Quantum Brachistochrone

16 Okuyama-KT Nonlinear Integrable Systems

17 Okuyama-KT C-STA

Issues

Q Commutator and C Poisson Bracket

- simple replacement?

CD Term for Q/C STA

- same or not

Q/C Adiabatic Theorems

- look different

Adiabatic Invariant (C)

- role in Q system

Yes

Not the same

Imply some relation(not completely understood)

Q Commutator and C Poisson Bracket

CD Term for Q/C STA

Q/C Adiabatic Theorems

Adiabatic Invariant (C)

06/28

CD DrivingBerry’s formula (2009)

Jarzynski (2013): Q

Jarzynski (2013): C

► microcanonical average on equienergy (H0=E0) plane► only for periodic trajectories?

► H0-eigenstate basis

CD term

Classical STA

► applicable to general systems► arbitrariness: {ξ,H0}={ξ’,H0}

: time derivative

: eq of motion

: instant energy of H0 “conserved”

Classical STA

► applicable to general systems► arbitrariness: {ξ,H0}={ξ’,H0}

: time derivative

: eq of motion

: instant energy of H0 “conserved”

Formula of CD driving in C-STA

Slightly different from Jarzynski formula

A different formula derived later

Scale-Invariant Systems

► translation (x0) + dilation (γ)

► same for Q and CJarzynski (2013)

Deffner-Jarzynski-del Campo (2014)

► energy independent of x0

Solvable Systems

All eigenvalues of F independent of t: Infinite conserved quantities

Lax equation

Lax pairLax (1968)

► Lax pairs known infinitelyKDV, Toda, Sine-Gordon, Nonlinear Schrödinger, …

Hierarchical structures: Scale-invariant systems in the simplest hierarchy

Okuyama-KT (2016)

Relation to C nonlinear integrable systems

KdV for C Nonlinear Systems and Q-STA

KdV Equation

Q-STAC Nonlinear Systems

► multiple solitons

► potential satisfies KdV► integrability of KdV

Toda Lattice and Q Spin

Toda equations

Isotropic XY spin chain

Toda lattice

Toda Lattice and Q Spin

Isotropic XY spin chain

Toda lattice

Toda equations Lax pair (L,M) ↔ CD driving (H0,ξ)

Integrability and adiabaticity

Simulate nonlinear systems in Q dynamics

From Q to C

KdV equation Dispersionless KdV equation

Q-STA C-STA

► E0=const

► different solutions

Dispersionless KdV Equation

DKdV solved by Hodograph Method

► f: arbitrary function► no soliton solutions

Dispersionless KdV Equation

DKdV solved by Hodograph Method

► f: arbitrary function► No Soliton Solutions

DKdV hierarchy for C-STA

Q-STA and C-STA are different

Exception: Scale-invariant systems

Other Applications?

TodaC-limit of discrete system?

Nonlinear Schrödingeruse NLS to solve NLS? (cf. Use KdV to solve Schrödinger eq in a KdV potential)

AKNStwo-block Hamiltonian

Q Nonlinear Integrable SystemsQ Lax

1D Heisenberg model ► mathematically interesting

► physics?

Q Commutator and C Poisson Bracket

CD Term for Q/C STA

Q/C Adiabatic Theorems

Adiabatic Invariant (C)

16/28

For ,

adiabatic state (instantaneous state with a phase)

is an approximate solution of the Schrödinger equation

if

Adiabatic Theorem: Q

► adiabatic condition removed by CD term (STA!)

Adiabatic Theorem: C

α: constant α(t): varied

Adiabatic invariant

is conserved if► periodic trajectories only

► condition removed by CD term?

α(t1)

α(t2)

Adiabatic Theorem Holds Exactly

► adiabatic condition removed by CD term

Adiabatic Theorem Holds Exactly

► adiabatic condition removed by CD term

Q/C adiabatic theorems look different

Relation?

C adiabatic invariant for general trajectory

Adiabatic Invariant for General Trajectories?

1D, periodic Hamilton’s characteristic function(Abbreviated action)► for general systems

► from Hamilton-Jacobi theory

Adiabatic Invariant for General Trajectories?

1D, periodic Hamilton’s characteristic function(Abbreviated action)► for general systems

► from Hamilton-Jacobi theory

Adiabatic invariant for general systems

Develop Hamilton-Jacobi theory of STA

Q and C systems related by HJ theory

Generalized Action

Time-dependent α(t): With CD term

Hamilton’s characteristic function (Abbreviated action)

► independent of history of x and α

Time-independent α: No CD term

► independent of history of x

► energy-conserved system

Action

Generalized Action

Hamilton-Jacobi Equation

Use of S

Use of S0

Time-dependent α(t): With CD term

Time-independent α: No CD term

Hamilton-Jacobi Equation

Use of S

Use of S0

Time-dependent α(t): With CD term

Time-independent α: No CD term

Fundamental formula of C-STA

Formula written instantaneously

Generalized action plays an important role

Scale-Invariant System

HJ eqsPotential

Dispersionless KdV

HJ eqsDKdV eq

► assume constant E0

► from HJ theory

Dispersionless KdV

HJ eqsDKdV eq

► assume constant E0

► from HJ theory

HJ theory useful for applications

Adiabatic Invariant

► DKdV system► oscillating α

► nonperiodic ► Ω(T)=J► proved by action-angle variables

Adiabatic Invariant

► nonperiodic ► Ω(T)=J► proved by action-angle variables

► DKdV system► oscillating α

Adiabatic invariant from nonperiodic trajectory

Semiclassical Approximation

► action at zeroth order► eq for generalized action in DKdV system

KdV system in Q

Schrödinger eq (STA)

Potential satisfies KdV

Semiclassical expansion

Semiclassical Approximation

KdV system in Q

Schrödinger eq (STA)

Potential satisfies KdV

Semiclassical expansion

► action at zeroth order► eq for generalized action in DKdV system

DKdV is C limit of KdV

Generalized action appears in wf phase

Role of Generalized Action in Q

► dependent on α (not t) except the dynamical phase

Adiabatic State

► Q analogue of generalized action

Role of Generalized Action in Q

Adiabatic State

► Q analogue of generalized action

► dependent on α (not t) except the dynamical phase

Q/C correspondence by Ω

CD term from Ω

Ω: Fundamental role in Q/C dynamics

Summary

► STA tested in C systems (easier than Q, hopefully)

► new aspects of dynamics: Hamiltonian consists of two parts

► many-body systems: work and heat?

STA extended from Q to C systemsfrom commutator to Poisson bracket

Solvable systems in Q/C STAKdV/DKdV hierarchy

Q-C correspondence of STAgeneralized action

M Okuyama and KT, Quantum-Classical Correspondence of Shortcuts to Adiabaticity, JPSJ 86, 043002 (2017) Editors’ Choice

M Okuyama and KT, From Classical Nonlinear Integrable Systems to Quantum Shortcuts to Adiabaticity, PRL 117, 070401 (2016)