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- Toward dissipation resistant transport -
JungYun Han
What makes quantum so special?
1/16
From https://Dilbert.com/
What makes quantum so special?
Classical bit
Quantum bit
Quantum Coherence
2/16
Mission: Protect Quantum Coherence
Disorder
Topology
Thermal Fluctuation
?
Goal: Understand topology under dissipation
Superconducting circuit
Cold atom
Figure credits to UCSB and NIST
3/16
Error-free quantum information
PhD. Journey
4/16
Topologically protected transport of light
E.g. transport along boundary
References: [1] Nat. Photonics 7, 1001–1005 (2013), [2] Rev. Mod. Phys. 91, 015006 (2019)
5/16
Helical coupled resonator optical waveguide (H-CROW)
Band Structure
6/16
Result: Wave Perspective
Q. Can we perform reliable quantum communication?
7/16
Result: Quantum perspective
Measure quantum coherence using coincidence probability of two identical particles
vs
BS: 50:50 Beam splitter
8/16
Non-Equilibrium Heat Current in nonlinear quantum system
Reference: [1] New J. Phys. 17, 055013 (2015), [2] Europhys. Lett. 125, 20007 (2019)
[3] Phys. Rev. X 10, 021022 (2020)
9/16
Non-linearly interacting cavities coupled to (harmonic) baths
1. Spontaneous Four wave mixing (SFWM)
2. Cross Phase Modulation (XPM)
a,b: Bosons I: heat current
10/16
Heat current supplied to the system
I + I
1
2
SFWM
XPM
11/16
I - I
Net current from hot to cold
1
2
Steady current
12/16
Future Plan: Topology of non-equilibrium quantum system
13/16
Topological invariant in open system
Uhlmann number [1]
Ex) Baths coupled to the SSH chain periodically
Q1. What is the role of baths in topological invariant?
Reference: [1] Phys. Rev. Lett. 112, 130401 (2014), [2] Phys. Rev. Lett. 113, 076408 (2014),
[3] O. Viyuela, PhD. Dissertation (2016)
14/16
Topological invariant in open system
Closed system
Open system
Q2. Geometric phase in Non-equilibrium?
Reference: [1] Phys. Rev. A 71, 012331 (2005), [2] Phys. Rev A 73, 062101 (2006),
[3] Phys. Rev. X 6, 041031 (2016)
15/16
Summary
Q. Can we find intriguing physical effect using topological invariant?
16/16
Extra slides
Disorder average Green's function
Diagrammatic expansion
Dyson equation
Self energy
Scattering time from 1st order of self energy
Localization length
Disorder average Master equation
Liouville equation
Decompose average + fluctuation
Weak disorder approximation yields
Disorder averaged master equation in momentum space
Correlation function
Coincidence probability
Output state
Beam splitter
Coincidence probability
Nonlinear system coupled to (harmonic) baths
1. SFWM
2. XPM
n: number of particles in the system
Non-equilibrium Thermodynamics
Lindblad equation
1st law of non-equilibrium thermodynamics
Derivative of internal energy
Landauer's formula
(for 1 linear mode)
Heat current
Interaction Hamiltonian between system and bath
Topological invariant in open system
Closed System
Open System
State
Pure state
(Eigenstate of Hamiltonian)
Mixed state
Gauge
Field
U(1) Gauge (Abelian)
U(N) Gauge (Non-Abelian)
Sarandy-Lidar phase
Berry phase
Topological
invariant
Uhlmann phase
Berry Phase
Uhlmann Phase
Periodicity in Generator
Periodicity in Hamiltonian
Uhlmann Winding Number
Thermal phase transition happens in topological nontrivial phase (w > v)
Steady state of SSH chain periodically coupled to the bath
Figure from PRL 112, 130401 (2014)
Geometric phase in Non-equilibrium
Extract the phase from 'generator' -> Sarandy-Lidar phase
Uniqueness of steady state
Reference: M. S. Sarandy and D. A. Lidar PRA 71, 012331 (2005)
Su-Schrieffer-Heeger (SSH) chain under dissipation
In the absence of heat baths, assuming periodicity gives
Non-trivial phase gives edge state at the boundary
Future plan 1: Dissipative SSH Chain
Edge driven
Q. What is the difference in transport (and local current) in two difference phases?
(v > w: trivial phase , w > v: non-trivial)
-> How do we obtain dissipative symmetry protection in SSH chain?
Reference: PRE 94, 062143 (2016), Sci Rep 7, 6350 (2017)
PRA 98, 013628 (2018), PRL 124,040401 (2020)