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

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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

Simula'on of Superconduc'ng Qubit Devices

Workshop on Microwave Cavities and Detectors for Axion Research

Nick Materise January 10, 2016

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§ 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

Outline

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§ 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)

Qubits

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§ DC Josephson Effect – B. Josephson, 19621

— 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

Source of Non-linearity: Josephson Junc'on

1 B.D. Josephson, Phys.Lea. 1, 7 (1962)

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§ DC Josephson Effect – B. Josephson, 19621

— 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

Source of Non-linearity: Josephson Junc'on

Superconductors Insulator

𝜔1(𝜒1) 𝜔2(𝜒2)

1 B.D. Josephson, Phys.Lea. 1, 7 (1962)

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IV Characteris'cs of Josephson Junc'ons

2 N.R. Werthamer, Phys. Rev. 147, 255 (1966)

§ The DC current in an SIS junc0on is given at zero temperature2 § where K0 is the zero-th order modified Bessel func0on of the first kind, Δ1, Δ2

are the superconduc0ng gap energies of the superconduc0ng leads

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IV Characteris'cs of Josephson Junc'ons

2 N.R. Werthamer, Phys. Rev. 147, 255 (1966)

§ The DC current in an SIS junc0on is given at zero temperature2 § where K0 is the zero-th order modified Bessel func0on of the first kind, Δ1, Δ2

are the superconduc0ng gap energies of the superconduc0ng leads

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Normalized DC Current, Idc Normalized Voltage Normalized IV Curve for Al-Al2O3-Al Josephson Junction 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Normalized Kramers-Kronig Current, Ikk Normalized Voltage Normalized Kramers-Kronig Curve for Al-Al2O3-Al Josephson Junction

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Josephson Junc'on Circuit Model

§ Josephson Junc0ons can be approximated by linear, passive

circuit elements shun0ng a non-linear inductance LJ

— 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.`

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Josephson Junc'on Circuit Model

§ Josephson Junc0ons can be approximated by linear, passive

circuit elements shun0ng a non-linear inductance LJ

— 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 Rn CJ LJ, EJ I V

3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.`

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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

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Cavity QED and Model Hamiltonians

§ Cavity QED: two level atomic system trapped in a mirrored, high

finesse resonant cavity

§ Follows the Jaynes-Cummings Hamiltonian3

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.

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Cavity QED and Model Hamiltonians

§ Cavity QED: two level atomic system trapped in a mirrored, high

finesse resonant cavity

§ Follows the Jaynes-Cummings Hamiltonian3

g κ γ⊥ Ttransit

from quantum optics group at CalTech

Atom trapped in a cavity with photon emission, atomic- cavity dipole coupling, and atom transit 0me shown4.

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.

2g = Vacuum Rabi Frequency κ = Cavity Decay Rate γ⊥ = Transverse Decay Rate Ttransit = Time for atom to leave cavity

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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 Small transi0on dipole moment Arbitrarily large transi0on dipole moment, e.g. strong coupling regime 1 / κ , 1 / γ T1 , T2

§ 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

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Cavity QED and Circuit QED, Device Comparison

Parameter Symbol Cavity QED3 Circuit QED3, 5 Resonator, Qubit Frequencies ωr, ωq / 2π ~ 50 GHz ~ 5 GHz Transi0on Dipole Moment d / ea0 ~ 1 ~ 104 Relaxa0on Time T1 30ms 60 μs Decoherence Time T2 ~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)

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Quan'zing Simple Circuits

§ Simplest model is an LC-resonator treated as a quantum harmonic

• scillator with classical Lagrangian, Hamiltonian, and quan0zed
• perators3

3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.

L C Cext EJ,𝜒J E0 E1 E2 E3 E E2 E3

ff

• 𝜚

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Quan'zing Simple Circuits

§ Simplest model is an LC-resonator treated as a quantum harmonic

• scillator with classical Lagrangian, Hamiltonian, and quan0zed
• perators3

3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.

L C Cext EJ,𝜒J E0 E1 E2 E3 E E2 E3

ff

• 𝜚

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Quan'zing Simple Circuits

§ Simplest model is an LC-resonator treated as a quantum harmonic

• scillator with classical Lagrangian, Hamiltonian, and quan0zed
• perators3

3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.

L C Cext EJ,𝜒J E0 E1 E2 E3 E E2 E3

ff

• 𝜚

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Quan'zing Simple Circuits

§ Simplest model is an LC-resonator treated as a quantum harmonic

• scillator with classical Lagrangian, Hamiltonian, and quan0zed
• perators3

3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.

L C Cext EJ,𝜒J E0 E1 E2 E3 E E2 E3

ff

• 𝜚

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Quan'zing Simple Circuits

§ Simplest model is an LC-resonator treated as a quantum harmonic

• scillator with classical Lagrangian, Hamiltonian, and quan0zed
• perators3

3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.

L C Cext EJ,𝜒J E0 E1 E2 E3 E E2 E3

ff

• 𝜚

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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) EJ

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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 form6

§ From this form, there are two synthesis approaches with two

quan0za0on schemes

— Lossy Foster approach (approximate circuit synthesis)7 — Brune exact synthesis approach8

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)

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Black box Circuit Quan'za'on—Lossy Foster

§ Taking the constant term d = 0 and excluding the pole at s = 1

• r sesng e = 0 leaves the ra0onal func0on with poles and

residues Rk , sk

7

• § Expanding the k-th component of Z(s) in par0al frac0ons and

taking the low loss limit, »k , bk ¿ 1 RLC Tank Circuit!

7 F. Solgun et al. Phys.Rev.B 90, 134504 (2014)

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Black box Circuit Quan'za'on—Lossy Foster

§ Main result of the Lossy Foster treatment is a set of uncoupled

harmonic oscillators as a series of RLC circuits

§ Circuit elements in terms of the real and imaginary components

• f the poles and residues7

R1 R2 R3 RM L1 L2 L3 LM C1 C2 C3 CM 𝜒 = Σm 𝜚m

7 F. Solgun et al. Phys.Rev.B 90, 134504 (2014) 9 J. Bourassa et al. Phys.Rev.A 86, 013814 (2012)

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Black box Circuit Quan'za'on—Lossy Foster

§ From the circuit elements, a lossless Hamiltonian is obtained by

taking the limit Rk→1 8k

§ The LC circuits are quan0zed as harmonic oscillators giving the

linear Hamiltonian8

§ A non-linear Hamiltonian accounts for the qubit and its coupling

to the harmonic modes

8 S.E. Nigg et al. Phys.Rev.Lea. 108, 240502 (2012)

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§ Early Charge qubit – Cooper Pair Box

(CPB)

— Cooper pairs tunnel across the junc0on

leading to a charge number operator

— Typical implementa0ons include a resonator

that plays the role of a cavity

— Hamiltonian in the charge basis for single

Josephson junc0on3

— “Split” CPB including RLC resonator and

coupling3

Types of Superconduc'ng Qubits: Charge Qubits

3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.

L R C Cg Cin EJ1 EC1 φ1 EJ2 EC2 φ2

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§ Early Charge qubit – Cooper Pair Box (CPB)

— Rota0ng wave approxima0on (RWA) and Jaynes-Cummings Hamiltonian

• Approximate number and charge number operators as Pauli operators3
• Expand voltage operator,V, apply RWA to coupling term and subs0tute the qubit

plasma frequency, !p = (LJCJ)-1/2

Types of Superconduc'ng Qubits: Charge Qubits

3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.

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§ Early Charge qubit – Cooper Pair Box (CPB)

— Rota0ng wave approxima0on (RWA) and Jaynes-Cummings Hamiltonian

• Approximate number and charge number operators as Pauli operators3
• Expand voltage operator,V, apply RWA to coupling term and subs0tute the qubit

plasma frequency, !p = (LJCJ)-1/2

Types of Superconduc'ng Qubits: Charge Qubits

3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.

**Reclaims Jaynes-Cummings Hamiltonian

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§ Transmon, an improved charge qubit

— Shunt capacitor reduces sensi0vity to charge noise — Flaaer energy levels, weakly anharmonic — Hamiltonian, qubit + resonator3

Types of Superconduc'ng Qubits: Transmon

3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.

L R C Cg Cin EJ1 EC1 φ1 Cs

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§ Transmon, an improved charge qubit

— Hamiltonian in the energy basis, anharmonic oscillator3 — Including resonator and coupling term follow a similar treatment as the

CPB qubit

— Kerr and cross Kerr terms may be included when expanding the fourth

power in b

Types of Superconduc'ng Qubits: Transmon

3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.

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§ Flux qubit

— Flux threading a loop is quan0zed; DC SQUID

biases the qubit

— Persistent current Ip forms in the

superconduc0ng loop

— Hamiltonian, with fixed gap Δ10, 11

§ Capaci0vely shunted flux qubit

— Shunt the flux qubit with a large capacitor,

similar to the transmon for charge qubits

— Hamiltonian, with resonator12

Types of Superconduc'ng Qubits: Flux Qubits

10 M.J. Schwarz et al. New Journal of Physics 15 (2013) 045001 11 T.P. Orlando, Phys.Rev.B 60, 15398 (1999) 12 F. Yei et al, Nature Communica0ons 7 (2016)

Φ

α EJ EJ EJ

Φ

α EJ EJ EJ L R C Cg Cin Cs

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§ Dipole moment, d , in terms of the magnitude of the applied voltage

V0 , CPW conductor width w , electric field magnitude E0 , and coupling of the qubit to the resonator13

Physical Designs: CPW + CPB = Cavity + Atom

Coplanar waveguide resonator and lumped circuit13

§ Implemented as Josephson Junc0on capaci0vely coupled to

transmission line resonator (coplanar waveguide, CPW)

— Transmission Line Resonator ~ Cavity — 2D Planar or 3D cavity couples qubit to drive and readout

13 A. Blais et al., Phys. Rev. A 69, 062320 (2004)

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Physical Designs: Transmon

§ Similar in design to CPB with the following modifica0ons

— Shunt capacitance implemented with an interdigitated capacitor or sufficiently

large gap of exposed substrate between conductors a) b) 100 µm 2 µm

Micrograph of resonator and transmon reproduced from3

3 D. I. Schuster, Circuit Quantum Electrodynamics. PhD thesis, Yale University, 2007.

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COMSOL RF Simula'on Building Blocks

§ Model systems used to develop more accurate descrip0ons of

the microwave circuits that cons0tute a qubit

§ Model Progression

• 1. Microstrip transmission line
• 2. Coplanar Waveguide (CPW)

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COMSOL RF Simula'on Building Blocks

§ Model systems used to develop more accurate descrip0ons of

the microwave circuits that cons0tute a qubit

§ Model Progression

• 2. Meanderline resonator
• 1. Interdigitated Capacitor (IDC)

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Microstripline Resonator

§ Electric field norm and characteris0c

impedance, Z0

§ Characteris0c impedance is given by

where h is the substrate thickness, t is the thickness of the microstrip, w is the width of the strip

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Coplanar Waveguide

§ Coplanar waveguide used as a

resonant coupling structure, i.e. cavity with the qubit

§ Characteris0c Impedance

from conformal mapping14:

h

εr

14 Rainee N Simons. Coplanar Waveguide Circuits, Components, and

Systems, chapter 2. Wiley Series in Microwave and Op0cal Engineering. Wiley, 2001.

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COMSOL RF Module Demo: Work planes

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COMSOL RF Module Demo: Work planes

Model builder pane includes the geometry, materials, and physics

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COMSOL RF Module Demo: Arrays

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COMSOL RF Module Demo: Arrays

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COMSOL RF Demo: Extrusions

Extrusion direc0on

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COMSOL RF Demo: Meshing

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COMSOL RF Demo: Meshing

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COMSOL RF Demo: Boundary Condi'ons, PEC

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COMSOL RF Demo: Boundary Condi'ons, PEC

PEC Boundaries

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COMSOL RF Demo: Boundary Condi'ons, Ports

Lumped Port 1 Terminal Sesngs

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COMSOL RF Demo: Sca_ering Boundary Condi'on

Applied to all non-PEC and non-port exterior boundaries

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COMSOL RF Demo: Frequency Sweep

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COMSOL RF Demo: E-Field Plot

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COMSOL RF Demo: E-Field Plot

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Closing Comments on cQED and COMSOL

§ Superconduc0ng qubits benefit from simple descrip0ons in

cQED by through analogies with cavity QED

§ Black box quan0za0on provides a systema0c method of

quan0zing the bulk features of devices as circuits

§ COMSOL provides a simula0on environment to model the

classical geometric features of qubits

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Acknowledgements

§ Jonathan Dubois, Eric Holland, Maahew Horsley, Vince Lordi,

Scoa Nelson, Yaniv Rosen, Nathan Woolea

§ This work was funded by the LLNL Laboratory Directed

Research and Development (LDRD) program, project number 15-ERD-051.

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Thank you—Ques'ons