Quantum Computing Using Harmonic Oscillators in the Micromaser

1. 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

2. 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

3. 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

4. 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, 1d=4  |3, (3+1) mod 4 = |3, 0

5. 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

6. 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

7. 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

8. States of interest Coherent State Fock State Phase State

9. States of interest Approximate Phase State Coherent State Fock State

10. n = photon number in cavity field g= atom-field coupling tint = interaction time Trapping occurs when: Trapping States

11. Generating Phase States • Pump parameter • Nex = effective pump rate •   1  spread in n is maximised  tune tint and Nex to produce phase state

12. Qudits in the Micromaser • Orthogonal, non-degenerate modes in a multimode cavity • Number (Fock) State • Phase State | • These are conjugate like x, p

13. 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)

14. 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

15. 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

16. 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 Criteria

17. The 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

18. 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 Cavities Whispering Gallery Mode Quantum Dot from: http://www.its.caltech.edu/~vahalagr/

19. 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.