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Measuring quantum geometry From superconducting qubits to spin chains Michael Kolodrubetz, Physics Department, Boston University Theory collaborators: Anatoli Polkovnikov (BU), Vladimir Gritsev (Fribourg) Experimental collaborators: Michael Schroer, Will Kindel, Konrad Lehnert (JILA)

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The quantum geometric tensor

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The quantum geometric tensor

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Geometric tensor The quantum geometric tensor

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Geometric tensor Real part = Quantum (Fubini-Study) metric tensor The quantum geometric tensor

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Geometric tensor Real part = Quantum (Fubini-Study) metric tensor Imaginary part = Quantum Berry curvature The quantum geometric tensor

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Outline Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariance of geometry Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit

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Outline Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariance of geometry Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit

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Outline Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariance of geometry Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit

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Outline Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariance of geometry Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit

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The quantum geometric tensor Metric Tensor Berry curvature

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The quantum geometric tensor Metric Tensor Berry curvature

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The quantum geometric tensor Real symmetric tensor Metric Tensor Berry curvature

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The quantum geometric tensor Real symmetric tensor Same as fidelity susceptibility Metric Tensor Berry curvature

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Measuring the metric tensor

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Generalized force

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Measuring the metric tensor Generalized force

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Measuring the metric tensor Generalized force

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Measuring the metric tensor Generalized force

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Measuring the metric tensor Generalized force

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Measuring the metric tensor Generalized force

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Measuring the metric tensor

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For Bloch Hamiltonians, Neupert et al. pointed out relation to current-current noise correlations [arXiv:1303.4643]

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Measuring the metric tensor For Bloch Hamiltonians, Neupert et al. pointed out relation to current-current noise correlations [arXiv:1303.4643] Generalizable to other parameters/non-interacting systems

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Measuring the metric tensor For Bloch Hamiltonians, Neupert et al. pointed out relation to current-current noise correlations [arXiv:1303.4643] Generalizable to other parameters/non-interacting systems

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Measuring the metric tensor

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REAL TIME

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Measuring the metric tensor REAL TIME IMAG. TIME

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Measuring the metric tensor REAL TIME IMAG. TIME

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Measuring the metric tensor REAL TIME IMAG. TIME

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Measuring the metric tensor Real time extensions:

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Measuring the metric tensor Real time extensions:

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Measuring the metric tensor Real time extensions:

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Measuring the metric tensor Real time extensions:

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Measuring the metric tensor Real time extensions: (related the Loschmidt echo)

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Outline Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariance of geometry Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit

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Outline Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariance of geometry Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit

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Outline Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariance of geometry Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit

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Outline Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariance of geometry Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit

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Visualizing the metric Transverse field Anisotropy

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Visualizing the metric Transverse field Anisotropy

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Visualizing the metric Transverse field Anisotropy Global z-rotation

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Visualizing the metric

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Outline Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariance of geometry Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit

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Outline Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariance of geometry Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit

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Visualizing the metric

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h- plane

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Visualizing the metric h- plane

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Visualizing the metric h- plane

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Visualizing the metric - plane

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Visualizing the metric - plane

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Visualizing the metric No (simple) representative surface in the h- plane - plane

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Geometric invariants Geometric invariants do not change under reparameterization

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Geometric invariants Geometric invariants do not change under reparameterization Metric is not a geometric invariant

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Geometric invariants Geometric invariants do not change under reparameterization Metric is not a geometric invariant Shape/topology is a geometric invariant

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Geometric invariants Geometric invariants do not change under reparameterization Metric is not a geometric invariant Shape/topology is a geometric invariant Gaussian curvature K Geodesic curvature k g http://cis.jhu.edu/education/introPatternTheory/ additional/curvature/curvature19.html http://www.solitaryroad.com/c335.html

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Geometric invariants Gauss-Bonnet theorem:

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Geometric invariants Gauss-Bonnet theorem:

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Geometric invariants Gauss-Bonnet theorem:

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Geometric invariants Gauss-Bonnet theorem: 1 0 1

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Geometric invariants - plane

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Geometric invariants - plane

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Geometric invariants - plane Are these Euler integrals universal? YES! Protected by critical scaling theory

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Geometric invariants - plane Are these Euler integrals universal? YES! Protected by critical scaling theory

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Singularities of curvature -h plane

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Integrable singularities KhKh h h KhKh

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Conical singularities

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Same scaling dimesions (not multi-critical)

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Conical singularities Same scaling dimesions (not multi-critical)

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Curvature singularities

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Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariant near phase transitions Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit Outline

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1 0 Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariant near phase transitions Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit Outline

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1 0 Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariant near phase transitions Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit Outline h KhKh

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1 0 Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariant near phase transitions Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit Outline h KhKh

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The quantum geometric tensor Real symmetric tensor Same as fidelity susceptibility Metric Tensor Berry curvature

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The quantum geometric tensor Real symmetric tensor Same as fidelity susceptibility Metric Tensor Berry curvature

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The quantum geometric tensor Real symmetric tensor Same as fidelity susceptibility Metric Tensor Berry curvature Adiabatic evolution

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The quantum geometric tensor Real symmetric tensor Same as fidelity susceptibility Metric Tensor Berry curvature Adiabatic evolution

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The quantum geometric tensor Real symmetric tensor Same as fidelity susceptibility Metric Tensor Berry curvature Adiabatic evolution

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The quantum geometric tensor Real symmetric tensor Same as fidelity susceptibility Metric Tensor Berry curvature Adiabatic evolution

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The quantum geometric tensor Real symmetric tensor Same as fidelity susceptibility Metric Tensor Berry curvature Adiabatic evolution

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The quantum geometric tensor Real symmetric tensor Same as fidelity susceptibility Metric Tensor Berry curvature

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The quantum geometric tensor Real symmetric tensor Same as fidelity susceptibility Metric Tensor Berry curvature

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The quantum geometric tensor Real symmetric tensor Same as fidelity susceptibility Metric Tensor Berry curvature

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The quantum geometric tensor Real symmetric tensor Same as fidelity susceptibility Metric Tensor Berry curvature

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The quantum geometric tensor Real symmetric tensor Same as fidelity susceptibility Metric Tensor Berry curvature

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The quantum geometric tensor Real symmetric tensor Same as fidelity susceptibility Metric Tensor Berry curvature Magnetic field in parameter space

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Topology of two-level system

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Chern number

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Topology of two-level system Chern number ( ) is a topological quantum number

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Topology of two-level system Chern number ( ) is a topological quantum number Chern number = TKNN invariant (IQHE)

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Topology of two-level system Chern number ( ) is a topological quantum number Chern number = TKNN invariant (IQHE) Gives invariant in topological insulators Split eigenstates into two sectors connected by time-reversal number is related to Chern number of each sector

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Topology of two-level system How do we measure the Berry curvature and Chern number?

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Topology of two-level system

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Ground state

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Topology of two-level system Ground state

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Topology of two-level system

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Ramp

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Topology of two-level system Ramp Measure

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Topology of two-level system Ramp Measure

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Topology of two-level system Ramp Measure

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Topology of two-level system

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How to do this experimentally?

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Superconducting transmon qubit [Paik et al., PRL 107, 240501 (2011)]

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Superconducting transmon qubit [Paik et al., PRL 107, 240501 (2011)]

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Superconducting transmon qubit [Paik et al., PRL 107, 240501 (2011)]

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Superconducting transmon qubit [Paik et al., PRL 107, 240501 (2011)]

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Superconducting transmon qubit [Paik et al., PRL 107, 240501 (2011)] Rotating wave approximation

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Superconducting transmon qubit [Paik et al., PRL 107, 240501 (2011)] Rotating wave approximation

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Superconducting transmon qubit [Paik et al., PRL 107, 240501 (2011)] Rotating wave approximation

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Topology of two-level system Ramp Measure

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Topology of transmon qubit

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Work in progress

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Topology of transmon qubit Can we change the Chern number? Work in progress

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Topology of transmon qubit Bx Bz By

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Topology of transmon qubit Bx Bz By

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Topology of transmon qubit Bx Bz By ch 1 =1

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Topology of transmon qubit Bx Bz By Bx Bz By ch 1 =1

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Topology of transmon qubit Bx Bz By Bx Bz By ch 1 =1

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Topology of transmon qubit Bx Bz By Bx Bz By ch 1 =1 ch 1 =0

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Topology of transmon qubit

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Topological transition in a superconducting qubit!

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1 0 Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariant near phase transitions Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit Outline h KhKh

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1 0 Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariant near phase transitions Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit Outline h KhKh

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1 0 Measuring the metric tensor Transport experiments Corrections to adiabaticity Classification of quantum metric geometry Invariant near phase transitions Classification of singularities Chern number of superconducting qubit Berry curvature from slow ramps Topological transition in a qubit Outline h KhKh

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Theory Collaborators Anatoli Polkovnikov (BU) Vladimir Gritsev (Fribourg) Experimental Collaborators Michael Schroer, Will Kindel, Konrad Lehnert (JILA) Funding BSF, NSF, AFOSR (BU) Swiss NSF (Fribourg) NRC (JILA) For more details on part 1, see PRB 88, 064304 (2013) Acknowledgments

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The quantum geometric tensor Berry connection Metric tensor Berry curvature