1. Josephson Junctions, What are they?
- A Superconductor-Insulator-Superconductor device, placed between two electrodes.
-Josephson Effect: the phase of the wavefunction of a superconducting
electron pair separated by an insulator maintains a fixed phase relation.
-This means that we can describe the wavefunction around the loop of a
Superconductor, with only a phase difference due to the presence of the insulating
Gap.
-This is the very basic form of quantum coherence. The wavefunction in one
branch is coherent with the wavefunction of the second branch. Thus if
we manipulate the state it will be continuous across the boundary with a only
phase difference.

2. Superconductors
A superconductor is a metal that allows a current to pass through it with no loss due to heat dissipation.
Typical values for the critical temperature range from mK to 100K
Using Superconductors we can preserve a wavefunction because the fact that the current wavefunction is not perturbed by its journey through the metal means that it will stay in a given state.
Metal
Critical T(K)
Aluminum
1.2K
Tin
3.7K
Mercury
4.2K
Niobium
9.3K
Niobium-Tin
17.9K
Tl-Ba-Cu-oxide
125K
The current can be seen as a wavefunction, and is thus
A probability distribution of different current values, this implies that clockwise and counter clockwise. It is this view of the current that enables us to create qubits from a simple loop of superconductor.

3. Superconductors II
-When a metal is cooled to the critical temperature, electrons in the metal form Cooper Pairs.
-Cooper Pairs are electrons which exchange phonons and become bound together.
-Bound electrons behave like bosons. Their wavefunctions don’t obey
Pauli exclusion rule and thus they can all occupy the same quantum state.
-The BCS theory of Superconductivity states that bound photons have slightly lower
energy, which prevents lattice collisions and thus eliminates resistance.
-As long as kT < binding energy, then a current can flow without dissipation.

4. Cooper Pairs
-Cooper pairs can tunnel together through the insulating layer of Josephson Junction.
-This process is identical to that of quantum barrier
penetration in quantum mechanics.
-Because of the superconducting nature (no
resistance) and the fact that Cooper pairs
can jointly tunnel through an insulator we can
maintain a quantum current through the Josephson Junction without an applied voltage.
-A changing magnetic field induces a current to flow in a ring of metal, this effect
can be used to detect flux quanta. Radio Astronomy uses these devices frequently.
-Thus a Josephson Junction can be used as a very sensitive voltage, current or
flux detector.

5. Josephson Junction Devices
-There are three primary Josephson Junction devices.
-The Cooper Pair box is the most basic device. We can envision it as a system with easily split levels, and use the degenerate lowest energy levels as a qubit.
-Similarly to the Cooper Pair box we can use inductors to adjust, a Josephson Junction, until the potential represented by the potential well is a degenerate double well. We can then use symmetric and anti- symmetric wavefunctions and their associated eigenvalues as |0> and |1>.

6. Josephson Junction Devices II
A current-biased Josephson Junction employs creates a “washboard” shaped potential.
Splitting in the wells indicates allows us to use the lowest two levels as qubit states.
The higher energy state |1> can be detected because the tunneling probability under a microwave probe will be 500 times as probable to induce a transition.
Creates a detectable voltage by “going downhill.” Thus we can know the state.

7. Why Josephson Junctions?
Microscopic implementations:
based on electron spins, nuclei spins, or other microscopic properties
(+)decohere slowly as naturally distinguishable from environment
(+)single ions can be manipulated with high precision
(-)hard to apply to many qubits
(-)difficult to implement with devices
Macroscopic Implementations: Solid State
Semiconductors: quantum dots, single donor systems
Superconductors: Josephson Junctions:
more success so far
Josephson tunnel junction is “the only non-dissipative, strongly non-linear circuit element available at low temperature “

8. Benefits of Josephson Junctions
Low temperatures of superconductor:
no dissipation of energyno resistanceno electron-electron interactions(due to energy gap of Cooper pairs)
low noise levels
Precise manipulation of qubits possible
Scalable theoretically for large numbers of qubits
Efficient use of resources: circuit implementation using existing integrated circuit fabrication technology
Nonlinear Circuit Element
Needed for quantum signal processing
“easy” to analyze electrodynamics of circuit
Current versus flux across Josephson Junction

9. Circuit Implementation Issues
Electrical measurements of circuit elements:
Classical Quantum =
Numerical values wavefunctions
- E.g. classical capacitor charge  superposition of positive and negative charge
C = 10 pF  |C > = a*|0> + b*|1>
Need to implement gate operations for transferring qubit information between junction and circuit via entanglement:
Read, Write, Control
But need to avoid introducing too much noise to system, want to isolate qubits from external electrodynamic environment

10. Problems Intrinsic decoherence due to entanglement
Statistical variations inherent in fabrication  transition frequencies and coupling strength determined and taken into account in algorithms
Noise from environment causes time dependent decoherence and relaxation
relaxation: bloch sphere latitude diffusing, state mixing-
decoherence: bloch sphere longtitude diffusing, dephasing -
Due to irreversible interaction with environment,
destroys superposition of states
-       change capacitor dielectric constant
low frequency parts of noise cause
resonance to wobble
diphase oscillation in circuit
noise with frequency of transition will cause transition between states energy relaxation

11. More Problems Unwanted transitions possible Spurious resonance states:
Can engineer energy difference between states to avoid this
Spurious resonance states:
Example: spurious microwave resonators inside Josephson tunnel barrier coupling destroys coherence by decreasing amplitude of oscillations
Measurement Crosstalk: entanglement of different qubits
Measuring 1 qubit affects state of other qubits
solve with single shot measurement of all qubits
2 qubits done, but multiple will be a challenge

12. Current Research in Superconducting Qubits
Identification and reduction of sources of decoherence
Improved performance of qubit manipulation

13. Decoherence In Josephson Phase Qubits from Junction Resonators
Microscopic two-level systems (resonators) found within tunnel barriers
Affect oscillation amplitude rather than timing

14. Decoherence In Josephson Phase Qubits from Junction Resonators

15. Simultaneous State Measurement of Coupled Josephson Phase Qubits
Previous studies rely on separate measurements of each qubit
Need simultaneous measurement to establish entanglement
Crosstalk necessitates faster measurement schemes

16. Simultaneous State Measurement of Coupled Josephson Phase Qubits

17. Faster Qubit Measurement Scheme
Allows for study of 2-qubit dynamics
~2-4ns measurement scheme is an order of magnitude faster than previous ones
Short bias current pulse reduces well depth

18. Superconducting Tetrahedral Quantum Bits

19. Superconducting Tetrahedral Quantum Bits
Enhanced quantum fluctuations allow junctions of higher capacitances
Quadratic susceptibility to flux, charge noise
Variety of manipulation schemes using magnetic or electric bias