Quantum-Classical Correspondence of Shortcuts T o Adiabaticity Kazutaka Takahashi Tokyo 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
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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)
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