Download Quantum number

Quantum simulation of 2D topological
physics in a 1D array of optical cavities
Xi-Wang Luo (罗希望)
Work with Zheng-Wei Zhou (周正威) &
Xingxiang Zhou (周幸祥)
Key lab of Quantum Information, USTC
冷原子物理博士生论坛 长春
Outline
• Introduction
– Orbital angular momentum of photon, degenerate
cavity
• 1D simulator
– Cavity chain, 2D gauge field
• 2D physics
– QHE, edge state, topological phase transition
• Summary
Orbital angular momentum (OAM)
Helical phase
Wave front
Spatial light
modulator
e
 il
Laguerre-Gaussian modes
p: radial mode index
Vertical field :
e
 ikz
Quantum
number l :
OAM
Unlimited
Science 340, 28 (2013)
Our idea: Quantum simulation.
Gauge field & Topological
physics
How?
Science 338, 2 (2012)
Degenerate cavity
Degenerate condition:
Unit ray matrix
E1 ( x1 , y1 )   G( x1, y1; x0 , y0 ) E0 ( x0 , y0 )
G depends on : ray matrix [A,B;C,D]
Delta function : Unit ray matrix
Resonance condition: kL  2n
Optical design
A single cavity
1D cavity array
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•Bloch theory
•Periodic & linear
iK y 
a j ,l 1  e
a j ,l ; a j 1,l  eiK x  a j ,l
•Transfer matrix
M BS
i
r

 it
 r
it 
 
r

i

r 
•Tight binding approach
 j ,l ; j 'l '   dr
  ei 2x
0
 (r )   0 (r  R j ',l ' )  E *j ,l E j ',l '
2
Effective optical circuit
Dispersion relation
0 | r |2
  0  2 cos(K x -2x )+cos(K y -2 y )  ,  
2
Hamiltonian
Abelian gauge field
Ax , y  2x , y
Gauge potential (field)
 y  j0
x  l0
•Non-Abelian gauge field
•Polarization--spin
•Wave plate
e
iAx , y
Ax , y  x , y  1, 2
2D physics
Butterfly
Edge state
0
E
Quantum spin Hall effect
Quantum Hall effect
e2
 xy  C
h
Probing scheme
Driving-damping dynamic
Input-output
  diag{ 1,  2 ,,  n ,}
Photonic transmission
System spectrum & edge state
Butterfly spectrum
Ttot 
j ',l ' 2
|
T
 j ,l 0 |
Cylindrical boundary
No backscattering
j ',l ', j
Unidirectional
Abelian
  2.2
  1
  2.2
Edge state transport
OAM distribution
Connection ?
Chern number
OAM displacement
le 

j ',l ' 2
l
'
|
T
 j ,0 |
jedge j ',l '
Cylindrical boundary
Quantum number k y
In the gap—edge state
Chern number
le   sgn( vm )  Ctot
m
Error bar: ,  , 
0  1 / 20
Non-Abelian
Abelian
0  1 / 6
Topological quantum phase transition
Band structure
0  0,0.075,0.125
0  1.5
Topological quantum phase transition
Polarized edge state
Topological insulator
0  0
  1.6
Normal insulator
0  0.125
Measurement of Chern number
Chern number
C
1


dk
dk


A
(k x , k y ) 
x
y  k

z
2 i
A  u  k u
u  u0 ,..., uq1 
umq q (kx , k y )  umq (kx , k y )
Gauge transformation
e
if ( k x , k y )
u
C is invariant
Global Stokes theorem ?
 
1
C
dk  A( k x , k y )

2i
C Always Zero
Brillouin zone : no boundary
However，the phase of
Bloch wave is not well
defined globally over
the Brillouin zone –
nontrivial topology.
No global Stokes theorem!
Measurement of Chern number
e
if ( k x , k y )
u
0  1 / 6
1
Regime B1, u0 is real.
1
Regime B2, u3 is real.
Phase mismatch at edge:
u
i ( kx ,k y )
B1
e
u
B2
Separate Stokes therom
u13  0 u10  0
Measurement of Chern number
j , ql l
T (k x , k y , lq )  F T0,0 q 
T (k x , k y )  i k x , k y E

  E  i / 2
E 0,0
u  T (k x , k y , 0),..., T (k x , k y , q  1) 
| T0,0j ,l |2
  arg T (k x , k y ,3)   arg T (k x , k y , 0) 
summary
• 2D physics in 1D simulator
– OAM
• Connection between transmission coefficient
and topological invariant
– Chern number
• Topological phase transition
– QSH
More detail : Nature Communications 6, 7704 (2015).
Thanks for your attention !
谢谢！