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)

The quantum geometric tensor

The quantum geometric tensor

Geometric tensor The quantum geometric tensor

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

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

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

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

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

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

The quantum geometric tensor Metric Tensor Berry curvature

The quantum geometric tensor Metric Tensor Berry curvature

The quantum geometric tensor Real symmetric tensor Metric Tensor Berry curvature

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

Measuring the metric tensor

Generalized force

Measuring the metric tensor Generalized force

Measuring the metric tensor Generalized force

Measuring the metric tensor Generalized force

Measuring the metric tensor Generalized force

Measuring the metric tensor Generalized force

Measuring the metric tensor

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

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

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

Measuring the metric tensor

REAL TIME

Measuring the metric tensor REAL TIME IMAG. TIME

Measuring the metric tensor REAL TIME IMAG. TIME

Measuring the metric tensor REAL TIME IMAG. TIME

Measuring the metric tensor Real time extensions:

Measuring the metric tensor Real time extensions:

Measuring the metric tensor Real time extensions:

Measuring the metric tensor Real time extensions:

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

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

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

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

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

Visualizing the metric Transverse field Anisotropy

Visualizing the metric Transverse field Anisotropy

Visualizing the metric Transverse field Anisotropy Global z-rotation

Visualizing the metric

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

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

Visualizing the metric

h-  plane

Visualizing the metric h-  plane

Visualizing the metric h-  plane

Visualizing the metric  -  plane

Visualizing the metric  -  plane

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

Geometric invariants Geometric invariants do not change under reparameterization

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

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

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 additional/curvature/curvature19.html

Geometric invariants Gauss-Bonnet theorem:

Geometric invariants Gauss-Bonnet theorem:

Geometric invariants Gauss-Bonnet theorem:

Geometric invariants Gauss-Bonnet theorem: 1 0 1

Geometric invariants  -  plane

Geometric invariants  -  plane

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

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

Singularities of curvature  -h plane

Integrable singularities KhKh h h KhKh

Conical singularities

Same scaling dimesions (not multi-critical)

Conical singularities Same scaling dimesions (not multi-critical)

Curvature singularities

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

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

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 KhKh

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 KhKh

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

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

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

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

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

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

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

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

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

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

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

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

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

Topology of two-level system

“Chern number”

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

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

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

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

Topology of two-level system

Ground state

Topology of two-level system Ground state

Topology of two-level system

Ramp

Topology of two-level system Ramp Measure

Topology of two-level system Ramp Measure

Topology of two-level system Ramp Measure

Topology of two-level system

How to do this experimentally?

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

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

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

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

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

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

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

Topology of two-level system Ramp Measure

Topology of transmon qubit

Work in progress

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

Topology of transmon qubit Bx Bz By

Topology of transmon qubit Bx Bz By

Topology of transmon qubit Bx Bz By ch 1 =1

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

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

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

Topology of transmon qubit

Topological transition in a superconducting qubit!

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 KhKh

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 KhKh

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 KhKh

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, (2013) Acknowledgments

The quantum geometric tensor Berry connection Metric tensor Berry curvature