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Coherent collective oscillations in disordered arrays of interacting superconducting Mikhail V. Fistul National University of Science and Technology MISiS Russian Quantum Center, Moscow Russia IBS Center for Theoretical of Complex Systems, Daejeon, Korea

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Outline • Motivation: • Macroscopic Qantum Dynamics • Superconducting Quantum Metamaterials (SQMs) • Quantum Modeling • Disorder vs. Interaction: Collective quantum effects

• Electrodynamics of SQMs: • Qubits, coherent quantum oscillations • Interaction between qubits • Disorder • Electromagnetic Waves Transmission

• Path Integral and Instanton (winding numbers) approach: • Coherent quantum oscillations of a single • Quantum time-dependent correlation function • Collective quantum effects: • Correlations in instanton (anti-instanton) numbers • Dynamic Anderson localization

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Motivation: Macroscopic SQMM S-QED MQC

MQT

I

I Quantum Quantum

1911 1981 2000 2005 2014

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Motivation: Superconducting

• Arrays of interacting qubits

From Fink J. M. et al. , PRL (2009) From G. Y. Chen et al. Scientific From P. Macha et al. Nature Reports (2012) Communication (2014)

• Single qubits (tunable two-levels systems)

From dwave.wordpress.com

From P. Macha, PhD thesis

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Motivation: Superconducting Quantum Metamaterial

• Interaction between qubits

• Nearest-neighbor interaction

• Long-range interaction

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Motivation: Quantum Modeling Flux qubits

g  i

• Interacting chain of spins E ex ˆ ˆ ˆ ˆ ˆ H   hi Sz,i  i Sx,i  g Sz,i Sz, j i i, j Egr •

• Complex quantum lattices

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Motivation: Quantum Modeling

Non-linear oscillator= qubit • Bose- (non- equilibrium lattice)

• Quantum XY spin model (BKT transition)

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Motivation: Quantum collective effects (QCE) Generic features of classical Generic features of QCE: collective effects (CCE): • Quantum oscillators (?) • Non-linear oscillators • Disorder and dissipation (decoherence) • Disorder and fluctuations • Interaction between quantum oscillators (?) • Interaction between oscillators • Measurements of the QCE • A set of differential equations • ??? • CCE • QCE

From P. Barbara et al., PRL (1999) From P. Macha et al. Nature Communication (2014)

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Josephson junction Single qubits

I  Ic sin(1 2 )  Ic sin d 2eV   m  C dt • Transmon qubits

• Flux qubits

• Phase qubit

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Single qubit: coherent quantum beatings

Generic features of QCE: • Quantum oscillators • Disorder and dissipation (decoherence) • Interaction between quantum oscillators (?) • Measurements of the QCE ∆ • ??? • QCE

• Experimental protocol to observe 

International Workshop: Dissipative 26.10.2017 From Poletto et al. New J. of Physics Quantum Chaos, PCS IBS , Daejeon (2009). Noninteracting qubits: coherent quantum beatings Generic features of QCE: • No interaction=“Gas” of qubits • Quantum oscillators • Disorder and dissipation (decoherence) • Interaction between quantum oscillators (?) • Measurements of the QCE • ??? • QCE

From M. Jerger et al., EPL, 96 (2011) 40012

 1 2 3 4 5 6 • Technological disorder • Gates inhomogeneities • Environment inhomogeneities • Interaction=? “Liquid “ of qubits

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Electrodynamic interaction between qubits • Magnetic interaction • Nearest neighbors interaction

 /(kBT ) 2 2 4 ( )     1 2 S  E d  i   i i  i     eff J   2 2 2 24 2 2 i i1 i 0  p  c

• Charge interaction -

• Long-range interaction

MVF, PRB (2007) International Workshop: Dissipative P. Volkov and MVF, PRB (2014), 26.10.2017 Quantum Chaos, PCS IBS , Daejeon S. Mukhin and MVF, Super. Sc. Techn. (2013) Measurements of coherent quantum oscillations Generic features of QCE: • Quantum oscillators • Disorder and dissipation (decoherence) • Interaction between quantum oscillators • Measurements of the QCE • ??? • QCE

• Experiment

From K. Shulga et al. , JETP Lett. (2017)

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon

EWs propagation in SQMs • Model Y • Transmission line mode q(y,t) 1 D( • Resonator mode Q(x,t) • Qubits modes i (t) D()=? • Dynamic equations

X • Hamiltonian of qubits:

Floquet states Hˆ (t)  Hˆ  i Q eitˆ 0 i 0 i Linear response

International Workshop: Dissipative From P. A. Volkov and M. V. Fistul, PRB 2014 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Linear Quantum Response

• The response function R. Kubo

C(t)   i (t)i (0) i

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Transmission of EWs in SQMs

1 D(

• Resonant drop: ~  C   i    2 2 i qi   • Identical qubits or QCE

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Resonant excitation of quantum oscillations

Generic features of QS: • Quantum oscillators • Disorder and dissipation (decoherence) • Interaction between quantum oscillators Measurements of the QCE • ??? • QS

2  2  2   2  2   4  2  R q R q res 2

• Resonant case:

• Splitting: res  2   N

• Next: Calculation of C(t) in SQMs

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Partition function, Hamiltonian and Instanton Analysis • Models • flux qubits+NNI • flux qubits+LRI • +NRI

• Path integrals method • Imaginary time

R. Feynman

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon

Single qubit: path integral method, imaginary time

Generic features of QCE: • Quantum oscillators • Disorder and dissipation (decoherence) • Interaction between quantum oscillators • Measurements of the QCE • Path integrals, Instanton/winding number approach • Time-dependent correlation function • QCE

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Single qubit: single instanton solution

• Instanton

W  eS0  

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Single qubit: large amount of instanton-(anti) instanton solutions • «Gas phase» of instantons and anti -instantons

 • Calculation of Z, C(t) = the sum of all trajectories with 2n  t t     2n 2    instantons and anti-instantons Z  dt dt .... dt  cosh 1    2n  2n1  1   n0    0 0 0 2kBT 

C()=

C(t)

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Partition function of non-interacting qubits

i Average number of instantons on qubits-Ni N  i k T k TN B Qubit frequencies   B i i 

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Partition function of interacting qubits: disordered quantum metamaterial

• Optimal path in qubit frequencies distribution

International Workshop: Dissipative 26.10.2017 MVF, Scientific Reports (2017) Quantum Chaos, PCS IBS , Daejeon The qubit frequencies correlations in the disordered SQMs

1D SQM

2D SQM

• Assumptions: K  ,    MVF, Scientific Reports (2017) International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Mapping to the Anderson localization

~ Ground and excited states of spin Hamiltonian K  

MVF, Scientific Reports (2017) International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon QCE in transmon qubits

• Experiments

• Model: interacting quantum rotators • Single qubit Hamiltonian m* ˆ  2 H   i  i  i 2

• Random variable: Winding number on qubits , Ni >>1 Ni  i /(kBT)

2 K | N  N | • Interaction between qubits: K ( N i  N j ) or i j

International Workshop: Dissipative 26.10.2017 [R. Fazio and G. Schoen, PRB (1991)] Quantum Chaos, PCS IBS , Daejeon Summary

• Superconducting quantum metamaterials: arrays of interacting qubits • Interaction: • direct magnetic/charge interactions • photon induced exchange interaction • EWs propagation in quantum metamaterials: singularities of the time-dependent quantum correlation function • Theoretical description: • path integrals approach • linear response theory • instanton/winding number approximation • Quantum collective effects: Qubit gas vs. Qubit liquid

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Thanks to collaborators!

• Bochum (P. Volkov) • Karlsruhe (A. Ustinov, N. Maleeva, P. Jung) • Moscow (S. Mukhin, M. Iontsev, K. Shulga)

International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon