Josephson Junction Qubits

Alex HegyiJustin Ellin

Andrew Chan

Classical Resistance (Review)

Metals 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 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

Superconductor Properties State characterized by zero (exactly) electrical resistance Meissner Effect – weak external fields only penetrate small

distances (London skin Depth) 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 Fermi Energy

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 fermionsThis can be derived by considering a 3-dimensional square box.

BSC- Bardeen, Cooper, and Schrieffer 1957 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 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

Current Lifetime – occasionally interactions may result that do go across the gap.

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 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

SQUIDs (superconducting quantum interference devices)

Precise Measurements

Voltage to Frequency Converter

Single-Electron Transistors

Flux Qubit

Quantum state is stored in the direction of the current |0> is counter-clockwise |1> is clockwise

Manipulate State

Requires a constant external magnetic flux Flux determines the energy difference

between the two states Apply a microwave pulse

Causes the flux qubit to oscillate between ground state (|0>) and excited state (|1>)

SQUID

Superconducting Quantum Interference Device

Critical Current

Below:

Current flows without voltage

Above:

Oscillating current develops

Measurement

Apply a current pulse to SQUID Collapses state

Magnetic flux through flux qubit determines critical current of SQUID

Qubit Interaction

Entanglement between two qubits is achieved by coupling their fluxes

Superconducting bus Transfers a quantum state from one

qubit to another by sending a single photon along a superconducting wire

“Additional” DiVincenzo Criteria

Conversion of stationary, flying qubits Optical Microcavities, Cavity QED

Transmission of flying qubits 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

Two-level quantum system coupled to electromagnetic cavity

“Strong Coupling” characterized as coherent exchange of excitation between cavity, quantum system 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

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