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?
Why Superinductors?
Superinductor:
dissipationless inductor
ℎ
Z >> 𝑅Q ≡
2 ≈ 6.5𝑘Ω
2𝑒
No extra dephasing
Potential applications:
- 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.
Impedance controls the scale of
zero-point motion in quantum
circuits:
Conventional “Geometric” Inductors
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:
𝑍 = 𝜔𝐿 ≈
𝜇0
= 8𝛼 × 𝑅𝑄 ~0.4𝑘Ω
𝜀0
the fine structure constant
 
1 e
2
2  0 hc

1
137
Tunable Nonlinear Superinductor
𝑬𝑱𝑳
𝒓≡
𝑬𝑱𝑺
Unit cell of the tested devices:
asymmetric dc SQUID threaded by
the flux .
Φ
Δ𝜙 = 2𝜋
Φ0
ℎ
Φ0 ≡
≈ 20𝐺 ∙ 𝜇𝑚2
2𝑒
Josephson energy of a two cell device (classical approx., 𝐸𝐽𝑆 ≪ 𝐸𝐽𝑆 )
𝐸𝐽 = −5 × 𝐸𝐽2 𝑐𝑜𝑠
𝜑
5
−1
𝜑
Φ
𝜑
− 𝐸𝐽1 𝑐𝑜𝑠 2𝜋 Φ − 3 5 − 𝐸𝐽1 𝑐𝑜𝑠 2𝜋 Φ + 3 5 .
For the optimal EJL/EJS, the energy becomes
“flat” at =1/20.
𝑑2 𝐸𝐽 𝜑
𝐿𝐾 
𝑑𝜑 2
Φ
- diverges, the
phase fluctuations
are maximized.
0
0
𝒓 = 𝟒. 𝟖
𝚽=𝟎
𝒓 = 𝟒. 𝟖
𝚽𝟎
𝚽=
𝟐
𝒓 < 𝟒. 𝟖
𝚽𝟎
𝚽=
𝟐
Kinetic Inductance
This limitation does not apply to superconductors whose kinetic inductance
𝐿𝐾 is associated with the inertia of the Cooper pair condensate.
Nanoscale superconducting wires:
ℎ ∆
Φ0
𝐸𝐽 = 2
=
8𝑒 𝑅𝑁
2𝜋
2
1
𝐿𝐾
Φ0
𝐿𝐾 =
2𝜋
2
1 ℏ𝑅𝑠𝑞
=
𝐸𝐽
𝜋∆
NbN films, d=5nm, R~0.9 k, L~1 nH
Annunziata et al., Nanotechnology 21, 445202 (2010).
InOx films, d=35nm, R~3 k, L~4 nH
Astafiev et al., Nature 484, 355 (2012).
Long chains of ultra-small
Josephson junctions:
(up to 0.3 H)
Manucharyan et at., Science 326, 113 (2009).
Tunable Nonlinear Superinductor (cont’d)
two-well
potential
I cell
2 cells
4 cells
6 cells
Optimal
𝑬𝑱𝑳
𝒓𝐨 ≡
𝑬𝑱𝑺
depends on the ladder length.
𝒐𝒑𝒕
Inductance Measurements
LC- resonator
LK
inductor
resonator
3-14 GHz
1-11 GHz
CK
L
LC
C
Two coupled (via LC) resonators:
- decoupling
feedline
from
the
MW
- two-tone measurements with
the LC resonance frequency
within the 3-10 GHz setup
bandwidth.
𝜔𝐿𝐶
≈ 6 − 7 𝐺𝐻𝑧
2𝜋
𝜔𝐾
≈ 1 − 20 𝐺𝐻𝑧
2𝜋
On-chip Circuitry
“Manhattan pattern”
nanolithography
Multi-angle deposition
of Al
Dev1
Dev2
Multiplexing:
several devices
with systematically
varied parameters.
Dev3
Dev4
Devices with 6 unit cells
Hamiltonian
diagonalization
𝑟
𝑟o
Device
𝐸𝐽𝑆 ,
K
𝐸𝐶𝑆 ,
K
𝐸𝐽𝐿 ,
K
𝐸𝐶𝐿 ,
K
1
3.5
0.46
15
0.15
2
3.5
0.46
14.3
0.15
𝑟o 𝑁 = 6 =
𝐸𝐽𝐿
𝐸𝐽𝑆
𝐸𝐽𝐿
𝐸𝐽𝑆
𝐿𝐾 Φ = 0 ,
𝐿𝐾 Φ = Φ0 /2 ,
nH
nH
4.3
4.5
3.7
150
4.1
4.3
3.8
310
𝑟≡
≈ 4.1 - for the ladders with six unit cells
opt
Rabi Oscillations
a 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 the flux noise:
Because the curvature
𝑑2 𝐸J 𝜑
𝑑𝜑2
(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.
Decoherence due to Aharonov-Casher effect:
fluctuations of offset charges on the islands + phase slips. The phase slip
rate
exp −𝑐
𝐸JL
𝐸CL
𝑐 ≅ 2.5 − 2.8
is negligible (for the junctions in the ladder backbone
𝐸JL
≅ 100
𝐸CL
).
Ladders with 24 unit cells
𝑟 ≈ 5.2 𝑟o 𝑁 = 24 ≈ 4.5
~ 100m
two-well
potential
almost linear inductor
𝐿𝐾 𝛷 = 𝛷0 /2 = 3𝜇𝐻
Ladders with 24 unit cells (cont’d)
𝑟 ≈ 4.6 𝑟o 𝑁 = 24 =
Number of
unit cells
𝐸JS , K
24
3.15
𝐸CS , K
𝐸𝐽𝐿
𝐸𝐽𝑆
𝐸JL ,
≈ 4.5
opt
𝐸CL , K
K
0.46
14.5
0.15
𝑵 = 𝟐𝟒
𝑟≡
𝐸JL
𝐸JS
4.6
𝐶𝐾 ,
𝐿𝐶 ,
𝐿K Φ = 0 ,
𝐿K Φ = Φ0 /2 ,
fF
nH
nH
nH
5
0.8
16
3 000
Ladders with 24 unit cells (cont’d)
quasi-classical
modeling
𝑳𝑲 𝜱 = 𝜱𝟎 /𝟐 = 𝟑𝝁𝑯
- this is the inductance of a 3meter-long wire!
𝑍 3𝐺𝐻𝑧 = 50𝑘Ω > 𝑅𝑄 ≡
ℎ
2𝑒
2
Φ0
Φ=
2
crit. point
Double-well potential
𝑟 ≈ 4.2 𝑟o 𝑁 = 24 ≈ 4.5
A new fully tunable platform for the
study of quantum phase transitions?
Summary
Our Implementation of the superinductor
𝑳𝑲 𝐮𝐩 𝐭𝐨 𝟑𝝁𝑯
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