From SQUID to Qubit Flux 1/f Noise: The Saga Continues

Fei Yan, S. Gustavsson, A. Kamal, T. P. Orlando Massachusetts Institute of Technology, Cambridge, MA

T. Gudmundsen, David Hover, A. Sears, J.L. Yoder, A. J. Kerman, W. D. Oliver

MIT Lincoln Laboratory, Lexington, MA

Jeffrey Birenbaum, JC University of California, Berkeley

Published in Nature Communications 3 November 2016

IWSSD2016 Tsukuba, Japan 15 November 2016

Support: Intelligence Advanced Research Projects Activity (IARPA) Army Research Office (ARO)

From SQUID to Qubit

• The Ubiquitous 1/f Noise • Three-Junction Flux Qubit

• Relaxation and Decoherence • The C-Shunt Flux Qubit: Origin

• The C-Shunt Flux Qubit: Recent Experiments • Concluding Remarks

The Ubiquitous 1/f Noise

• Vacuum tubes • Carbon resistors • Semiconductor devices • Metal films • Superconducting devices

log f

log

S x(f

)

Spectral density: Sx(f) ∝ 1/fα, α ~1

time (t)

X(t)

Random Telegraph Signals (RTS) and l/f Noise

• For a single characteristic time τ: SRTS(f) ∝ τ/[1 + (2πfτ)2]

• The superposition of Lorentzians from uncorrelated processes with a broad distribution of τ yields 1/f noise (Machlup 1954)

• To generate 1/f noise at frequency f0, the particle must reside in a well for time 1/f0

• For example, for f0 = 10−4 Hz, 1/f0 ~ 3 h

time (t)

X(t)

log f lo

g S x

(f)

1/f 1/f2

For example, electron hops between traps in a semiconductor

Intensity Fluctuations in Music and Speech

Richard Voss and JC Nature 1975

log 1

0[S i

nten

sity

(f)]

log10(f)

Spectra have been offset vertically

1/f Noise in Superconducting Devices: Three basic mechanisms

1. Critical current noise: Trapping and release of electrons in tunnel barriers modify the transparency of the junction, causing its resistance and critical current to fluctuate. At low temperatures, the process may involve quantum tunneling.

2(a). Charge noise: Hopping of electrons between traps induces fluctuating charges

on to nearby films and junctions 2(b).“Ohmic” charge noise: At high frequencies, so called ohmic charge noise

arises from transitions between the quantized energy levels in the traps that produce 1/f charge noise at low frequencies. (“Ohmic” because in the original Caldeira-Leggett model of relaxation and decoherence in Josephson junctions the loss mechanism was a resistor modeled by an array of quantum oscillators.)

Ohmic Charge Noise Shnirman et al. PRL 2005

Courtesy Gerd Schön

Charge qubit

η η

η

3. Flux noise: Flux-sensitive devices (SQUIDs, flux qubits….) exhibit “flux noise”. This arises from the random reversal of spins at the surface of the superconductor thereby coupling magnetic flux into the loop of the SQUID or flux qubit.

V

0 1 2 Φ 0

Φ

δV

δΦ

I

V Φ

What do we see when we bias here?

DC SQUIDs: Flux 1/fα Noise

Nb washer T = 90 mK

V Φ

α ≈ 0.8 (spectral density)

First observed in 1982

From SQUID to Qubit

• The Ubiquitous 1/f Noise • Three-Junction Flux Qubit

• Relaxation and Decoherence • The C-Shunt Flux Qubit: Origin

• The C-Shunt Flux Qubit: Recent Experiments • Concluding Remarks

Three-Junction Flux Qubit DC SQUID + One Smaller Junction

Iq

Φ

|0⟩ Iq

L Φ

|1⟩

L

Degeneracy point: Applied flux Φq = Φ0/2

U

Φq Φ0/2

(Φ0 ≡ h/2e)

J.E. Mooij et al., Science 285, 1036 (1999) C.H. Van der Wal et al., Science 290, 773 (2000)

|Ψ⟩ = (|0⟩ ± |1⟩)/√2

Superposition resolves degeneracy

Small junction area/ large junction area = γ

Energies of the Flux Qubit

ε = 2Iq(Φq – Φ0/2)

ν = ∆ at degeneracy

dε/dΦq = 2Ιq

dν/dΦq = (dν/dε)(dε/dΦq)

= 2(ε/ν)Ιq

= 0 at degeneracy

• Josephson coupling energy: Ej ≡ I0Φ0/2π, I0 = critical current

• Junction charging energy: Ec ≡ e2/2C, C is junction capacitance

• Ej/Ec is large, typically ~ 100

• γ = small junction I0/large junction I0 ~ 0.7

• Excellent approximation to two-level system near Φ0/2

From SQUID to Qubit

• The Ubiquitous 1/f Noise • Three-Junction Flux Qubit

• Relaxation and Decoherence • The C-Shunt Flux Qubit: Origin

• The C-Shunt Flux Qubit: Recent Experiments • Concluding Remarks

Relaxation and Decoherence T1 Relaxation time:

• Time to relax from first excited state |e> to ground state |g> • Energy loss involved • The classic Caldeira-Leggett spin-boson model treats linear dissipation as a

bath of harmonic oscillators and calculates the Johnson-Nyquist noise from the complex impedance of the environment. The power spectrum of this “ohmic dissipation” scales with frequency.

T2 Decoherence time:

• Time for phase difference between two eigenstates to become randomized

1/T2 = 1/(2T1) + 1/τφ τφ is the pure dephasing time (fluctuations in energy level splitting) Maximum value of T2 is 2T1

Previously Known Relaxation Processes

• Purcell effect (Purcell 1946). With the qubit embedded in a microwave

transmission line cavity, spontaneous emission of photons from the qubit into the cavity produces relaxation.1

• High frequency ohmic charge noise arises from quantized energy levels

in the two-level systems (TLSs) that at low frequencies produce 1/f charge noise.2,3

• Excess quasiparticles (above the thermal population) generate additional dissipation at microwave frequencies, reducing T1.4,5

[1Houk et al., Phys. Rev Lett. 101, 808502 (2008). 2stafiev et al., Phys. Rev Lett. 93, 267007 (2004). 3Shnirman et al., Phys. Rev Lett. 94, 127002 (2005). 4Martinis et al., Phys. Rev Lett. 103, 097002 (2009). 5Catelani et al., Phys. Rev Lett. 106, 077002 (2011).]

Typical Values of T1 and T2

• For many years, T1, T2ECHO were typically a few µs at degeneracy

• Famous, solitary exception: flux qubit fabricated at NEC and measured at MIT-LL1: T1 = 12 µs, T2ECHO = 23 µs • More recently, six flux qubits in a 3-D cavity yielded2

T1 = 6 - 20 µs,T2*(Ramsey) = 2 – 8 µs

1Bylander et al. (2011) 2Stern et al. (2014)

From SQUID to Qubit

• The Ubiquitous 1/f Noise • Three-Junction Flux Qubit

• Relaxation and Decoherence • The C-Shunt Flux Qubit: Origin

• The C-Shunt Flux Qubit: Recent Experiments • Concluding Remarks

The C-Shunt Flux Qubit: Origin You, Hu, Ashab & Nori, Phys. Rev. B 140515 (2007)

• Charge noise arises from TLSs close to small junction • Very small area of junction is not so easy to control, producing variations in Cj and Ej, and thus in qubit characteristics.

Steffen et al. (IBM): T1 ≈ 5.7 µs, T2E = 9.6 µs ≈ 2T1

γ γ

γ γ

Φq

• Adding Cs allows more flexibility in design and lower values of Iq. • In the presence of flux noise, since T2 ∝ 1/(dν/dΦq)2

∝ 1/Ιq

2, reducing Iq increases T2 while retaining high anharmonicity. • Since Cs >> γCj, most of the electric field energy is stored in Cs, which can be designed to have a very high Q.

Φq

Energy Spectrum vs. Applied Flux

• States |0> and |1> provide an excellent approximation to a two-level system near Φ0/2 • Anharmonicity: A = f12 − f01 • Higher anharmonicity results in faster gates since the spread of frequencies around f01 in a microwave pulse can be greater without exciting higher states

Importance of Anharmonicity

Φq/Φ0

Freq

uenc

y (G

Hz)

From SQUID to Qubit

• The Ubiquitous 1/f Noise • Three-Junction Flux Qubit

• Relaxation and Decoherence • The C-Shunt Flux Qubit: Origin

• The C-Shunt Flux Qubit: Recent Experiments • Concluding Remarks

slide-21 CSQ 11/28/2016

Resonator & Qubits

QB#1

QB#2

λ/2 CPW resonator

IDC Qubit

shunt capacitor

small JJ

qubit loop

2.5 mm

100 μm

10 μm

Design Template: Interdigitated Capacitor

• Two qubits capacitively coupled to the same coplanar waveguide (CPW) resonator via their shunt capacitors

• Resonator length = 8 mm, fR = 8.27 GHz

CPW

strip ground ground

slide-22 CSQ 11/28/2016

Resonator & Qubits

QB#1

QB#2

λ/2 CPW resonator

Shunt capacitor

small JJ

qubit loop

2.5 mm

100 μm

10 μm

IDC Qubit

Fabricated IDC and Qubits

Square-Plate Capacitor for Qubits

Capacitor plates: 200 µm × 200 µm Cs ≈ 51 fF

Small junction: 150 nm × 150 nm Cjsmall ≈ 1 – 2 fF γ ≈ 0.42 Area of the capacitor plates ≈ 106 times greater than the area of the small junction

Small junction

CPW strip

Fabrication

• All structures except junctions deposited with: Molecular beam epitaxy (MBE) of Al or Electron beam evaporation of Al • Junctions: Electron beam evaporation of Al • Substrate:

Sapphire annealed at 900 ° for outgassing and surface reconstruction

C

Dispersive cQED Readout

Readout (resonator) frequency ωro/2π = 8.27 GHz Qubit drive frequency ωd

HEMT

Directional couplers

• Magnetic field B applied to the qubit changes f01 and thus the resonant frequency of the resonator • This changes the amplitude and phase of the transmitted readout pulse, yielding f01 • Average over typically 10,000 pulses

Dilution refrigerator typically at 20 mK

Spectra of Qubits A and B

Parameters chosen to yield ∆ ∼ 5 GHz

∆A = 4.36 GHz ∆B = 4.70 GHz

Flux Bias Current (µA)

ωq/2

π (G

Hz)

Flux Qubits A & B: T1 at Degeneracy

∆A = 4.36 GHz ∆B = 4.70 GHz Solid lines are exponential fits

Exc

ited-

stat

e po

pula

tion

ρ e

τ(µs)

• Devices fabricated simultaneously on the same wafer with identical geometries, and measured during the same cool down

Calculation of Relaxation Processes 1/T1 = ħ−2| 𝑔 �̂�Φ 𝑒 |2SΦ(ωq) SΦ(ωq) spectral density at qubit frequency Flux noise 𝑑�Φ

is transition magnetic dipole. + ħ−2| 𝑔 �̂�𝑄 𝑒 |2SQ(ωq) SQ(ωq) spectral density at qubit frequency Ohmic charge noise �̂�Q

is transition electric dipole. + g2𝜅| 𝑔�1𝛾 σ�𝑦 𝑎�† + 𝑎� �̃�1𝛾 |2 Energy lost by qubit due to coupling Purcell effect to the resonator with decay rate 𝜅; g is the coupling strength. States |𝑔�1𝛾� and |�̃�1𝛾� are the dressed states of the coupled qubit- resonator system. The Pauli operator for the qubit is σ�𝑦.

Qubit B ∆ = 4.70 GHz • T1 limited predominantly by

ohmic charge noise at lower frequencies

• T1 dominated by Purcell at higher frequencies

• Flux noise is relatively

unimportant

• Ohmic charge noise: Calculated from SQ(ω) = AQω/(2π×1 GHz), with AQ fitted to data

• Purcell: Obtained from measured resonator frequency and linewidth and calculated coupling to the qubit

• Flux noise: Assumes SΦ(f) = [(1 µΦ0)2/Hz](2π×1 Hz/ω)0.8 (next slide)

Qubit B: T1 vs. Frequency

Flux Qubit C (Small ∆): T1 vs. Frequency

• Flux noise: Assumes SΦ(f) = [(1 µΦ0)2/Hz](2π×1Hz/ω)0.8 (From SQUID spectra)

• Purcell: Obtained from measured resonator frequency and linewidth and calculated coupling to the qubit • Ohmic charge noise: Calculated from SQ(ω) = AQω/(2π×1 GHz), with AQ fitted to data from qubit B

Parameters chosen to yield small ∆: ∆C = 0.88 GHz • T1 limited totally by flux

noise below about 4 GHz (∆C ≈ 0.2 ∆B) • Flux noise observed at frequencies up to 4 GHz • T1 dominated by Purcell at

higher frequencies • Charge noise is relatively

unimportant

Qubit C ∆ = 0.88 GHz

Dependence of T1 on Qubit Geometry

Interdigitated Parallel bars Squares

T1 (µs) T1 (µs) T1 (µs) Qubit A 23 21 44 Qubit B 22 17 55

• Qubit A & B geometries identical

• All devices fabricated simultaneously on same wafer

• Measured during same cool down

• Square geometry wins!

Dependence of AQ on Qubit Geometry

Interdigitated Squares

AQ = (15.4 × 10−9 e)2/Hz AQ = (7.4 × 10−9 e)2/Hz

• Large area capacitor (Cs >> Cj) stores most of electric field energy with greatly reduced electric field strength compared with that of the junctions.

• Reduced electric field participation of surface and interface losses. • Large shunt capacitor improves qubit reproducibility: effect of variation in

junction capacitance is reduced, and unwanted stray capacitances are overwhelmed.

Parameters of 22 Qubits

Parameters of 22 Qubits

Longest T1’s Anharmonicity A 800 – 910 MHz

Measured & Predicted Values of T1 at Degeneracy: 22 Qubits with Different Parameters & Cs

• Values of T1 at degeneracy calculated from known qubit parameters using fixed values of flux and charge noise. • Good agreement given wide variation in design and critical current density over five fabrication runs

Cs: 18 – 51 fF Ec: 0.38 – 1.1 GHz Ej: 36 – 140 GHz Frequency at deg: 0.5 – 5 GHz Anharmonicity for longest T1’s 800 - 910 MHz

Flux Noise: α = 0.8

• It appears that α = 0.8 is maintained rather accurately over the entire frequency range of flux noise below temperatures of, say, 100 mK.

• A minor deviation from α = 0.8 would have a major impact. As an example, consider two flux noise spectral densities:

S1(f) ∝ 1/f, S2(f) ∝ 1/f0.8 And S1(1 Hz) = S2(1 Hz)

At 5 GHz, S2(5 GHz)/S1(5 GHz) = 87!

• Flux 1/f noise persists from 10−4 Hz (SQUID) to 4 GHz (flux qubit) with constant slope.

slide-37 CSQ 11/28/2016

• T1 = 44 µs

• Maximum T2Echo = 2T1 = 88 µs

• Using Carr-Purcell-Meiboom-Gill (CPMG) Sequence:

• T2CPMG ≈ 85 µs ≈ 2T1

Qubit A T2: Spin-Echo at Degeneracy

From SQUID to Qubit

• The Ubiquitous 1/f Noise • Three-Junction Flux Qubit

• Relaxation and Decoherence • The C-Shunt Flux Qubit: Origin

• The C-Shunt Flux Qubit: Recent Experiments • Concluding Remarks

C-Shunt Flux Qubit: Concluding Remarks • Planar device. Combination of C-shunt, which increases design flexibility, and low loss materials yields: T1 ≈ 55 µs.

• Qubits with highest values of T1 and T2 have an anharmonicity as high as 800 - 910 MHz.

• C-shunt—which stores virtually all of the electrical energy—increases reproducibility and largely eliminates the effects of stray capacitances near the small junction.

• Large capacitor (Cs >> Cj) reduces sensitivity to charge noise. Large area increases T1 & T2 by reducing the electric field at its edges, and hence the coupling of the qubit to two-level systems.

• Flux 1/f noise extends over 14 decades of frequency, from 10−4 Hz (SQUIDs) to 4×109 Hz (flux qubits). Future

• Simulations of the electric field losses around different capacitor designs are in progress. • The C-shunt flux qubits made to date are not necessarily optimized. The trade offs between ohmic charge noise, flux noise and the Purcell effect are subtle, and detailed simulations are required to optimize the performance.