Full distribution function of quantum noise: from interference experiments to string theory Full distribution function of quantum noise: from interference experiments to string theory Vladimir Vladimir Gritsev Gritsev Collaboration: Ehud Ehud Altman Altman - - Weizmann Weizmann Eugene Eugene Demler Demler - - Harvard Harvard Adilet Adilet Imambekov Imambekov - Harvard Yale Harvard Yale Anatoli Anatoli Polkovnikov Polkovnikov - - Boston Uni. Boston Uni.
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Full distribution function of quantum noise: from interference
experiments to string theory
Full distribution function of quantum noise: from interference
experiments to string theoryVladimir Vladimir GritsevGritsev
QA L a z a dz∫Interference amplitude contains information about fluctuations Interference amplitude contains information about fluctuations within each condensate.within each condensate.
2int int
2 † †1 1 2 1 2 2 1 2 1 20 0
( ) ( ) cos ( )
( ) ( ) ( ) ( )
Q
L L
Q
x y A Q x y
A a z a z a z a z dz dz
ρ ρ = −
∫ ∫
Interference of spatially extended condensates
Higher MomentsHigher Moments2 † †
1 1 1 2 2 1 2 2 1 20 0( ) ( ) ( ) ( )
L L
QA a z a z a z a z dz dz⎡ ⎤⎣ ⎦∫ ∫ is an observable is an observable quantum operatorquantum operator
22 †
1 2 1 1 1 20 0( ) ( )
L L
QA dz dz a z a z∫ ∫Identical condensates. Mean:Identical condensates. Mean:
Shot to shotShot to shot fluctuations fluctuations --quantum noisequantum noise22 † †
( )1/ 22 2 1/ 1/ , Interference contrast / KK KQ h hA L Lξ ξ− ∝
Repulsive bosons with short range interactions: Repulsive bosons with short range interactions: 2 2
2
Weak interactions 1
Strong interactions (Fermionized regime) 1
Q
Q
K A L
K A L
→
→
Finite temperature:Finite temperature:1 1/2
2 2 1K
Q hh
A Lm T
ξ ρξ
−⎛ ⎞⎜ ⎟⎝ ⎠
h
Angular Dependence.Angular Dependence.
† ( tan )int 1 20
†1 20
( ) ( ) ( ) c.c.
exp( ) ( ) ( ) +c.c., tan
L iQ x z
L iqz
x a z a z e dz
iQx a z a z e dz q Q
ϑρ
ϑ
−
−
+
= =
∫∫
( )2 † †1 1 1 2 2 1 2 2 2 1 1 20 0
( ) ( ) ( ) ( ) ( ) cos ( )L L
QA q a z a z a z a z q z z dz dz⎡ ⎤ −⎣ ⎦∫ ∫q is equivalent to the relative momentum of the two condensates q is equivalent to the relative momentum of the two condensates (always present e.g. if there are dipolar oscillations).(always present e.g. if there are dipolar oscillations).
ϑ
z
x(z 1
)
x(z 2
)
(for the imaging beam (for the imaging beam orthogonal to the orthogonal to the page, page, ϑϑ
is the angle of is the angle of
the integration axis the integration axis with respect to z.)with respect to z.)
ϑ
1D condensates at zero temperature:1D condensates at zero temperature:Low energy action:Low energy action:
ThenThen1/ 2
( ) †( ) , ( ) ( )K
i y hc ca y e a y a y
y yπφ ξρ ρ
⎛ ⎞⎜ ⎟⎜ ⎟−⎝ ⎠
%%
SimilarlySimilarly1/ 22
1 2 1 2† † 21 2 1 2
1 2 1 2 1 1 2 2
( ) ( ) ( ) ( )K
hc
y y y ya y a y a y a y
y y y y y y y yξ
ρ⎛ ⎞− −⎜ ⎟⎜ ⎟− − − −⎝ ⎠
% %% %
% % % %
Easy to generalize to all orders.Easy to generalize to all orders.
Changing open boundary conditions to periodic findChanging open boundary conditions to periodic find
( )2 1/ 2 1 1/ 22
nn K KQ c h nA C L Zρ ξ −≅
These integrals can be evaluated using Jack polynomials These integrals can be evaluated using Jack polynomials ((Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995))
Explicit expressions are cumbersome (slowly converging series Explicit expressions are cumbersome (slowly converging series of products).of products).
Weakly interacting BEC like regime.Weakly interacting BEC like regime.
Connection to the impurity in a Connection to the impurity in a LuttingerLuttinger liquid problem.liquid problem.Boundary SineBoundary Sine--Gordon theory:Gordon theory:
( )
( ) ( )( )2 2
0 0
exp ,
2 cos 2 (0, )2 x
Z D S
KS dx d g dβ β
τ
ϕ
π τ ϕ ϕ τ πϕ τ∞
−∞
= −
= ∂ + ∂ +
∫
∫ ∫ ∫
( )( )
21/ 2
22( ) , 2 ,!
nK
nn
xZ x Z x gn
β π κβ= =∑Same integrals as in the expressions forSame integrals as in the expressions for 2n
QA
2 20 00
( ) ( ) (2 / ) ,Z x W A I Ax A dA∞
= ∫1/ 2 1 1/ 2
0K K
c hA C Lρ ξ −=
P. Fendley, F. Lesage, H. Saleur (1995).
20 02 0
0
2( ) ( ) (2 / ) ,W A Z ix J Ax A xdxA
∞= ∫
( )Z x can be found using can be found using BetheBethe ansatzansatz methods for half integer K.methods for half integer K.
Then Then in principlein principle one can find one can find WW::Difficulties:Difficulties:
••have to do analytic continuation have to do analytic continuation ••the problem becomes increasingly harder as the problem becomes increasingly harder as K K increases.increases.
TBA approachTBA approach
Relation to PTRelation to PT--symmetric quantum mechanicssymmetric quantum mechanics
( ) ( )vacZ ix Q λ= sin / 2Kx πλπ
=
( )Q λ is the Baxter is the Baxter QQ--operator, related to the transfer matrix of operator, related to the transfer matrix of integrableintegrable conformal field theories with the central conformal field theories with the central charge c<1:charge c<1:
Yang-Lee singularity
2D quantum gravity,non-intersecting loops on 2D lattice
( )22 11 3
Kc
K−
= −
BLZBLZ theory: CFT, theory: CFT, KdVKdV , Baxter’s T, Baxter’s T--T and TT and T--Q relationsQ relations
Zinn-Justin, Bessis (92): real spectrum
Bender, Boettcher (98): real spectrum
Dorey, Tateo (99): real spectrum
FDF via FDF via spectral determinantspectral determinant of of SchrodingerSchrodinger equationequation
Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999);
Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001)
0
( ) 1n n
ED EE
∞
=
⎛ ⎞= −⎜ ⎟
⎝ ⎠∏
sin / 2Kx πλπ
=
for more details see V. Gritsev, et.al. Nature Physics (2006) (cond-mat/0602475), and cond-mat/0703766 (review)
2 4 22
( 1)( ) ( ) ( ) ( )Ky
l ly y y E yy
− +−∂ Ψ + + Ψ = Ψ
14 ; tan2 2
m dLl pK pt
θπ
= − =h
See section 5See section 5Bethe Ansatz -- ODE correspondence
2( ) ( )Z ix D ρλ=
Evolution of the distribution function for periodic boundary conditions
Interference of 1d condensates at finite temperature. Distribution function of the fringe contrast
S. Hofferberth, I. Lesanovsky, T. Schumm, J. Schmiedmayer, A. Imambekov, V. Gritsev, E. Demler arXiv:0710.1575
4. FDF and other problems
FDF concept: if it is difficult to find n-th order correlation functions, try to find FDF
• Spin-boson type problems (non-mean-field regimes, Vojta’07 ) via FDF
• Systems with quantum phase transition (e.g. talk by A. Lamacraft)• Spin systems with complicated order (e.g. topological)• Scattering of light in disordered media
• Turbulence(?)
• …..
( ) ( )zP t S t= ⟨ ⟩
Kane-Fisher (impurity in a Luttinger liquid) problem, FQHE edge tunneling
2 4 22
( 1)( ) ( ) ( ) ( )Ky
l ly y y E yy
− +−∂ Ψ + + Ψ = Ψ
DualityP. Fendley
H. Saleur
2
2
( )log
( )p
p
ZI V i T
Zμ
μπ μ
μ−
⎛ ⎞= + ∂ ⎜ ⎟⎜ ⎟
⎝ ⎠
1
2
ggg Tπμ λ
π
−⎛ ⎞= ⎜ ⎟⎝ ⎠ 4
gp iVTπ
= −
2 1/K g=
FCS can be computed from the spectral determinant( )χ θ
Mesoscopics
1D strongly coupled quantum optics (or plasmonics) (D. Chang)
• Strong coupling – mapping to the (anisotropic) Kondo model (LeClair)
• Kondo problem boundary SG• Evolution problem can be solved using BLZ
approach and correlation functions can be derived via spec. det. approach
Vortices and single columnar pin in a magnetic field
Ian Affleck, Walter Hofstetter, David R. Nelson, Ulrich Schollwock (04)
i FJL h− ∂
=∂
Ambegaokar-Eckern-Schoen type problems
( )Z D κ
Lukyanov (04-07)
5. FDF and AdS/CFT correspondence
FDF and Langlands duality
FDF
AdS/CFT Langlands duality
FDF AdS/CFT• Maldacena’ 97 (hep-th/9711200) Ν=4 SU(N) Yang-Mills at large N in d=4 (CFT) is dual to string
theory on AdS5 (whose weak coupling theory is gravity) Weak coupling gravity is dual to Strong coupling large N SQCD• Allows various generalizations: CFTd /AdSd+1; finite T;
non-CFT/non-AdS
• AdS with black hole (doubling of fields) dual to theory with Keldysh propagators for finite T => nonequilibrium FDF
0 0exp ( ) ( ) [ ]d
CFT GRS
x O x Zϕ ϕ⎛ ⎞
⟨ ⟩ =⎜ ⎟⎜ ⎟⎝ ⎠∫
Langlands correspondence (Grand Unification scheme of mathematics)
Number theory <-> representation
theory
Functions and operators <->geometry
QFT
Langlands(automorphic forms (70))
Taniyama- Shimura conj.
Fermat’s last theorem
Goddard, Nuyts,Olive
(77)
Kapustin: t’Hooft-Wilson
duality
Witten 05-…
?
Reviews: E. Frenkel, E. Witten, (DARPA project)
• ODE/BA correspondence is a particular case of the (geometric) Langlands duality (Feigin,Frenkel’07)
• Regular way of construction of ODE/BA correspondences
Langlands FDF
• 6-vertex model, XXZ spin chain, boundary SG
• Spin-j su(2) quantum spin chains and boundary parafermionic theories
• Perk-Schulz type models, hairpin boundary interaction
• Paperclip models
List of models for which FDF functions can be constructed via spectral
determinant of ODE
List of models for which FDF functions can be constructed via spectral
determinant of ODE (continuation, not complete)
• SU(n) vertex models
• SO(n) vertex models
• SP(n) vertex models
Questions
• AdS/CFT FDF ?• Langlands AdS/CFT ?• Measurement of FDF for non-cold-atoms