Coherent collective quantum oscillations in disordered arrays of interacting superconducting qubits Mikhail V. Fistul National University of Science and Technology MISiS Russian Quantum Center, Moscow Russia IBS Center for Theoretical Physics 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 qubit • 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 Quantum Dynamics SQMM S-QED MQC MQT I I Quantum Complexity Quantum 1911 1981 2000 2005 2014 International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Motivation: Superconducting Quantum Metamaterial • 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 • Quantum Annealing • Complex quantum spin lattices International Workshop: Dissipative 26.10.2017 Quantum Chaos, PCS IBS , Daejeon Motivation: Quantum Modeling Non-linear oscillator= transmon qubit • Bose-Hubbard model (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 i1 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 eitˆ 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 • transmons+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 eS0 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 2n1 1 n0 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 .