ramsey interference as a probe of - harvard...
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Ramsey interference as a probe ofRamsey interference as a probe of “synthetic” condensed matter systems
Takuya Kitagawa (Harvard)Di Ab i (H d)Dima Abanin (Harvard)Mikhael Knap (TU Graz/Harvard)E D l (H d)Eugene Demler (Harvard)
Supported by NSF, DARPA OLE, AFOSR MURI, ARO MURI
Ramsey interferencep/2l
p/2lpulse pulse
1
t0
Ramsey interference in atomic clocks
Atomic clocks and Ramsey interference:Working with N atoms improves the precision by .
y
Using interactions to improve the clock precisionKitagawa, Ueda, PRA 47:5138 (1993)
Classical BECSingle modeSingle mode approximation
Ramsey interference as a probe of 1d dynamics
Interaction induced collapse of Ramsey f i se
y fri
nge
ityfringes. timeR
ams
visi
bili
Only weak spin echo was observed in experiments
Experiments:Widera et al. PRL (2008)PRL (2008)
Decoherence of Ramsey fringes due to many-body dynamics of low dimensional BECsof low dimensional BECsNew probe of dynamics of 1d many-body systems
This talk:This talk:Exploring “synthetic” condensed matter systemswith Ramsey interferencewith Ramsey interference
1. Measuring topological properties of band structures. Berry and Zak phases
2. Orthogonality catastrophe
Measuring topological propertiesMeasuring topological properties of band structures. Berry and Zak phasesBerry and Zak phases
Topological aspects of band structures
All evidence is indirect:(transport, edge states)Can there be a direct measurement ofBerry phase of Bloch states ?
Example of band structure with Berry phase.H l ( h ) l iHexagonal (graphene) lattice
Tight binding model on a hexagonal lattice
Dirac fermions in optical lattices, Tarruell et al., Nature 2012
Berry phase in hexagonal lattice
• Eigenvectors lie in the XY plane• Aro nd each Dirac point eigen ector• Around each Dirac point eigenvector makes 2p rotation
• Integral of the Berry phase is p
How to measure Berry phase in hexagonal latticeNaïve approach:Naïve approach:
Move atom on a closed trajectoryaround Dirac pointMeasure accumulated phase
Problems with this approach: Need to move atom on a complicated curved trajectoryNeed to separate dynamical phase
Integral of the Berry phase is only well defined on a closed trajectory
From Berry phase to Zak phaseIntegral of the Berry phase is only well defined on a closed trajectory
gauge invariantintegral of
Brillouin zone is a torus There are two types of closed trajectories
is not gauge invariantC Berry curvature
Brillouin zone is a torus. There are two types of closed trajectories
How to measure Zak phase usingR i fRamsey interference sequence
Two hyperfine spin states experience the same optical potentialyp p p p p
AdvantagesRequires only straight trajectoryDynamical phase cancels between two spin states
How to measure Zak phase usingRamsey interference sequenceRamsey interference sequence
Dynamically induced topological phases in a hexagonal lattice T Kitagawa et alin a hexagonal lattice T. Kitagawa et al.,
Phys. Rev. B 82, 235114 (2010)
Dynamically induced topological phases in a hexagonal latticein a hexagonal latticeFloquet spectrum on a strip
Edge states indicate theappearance of topologicallynon-trivial phases
Equivalent to Haldane model
Detection of topological band structure
Detection of topological band structure
Zak phase as a probe of band topologyd i1d version
Su‐Schrieffer‐Heeger model of polyacetylene
Analogous to bichromatic optical lattice potential
LMU/MPQ
Dimerized model
A B A AB
Topology of the band shows up in the winding of the eigenstateZak phase is equal to p
Characterizing SSH model using Zak phase T h fi i i h i l i lTwo hyperfine spin states experience the same optical potential
a
p/2a-p/2a 0
Zak phase is equal to p
Orthogonality catastrophe
Single impurity problems in condensed matter physics
Rich many-body physics
a e p ys cs
-Edge singularities in the
X-ray absorption spectra-Kondo effect: entangled
state of impurity spin and y p p(exact solution of non-equilibrium
many-body problem)
state of impurity spin and
fermions
Influential area, both for methods (renormalization group) and for strongly correlated materials
Probing impurity physics in solids is limited -Many unknowns;Simple models hard to test( l d b d k(complicated band structure, unknown
impurity parameters, coupling to phonons)
Probing impurity physics in solids is limited -Many unknowns;Simple models hard to test( l d b d k(complicated band structure, unknown
impurity parameters, coupling to phonons)
X-ray absorption in Na
Probing impurity physics in solids is limited -Many unknowns;Simple models hard to test( l d b d k(complicated band structure, unknown
impurity parameters, coupling to phonons)
-Limited probesLimited probes(usually only absorption spectra)
-Dynamics beyond linear response
out of reach (relevant time scales GHz-THz,
i t ll diffi lt)X-ray absorption in Na
experimentally difficult)
Cold atoms: new opportunities for studying impurity physics
-Parameters known
pu y p ys cs
allow precise tests of theoryexact solutions available
-Tunable by the Feshbach resonanceresonance
f (k) a1 ika
-Fast control of microscopic parameters(compared to many-body scales)
-Rich toolbox for probing many-body states
Cold atoms: new opportunities for studying impurity physics
-Parameters known
pu y p ys cs
allow precise tests of theoryexact solutions available
New challenges for theory:
Describe dynamics
-Tunable by the Feshbach resonance
-Describe dynamics-Characterize complicated transient many-body states
resonancef (k) a
1 ika-Explore new many-body phenomena
-Fast control of microscopic parameters(compared to many-body scales)
-Rich toolbox for probing many-body states
Introduction to Anderson orthogonalitycatastrophe (OC)ca as op e (OC)
-Overlap S FS | FS '
- as system size , “orthogonality catastrophe”S 0 L
-Infinitely many low-energy electron-hole pairs produced
Fundamental property of the Fermi gas
Orthogonality catastrophe in X-ray absorption spectraabso p o spec a
Without impurity
With impurity
-Relevant overlap: S(t) FS | eiH0teiH f t | FS t2 / 2
-- scattering phase shift at Fermi energy, tan kFa
-Manifests in a power-law singularity in the absorption spectrum
1A() exp(it)S(t)dt 112 / 2
Orthogonality catastrophe in X-ray absorption spectraabso p o spec a
Without impurity
With impurity
-Relevant overlap: S(t) FS | eiH0teiH f t | FS t2 / 2
-- scattering phase shift at Fermi energy, tan kFa
-Manifests in a power-law singularity in the absorption spectrum
1
OBSERVATION IN SOLIDS CHALLENGING
A() exp(it)S(t)dt 112 / 2
Orthogonality catastrophe with cold atoms:Setup
-Fermi gas+single impurity
Se up
-Two pseudospin states of impurity and impurity, and
- -state scatters fermions
state scatte s e o s-state does not
-Scattering length
a-Fermion Hamiltonian for pseudospin -- , H0, H f
Earlier theoretical work on Kondo in cold atoms: Zwerger, Lamacraft, …
Ramsey fringes – new manifestation of OC-Utilize control over spin -Access coherent coupled dynamics of spin and Fermi gas-Ramsey interferometry
1) p/2 pulse FS 1 FS 1 FS1) p/2 pulse
2) Evolution
FS 2 FS
2 FS
1 eiH0t FS 1 eiH f t FS2) Evolution
3) Use p/2 pulse to measure )](Re[ tSSx 2 e FS
2 e FS
S(t) FS | eiH0teiH f t | FS
Direct measurement of OC in the time domain
Ramsey fringes as a probe of OCFirst principle calculationsFirst principle calculations
Spin echo: probing non-trivial dynamics of the Fermi gase e gas
-Response of Fermi gas to process in which impurity
switches between different states several times
iH t iH t
-Such responses play central role in analysis of Kondo
S1(t) FS | eiH0teiH f teiH0teiH f t | FSp p y y
problem but could not be studied in solid state systems
-Advantage: insensitive to slowly fluctuating magnetic fields
(unlike Ramsey)
Spin echo response: features
32 / 2
-Power-law decay at long times with an enhanced exponent
S1(t) t32 / 2
-Unlike the usual situation
(spin-echo decays slower
than Ramsey)
Cancels magnetic field-Cancels magnetic field
flutuations
-UniversalUniversal
-Generalize to n pi-pulses
to study even more complex
response functions
RF spectroscopy of impurity atomFree atom
Atom in a Fermi sea – many-body effects completely changey y p y gabsorption function
RF spectra can be calculated exactly
Edge singularities in the RF spectra:Intuitive pictureu ve p c u e
Photon pseudospin flipped+massive production of Photon pseudospin flipped+massive production of
excitations
Exact RF spectra
a<0; no impuritybound state
a>0; bound stateTwo thresholds
Single thresholdin absorption
-bound state filled after absorption
b d t t t-bound state empty
Generalizations: non-equilibrium OC, non-abelian Riemann-Hilbert problemo abe a e a be p ob e
-Impurity coupled to several Fermi seas at different chemical
i lpotentials
-Theoretical works in the context of quantum transportq p
-Mathematically, reduces to non-abelian Riemann-Hilbert problem (challenging)
Muzykantskii et al’03Abanin Levitov ‘05 (challenging)
-Experiments lacking
Abanin, Levitov 05
Generalizations: non-equilibrium OC, non-abelian Riemann-Hilbert problemo abe a e a be p ob e
-Multi-component Fermi gas coupled
to impurityp y
-Imbalance different species
-Mix them by pi/2 pulses
-Realization of non-equilibrium OC problem
-”Simulator” of quantum transport Simulator of quantum transport
and non-abelian Riemann-Hilbert problem
-Charge full counting statistics can be probed
Hyperfine state 1
Hyperfine t t 2state 1 state 2
Other directions
-Multi-component Fermi gas: non-equilibrium orthogonality catastrophe, non-abelian Riemann-Hilbert problemp , p
-Spinful fermions dynamics in the Kondo problem
-Dynamics: many-body effects in Rabi oscillations ofimpurity spin
Very different physics for bosons -Very different physics for bosons
Exploring “synthetic” condensed matter systemsExploring synthetic condensed matter systemswith Ramsey interference
Measuring topological properties of band structures. Berry and Zak phases
Orthogonality catastropheOrthogonality catastrophe
Interaction induced collapse of Ramsey fringesTwo component BEC. Single mode approximation
Ramsey fringe visibility
time
Experiments in 1d tubes:A. Widera et al. PRL 100:140401 (2008)
Spin echo. Time reversal experiments
Single mode approximation
The Hamiltonian can be reversed by changing a12
Predicts perfect spin echo
Spin echo. Time reversal experiments
Expts: A. Widera, I. Bloch et al.
Experiments done in array of tubes. S fl i i 1d
No revival?Strong fluctuations in 1d systems.Single mode approximation does not apply.Need to analyze the full model
Interaction induced collapse of Ramsey fringesin one dimensional systems
Only q=0 mode shows complete spin echoOnly q=0 mode shows complete spin echoFinite q modes continue decay
The net visibility is a result of competition between q=0 and other modes
Decoherence due to many‐body dynamics of low dimensional systems
New probe of dynamicsof 1d many‐body systems
Manifestations of topological character of band structurescharacter of band structures
All evidence is indirect:(transport, edge states)(transport, edge states)Can there be a direct measurement ofBerry phase of Bloch states ?Berry phase of Bloch states ?
Topologically different dimerizations
Dimerized tunneling Dimerized on‐site potential
topological non‐topological
Zak phase is equal to p Zak phase is equal to 0
Dimerized without inversion symmetry
Zak phase can be arbitrary