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Hamiltonian engineering for many-body quantum systems by Shortcuts To Adiabaticity
Trieste, August 23, 2016
Kazutaka Takahashi Tokyo Tech
01/23
Hamiltonian engineering for many-body quantum systems by Shortcuts - - PowerPoint PPT Presentation
Hamiltonian engineering for many-body quantum systems by Shortcuts - - PowerPoint PPT Presentation
Hamiltonian engineering for many-body quantum systems by Shortcuts To Adiabaticity Kazutaka Takahashi Tokyo Tech Trieste, August 23, 2016 01/23 Motivation: QA Target Hamiltonian Best driver Hamiltonian? Best schedule? Hamiltonian
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Shortcuts to Adiabaticity Tracking adiabatic passage of reference H
Slow Adiabatic Fast Nonadiabatic
Red: without STA Black: with STA Blue: fidelity
Demirplak Rice 2003 2005 Berry 2009 Chen Ruschhaupt Schmidt del Campo Guéry-Odelin Muga 2010
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Aim Use of STA for Hamiltonian Engineering Inverse Engineering / Counterdiabatic Driving for Many-body Spin Systems
► Fundamental equation for QM ► 3 meanings
Principle:
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Result 1
0.0 0.5 1.0 0.5 1.0
nz t/T
T=1 T=2 T=5 T=10 T=20 T=50 T=100 T=1.0 T=2.0 T=5.0 T=10.0
0.0 0.5 1.0 0.5 1.0
nz t/T
1.0 2.0 3.0
T=0.8 T=1.0 T=2.0 T=5.0 T=10.0
0.0 0.5 1.0
f t/T
1.0 2.0
T=0.8 T=1.0 T=2.0 T=5.0 T=10.0
0.0 0.5 1.0
Γ t/T
Linear schedule Inverse Engineering
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Result 2
N=3 Infinite system: Soliton Counterdiabatic Driving
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Shortcuts to Adiabaticity
07/23
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Equation of Motion ► Equation for density operator ► Equivalent to Schrödinger eq. ► ρ obtained for a given H (and initial state) von Neumann equation
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Dynamical Invariant
Lewis Riesenfeld 1969
Eigenstates of F Time independent ► Solution of Schrödinger eq = Adiabatic state of F (not of H!) ► Time evolution described by eigenvalue eq.
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Shortcuts to Adiabaticity
Had Hcd
Counterdiabatic Driving
- Had = Target H ≠ Total H
- Hcd: Assist term for a given Had
- State follows an adiabatic passage of Had, not of H
Inverse Engineering
- Total H = Target H
- F engineered to design H
- State follows an adiabatic passage of F, not of H
Eigenstates of F
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How to Solve?
Reduction to Simple Systems
- Mean-field approximation
- Reduction of Hilbert space
Use of Classical Nonlinear Systems
- Quantum STA is equivalent to
Classical nonlinear integrable systems
- Integrability: existence of dynamical invariant
Okuyama and KT 2016
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Inverse Engineering of Transverse-field Ising Model
12/23
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Mean-field Solution
► F solved by mean-field ansatz ► Effective 1-body 2-level system ► Solutions characterized by Bloch vector n(t)
Mean field
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Inverse Engineering
► Schedule determined by Bloch vector ► Initial and final conditions fixed ► No need to solve differential equations ► Follow adiabatic passage of F, not of H
x y z F H
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Results
► Interpolating boundary conditions ► Non-unique schedule
1 2 3 0.2 0.4 0.6 0.8 1
f t/T
T=0.8 T=1.0 T=2.0 T=5.0 T=10.0 1 2 0.2 0.4 0.6 0.8 1
Γ t/T
T=0.8 T=1.0 T=2.0 T=5.0 T=10.0 2 4 6 0.2 0.4 0.6 0.8 1
f t/T
T=0.5 T=1.0 T=2.0 T=5.0 1 2 3 0.2 0.4 0.6 0.8 1
Γ t/T
T=0.5 T=1.0 T=2.0 T=5.0
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
nz t/T
T=1.0 T=2.0 T=5.0 T=10.0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
t/T
T=1.0 T=2.0 T=5.0
nz
Schedule 1 Schedule 2
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Generalization
How about Non-MF Hamiltonian?
► Solvable for Mattis model (No Frustration): No need to know ξi ► Mean-field approximation ► Local manipulation required
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From Classical Nonlinear Integrable Systems to Quantum Shortcuts to Adiabaticity
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Lax Formalism Lax equation Lax 1968 Lax pair
► Expression of integrability ► Time-independent eigenvalues of L: Infinite conserved quantities ► KdV, Toda, Sine-Gordon, Nonlinear Schrödinger, …
Classical Nonlinear Dynamics Quantum Dynamics
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KdV Equation
► Particle transported by multi-soliton potentials
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Toda Equation in XY Spin Chain
Toda Equations ► Lax pair in tri-diagonalized form ► Gauge transformation in a more convenient form ► Equivalent to free fermion
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Solutions of Toda Equation: Finite Systems
N=3 ► Continuous deformation from XY to noninteracting H ► Isospectral deformation
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Solutions of Toda Equation: Infinite Systems
Single soliton Double solitons ► Local manipulation of coupling functions ► Flipped-spin transferred by solitons
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Summary
KT, Inverse engineering for many-body spin systems, To appear soon… M Okuyama and KT, From Classical Nonlinear Integrable Systems to Quantum Shortcuts to Adiabaticity, PRL 117, 070401 (2016)
Time evolution of quantum state Dynamical Invariant: Basis for STA Integrability for classical nonlinear systems ► States follow an adiabatic passage of a reference H ► Importance of local manipulations
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