1. 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)

2. The quantum geometric tensor

3. The quantum geometric tensor

4. Geometric tensor The quantum geometric tensor

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

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

7. 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

8. 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

9. 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

10. 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

11. The quantum geometric tensor Metric Tensor Berry curvature

12. The quantum geometric tensor Metric Tensor Berry curvature

13. The quantum geometric tensor Real symmetric tensor Metric Tensor Berry curvature

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

15. Measuring the metric tensor

18. Generalized force

19. Measuring the metric tensor Generalized force

20. Measuring the metric tensor Generalized force

21. Measuring the metric tensor Generalized force

22. Measuring the metric tensor Generalized force

23. Measuring the metric tensor Generalized force

24. Measuring the metric tensor

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

30. 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

31. 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 ◦

32. Measuring the metric tensor

33. REAL TIME

34. Measuring the metric tensor REAL TIME IMAG. TIME

35. Measuring the metric tensor REAL TIME IMAG. TIME

36. Measuring the metric tensor REAL TIME IMAG. TIME

37. Measuring the metric tensor Real time extensions:

38. Measuring the metric tensor Real time extensions:

39. Measuring the metric tensor Real time extensions:

40. Measuring the metric tensor Real time extensions:

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

42. 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

43. 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

44. 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

45. 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

46. Visualizing the metric Transverse field Anisotropy

47. Visualizing the metric Transverse field Anisotropy

48. Visualizing the metric Transverse field Anisotropy Global z-rotation

49. Visualizing the metric

50. 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

51. 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

52. Visualizing the metric

53. h-  plane

54. Visualizing the metric h-  plane

55. Visualizing the metric h-  plane

56. Visualizing the metric  -  plane

57. Visualizing the metric  -  plane

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

59. Geometric invariants Geometric invariants do not change under reparameterization

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

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

62. 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

63. Geometric invariants Gauss-Bonnet theorem:

64. Geometric invariants Gauss-Bonnet theorem:

65. Geometric invariants Gauss-Bonnet theorem:

66. Geometric invariants Gauss-Bonnet theorem: 1 0 1

67. Geometric invariants  -  plane

68. Geometric invariants  -  plane

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

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

71. Singularities of curvature  -h plane

72. Integrable singularities KhKh h h KhKh

73. Conical singularities

74. Same scaling dimesions (not multi-critical)

75. Conical singularities Same scaling dimesions (not multi-critical)

76. Curvature singularities

77. 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

78. 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

79. 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

80. 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

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

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

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

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

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

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

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

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

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

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

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

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

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

94. Topology of two-level system

104. “Chern number”

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

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

107. 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

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

109. Topology of two-level system

110. Ground state

111. Topology of two-level system Ground state

112. Topology of two-level system

113. Ramp

114. Topology of two-level system Ramp Measure

115. Topology of two-level system Ramp Measure

116. Topology of two-level system Ramp Measure

117. Topology of two-level system

121. How to do this experimentally?

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

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

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

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

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

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

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

129. Topology of two-level system Ramp Measure

130. Topology of transmon qubit

135. Work in progress

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

137. Topology of transmon qubit Bx Bz By

138. Topology of transmon qubit Bx Bz By

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

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

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

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

143. Topology of transmon qubit

146. Topological transition in a superconducting qubit!

147. 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

148. 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

149. 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

150. 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

151. The quantum geometric tensor Berry connection Metric tensor Berry curvature