Superinductor with Tunable Non-Linearity

M.E. Gershenson

M.T. Bell, I.A. Sadovskyy, L.B. Ioffe, and A.Yu. Kitaev*

Department of Physics and Astronomy, Rutgers University, Piscataway NJ

* Caltech, Institute for Quantum Information, Pasadena CA

Outline:

Superinductor: why do we need it?

Our Implementation of the superinductor

Microwave Spectroscopy and Rabi oscillations

Potential Applications

- A new fully tunable platform for the study of quantum phase transitions?

Impedance controls the scale of zero-point motion in quantum

circuits:

- reduction of the sensitivity of Josephson qubits to the charge noise,

- Implementation of fault tolerant computation based on pairs of Cooper pairs and pairs of flux quanta (Kitaev, Ioffe),

- ac isolation of the Josephson junctions in the electrical current standards based on Bloch oscillations.

Superinductor:dissipationless inductor

Z >> No extra dephasing

Potential applications:

Why Superinductors?

Conventional “Geometric” Inductors

𝑍=𝜔𝐿≈√ 𝜇0

𝜀0=8𝛼×𝑅𝑄 0.4 𝑘Ω

the fine structure constant2

0

1 12 137

ehc

Geometrical inductance of a wire: ~ 1 pH/m.

Hence, it is difficult to make a large (1 H 6 k

@ 1 GHz) L in a planar geometry.

Moreover, a wire loop possesses not only geometrical

inductance, but also a parasitic capacitance, and its microwave

impedance is limited:

Tunable Nonlinear Superinductor

𝐿𝐾ቆ𝑑2𝐸𝐽ሺ𝜑ሻ𝑑𝜑2 ቇ

−1

Δ𝜙=2𝜋 ΦΦ0Φ0≡

h2𝑒 ≈20𝐺 ∙𝜇𝑚2

For the optimal EJL/EJS, the energy becomes “flat” at =1/20.

- diverges, the phase fluctuations are maximized.

Josephson energy of a two cell device (classical approx., )

𝒓 ≡ 𝑬 𝑱𝑳

𝑬 𝑱𝑺

Unit cell of the tested devices: asymmetric dc SQUID threaded by

the flux .

𝐸𝐽= −5× 𝐸𝐽2𝑐𝑜𝑠ቀ𝜑5ቁ−𝐸𝐽1𝑐𝑜𝑠ቀ2𝜋ΦΦ0 −3𝜑5ቁ−𝐸𝐽1𝑐𝑜𝑠ቀ2𝜋ΦΦ0 +3𝜑5ቁ.

Kinetic InductanceThis limitation does not apply to superconductors whose kinetic inductance

is associated with the inertia of the Cooper pair condensate.

Manucharyan et at., Science 326, 113 (2009).

Long chains of ultra-small Josephson junctions:

(up to 0.3 H)

Nanoscale superconducting wires:

InOx films, d=35nm, R~3 k, L~4 nH Astafiev et al., Nature 484, 355 (2012).

NbN films, d=5nm, R~0.9 k, L~1 nH Annunziata et al., Nanotechnology 21, 445202 (2010).

𝐸 𝐽=h

8𝑒2∆𝑅𝑁

=( Φ0

2𝜋 )2 1𝐿𝐾

𝐿𝐾=( Φ0

2𝜋 )2 1𝐸 𝐽

=ℏ𝑅𝑠𝑞

𝜋∆

Tunable Nonlinear Superinductor (cont’d)

I cell2 cells

4 cells6 cells

𝒓𝐨≡( 𝑬 𝑱𝑳

𝑬 𝑱𝑺 )𝒐𝒑𝒕Optimal depends on the ladder length.

two-wellpotential

Inductance Measurements

CK

LC

LC

LC- resonatorinductor

resonatorLK

Two coupled (via LC) resonators:- decoupling from the MW

feedline- two-tone measurements with

the LC resonance frequency within the 3-10 GHz setup bandwidth.

𝜔𝐿𝐶

2𝜋 ≈6−7𝐺𝐻𝑧𝜔𝐾

2𝜋 ≈1−20𝐺𝐻𝑧

1-11

GH

z 3-

14 G

Hz

MW feedline

Dev1Dev2

Dev3Dev4

Multiplexing: several devices

with systematically varied parameters.

“Manhattan pattern” nanolithography

Multi-angle deposition

of Al

On-chip Circuitry

Devices with 6 unit cells

Device

𝐸𝐽𝑆, K

𝐸𝐶𝑆, K

𝐸𝐽𝐿, K

𝐸𝐶𝐿, K 𝑟≡ 𝐸𝐽𝐿𝐸𝐽𝑆

𝐿𝐾ሺΦ = 0ሻ, nH

𝐿𝐾ሺΦ = Φ0/2ሻ, nH

1 3.5 0.46 15 0.15 4.3 3.7 150

2 3.5 0.46 14.3 0.15 4.1 3.8 310

Hamiltonian diagonalization

𝑟 o (𝑁=6 )=(𝐸 𝐽𝐿

𝐸 𝐽𝑆)opt ≈4.1 - for the ladders with six unit cells

4.5

4.3

𝑟 o

𝑟

Rabi Oscillationsa non-linear quantum system in the presence of an resonance driving field.

1

The non-linear superinductor shunted by

a capacitor represents a Qubit.

Damping of Rabi oscillations is due to the

decay (coupling to the LC resonator and the feedline).

Mechanisms of Decoherence

Decoherence due to Aharonov-Casher effect:

fluctuations of offset charges on the islands + phase slips. The phase slip

rate

is negligible (for the junctions in the ladder backbone ).

exp (−𝑐√ 𝐸 JL

𝐸CL) 𝑐≅ 2.5−2.8

𝐸 JL

𝐸CL(≅ 100)

Decoherence due to the flux noise:

Because the curvature (which controls the position of energy levels) has a

minimum at full frustration, one expects that the flux noise does not affect

the qubit in the linear order.

Ladders with 24 unit cells

𝐿𝐾 (𝛷=𝛷0/2 )=3𝜇𝐻almost linear inductor

~ 100m

𝑟 ≈5.2𝑟o (𝑁=24 )≈ 4.5

two-wellpotential

Ladders with 24 unit cells (cont’d)

Number of unit cells

, K , K , K

, KfF nH

, nH , nH

24 3.15 0.46 14.5 0.15 4.6 5 0.8 16 3 000

𝑟 ≈ 4.6𝑟o (𝑁=24 )=(𝐸 𝐽𝐿

𝐸 𝐽𝑆)opt≈ 4.5𝑵=𝟐𝟒

Ladders with 24 unit cells (cont’d)

𝑳𝑲 (𝜱=𝜱𝟎 /𝟐 )=𝟑𝝁𝑯- this is the inductance of a 3-

meter-long wire!

𝑍 (3𝐺𝐻𝑧 )=50 𝑘Ω>𝑅𝑄≡h

(2𝑒 )2

quasi-classical modeling

Double-well potential

𝑟 ≈ 4.2𝑟 o (𝑁=24 )≈ 4.5

Φ=Φ

0

2

crit.

poi

nt

A new fully tunable platform for the study of quantum phase transitions?

Summary

Our Implementation of the superinductor

Microwave Spectroscopy and Rabi oscillations

- Rabi time up to 1.4 s, limited by the decay

Potential Applications

- Quantum Computing

- Current standards

- Quantum transitions in 1D

𝑳𝑲 𝐮𝐩𝐭𝐨𝟑𝝁𝑯