Download Long-Range Interactions and Entanglement

Efi Shahmoon, Igor Mazets, Gershon Kurizki
Taipei L1
Outline
- “Transmitting” vacuum forces:
Giant vdW/Casimir via transmission lines
- Long-range dipolar interactions in fiber-grating
Deterministic excitation transfer ( entanglement)
Laser-induced forces
 Highly nonlocal NL optics: EIT + dipolar forces
Geometry dependence of vacuum forces
Point dipoles: vacuum fluctuations + scattering
- quasistatic (vdW)
r  dipole : U  1 / r 6
- retarded (Casimir-Polder)
Dielectric/metallic mirrors:
Esc
r  dipole : U  1 / r 7
U  1/ d
1
3
Key point: interaction via virtual-photons
2 Evac
Evac 1
d
(from vacuum)
 v. photons propagation & scattering  space dependence
object geometry
environment geometry
Our approach: objects: dipoles (vdW), environment: 1d space!
Giant vdW & Casimir in 1d
2
1. Giant vdW/Casimir
Transmission Lines
TL Geometries:
(a) Coaxial cable
(b) coplanar waveguide
microstrips, …
a
f co  1 / a
Transverse modes: TEM + other modes with cutoffs
(e.g.
TEmn TM mn
for coax)
TEM mode  1d photon vacuum modes!
Dispersion:
k  c k
Mode function: u k ( r ) 
TEM mode  “free space” in 1d
e ikz
A( x, y ) L
1. Giant vdW/Casimir
Analytical approaches
Shahmoon, Mazets & Kurizki, PNAS 111, 10485 (2014)
z
TEM-mediated vdW/Casimir
a~
1. QED perturbation theory
D. Craig & T. Thirunamachandran, “Molecular QED” (1984)
2. Scattering of vac. fluctuations
P. Milonni, “The Quantum Vacuum” (1994)
Spruch & Kelsey, PRA (1978)
1. Giant vdW/Casimir
Interaction energy mediated by TEM mode
Both analytical approaches yield identical results
e
dipole:
e 
Ee
g
obtain:
U ( z) 
vdW regime:
F ( z )    16
1
a 4 e
2
hc
Ee
F ( z / e )
z  e
 z
ln
e  e
z
z
a~



U
fs
 1/ z6

Casimir-Polder (retarded) regime: z  e
1 e
F ( z) 
8 3 z 3
3
U
fs
 1/ z7

Shahmoon, Mazets & Kurizki, PNAS 111, 10485 (2014)
1. Giant vdW/Casimir
Comparison with free-space result
- TEM-mediated energy:
- free-space counterpart:
-
For
a  10 4 e
from
vdW regime
U
U fs
U ~ const.  z ln z
(e.g. circuit QED)
z  10 3 e :
z / e
Giant enhancement
w.r.t free-space
Free-space regime:
z  a
z
Casimir-Polder regime
U ~ 1/ z3
a~
- include transverse mode other then TEM
- Free-space result is restored
(e.g.
TEmn TM mn
(numerically verified for coax & metal waveguide)
E. Shahmoon, & G. Kurizki, PRA 87, 062105 (2013)
Shahmoon, Mazets & Kurizki, PNAS 111, 10485 (2014)
1. Giant vdW/Casimir
Possible realization: Circuit QED
TL: superconducting coplanar waveguide
Dipole: superconducting qubit/“transmon”
z
a ~ 5m
Measurement of interaction energy for a single pair of dipoles!
Observable: energy shift of dipole level
Typical parameters [1-3]:
Ee  2GHz
d/
A  1.35  10 -19 C
deph  0.05MHz
vdW limit
z  0.01e  1.5mm
U  28MHz
(level width)
[1] Shahmoon, Mazets & Kurizki, PNAS (2014)
[2] Baur et al. Phys. Rev. Lett. 108, 040502 (2012)
[3] Stojanovi’c et al. Phys. Rev. B 85, 05404 (2012)
U  deph
detectable!
Wallraff (ETH)
Casimir-Polder regime
z  2e  30cm
U  6.62 KHz
U  deph
very challenging
(with present tech.)
2. Long-range DDI
Coherent dipole-dipole interaction (DDI)
“incoherent” interaction atom 1 spontaneously emits photon to atom 2
- dissipative
atoms  
  Im G
g1
- probabilistic excitation exchange
- non-conservative forces (via scattering) diffusion
sin kr
- scaling: 3d:
1d: cos kz
r
e1
e2

g2
“coherent” interaction via evanescent/“near” fields, virtual photons
- dispersive
atoms  
  Re G
- deterministic (Hamiltonian) excitation exchange
H DD  (ˆ  ,1ˆ , 2  h.c.)
- conservative forces (potentials)
- scaling: 3d:
1 cos kr
;
r3
r

0’ Dell, Giovanazzi, GK, PRL (00’, 02’)
1d: sin k z
Goal: coherent & long-range interaction
1
  
2

2. Long-range DDI
Long-range coherent interactions via fiber-grating
Fiber-grating (FG) = 1d photonic crystal
bandgap
density k
of states 
(=DOS)

u
d
bandgap

DDI & scat. for  atom inside gap:
1
0.4
atomu
2

 ( ) 
4
6
8
10

k
1
1


 v group
( / u )  1
, 
   free exp( z /  )
1
1  atom / u
(due to DOS)
   free
Applications:
0.6
0.2
k
2
Close to bandgap edge:
0.8
atom  u
1. strong:
  1
  
1. Excitation exchange  long-range deterministic entanglement
2. Laser-induced inter-atomic force  non-extensive thermodyn.
2. long-range:
  1
Shahmoon & Kurizki, PRA 87, 033831 (2013)
Shahmoon & Kurizki, PRA 89, 043419 (2014)
Douglas et al. Nature Photon. (2015)
Shahmoon, Mazets & Kurizki, Opt. Lett. (2014)
2. Long-range DDI
DD interaction: non-Markovian theory
The theory: *Schrodinger Eq. for 2 atoms+waveguide vacuum
*Solved using Laplace transform
excited-level lifetime
Fixed distance z, a near cutoff:
11 
- interaction strength gets larger (divergence)
- population loss (incomplete decay)  less entanglement
co / 11  500 a / 11  490 z  a
1
in free-space
H DD  (ˆ  ,1ˆ , 2  h.c.)

1
1  (a / co ) 2
2. Long-range DDI
Inter-atomic forces: laser-induced DDI (LI-DDI)
LIDDI [1]:
U12 ~ I L1 2 Re[G ( L , r )]  
EL
1
Examples:
Free-space [2]
Re G ~ cos(k L r ) / r
Fiber-mediated [3]
Re G ~ sin(k L | z |)
 long-range interaction!
EL
2
(1)
Esc  G ( L , r )
photon Green’s function
tapered-fiber
(Rauschenbeutel, Kimble)
hollow-fiber
(Lukin, Gaeta)
Problem: spontaneous emission/scattering (e.g. Rayleigh)  diffusion
1
Rsc ~ I L 2 Im G  
2
Esc
Esc
(real )
(vir )
can have
U12  Rsc ?
   ?
[2] O’Dell, Giovanazzi, Kurizki & Akulin, PRL (2000)
[1] ES & Kurizki, PRA 89, 043419 (2014) [3] Chang, Cirac & Kimble, PRL 110, 113606 (2013)
3. Highly nonlocal NLO
EIT + DDI (dipolar int.) = nonlocal NLO
NLO=nonlinear optics
Fleischhauer & Lukin (2002)
photon
E-field
E
e
Atom excitation
 gd  g d g-d spin wave
dipolar interaction of atoms
interaction of photons

E
d
g
U ( z  z ' ) dd ( z ) dd ( z ' )
U ( z  z' )I ( z)I ( z' )
(for atoms in state d)
(I=intensity)
Friedler, Petrosyan,
Kurizki & Fleischhauer
(PRA 2005)
 NL optics with EIT: nonlocal Kerr effect
H NL   d z  dz U
' ( z  z' ) E( z) E( z' )
2
implications:
d  Rydberg U (z )  Extremely Strong NLO
2
Lukin (Harvard), Firstenberg (WIS)
Adams, Hofferberth, Rempe,…
In fibers: et al. (PRA 83, 2011)
U ( z )  modified long-range DDI interactions?
 Extremely nonlocal NLO
Shahmoon, Grisins, Stimming, Mazets & Kurizki,
arXiv:1412.8331;
Optica 3, 725 (2016)
3. Highly nonlocal NLO
Proposed scheme
1. Laser-induced DDI potential between atoms in
2. Use level
d
in an EIT scheme
 Nonlinear level shift and detuning:
na  atom density
Pd (z )  occupation of level d for atom at z
d
3. Highly nonlocal NLO
Resulting nonlocal nonlinear propagation
What propagates in the EIT medium?
EIT polariton = superposition of light and spin waves
Resulting propagation:

wave
frequency shift
- linear shift
- nonlinear shift (SPM)
dispersion (“mass”)
resulting from freq. shift
Nonlocal nonlinear shift:
Nonlocal + nonlinear dispersion 
 New regime of nonlinear optics!
Optica 3, 725 (2016)
Origin of nonlinear dispersion
frequency shift
3. Highly nonlocal NLO

 resulting from freq. shift
Quadratic dispersion of probe appears for  C  0 :
3. Highly nonlocal NLO
Predictions (1): optical roton
Long-range interaction via nano-waveguide + grating:
U ( z )   cos(k L z ) cos(


z )e  z / 
 ~ 3000L
Dispersion relation of wave excitations about a CW (Bogoliubov):
n p  Photon density of CW
U k  FT [U ( z )]
quadratic dispersion:
 c  0 : optical “anti-roton”
 c  0 : optical roton
blue: long-range int.
red: local int.
roton extrema at k R  k L   /   spatial ordering at 2 / k R !
3. Highly nonlocal NLO
Predictions (2): emergence of spatial self-order
Instability: imaginary dispersion peaked at k R  k L   / 
 pair generation: peaked around  k R
 emergence of order in field intensity with period ~ 2 / k R
I ( z / v)
delta-corr. field
local vs. nonlocal
L
photon detector
g ( 2) ~ I ( z / v) I ( z ' / v)
Green: intensity corr. at waveguide output
Other colors: intensity corr. at different stages t during propagation in waveguide
3. Highly nonlocal NLO
Instability: spectrum
ES, Grisins, Stimming, Mazets & Kurizki (in preparation)
Im. part of Bogoliubov spectrum
Mode occupation
N k  ak ak
Squeezing spectrum
3. Highly nonlocal NLO
Measuring the optical roton spectra
ES, Grisins, Stimming, Mazets & Kurizki (in preparation)
U k  FT [U ( z )]
n p  Photon density of CW
optical “anti-roton”
homodyne detection
2. Long-range DDI
Many-body physics with long-range interactions
E1   U ij
2
1
ij1
E2   U ij
ij2
interaction/”surface” energy
total energy
Etot   U ij
short-range int.
long-range int.
Etot  E1  E2  E12
ij
E12  E1 , E2
E12 ~ E1 , E2
Long-range interactions:
E12 
U
i1, j2
ij
Etot  E1  E2
Etot  E1  E2 non-additive!
 drU (r)  V
V
Stat. physics + long range int. (=Non-additive sys.)
 New concepts & predictions: - inequivalence of stat. ensembles
No (controlled) experiments!
- Negative specific heat
- Breaking of ergodicity
Mukamel, arXiv: 0905.1457
Campa, Dauxois & Ruffo, Phys. Rep. (2009)
Summary
g1
- Dipolar interactions in confined geometries
Confined geometries  modify modes  enhance dipolar int.
e2
k
e1
uk (r )
mode
g 2 function
simple idea, drastic consequences!
Esc  G (k , r )
(1)
1. Transmission lines: z  a  TEM “1d”-mode
 Giant Casimir effects in 1d  realization: circuit QED
2. Cutoffs & gaps in photon DOS
a
 enhanced dispersive int. ; inhibited dissipative effects
- Long-range deterministic entanglement generation
- Long-range laser-induced forces: nonlocal NL optics, non-extensive thermo.
3. Highly nonlocal NLO
Imperfections & scattering
1. Scattering
Rsc
due to laser-induced DDI
U
Bandgap: scat. only to non-guided  weak!
U  Rsc  Rsc
1
Rsc U  Rsc
2. Scattering from defects in WG
RD 
L
C  10

U
C
4
 ~ 3000L  RD ~ U
k ~ n pU

Douglas et al. Nature Photon. (2015)
single-atom effect
but our observable k is cooperative!
3. EIT absorption
2
1
1   atom / u
1. Giant vdW/Casimir
Analytical approaches
ES, I. Mazets & G. Kurizki, arXiv: 1304:2028 (2013)
1. QED perturbation theory
2. Scattering of vac. fluctuations
4th order energy correction for
G  g1, g2 , vac :
z
a~
(1d )
(1)
Uˆ   2 Eˆ vac ( z2 ) Eˆ sc ( z2 )
Find scattered field in 1d (TEM):
(1)
Eˆ ( z )
sc
Dipole model:
n 
g
Vacuum modes: TEM u k  e ikz
D. Craig & T. Thirunamachandran, “Molecular QED” (1984)
2
Averaging over vacuum:
U  vac Uˆ vac
P. Milonni, “The Quantum Vacuum” (1994)
Theory
QED Hamiltonian
TEM-mediated interaction energy
de 
dipole matrix element
1. Giant vdW/Casimir
2. Long-range DDI
Energy transfer: deterministic entanglement
Realization: nano-waveguide + Bragg grating
1. In bandgap: incoherent radiation to non-guided modes ~  free
2. @ bandgap edge: v group  0
 free

1  (a / u )
atoms:
a  780nm
 free  2  6.07
MHz
u /  free  10
8
u
d
0.6
0.2
k
 a u
2
4
6
8
10

z  20a
~ 16 m
0.6
C  0.9663
0.4
0.2
0
0
bandgap
0.8
0.4
 /
0.8
a /  free  108  2000
1 k

v g 
bandgap
1
|e population
Rb
  free strong
1
long-distance
1  (a / u )

87

tent ~ 1.8ns
0.05
0.1
t
0.15
0.2
Rauschenbeutel, Kimble
ES & Kurizki, PRA 87, 033831 (2013)
See also works by Darrick Chang
2. Long-range DDI
Mapping to 2d spin (XY) model
LI-DDI for may-atoms( size   ) : U12   cos[4 ( z1  z 2 ) / L ]
atom position
2d spin orientation
L / 2 L / 2 L / 2
 i  2
zi
L / 2
ordered state
z
 known spin (XY) model!
 slow relaxation in -canonical [*]
 c  (1 / 9)N 1.7
~ system size!
never observed!
ES, Mazets & Kurizki, Opt. Lett. 39, 3674 (2014)
[*] Yamaguchi et al., Physica A 337, 36 (2004)
2. Long-range DDI
Battle of relaxations
..
.
U
Dynamics:
i   A
  i  Af i
 i
-canonical (slow) relaxation
via: atomic “collisions” (LI-DDI)
vs.
to: equilibrium determined
by sys. initial state
time:  c ( N )  1 N 1.7~ system size!
9
For Cs atoms:
1
2
Rsc U  Rsc
canonical relaxation
via: photon scattering
to: equilibrium determined by
D
effective T
time:  c   1
Teff 
m
 c ( N )   c
 Can experimentally probe slow
relaxation!
 Probe this ensemb. inequivalence
Probe many-body effect
of long-range int.!
ES, Mazets & Kurizki, Opt. Lett. 39, 3674 (2014)
2. Long-range DDI
Mapping to spin model
ES, Mazets & Kurizki, arXiv:1309.0555 (2013)
- LIDDI + kinetic energy lead to ( z   ) :
U12  I L 2 cos 2 ( k L z )e  z / 
Rsc
zi , pi & m  position, momentum & mass of atom i
L  laser wavelength
1.7
 known XY model!  slow relaxation in -canonical [1]  c  (1 / 9)N
in our case:  
BUT:
m L
J 4
- Scattering  friction + diffusion = effective bath  canonical ensemble…
 Dynamics= XY model H + friction
i  
 + diffusion D (Langevin):
4 H
4
  
fi
mL  i
mL
[1] Yamaguchi et al., Physica A 337, 36 (2004)
f i (t ) f j (t ' )  2 D ij (t  t ' )
 & D : see Cohen-Tannoudji
free