Measuring quantum geometry - Boston University Physics Department.

Report
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

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
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
Metric Tensor

Real symmetric tensor
Berry curvature
The quantum geometric tensor
Metric Tensor
Real symmetric tensor
 Same as fidelity susceptibility

Berry curvature
Measuring the metric tensor
Measuring the metric tensor
Measuring the metric tensor
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
Measuring the metric tensor
Measuring the metric tensor
Measuring the metric tensor
Measuring the metric tensor
Measuring the metric tensor

For Bloch Hamiltonians, Neupert et al. pointed out relation to
current-current noise correlations [arXiv:1303.4643]
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
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
◦
Measuring the metric tensor
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
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
- plane
No (simple)
representative surface in
the h- 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
http://cis.jhu.edu/education/introPatternTheory/
additional/curvature/curvature19.html

Geodesic curvature kg
http://www.solitaryroad.com/c335.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
Are these Euler
integrals universal?
YES!
Protected by critical
scaling theory
- plane
Geometric invariants
Are these Euler
integrals universal?
YES!
Protected by critical
scaling theory
- plane
Singularities of curvature
-h plane
Integrable singularities
Kh
Kh
h
h
Conical singularities
Conical singularities
Same scaling dimesions
(not multi-critical)
Conical singularities
Same scaling dimesions
(not multi-critical)
Curvature singularities
Outline

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

Measuring the metric tensor
◦ Transport experiments
◦ Corrections to adiabaticity

Classification of quantum metric geometry
◦ Invariant near phase transitions
◦ Classification of singularities

1
0
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
◦ Invariant near phase transitions
◦ Classification of singularities

Kh 
h
1
0
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
◦ Invariant near phase transitions
◦ Classification of singularities

Kh 
h
1
0
Chern number of superconducting qubit
◦ Berry curvature from slow ramps
◦ Topological transition in a qubit
The quantum geometric tensor
Metric Tensor
Real symmetric tensor
 Same as fidelity susceptibility

Berry curvature
The quantum geometric tensor
Metric Tensor
Real symmetric tensor
 Same as fidelity susceptibility

Berry curvature
The quantum geometric tensor
Metric Tensor
Berry curvature
Adiabatic evolution
Real symmetric tensor
 Same as fidelity susceptibility

The quantum geometric tensor
Metric Tensor
Berry curvature
Adiabatic evolution
Real symmetric tensor
 Same as fidelity susceptibility

The quantum geometric tensor
Metric Tensor
Berry curvature
Adiabatic evolution
Real symmetric tensor
 Same as fidelity susceptibility

The quantum geometric tensor
Metric Tensor
Berry curvature
Adiabatic evolution
Real symmetric tensor
 Same as fidelity susceptibility

The quantum geometric tensor
Metric Tensor
Berry curvature
Adiabatic evolution
Real symmetric tensor
 Same as fidelity susceptibility

The quantum geometric tensor
Metric Tensor
Real symmetric tensor
 Same as fidelity susceptibility

Berry curvature
The quantum geometric tensor
Metric Tensor
Real symmetric tensor
 Same as fidelity susceptibility

Berry curvature
The quantum geometric tensor
Metric Tensor
Real symmetric tensor
 Same as fidelity susceptibility

Berry curvature
The quantum geometric tensor
Metric Tensor
Real symmetric tensor
 Same as fidelity susceptibility

Berry curvature
The quantum geometric tensor
Metric Tensor
Real symmetric tensor
 Same as fidelity susceptibility

Berry curvature
The quantum geometric tensor
Metric Tensor
Berry curvature
“Magnetic field” in
parameter space
Real symmetric tensor
 Same as fidelity susceptibility

Topology of two-level system
Topology of two-level system
Topology of two-level system
Topology of two-level system
Topology of two-level system
Topology of two-level system
Topology of two-level system
Topology of two-level system
Topology of two-level system
Topology of two-level system
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 (

Chern number = TKNN invariant (IQHE)
◦
) is a “topological quantum number”
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
Topology of two-level system
Ground
state
Topology of two-level system
Ground
state
Topology of two-level system
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
Topology of two-level system
Topology of two-level system
Topology of two-level system
Topology of two-level system
How to do this
experimentally?
Superconducting transmon qubit
[Paik et al., PRL 107, 240501 (2011)]
Superconducting transmon qubit
[Paik et al., PRL 107, 240501 (2011)]
Superconducting transmon qubit
[Paik et al., PRL 107, 240501 (2011)]
Superconducting transmon qubit
[Paik et al., PRL 107, 240501 (2011)]
Superconducting transmon qubit
Rotating wave
approximation
[Paik et al., PRL 107, 240501 (2011)]
Superconducting transmon qubit
Rotating wave
approximation
[Paik et al., PRL 107, 240501 (2011)]
Superconducting transmon qubit
Rotating wave
approximation
[Paik et al., PRL 107, 240501 (2011)]
Topology of two-level system
Ramp
Measure
Topology of transmon qubit
Topology of transmon qubit
Topology of transmon qubit
Topology of transmon qubit
Topology of transmon qubit
Topology of transmon qubit
Work in
progress
Topology of transmon qubit
Work in
progress
Can we change the
Chern number?
Topology of transmon qubit
Bz
Bx
By
Topology of transmon qubit
Bz
Bx
By
Topology of transmon qubit
Bz
ch1=1
Bx
By
Topology of transmon qubit
Bz
Bz
ch1=1
Bx
Bx
By
By
Topology of transmon qubit
Bz
Bz
ch1=1
Bx
Bx
By
By
Topology of transmon qubit
Bz
Bz
ch1=1
Bx
Bx
ch1=0
By
By
Topology of transmon qubit
Topology of transmon qubit
Topology of transmon qubit
Topology of transmon qubit
Topological transition in
a superconducting qubit!
Outline

Measuring the metric tensor
◦ Transport experiments
◦ Corrections to adiabaticity

Classification of quantum metric geometry
◦ Invariant near phase transitions
◦ Classification of singularities

Kh 
h
1
0
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
◦ Invariant near phase transitions
◦ Classification of singularities

Kh 
h
1
0
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
◦ Invariant near phase transitions
◦ Classification of singularities

Kh 
h
1
0
Chern number of superconducting qubit
◦ Berry curvature from slow ramps
◦ Topological transition in a qubit
Acknowledgments

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)
The quantum geometric tensor
Berry connection
Metric tensor
Berry curvature

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