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Josephson Junction Qubits
Alex Hegyi
Justin Ellin
Andrew Chan
Classical Resistance (Review)
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Metals
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In a metal, the electrons are shared by atoms in a lattice.
This sea of electrons is free to travel along the entire lattice.
Dissipation
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Caused by inter-electron/ion interactions or other atoms, resulting
in heat
(dV) = (dI)R, R = pL/A (p resistivity, length, cross-sectional area)
P = IV
Prevents indefinite propagation of currents, analogous to friction
Superconductors
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Superconductor Properties
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State characterized by zero (exactly) electrical resistance
Meissner Effect – weak external fields only penetrate
small distances (London skin Depth)
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Type I – Superconductivity destroyed abruptly when field
reaches critical value
Type II – additional critical temperature which permits
magnetic flux but still no electrical resistivity
Generation of a current to cancel external field
BSC Theory
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Fermi Energy
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The lowest energy of the highest occupied quantum state at
absolute zero was considered to be the Fermi Energy
Where N/V is the density of fermions
This can be derived by considering a 3-dimensional square box.
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BSC- Bardeen, Cooper, and Schrieffer 1957
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The theory essentially accounts for an energy level even
below this threshold.
The gap between this energy level and the fermi energy
accounts for many of the properties of superconductors
Whereas before the electron could be excited in a continuous
spectrum of possible energy interactions (and
interchange/lose energy with lattice and other electrons),
there is now a discrete energy gap.
The excitations become forbidden and the electron sees no
“obstacles” or no resistance! But what accounts for this gap?
Cooper Pairs
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The atoms in a lattice are not fixed
Free electrons are repulsed from other electrons but are able to
attract and distort the positively charged nucleus. This distortion in
turn attracts other electrons.
Coupling (on the order of fractions of an eV) usually broken by
thermal energy or coulomb interaction.
When the thermal energy is low, T ~ 5K, this dominates effectively
linking electrons in pairs to each other even over “large” distances .
The electrons pair up with those of opposite spin.
Exclusion principle no longer applies. All electron pairs condense into
this bound state energy.
Two Notes on Modern Superconductors
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Current Lifetime – occasionally interactions may result that
do go across the gap.
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Experimentally, currents on superconductors can perpetuate for
upwards of tens of thousands of years.
Theoretically, could last longer than the known age of the
universe.
High Temperature Superconductors –superconductors that
can’t be explained by BCS because state achieved well above
fermi levels
 (Sn5In)Ba4Ca2Cu10Oy: superconducting at ~200K (Dry ice
is about this range)
 How do they work?
Josephson Junction
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Brian David Josephson proposed (1964) sandwiching an insulator
between two superconductors.
Provided separation is small, current will tunnel through the
barrier
However when the current reaches a certain critical value then a
voltage will develop across the junction which will in turn increase
the voltage further.
The frequency of this oscillation is ~ 100 GHz
Below this critical current, no voltage. Above, oscillating voltage.
Some Uses of Junctions
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SQUIDs (superconducting quantum interference devices)
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Precise Measurements
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Voltage to Frequency Converter
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Single-Electron Transistors
Flux Qubit
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Quantum state is stored in the direction
of the current
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|0> is counter-clockwise
|1> is clockwise
Manipulate State
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Requires a constant external
magnetic flux
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Flux determines the energy difference
between the two states
Apply a microwave pulse
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Causes the flux qubit to oscillate
between ground state (|0>) and
excited state (|1>)
SQUID
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Superconducting Quantum
Interference Device
Critical Current
Below:
Current flows without voltage
Above:
Oscillating current develops
Measurement
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Apply a current pulse to SQUID
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Collapses state
Magnetic flux through flux qubit
determines critical current of SQUID
Qubit Interaction
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Entanglement between two qubits is
achieved by coupling their fluxes
Superconducting bus
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Transfers a quantum state from one
qubit to another by sending a single
photon along a superconducting wire
“Additional” DiVincenzo Criteria
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Conversion of stationary, flying
qubits
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Transmission of flying qubits
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Optical Microcavities, Cavity QED
Fiber Optics
Microwave transmission lines
(Circuit QED)—way to accomplish
the above in case of
superconducting qubits*
*Wallraff et al., Nature, 431, 9 Sept. 2004
Strong Coupling/Cavity QED
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Two-level quantum system coupled
to electromagnetic cavity
“Strong Coupling” characterized as
coherent exchange of excitation
between cavity, quantum system
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i.e., coherent conversion between
stationary, flying qubit
Model—Two SHOs connected by
weak spring
Microwave Resonator/Qubit System
*Schoelkopf and Girvin,
Nature, 451, 7 Feb. 2008
Quantum Communication
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If energy difference between |0>
and |1> resonant with cavity,
energy exchanged (Rabi rotation)
If off-resonant (dispersive) energy
not exchanged
Align qubits along transmission line,
tune energy difference (using gate
bias, flux bias) to control interaction
with line
Microwave Resonator/Qubit System
*Wallraff et al., Nature, 431, 9 Sept. 2004