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Controllinglightandma.erusingcoopera4veradia4on
PartI:Dickestates,coopera4veeffects,entanglement
SusanneYelin
UniversityofConnec4cutHarvardUniversity
Herrsching,March6,2019 �1
�2
Roy Glauber, 1925 - 2018
Goalsofthelectures
• Use2Darray…
�3
�4
• …toreflectlight…
• …emitincontrolledway…
Increase(impurity)crosssec4on?
Increase(impurity)crosssec4on?
• …edgestateswithphotons?
wikipedia
• 2DmaterialslikegrapheneorTMDs
• …toswitch(singlephotons)
• …toswitch(singlephotons)
Quantummirror:Refrac4onsuperposi4on
�10whyquantumphysicists.com
Why “cooperative effects”??
Coopera4veradia4on:superradiance
�12
For want of a better term, a gas which is radiating strongly because of coherence will be called “superradiant.”
— Robert H Dicke, 1954.
Dicke, Phys. Rev. 93, 99 (1954).
What is an atom?
levels
What is an atom?
levels
(sub)levels
How does it couple to light?
photon energy: hνatomic transition energy: Eup-Edown
hνEup
Edown
How does it couple to light?
photon energy: hνatomic transition energy: Eup-Edown
hνEup
Edown
time
inte
nsity
Cooperative effects in radiation
single atom
time
inte
nsity
Cooperative effects in radiation
two far-away atoms
time
inte
nsity
Cooperative effects in radiation
two close atoms
time
inte
nsity
Cooperative effects in radiation• Superradiance∝N2
duetoconstruc4veinterference
• Build-upof“collec4vedipoles”
many close atoms
Coopera4veEffects
• Twoimportantaspects:– collec4veeffects(manypar4cles)
�27
Coopera4veEffects
• Twoimportantaspects:– collec4veeffects(manypar4cles)
• e.g.,muchhigherchanceforphotonstointeractwithmanyatomsthanone
– “exchange”duetodipole-dipoleinterac4on• excita4onsareexchanged
• “Coopera4ve”ismorethanjust“collec4ve”!
�27
Collective effects: Example of quantized field
• Interaction Hamiltonian with classical light (Rabi frequency Ω)
• Interaction Hamiltonian with quantized light (coupling element g, annihilation operator a, number of atoms N)
!28
H/~ = g
pN�|eihg| a+ a
+ |gihe|�
H/~ = ⌦ (|eihg|+ |gihe|)
Coopera4veeffects
• simplestformof“exchangeinterac4on”
�29
Coopera4veeffects
• Tradi4onalexample:superradiance
�30
time
inte
nsity
Cooperative effects in radiation• Superradiance∝N2
duetoconstruc4veinterference
• Build-upof“collec4vedipoles”
many close atoms
Whatissuperinsuperradiance?
�32
man(Clark)
manymen(Clarks)
Whatissuperinsuperradiance?
�32
man(Clark)
manymen(Clarks)
SUPERMAN
Whatissuperinsuperradiance?
�32
man(Clark)
manymen(Clarks)
SUPERMAN
�33
Examples
Superradiantlaser
�34
Bohnet,Chen,Weiner,Meiser,Holland,Thompson,Nature484,78(2012)
JILA
#Atoms ≫ #Photons
⟱
much more coherent lasing
�35
Super- radiance in BECs
Inouye, Chikkatur, Stamper-Kurn, Stenger, Pritchard, KetterleScience 23 (1999)
momentum conservation
+interference of matter waves
�36Torres, Patrick, Coutant, Richartz, Tedford, WeinfurtnerNature Physics 13, 833 (2017).
Physics World
rotation energy amplifies waves if Ωrot > ωwave
Black hole superradiance
�37Pani, Brito, Cardoso, Class. Quantum Grav. 32 134001 (2015)
(basically same as for water waves)
spinning black hole….
… amplifies graviational waves
Subradiance
�38Guerin, Araujo, Kaiser, PRL 116, 083601 (2016)
pulseno pulse
Theselectures
• Coopera4veeffectsincomplexsystems
• Newapplica4on:atomicallythinmirrors
Theselectures
• Coopera4veeffectsincomplexsystems
• Newapplica4on:atomicallythinmirrors
‣ Collec4ve(Lamb)levelshifs
‣ Subradiance
‣ Entanglement
Theselectures
• Coopera4veeffectsincomplexsystems
• Newapplica4on:atomicallythinmirrors‣ Coopera4veresonances
‣ Applica4ons:
Theselectures
• Coopera4veeffectsincomplexsystems
• Newapplica4on:atomicallythinmirrors‣ Coopera4veresonances
‣ Applica4ons:
• topologywithphotons• nonlinearquantumop4cs
• Quantummetasurfaces
Superradiance:recentexperiment
Grimes, Coy, Barnum, Zhou, Yelin, Field, PRA 95, 043818 (2017)
Superradiance:recentexperiment
Grimes, Coy, Barnum, Zhou, Yelin, Field, PRA 95, 043818 (2017)
Ba
SuperradianceinBaRydbergs
typical superradiance
time profile
shift from red…
Collective shift
SuperradianceinBaRydbergs
typical superradiance
time profile …to blue
Collective shift
�45
|ee⇥ � |1, 1⇥
|gg⇤ ⇥ |1,�1⇤
Coopera4veradia4on:twoatoms
|egi |gei
atoms distinguishable
�47
|gg⇤ ⇥ |1,�1⇤
1⇤2
(|eg⇥+ |ge⇥) � |1, 0⇥ 1⌅2
(|eg⇤ � |ge⇤) ⇥ |0, 0⇤
atoms indistinguishable
Coopera4veradia4on:twoatoms
|ee⇥ � |1, 1⇥
�47
|gg⇤ ⇥ |1,�1⇤
1⇤2
(|eg⇥+ |ge⇥) � |1, 0⇥ 1⌅2
(|eg⇤ � |ge⇤) ⇥ |0, 0⇤
atoms indistinguishable
Coopera4veradia4on:twoatoms
|ee⇥ � |1, 1⇥
�47
|gg⇤ ⇥ |1,�1⇤
1⇤2
(|eg⇥+ |ge⇥) � |1, 0⇥ 1⌅2
(|eg⇤ � |ge⇤) ⇥ |0, 0⇤
atoms indistinguishable
Coopera4veradia4on:twoatoms
|ee⇥ � |1, 1⇥
destructive interference
�48
|ee⇥ � |1, 1⇥
|gg⇤ ⇥ |1,�1⇤
1⇤2
(|eg⇥+ |ge⇥) � |1, 0⇥
1⌅2
(|eg⇤ � |ge⇤) ⇥ |0, 0⇤
dipole-dipole exchange interaction
atoms indistinguishable
Coopera4veradia4on:twoatoms
�49dipole-dipole exchange interaction
|ee⇥ � |1, 1⇥
|gg⇤ ⇥ |1,�1⇤
1⇤2
(|eg⇥+ |ge⇥) � |1, 0⇥
1⌅2
(|eg⇤ � |ge⇤) ⇥ |0, 0⇤
atoms indistinguishable
Coopera4veradia4on:twoatoms
�51dipole-dipole exchange interaction
|ee⇥ � |1, 1⇥
|gg⇤ ⇥ |1,�1⇤
1⇤2
(|eg⇥+ |ge⇥) � |1, 0⇥
1⌅2
(|eg⇤ � |ge⇤) ⇥ |0, 0⇤
atoms indistinguishable
Coopera4veradia4on:twoatoms
exchange interaction:• usually dipole-dipole mediated• creates shift and broadening
(Kramers-Kronig) •
Whatis“superradiance”?
�52
1.EverythingthatinvolvesDickestates- (e.g.,collec4ve√Neffects,-bad-cavitylimit,
-…)
Dickestates
�53
Fullysymmetricstateofnexcita4onsinNpar4cles,forexample
N-par4cleDickestatesdecaywithuptoN2speedup
|2i4 =1p6
(|1100i+ |1010i+ |1001i+
|0110i+ |0101i+ |0011i)
Dickestates
• Ques4on:WhendowehaveasystemthatconsistsonlyofDickestates?
• Answer1:Whenthereexistsnomechanismtodis4nguishatoms.Example:
• Problem:interac4ons(dip-dip)drivesystemoutofpurelysymmetricstate!Howtodeal?
• (Answer2:Whentheexchangeinterac4onisinfinitelyhigh.Inthiscase,otherstatescannotbereached.Example:“many-bodyprotectedmanifolds”)
�54
λ
�55
Describingsuperradiance
�55
Describingsuperradiance
Useangularmomentumform:|J,mdenotessystemwithmexcita4ons,m=-J…J.
Angularmomentumstates
• Formof3-atomDickestate?
• Whataretheotherstates?
• 4atoms?
• 40?
�56
Popula4onofsymmetricstates
• Whatpossiblepathcouldasystemstar4ngin|11111…>take,
• withonlydecay?
• whenthereisdipole-dipoleinterac4on?
�57
Stateconnec4ons
�58
J =N
2J =
N
2� 1 J = 0
Spontaneous superradiant decay
Stateconnec4ons
�58
J =N
2J =
N
2� 1 J = 0
Spontaneous superradiant decay
dipole-dipole interaction
Formofdipole-dipoleinterac4on
�59
Hdip�dip =X
i 6=j
Vij �+i �
�j
“Exchange”term• Howwouldthisshowupinamasterequa4on?
�60
Formofdipole-dipoleinterac4on
�61
dipole-dipole interaction: ‣distance dependence‣angle dependence‣ real + imaginary part
real virtual photon exchange
flip-flop“exchange”
shift/dephasing
Atom-atomcorrela4onsinsuperradiance:Classicexample
•Superradiance λ
Gross,Haroche,Phys.Rep.93,301(‘82) �62
�63
inte
nsity
/ at
om
t/γ-1
Same parameters as before: C=10
with exchange
no exchange(“amplified spontaneous emission”)
Fleischhauer, Yelin, PRA 59, 2427 (99); Lin, Yelin, Adv. Atom. Mol. Opt. Phys. 61, 295 (2012)
Whatis“superradiance”?
�64
1.EverythingthatinvolvesDickestates- (e.g.,collec4ve√Neffects,-bad-cavitylimit,
-…)2.Onlysystemsinvolvingcoopera4ve(andnonlinear)effects
- i.e.,effectofexchangeinterac4on-morethansingleexcita4on
Whatis“superradiance”?
�64
1.EverythingthatinvolvesDickestates- (e.g.,collec4ve√Neffects,-bad-cavitylimit,
-…)2.Onlysystemsinvolvingcoopera4ve(andnonlinear)effects
- i.e.,effectofexchangeinterac4on-morethansingleexcita4on
only for purists
Fulldynamics(alldegreesoffreedomofatoms,fields)
Dynamicsofatomsindensemedia-Schwinger-Keldysh&DysonEq.
twoprobeatoms+
surroundingatoms
Twoatoms+field
effec4vetwo-atomdescrip4on�65
Fleischhauer, Yelin, PRA 59, 2427 (99); Lin, Yelin, Adv. Atom. Mol. Opt. Phys. 61, 295 (2012)
Fulldynamics(alldegreesoffreedomofatoms,fields)
Dynamicsofatomsindensemedia-Schwinger-Keldysh&DysonEq.
Gauss: fielddegreesoffreedom
TwoatomMasterequa4on
twoprobeatoms+
surroundingatoms
Twoatoms+field
effec4vetwo-atomdescrip4on
Twoatoms+field
Vprobe =�
i=1,2
piEiH � S = Te�i�
Rd�Vprobe(�)
effec4vetwo-atomdescrip4on
Fulldynamics(alldegreesoffreedomofatoms,fields)
H = Hatoms + Hfield ��
i=1,2
piEi
�65Fleischhauer, Yelin, PRA 59, 2427 (99); Lin, Yelin, Adv. Atom. Mol. Opt. Phys. 61, 295 (2012)
�66
∫ DDEi(t1)Ej(t2)
EE
trace out field degrees of freedom
�68
Canoneexpectsuperradiance?
Theimportantparameteris
n�2r
n:density,λ:wavelength,r:systemsize
op4caldepth
Lin, Yelin, Adv. Atom. Mol. Opt. Phys. 61, 295 (2012)
�68
Canoneexpectsuperradiance?
Theimportantparameteris
n�2r
n:density,λ:wavelength,r:systemsize
op4caldepth
Lin, Yelin, Adv. Atom. Mol. Opt. Phys. 61, 295 (2012)
or nλ3 ?
MasterEqua4on
�69
MasterEqua4on
�69
DDE(t)
EE
MasterEqua4on
�69
DDE(t)
EE
DDE1(t1)E2(t2)
EE
Newexperimentalsystems:example
•UltracoldRydbergatoms
(PhilGould,EdEyler,Uconn)…
…
40P
39S
38S
5S
39D
38D
5D
Rb
�70
Effec4vedecay4mesfrom40PintonS
principlequantumnumbern
τ eff/µs(in
verseEinsteinA)
vacuum
(DanielVrinceanu)
densegas
Invacuum:decayintolownisfavoredIndensegas:decayintohighnisfavored➱λlarge,nλ2rlarge!
superradiantdecay! �71
Effec4vedecay4mesfrom40PintonS
principlequantumnumbern
τ eff/µs(in
verseEinsteinA)
vacuum
(DanielVrinceanu)
densegas
Invacuum:decayintolownisfavoredIndensegas:decayintohighnisfavored➱λlarge,nλ2rlarge!
superradiantdecay! �71
long wavelengths are much favored ⇒
experiments with Rydberg, molecular,
microwave transitions likely involve
superradiance
Wang, et al.,PRA 75, 033802 (2007); Lin & Yelin Mol. Phys. 111, 1917 (2013).
ExperimentalProof!
�72Carr,Ri.er,Wade,Adams,Weatherill,PRL111,113901(2013)
lowdensity
highdensity
SuperradianceinRydbergsystems
start of measurement
experiment
time (µs)
nu
mb
er
of
Ryd
be
rg a
tom
s
theory
200
500
1000
1500
2000
2500
0 5 10 15
�73
Theselectures
• Coopera4veeffectsincomplexsystems
• Newapplica4on:atomicallythinmirrors
‣ Collec4ve(Lamb)levelshifs
‣ Subradiance
‣ Entanglement
‣ Collec4ve(Lamb)levelshifs
Decay dynamics
t/γ-1
excit
ed le
vel p
opul
atio
n fast
slow
cf. two atoms
�77
Subradiance?
super-radiance
sub-radiance
radiationtrapping
Subradiantstates
�78
J =N
2J =
N
2� 1 J = 0
Spontaneous superradiant decay
dipole-dipole interaction
�79
Transitions
excited state population
two-atomcoherence
Subradiance:Outlook
• Dynamicsofsubradiance=transi4ontomany-bodylocalizedstate?
➡ Create,manipulatelocaliza4on
• Engineeredsubradiancetocreatestablestateswithoutspontaneousdecay
➡ Create,stabilizemany-bodyentangledstate(Dissipa4venon-equilibriumphysics)
Collec4veLambshif“Lambshif”istheresultofinterac4onwiththevacuumfluctua4ons
Inthecaseofaltereddensityofstatesofthe“vacuum”(i.e.,thesurroundingspace),thevalueoftheshifchanges
Withahigh(superradiant)densityofradiators,thedensityofstatesinsidethemediumcanbeconsiderablyaltered
“Collec4veLambshif”Putnam, Lin, Yelin, arXiv:1612.04477
hasspontaneouspart....�ij(!) =
}2
~2
Zd⌧
DhE�i (t),E+
j (t + ⌧)iE
ei!⌧
�(ij)spont =
1
2⇡P
Zd!0 �ij(!0)
! � !0
⌦⇥E�,E+
⇤↵/
⌦⇥a, a†
⇤↵= 1
independentonnumberofphotons
Collective Shift
hasspontaneouspart....�ij(!) =
}2
~2
Zd⌧
DhE�i (t),E+
j (t + ⌧)iE
ei!⌧
�(ij)spont =
1
2⇡P
Zd!0 �ij(!0)
! � !0
⌦⇥E�,E+
⇤↵/
⌦⇥a, a†
⇤↵= 1
independentonnumberofphotons
Collec4
veLamb
Shif
Collective Shift
Collective Lamb Shift
Keaveney et al.,PRL 108, 173601 (2012), theory: Lin, Li, Yelin, in prep.
cavitywith variable
thickness
~ λ
~ λ/8
Collective Lamb Shift
Keaveney et al.,PRL 108, 173601 (2012), theory: Lin, Li, Yelin, in prep.
cavitywith variable
thickness
~ λ
~ λ/8
Collective Lamb Shift
Keaveney et al.,PRL 108, 173601 (2012), theory: Lin, Li, Yelin, in prep.
cavitywith variable
thickness
~ λ
~ λ/8
In Fig. 4 we plot the gradient of the line shift as afunction of cell thickness. For the Rb D2 resonance,!LL=N ¼ "2!"k"3, where we have used the relationshipbetween the dipole moment for the s1=2 ! p3=2 transit-
ion and the spontaneous decay rate, d ¼ffiffiffiffiffiffiffiffi2=3
phLe ¼
1jerjLg ¼ 0i (see Ref. [27]). We extract the collisional
shift by comparing the data to Eq. (8) with !col the onlyfree parameter. The amplitude and period of the oscillatorypart are fully constrained by Eq. (7). We find the collisionalshift to be !col=2! ¼ ð"0:25$ 0:01Þ & 10"7Hzcm3,similar to previous measurements on potassium vapor[24]. In this high density limit, the collisional shift is alsoindependent of hyperfine splitting. The solid line is theprediction of Eq. (7), and the agreement between themeasured shifts and the theoretical prediction is remark-able (the reduced "2 for the data is 1.7). As well asmeasuring the thickness dependence of the cooperative
Lamb shift, our data also provide a determination of theLorentz shift which can only be measured in the limit ofzero thickness. An important advance on previous studies[8] is that the results clearly show the oscillations in theshift versus the thickness which arises due to the relativephase of the reradiated dipolar field.The demonstration of the cooperative Lamb shift and
coherent dipole-dipole interactions in media with thickness'#=4 opens the door to a new domain for quantum optics,analogous to the strong dipole-dipole nonlinearity inblockaded Rydberg systems [29,30]. As the cooperativeLamb shift depends on the degree of excitation [5], exoticnonlinear effects such as mirrorless bistability [31,32] arenow accessible experimentally. In addition, given thefundamental link between the cooperative Lamb shift andsuperradiance, subquarterwave thickness vapors offer anattractive system to study superradiance in the smallvolume limit. Finally, we note that the measured coopera-tive Lamb shift is the average dipole-dipole interactionfor a homogeneous gas which contains both positive andnegative contributions. It could therefore be enhanced byeliminating directions that contribute with the undesiredsign, for example, by patterning the distribution of dipoles.These topics will form the focus of future research.We would like to thank M. P. A. Jones for stimulating
discussions. We acknowledge financial support fromEPSRC and Durham University.
*[email protected][1] W. E. Lamb and R. C. Retherford, Phys. Rev. 72, 241
(1947).[2] P. Goy, J.M. Raimond, M. Gross, and S. Haroche, Phys.
Rev. Lett. 50, 1903 (1983).[3] W. Jhe, A. Anderson, E. A. Hinds, D. Meschede, L. Moi,
and S. Haroche, Phys. Rev. Lett. 58, 666 (1987).[4] J. Eschner, C. Raab, F. Schmidt-Kaler, and R. Blatt,
Nature (London) 413, 495 (2001).[5] R. Friedberg, S. R. Hartmann, and J. T. Manassah, Phys.
Rep. 7, 101 (1973).[6] M.O. Scully and A.A. Svidzinsky, Science 328, 1239
(2010).[7] M. Gross and S. Haroche, Phys. Rep. 93, 301 (1982).[8] W.R. Garrett, R. C. Hart, J. E. Wray, I. Datskou, and
M.G. Payne, Phys. Rev. Lett. 64, 1717 (1990).[9] R. Rohlsberger, K. Schlage, B. Sahoo, S. Couet, and
R. Ruffer, Science 328, 1248 (2010).[10] S. Briaudeau, S. Saltiel, G. Nienhuis, D. Bloch, and
M. Ducloy, Phys. Rev. A 57, R3169 (1998).[11] D. Sarkisyan, D. Bloch, A. Papoyan, and M. Ducloy, Opt.
Commun. 200, 201 (2001).[12] G. Dutier, A. Yarovitski, S. Saltiel, A. Papoyan,
D. Sarkisyan, D. Bloch, and M. Ducloy, Europhys. Lett.63, 35 (2003).
[13] G. Dutier, S. Saltiel, D. Bloch, and M. Ducloy, J. Opt. Soc.Am. B 20, 793 (2003).
FIG. 3 (color online). Shift of resonance lines with density.Measured shift of resonance lines with density and fit to thelinear, high density region for ‘ ¼ 90 nm (red squares, dashedline) and ‘ ¼ 250 nm (blue circles, solid line).
FIG. 4 (color online). Experimental verification of the coop-erative Lamb shift. The gradient of the dipole-dipole shift !dd=Nis plotted against cell thickness ‘. A collisional shift !col=2! ¼"0:25& 10"7 Hz cm3 has been subtracted. The solid black lineis Eq. (7) with no other free parameters. The conversion betweenexperimental and universal units is outlined in the main text.
PRL 108, 173601 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending
27 APRIL 2012
173601-4
shift
/ "
layer thickness
Collective Lamb Shift
�86
Optical depth = lowest
Optical depth = highest
In Fig. 4 we plot the gradient of the line shift as afunction of cell thickness. For the Rb D2 resonance,!LL=N ¼ "2!"k"3, where we have used the relationshipbetween the dipole moment for the s1=2 ! p3=2 transit-
ion and the spontaneous decay rate, d ¼ffiffiffiffiffiffiffiffi2=3
phLe ¼
1jerjLg ¼ 0i (see Ref. [27]). We extract the collisional
shift by comparing the data to Eq. (8) with !col the onlyfree parameter. The amplitude and period of the oscillatorypart are fully constrained by Eq. (7). We find the collisionalshift to be !col=2! ¼ ð"0:25$ 0:01Þ & 10"7Hzcm3,similar to previous measurements on potassium vapor[24]. In this high density limit, the collisional shift is alsoindependent of hyperfine splitting. The solid line is theprediction of Eq. (7), and the agreement between themeasured shifts and the theoretical prediction is remark-able (the reduced "2 for the data is 1.7). As well asmeasuring the thickness dependence of the cooperative
Lamb shift, our data also provide a determination of theLorentz shift which can only be measured in the limit ofzero thickness. An important advance on previous studies[8] is that the results clearly show the oscillations in theshift versus the thickness which arises due to the relativephase of the reradiated dipolar field.The demonstration of the cooperative Lamb shift and
coherent dipole-dipole interactions in media with thickness'#=4 opens the door to a new domain for quantum optics,analogous to the strong dipole-dipole nonlinearity inblockaded Rydberg systems [29,30]. As the cooperativeLamb shift depends on the degree of excitation [5], exoticnonlinear effects such as mirrorless bistability [31,32] arenow accessible experimentally. In addition, given thefundamental link between the cooperative Lamb shift andsuperradiance, subquarterwave thickness vapors offer anattractive system to study superradiance in the smallvolume limit. Finally, we note that the measured coopera-tive Lamb shift is the average dipole-dipole interactionfor a homogeneous gas which contains both positive andnegative contributions. It could therefore be enhanced byeliminating directions that contribute with the undesiredsign, for example, by patterning the distribution of dipoles.These topics will form the focus of future research.We would like to thank M. P. A. Jones for stimulating
discussions. We acknowledge financial support fromEPSRC and Durham University.
*[email protected][1] W. E. Lamb and R. C. Retherford, Phys. Rev. 72, 241
(1947).[2] P. Goy, J.M. Raimond, M. Gross, and S. Haroche, Phys.
Rev. Lett. 50, 1903 (1983).[3] W. Jhe, A. Anderson, E. A. Hinds, D. Meschede, L. Moi,
and S. Haroche, Phys. Rev. Lett. 58, 666 (1987).[4] J. Eschner, C. Raab, F. Schmidt-Kaler, and R. Blatt,
Nature (London) 413, 495 (2001).[5] R. Friedberg, S. R. Hartmann, and J. T. Manassah, Phys.
Rep. 7, 101 (1973).[6] M.O. Scully and A.A. Svidzinsky, Science 328, 1239
(2010).[7] M. Gross and S. Haroche, Phys. Rep. 93, 301 (1982).[8] W.R. Garrett, R. C. Hart, J. E. Wray, I. Datskou, and
M.G. Payne, Phys. Rev. Lett. 64, 1717 (1990).[9] R. Rohlsberger, K. Schlage, B. Sahoo, S. Couet, and
R. Ruffer, Science 328, 1248 (2010).[10] S. Briaudeau, S. Saltiel, G. Nienhuis, D. Bloch, and
M. Ducloy, Phys. Rev. A 57, R3169 (1998).[11] D. Sarkisyan, D. Bloch, A. Papoyan, and M. Ducloy, Opt.
Commun. 200, 201 (2001).[12] G. Dutier, A. Yarovitski, S. Saltiel, A. Papoyan,
D. Sarkisyan, D. Bloch, and M. Ducloy, Europhys. Lett.63, 35 (2003).
[13] G. Dutier, S. Saltiel, D. Bloch, and M. Ducloy, J. Opt. Soc.Am. B 20, 793 (2003).
FIG. 3 (color online). Shift of resonance lines with density.Measured shift of resonance lines with density and fit to thelinear, high density region for ‘ ¼ 90 nm (red squares, dashedline) and ‘ ¼ 250 nm (blue circles, solid line).
FIG. 4 (color online). Experimental verification of the coop-erative Lamb shift. The gradient of the dipole-dipole shift !dd=Nis plotted against cell thickness ‘. A collisional shift !col=2! ¼"0:25& 10"7 Hz cm3 has been subtracted. The solid black lineis Eq. (7) with no other free parameters. The conversion betweenexperimental and universal units is outlined in the main text.
PRL 108, 173601 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending
27 APRIL 2012
173601-4
shift
/ "
layer thickness
Keaveney et al.,PRL 108, 173601 (2012), theory: Lin, Li, Yelin, in prep.
�87
hasspontaneouspart....�ij(!) =
}2
~2
Zd⌧
DhE�i (t),E+
j (t + ⌧)iE
ei!⌧
�(ij)spont =
1
2⇡P
Zd!0 �ij(!0)
! � !0
...and“s4mulated”part
�ij(!) =}2
~2
Zd⌧
DDE�i (t)E+
j (t + ⌧)EE
ei!⌧
�(ij)stim =
1
2⇡P
Zd!0 �ij(!0)
! � !0
dependentonnumberofphotons
⌦⌦E�E+
↵↵ /⇠⌦⌦
a† a↵↵
= n
Collective Shift
�87
hasspontaneouspart....�ij(!) =
}2
~2
Zd⌧
DhE�i (t),E+
j (t + ⌧)iE
ei!⌧
�(ij)spont =
1
2⇡P
Zd!0 �ij(!0)
! � !0
...and“s4mulated”part
�ij(!) =}2
~2
Zd⌧
DDE�i (t)E+
j (t + ⌧)EE
ei!⌧
�(ij)stim =
1
2⇡P
Zd!0 �ij(!0)
! � !0
dependentonnumberofphotons
⌦⌦E�E+
↵↵ /⇠⌦⌦
a† a↵↵
= n
Induced
shif
Collective Shift
�88
Collective Shift: decay of inverted TLS
Putnam, Lin, Yelin, arXiv:1612.04477
Theselectures
• Coopera4veeffectsincomplexsystems
• Newapplica4on:atomicallythinmirrors
‣ Collec4ve(Lamb)levelshifs
‣ Subradiance
‣ Entanglement‣ Entanglement
�90
SuperradianceandEntanglement
Does(Dicke)superradianceneed/createentanglement?
NO
Wolfe,Yelin,PRL112,140402(’14)
Example:2-atomDicke
• PPT(Peres-Horodecki)criterion:
Eigenvaluesofpar4alposi4vetranspose≥0
�91
⇢ =X
ijkl
pijkl|iihj|⌦ |kihl|
⇢PPT =X
ijkl
pijkl|iihj|⌦ |lihk|
H Barnum, E Knill, G Ortiz, R Somma, L Viola Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: Quantification, sharing structure, and decoherence G Adesso, A Serafini, F Illuminati Energy and multipartite entanglement in multidimensional and frustrated spin models O Gühne, G Toth Multipartite entanglement in four-qubit cluster-class states YK Bai, ZD Wang Exact and asymptotic measures of multipartite pure-state entanglement CH Bennett, S Popescu, D Rohrlich, JA Smolin Geometric measure of entanglement and applications to bipartite and multipartite quantum states TC Wei, PM Goldbart Scalable multiparticle entanglement of trapped ions H Häffner, W Hänsel, CF Roos, J Benhelm, M Chwalla
�92
Howtodefine/calculatemany-par4cleentanglement?
SuperradianceandEntanglement
~ 10,000 for “definitio
n of
multipartit
e entanglement”
�93
SuperradianceandEntanglement
Does(Dicke)superradianceneed/createentanglement?(Ini4alstate:noentanglement)
Dickesuperradiant4meevolu4on
separablestates=
construc+veproof
Wolfe,Yelin,PRL112,140402(’14)
Dickesuperradiantstates
separablestates
�94
SuperradianceandEntanglement
=
Does(Dicke)superradianceneed/createentanglement?(Ini4alstate:noentanglement)
Wolfe,Yelin,Wolfe,Yelin,PRL112,140402(’14)
Dickesuperradiantstates
separablestates
�94
SuperradianceandEntanglement
oursystem:mixedstateofN-atomDickestateswithN+1knownindependent
coefficientspi
comparetomixtureofsymmetricproductstatesofN(two-level)atoms(needs
N+1coefficientsyi)
=
(N+1)-dim.equa4onsystem
Does(Dicke)superradianceneed/createentanglement?(Ini4alstate:noentanglement)
Wolfe,Yelin,Wolfe,Yelin,PRL112,140402(’14)
Formofequa4ons
• GeneralDickestates:
• Separablediagonallysymmetric:
�95
�96
SuperradianceandEntanglement
oursystem:mixedstateofN-atomDickestateswithN+1knownindependent
coefficientspi
comparetomixtureofsymmetricproductstatesofN(two-level)atoms(needs
N+1coefficientsyi)
=
(N+1)-dim.equa4onsystem
condi4
on:
allcoeffi
cients
0≤pi≤
1
Does(Dicke)superradianceneed/createentanglement?(Ini4alstate:noentanglement)
Wolfe,Yelin,PRL112,140402(’14)
�97
SuperradianceandEntanglement
oursystem:mixedstateofN-atomDickestateswithN+1knownindependent
coefficientspi
comparetomixtureofsymmetricproductstatesofN(two-level)atoms(needs
N+1coefficientsyi)
=
(N+1)-dim.equa4onsystem
condi4
on:
allcoeffi
cients
0≤pi≤
1
!
Does(Dicke)superradianceneed/createentanglement?(Ini4alstate:noentanglement)
Wolfe,Yelin,PRL112,140402(’14)
(works
!)
�98
SuperradianceandEntanglement
Drivensuperradiantsystem:
• Drivingalonedoesnotentangleatoms
• Superradiancealonedoesnotentangleatoms
• Drivingandsuperradiancetogetherentangleatoms!
FuzzyBunny?
�99
SpinSqueezing
• Correlated(“squeezed”)spinscouldimproveresolu4oninonedirec4on(“quadrature”).
�100
(Spin)Squeezing
• Howtomeasuresqueezing/measurementimprovement?
�101Kitagawa, Ueda, PRA 47, 5138 (93)
Spinsqueezing
Oldproblem:Howtoimprovemetrologybyspinsqueezingensembles
➡GroupsofBigelow,Kuzmich,Lewenstein,Mølmer,Polzik,Sanders,Sørensen,Vule4c,Wineland,…
�102
SpinSqueezing
• Correlated(“squeezed”)spinscouldimproveresolu4oninonedirec4on(“quadrature”).
�103Kitagawa, Ueda, PRA 47, 5138 (93)
(Spin)Squeezing
• Howtomeasuresqueezing/measurementimprovement?
�104(J1, J2, are uncertainties in the two directions orthogonal to the total spin J)
SuperradiantSpinSqueezing
�105
Out[31]=
N= 5 10 15 20
0 1 2 3 4 5 60.40.450.50.550.60.650.70.750.80.850.90.951.
W
x2
(driving field)Wolfe, Yelin, arXiv:1405.5288, González Tudela, Porras, PRL 110, 080502 (’13)
SuperradiantSpinSqueezing
�106(driving field)Wolfe, Yelin, arXiv:1405.5288, González Tudela, Porras, PRL 110, 080502 (’13)
BestcaseforDickeensemble
�107Wolfe, Yelin, arXiv:1405.5288, González Tudela, Porras, PRL 110, 080502 (’13)
Whataboutrealis4csystems?
• Dicke:3parameters(N,Γ,Ω)
• Realis4csystems:(OD,rel.density,Γ,Ω,γij,Δ,δij)
• Isitpossibletofindparallels?
• minimize“incoherent”aspects?
➡key:spontaneousdecay,shifinsteadofinduced!
�108
Spinsqueezinginrealis4csystems?
�109
Realistic Case (with dipole-dipole, re-absorption/re-
�110
Realistic Case (with dipole-dipole, re-absorption/re-
Out[31]=
N= 5 10 15 20
0 1 2 3 4 5 60.40.450.50.550.60.650.70.750.80.850.90.951.
W
x2Spinsqueezinginrealis4csystems?
Thank you!
�112