Superinductor with Tunable Non
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Transcript Superinductor with Tunable Non
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
โ
Z >> ๐
Q โก
2 โ 6.5๐ฮฉ
2๐
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 โGeometricโ Inductors
Geometrical inductance of a wire: ~ 1 pH/๏ญm.
Hence, it is difficult to make a large (1 ๏ญH ๏ฎ 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:
๐ = ๐๐ฟ โ
๐0
= 8๐ผ × ๐
๐ ~0.4๐ฮฉ
๐0
the fine structure constant
๏ก ๏ฝ
1 e
2
2 ๏ฅ 0 hc
๏ป
1
137
Tunable Nonlinear Superinductor
๐ฌ๐ฑ๐ณ
๐โก
๐ฌ๐ฑ๐บ
Unit cell of the tested devices:
asymmetric dc SQUID threaded by
the flux ๏.
ฮฆ
ฮ๐ = 2๐
ฮฆ0
โ
ฮฆ0 โก
โ 20๐บ โ ๐๐2
2๐
Josephson energy of a two cell device (classical approx., ๐ธ๐ฝ๐ โช ๐ธ๐ฝ๐ )
๐ธ๐ฝ = โ5 × ๐ธ๐ฝ2 ๐๐๐
๐
5
โ1
๐
ฮฆ
๐
โ ๐ธ๐ฝ1 ๐๐๐ 2๐ ฮฆ โ 3 5 โ ๐ธ๐ฝ1 ๐๐๐ 2๐ ฮฆ + 3 5 .
For the optimal EJL/EJS, the energy becomes
โflatโ at ๏=1/2๏0.
๐2 ๐ธ๐ฝ ๐
๐ฟ๐พ ๏ต
๐๐ 2
ฮฆ
- diverges, the
phase fluctuations
are maximized.
0
0
๐ = ๐. ๐
๐ฝ=๐
๐ = ๐. ๐
๐ฝ๐
๐ฝ=
๐
๐ < ๐. ๐
๐ฝ๐
๐ฝ=
๐
Kinetic Inductance
This limitation does not apply to superconductors whose kinetic inductance
๐ฟ๐พ is associated with the inertia of the Cooper pair condensate.
Nanoscale superconducting wires:
โ โ
ฮฆ0
๐ธ๐ฝ = 2
=
8๐ ๐
๐
2๐
2
1
๐ฟ๐พ
ฮฆ0
๐ฟ๐พ =
2๐
2
1 โ๐
๐ ๐
=
๐ธ๐ฝ
๐โ
NbN films, d=5nm, R๏ฏ~0.9 k๏, L๏ฏ~1 nH
Annunziata et al., Nanotechnology 21, 445202 (2010).
InOx films, d=35nm, R๏ฏ~3 k๏, L๏ฏ~4 nH
Astafiev et al., Nature 484, 355 (2012).
Long chains of ultra-small
Josephson junctions:
(up to 0.3 ๏ญH)
Manucharyan et at., Science 326, 113 (2009).
Tunable Nonlinear Superinductor (contโd)
two-well
potential
I cell
2 cells
4 cells
6 cells
Optimal
๐ฌ๐ฑ๐ณ
๐๐จ โก
๐ฌ๐ฑ๐บ
depends on the ladder length.
๐๐๐
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.
๐๐ฟ๐ถ
โ 6 โ 7 ๐บ๐ป๐ง
2๐
๐๐พ
โ 1 โ 20 ๐บ๐ป๐ง
2๐
On-chip Circuitry
โManhattan patternโ
nanolithography
Multi-angle deposition
of Al
Dev1
Dev2
Multiplexing:
several devices
with systematically
varied parameters.
Dev3
Dev4
Devices with 6 unit cells
Hamiltonian
diagonalization
๐
๐o
Device
๐ธ๐ฝ๐ ,
K
๐ธ๐ถ๐ ,
K
๐ธ๐ฝ๐ฟ ,
K
๐ธ๐ถ๐ฟ ,
K
1
3.5
0.46
15
0.15
2
3.5
0.46
14.3
0.15
๐o ๐ = 6 =
๐ธ๐ฝ๐ฟ
๐ธ๐ฝ๐
๐ธ๐ฝ๐ฟ
๐ธ๐ฝ๐
๐ฟ๐พ ฮฆ = 0 ,
๐ฟ๐พ ฮฆ = ฮฆ0 /2 ,
nH
nH
4.3
4.5
3.7
150
4.1
4.3
3.8
310
๐โก
โ 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
๐2 ๐ธJ ๐
๐๐2
(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 โ๐
๐ธJL
๐ธCL
๐ โ
2.5 โ 2.8
is negligible (for the junctions in the ladder backbone
๐ธJL
โ
100
๐ธCL
).
Ladders with 24 unit cells
๐ โ 5.2 ๐o ๐ = 24 โ 4.5
~ 100๏ญm
two-well
potential
almost linear inductor
๐ฟ๐พ ๐ท = ๐ท0 /2 = 3๐๐ป
Ladders with 24 unit cells (contโd)
๐ โ 4.6 ๐o ๐ = 24 =
Number of
unit cells
๐ธJS , K
24
3.15
๐ธCS , K
๐ธ๐ฝ๐ฟ
๐ธ๐ฝ๐
๐ธJL ,
โ 4.5
opt
๐ธCL , K
K
0.46
14.5
0.15
๐ต = ๐๐
๐โก
๐ธJL
๐ธJS
4.6
๐ถ๐พ ,
๐ฟ๐ถ ,
๐ฟK ฮฆ = 0 ,
๐ฟK ฮฆ = ฮฆ0 /2 ,
fF
nH
nH
nH
5
0.8
16
3 000
Ladders with 24 unit cells (contโd)
quasi-classical
modeling
๐ณ๐ฒ ๐ฑ = ๐ฑ๐ /๐ = ๐๐๐ฏ
- this is the inductance of a 3meter-long wire!
๐ 3๐บ๐ป๐ง = 50๐ฮฉ > ๐
๐ โก
โ
2๐
2
ฮฆ0
ฮฆ=
2
crit. point
Double-well potential
๐ โ 4.2 ๐o ๐ = 24 โ 4.5
A new fully tunable platform for the
study of quantum phase transitions?
Summary
Our Implementation of the superinductor
๐ณ๐ฒ ๐ฎ๐ฉ ๐ญ๐จ ๐๐๐ฏ
Microwave Spectroscopy and Rabi oscillations
- Rabi time up to 1.4 ๏ญs, limited by the decay
Potential Applications
- Quantum Computing
- Current standards
- Quantum transitions in 1D