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When Ms. Topology meets Mr. Bath

- Toward dissipation resistant transport -

JungYun Han

Center for Theoretical Physics of Complex Systems (PCS),

Institute for Basic Science (IBS),

University of Science and Technology (UST), Korea

2020. 09. 25 @ PCS IBS, Comprehensive exam

What makes quantum so special?

Topic

1/16

From https://Dilbert.com/

What makes quantum so special?

Classical bit

Quantum bit

Topic

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

Outline

Error-free quantum information

PhD. Journey

4/16

Topologically protected transport of light

  • Understand the effect of the topological phase in transport under fluctuation (disorder).
  • So far, disorder protected transport has been demonstrated for two or higher dimensional systems.

E.g. transport along boundary

Topologically-protected transport

  • Protect quantum coherence in the one-dimensional disordered system.
  • JY. Han, C. Gneiting, D. Leykam, Phys. Rev. B 99, 224201 (2019)
  • JY. Han, A. Sukhorukov, D. Leykam, Photonics Res. 399919 (to be published)

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

  • Enhancement of localization length

  • Dephasing suppression -> Still imperfect. Phase fluctuation exists.

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

  • Quantum coherence is protected in H-CROW as group velocity fluctuation is suppressed

8/16

Non-Equilibrium Heat Current in nonlinear quantum system

  • Comprehend quantum transport in thermal non-equilibrium for the nonlinear system.
  • Linear & Hubbard interaction cannot cool down the system.

Non-Equilibrium Heat transport in nonlinear Quantum system

  • Two photons hopping interaction can yield the transition of the current.
  • JY. Han, D. Leykam, J. Thingna, coming soon!

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)

  • Two photons oscillation
  • Non-Kerr type interaction

2. Cross Phase Modulation (XPM)

  • Number-number interaction
  • Kerr type interaction

a,b: Bosons I: heat current

10/16

Heat current supplied to the system

I + I

1

2

SFWM

XPM

  • Initial state: vacuum
  • System-Bath interaction: Dipole coupling
  • : Decay constant

  • Transition of current occurs in the presence of SFWM
  • Metastable state arises in the cooling phase

11/16

I - I

Net current from hot to cold

1

2

Steady current

  • New hopping channel opens up in the presence of SFWM

  • Strong non-linear interaction prevents the transport.

12/16

Future Plan: Topology of non-equilibrium quantum system

  • Understand quantum transport with topological phase in non-equilibrium

  • Can we define topological invariant in the dissipative non-equilibrium system?

Future plan: Topology of non-equilibrium quantum system

13/16

Topological invariant in open system

Uhlmann number [1]

Topological invariant in open system

Ex) Baths coupled to the SSH chain periodically

Q1. What is the role of baths in topological invariant?

  • In a closed system (e.g. SSH chain), we have symmetry protected topological phase.

  • Topological invariant in open system can be defined by preserving symmetry along with dissipators.

  • Topological invariant from the steady state: Uhlmann phase

  • What if we deviate temperatures at site Bs? -> Non-Equilibrium

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?

  • In a closed system, Hamiltonian can provide the information of topological invariant.

  • Can we define topological invariant from generator governing open system dynamics?

  • Sarandy-Lidar phase is a geometric phase obtained from the generator

  • Can we connect Uhlmann phase (from state) to the Sarandy-Lidar phase (from generator) in both equilibrium and non-equilibrium cases?

  • Can we describe the transition of topological invariant (Berry to Uhlmann) using the scheme?

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

Topic

  • We studied how topology protects quantum state against disorder

  • We studied the evolution of quantum state in the presence of heat bath

  • Future plan: Study topological invariant in thermodynamic non-equilibrium system.

Q. Can we find intriguing physical effect using topological invariant?

16/16

Extra slides

Topic

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

  • Diagonalize it by two particles basis
  • Energy per single excitation

2. XPM

  • Number operators are positive
  • Energy per single excitation

n: number of particles in the system

Non-equilibrium Thermodynamics

Lindblad equation

1st law of non-equilibrium thermodynamics

Derivative of internal energy

Topic

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

Topic

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

Topic

  • Berry phase can be extracted from Hamiltonian itself!

Extract the phase from 'generator' -> Sarandy-Lidar phase

  • Non-zero phase appears for degenerate and a kinked eigenstate of generator. How do we obtain non-trivial 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

  • v > w: Topological Trivial Phase (n = 0)

  • w > v: Topological Non-trivial Phase (n = 1)

Non-trivial phase gives edge state at the boundary

  • SJ. Park, J. Thingna, JY. Han (in preparation)

Future plan 1: Dissipative SSH Chain

Edge driven

  • Understand the topological thermal current in non-equilibrium

Q. What is the difference in transport (and local current) in two difference phases?

(v > w: trivial phase , w > v: non-trivial)

  • How can we find and define the topological invariant from the current?

-> 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)

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