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Quantum Computing Using Harmonic Oscillators in the Micromaser

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### Quantum Computing Using Harmonic Oscillatorsin the Micromaser

Dr. Ben Varcoe,

Martin Jones, Gary Wilkes

University of Sussex

Department of Physics and Astronomy

Atomic, Molecular and Optical Physics group

Qubits Qudits

- Qudit = d-dimensional system
- computational basis:
- qubit: {|0, |1}
- qudit: {|s: s = 0, 1, ... , d-1}
- Dimensions of Hilbert Space
- qubit 2n
- qudit dn
- d Continuous Variable computation

}

for n quantum systems

Physical Qudits

- Harmonic Oscillators
- Position and Momentum of a particle
- Gottesman, Kitaev and Preskill: PRA 64, 012310
- Amplitude and Phase of a field
- Bartlett, de Guise and Sanders: PRA 65, 052316

The SUM Gate

- USUM|a, b |a, (a+b) mod d “addition modulo-d”
- e.g. for d = 4; |a = |3; |b = |1

USUM|3, 1d=4 |3, (3+1) mod 4 = |3, 0

Special Case: d = 2

- d = 2

USUM|00 |00 USUM|01 |01 USUM|10 |11 USUM|11 |10 USUM UCN (for d = 2)

SUM gate is a generalised CNOT gate

The Micromaser

- Single atoms and single modes of the field interact via Jaynes-Cummings Hamiltonian

from: http://prola.aps.org/figure/PRA/v46/i1/p567_1/fig1

Micromaser Basics

- Rubidium-85 excitedby three step laser to upper Rydberg level
- Transition between two levels is resonant with microwave cavity mode
- Detection of atoms provides information about field

n = photon number in cavity field

g= atom-field coupling

tint = interaction time

Trapping occurs when:

Trapping StatesGenerating Phase States

- Pump parameter
- Nex = effective pump rate
- 1 spread in n is maximised

tune tint and Nex to produce phase state

Qudits in the Micromaser

- Orthogonal, non-degenerate modes in a multimode cavity
- Number (Fock) State
- Phase State |
- These are conjugate like x, p

Scaling in the Micromaser

- Nex defines maximum Fock state in the phase state superposition; e.g. for Nex = 5:

| = a|0 + b |1 + c |2 + d |3 + e |4 + f |5

- So two qudits give dn = 36 states
- Maximum Nex 1500
- n = 2 (1500)2 = 2.25 million states!!(compared to 4 for qubits)

SUM Gate in the Micromaser

- Couple two modes via non-linear Kerr media,
- e.g. a suitable atom
- Gives:
- So if t= -1, interaction is a SUM gate
- possible in the micromaser (t 1/g)

(n+1)P3/2

(n+1)S1/2

nP3/2

Single Qudit Operations

- Arbitrary unitary transformations by injecting sequences of appropriate atoms
- linear displacement of cavity mode
- squeezing of the field state
- non-linear Kerr transformations
- Fourier transform converts between Fock and phase eigenstates

A scalable physical system with well-characterised qudits

The ability to initialise the qudit state

Decoherence times much longer than the quantum gate operation time

A universal set of quantum gates

The ability to measure specific qudits

?

DiVincenzo CriteriaThe Future

- Desktop Micromaser Quantum PCs?
- Micromaser theory can be used in some quantum dot proposals
- allows miniaturisation
- better control over “atoms” (e.g. tint)
- very strong “atom” – field interaction

Higher frequency (100THz vs. GHz)

lower mode volume (m3 vs. cm3)

More than compensates for reduced lifetime

Single quantum dot

Whispering Gallery Mode

Reduced photon lifetime

Microdisk CavitiesWhispering Gallery Mode

Quantum

Dot

from: http://www.its.caltech.edu/~vahalagr/

Summary

- Qudits offer a new and potentially more efficient alternative to qubits.
- The micromaser is a promising candidate for quantum information applications.
- Implementation of a qudit QC in the micromaser looks possible.
- Possibility of future incorporation into solid state architectures.

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