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
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Ramsey interferencep/2l
p/2lpulse pulse
1
t0
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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
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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
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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
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Measuring topological propertiesMeasuring topological properties of band structures. Berry and Zak phasesBerry and Zak phases
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Topological aspects of band structures
All evidence is indirect:(transport, edge states)Can there be a direct measurement ofBerry phase of Bloch states ?
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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
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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
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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
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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
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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
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How to measure Zak phase usingRamsey interference sequenceRamsey interference sequence
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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)
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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
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Detection of topological band structure
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Detection of topological band structure
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Zak phase as a probe of band topologyd i1d version
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Su‐Schrieffer‐Heeger model of polyacetylene
Analogous to bichromatic optical lattice potential
LMU/MPQ
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Dimerized model
A B A AB
Topology of the band shows up in the winding of the eigenstateZak phase is equal to p
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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
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Orthogonality catastrophe
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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
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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)
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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
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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)
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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
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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
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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
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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
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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
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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, …
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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
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Ramsey fringes as a probe of OCFirst principle calculationsFirst principle calculations
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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)
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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
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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
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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
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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
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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
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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
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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
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Exploring “synthetic” condensed matter systemsExploring synthetic condensed matter systemswith Ramsey interference
Measuring topological properties of band structures. Berry and Zak phases
Orthogonality catastropheOrthogonality catastrophe
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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)
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Spin echo. Time reversal experiments
Single mode approximation
The Hamiltonian can be reversed by changing a12
Predicts perfect spin echo
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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
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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
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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 ?
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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