Simula'on of Superconduc'ng Qubit Devices Workshop on Microwave - PowerPoint PPT Presentation

Simula'on of Superconduc'ng Qubit Devices Workshop on Microwave Cavities and Detectors for Axion Research Nick Materise January 10, 2016 LLNL-PRES-676622 This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC

Outline § Defini0on of a qubit § Non-linearity in superconduc0ng qubits and Josephson junc0ons § Cavity QED and Circuit QED § Black box Circuit Quan0za0on § Types of Superconduc0ng Qubits § Physical realiza0on of superconduc0ng circuits § Simula0ng RF components of qubits in COMSOL 2 LLNL-PRES-676622

Qubits § A quantum “bit” or two level system / effec$ve two level system with addressable energy levels § In some cases, a qubit can be treated as a harmonic oscillator with non-linearly spaced levels § Level spacing due to anharmonicity from non-linearity(ies), allows for designs that minimize leakage to higher excited states of the qubit(s) 3 LLNL-PRES-676622

Source of Non-linearity: Josephson Junc'on § DC Josephson Effect – B. Josephson, 1962 1 — Non-zero periodic current, due to tunneling Cooper Pairs across an SIS (superconductor-insulator-superconductor) junc0on — The current varies periodically in the phase difference across the junc0on, ac0ng as a macroscopic quantum variable — Josephson Current and Voltage Equa0ons 1 B.D. Josephson, Phys.Lea. 1 , 7 (1962) 4 LLNL-PRES-676622

Source of Non-linearity: Josephson Junc'on § DC Josephson Effect – B. Josephson, 1962 1 — Non-zero periodic current, due to tunneling Cooper Pairs across an SIS (superconductor-insulator-superconductor) junc0on — The current varies periodically in the phase difference across the junc0on, ac0ng as a macroscopic quantum variable — Josephson Current and Voltage Equa0ons 𝜔 1 ( 𝜒 1 ) 𝜔 2 ( 𝜒 2 ) Insulator Superconductors 1 B.D. Josephson, Phys.Lea. 1 , 7 (1962) 5 LLNL-PRES-676622

IV Characteris'cs of Josephson Junc'ons § The DC current in an SIS junc0on is given at zero temperature 2 § where K 0 is the zero-th order modified Bessel func0on of the first kind, Δ 1 , Δ 2 are the superconduc0ng gap energies of the superconduc0ng leads 2 N.R. Werthamer, Phys. Rev. 147 , 255 (1966) 6 LLNL-PRES-676622

IV Characteris'cs of Josephson Junc'ons § The DC current in an SIS junc0on is given at zero temperature 2 § where K 0 is the zero-th order modified Bessel func0on of the first kind, Δ 1 , Δ 2 are the superconduc0ng gap energies of the superconduc0ng leads Normalized IV Curve for Al-Al 2 O 3 -Al Josephson Junction Normalized Kramers-Kronig Curve for Al-Al 2 O 3 -Al Josephson Junction 1.8 1.6 Normalized Kramers-Kronig Current, I kk 1.6 1.4 Normalized DC Current, I dc 1.4 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Normalized Voltage Normalized Voltage 2 N.R. Werthamer, Phys. Rev. 147 , 255 (1966) 7 LLNL-PRES-676622

Josephson Junc'on Circuit Model § Josephson Junc0ons can be approximated by linear, passive circuit elements shun0ng a non-linear inductance L J — RCSJ Model (Resistance and Capaci0ve Shunted Junc0on) 3 — Useful model for including simple non-linear behavior in classical simula0ons, e.g. COMSOL — From Kirchhoff's current law, the current flowing through each element in the circuit is given by 3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.` 8 LLNL-PRES-676622

Josephson Junc'on Circuit Model § Josephson Junc0ons can be approximated by linear, passive circuit elements shun0ng a non-linear inductance L J — RCSJ Model (Resistance and Capaci0ve Shunted Junc0on) 3 — Useful model for including simple non-linear behavior in classical simula0ons, e.g. COMSOL — From Kirchhoff's current law, the current flowing through each element in the circuit is given by V I R n C J L J , E J 3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.` 9 LLNL-PRES-676622

Circuit Quantum Electrodynamics (cQED) § Use Josephson Junc0ons as a source of non-linearity to realize macroscopic quantum systems § Borrow concepts from the op0cs community, e.g. cavity QED to implement familiar systems § Atom in a resonant cavity is the most basic model 10 LLNL-PRES-676622

Cavity QED and Model Hamiltonians § Cavity QED: two level atomic system trapped in a mirrored, high finesse resonant cavity § Follows the Jaynes-Cummings Hamiltonian 3 3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007. 4 R. J. Schoelkopf and S. M. Girvin, Nature, vol. 451, pp. 664– 669, 02 2008. 11 LLNL-PRES-676622

Cavity QED and Model Hamiltonians § Cavity QED: two level atomic system trapped in a mirrored, high finesse resonant cavity § Follows the Jaynes-Cummings Hamiltonian 3 from quantum optics group at CalTech 2 g = Vacuum Rabi Frequency κ = Cavity Decay Rate κ g γ ⊥ = Transverse Decay Rate T transit = Time for atom to leave cavity γ ⊥ T transit 3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Atom trapped in a cavity with photon emission, atomic- Yale University, 2007. cavity dipole coupling, and atom transit 0me shown 4 . 4 R. J. Schoelkopf and S. M. Girvin, Nature, vol. 451, pp. 664– 669, 02 2008. 12 LLNL-PRES-676622

Cavity QED and Circuit QED, from op'cs to RF Cavity QED Circuit QED Two Level Atom Ar0ficial atom, truncated to two levels High Finesse Cavity High Q Cavity / Planar Resonator Arbitrarily large transi0on dipole Small transi0on dipole moment moment, e.g. strong coupling regime 1 / κ , 1 / γ T 1 , T 2 § Large dipole moment couples the qubit well to the cavity in superconduc0ng qubits: coupling strength and energy levels are tunable by design or in situ 13 LLNL-PRES-676622

Cavity QED and Circuit QED, Device Comparison Parameter Symbol Cavity QED 3 Circuit QED 3, 5 Resonator, Qubit Frequencies ω r , ω q / 2 π ~ 50 GHz ~ 5 GHz Transi0on Dipole Moment d / ea 0 ~ 1 ~ 10 4 Relaxa0on Time T 1 30ms 60 μs Decoherence Time T 2 ~1 ms ~10-20 μs § Large dipole moment couples the qubit well to the cavity in superconduc0ng qubits: coupling strength and energy levels are tunable § Trapped atoms in cavi0es have longer coherence $mes , not tunable , weakly coupled to the cavity for measurement 3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007. 5 H. Paik, et al., Phys. Rev. Lea. 107 , 240501 (2011) 14 LLNL-PRES-676622

Quan'zing Simple Circuits § Simplest model is an LC-resonator treated as a quantum harmonic oscillator with classical Lagrangian, Hamiltonian, and quan0zed 𝜚 operators 3 E 3 E 2 L C E 1 E 0 E 3 C ext E J, 𝜒 J E 2 E ff � 3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007. 15 LLNL-PRES-676622

Quan'zing Simple Circuits § Simplest model is an LC-resonator treated as a quantum harmonic oscillator with classical Lagrangian, Hamiltonian, and quan0zed 𝜚 operators 3 E 3 E 2 L C E 1 E 0 E 3 C ext E J, 𝜒 J E 2 E ff � 3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007. 16 LLNL-PRES-676622

Quan'zing Simple Circuits § Simplest model is an LC-resonator treated as a quantum harmonic oscillator with classical Lagrangian, Hamiltonian, and quan0zed 𝜚 operators 3 E 3 E 2 L C E 1 E 0 E 3 C ext E J, 𝜒 J E 2 E ff � 3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007. 17 LLNL-PRES-676622

Quan'zing Simple Circuits § Simplest model is an LC-resonator treated as a quantum harmonic oscillator with classical Lagrangian, Hamiltonian, and quan0zed 𝜚 operators 3 E 3 E 2 L C E 1 E 0 E 3 C ext E J, 𝜒 J E 2 E ff � 3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007. 18 LLNL-PRES-676622

Quan'zing Simple Circuits § Simplest model is an LC-resonator treated as a quantum harmonic oscillator with classical Lagrangian, Hamiltonian, and quan0zed 𝜚 operators 3 E 3 E 2 L C E 1 E 0 E 3 C ext E J, 𝜒 J E 2 E ff � 3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007. 19 LLNL-PRES-676622

Black box Circuit Quan'za'on § Idea is to extract all linear components of the qubit and microwave circuitry by synthesizing an equivalent passive electrical network § The network is obtained by compu0ng the S-parameters of a device using FEM sorware (COMSOL, HFSS) and conver0ng them to an impedance, Z ( j ! ) � Z( ȷω ) E J 20 LLNL-PRES-676622

Black box Circuit Quan'za'on—Vector Fit § The impedance func0on is fit to a pole-residue expansion following the Vector Fit procedure, a least squares fit to a ra0onal func0on of the form 6 § From this form, there are two synthesis approaches with two quan0za0on schemes — Lossy Foster approach (approximate circuit synthesis) 7 — Brune exact synthesis approach 8 6 B. Gustavsen et al., IEEE Tran on Power Delivery, 14(3):1052–1061, Jul 1999 7 F. Solgun et al. Phys.Rev.B 90 , 134504 (2014) 8 S. E. Nigg et al. Phys.Rev.Lea. 108 , 240502 (2012) 21 LLNL-PRES-676622