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