Tim Freegarde

Dipartimento di Fisica, Università di Trento, 38050 Povo, Italy

Quantum Optics & Laser Science, Imperial College, London SW7 2BZ, UK

Danny Segal

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Matrices

• Non-zero elements cluster around leading diagonal

• mi,j and mi+2n,j+2n differ only through momentum dependence

• Matrices therefore summarized as 4x4 elements:

G(t)

level name description sequence

basic G(t/) W-(,0) . FG(t/4) . W-(,0) . FG(t/4) . W+(,0) . FG(t/4) . W+(,0) . FG(t/4)

1 qubit NOT(0) invert lsb F(/2) . W+(/2,0) . F(/2)

CP1(0) if state=0, invert phase F() . W+(,0)

HAD(0) Hadamard on Q0 W+(/4, /4) . F() W+(,0)

2 qubit EX(1,0) exchange Q1, Q0 F(/2) . W-(/4, ) . G(/4) . W-(/4, /4) . F(5/4)

XOR(1,0)

CNOT Q1, Q0 F(/2) . W+(/4, ) . G(/4) . W+(/4, /4) . F(5/4)

CP2(0) if state=0, invert phase F(3/4) . G(/4) . W+(, )

HAD(1,0)

Hadamard on Q1, Q0 EX(1,0) . HAD(0) . EX(1,0) . HAD(0)

• Laser cooling may be achieved through the coherent manipulation of two-level atoms between discrete one-dimensional momentum states

• This is formally equivalent to a 'momentum state quantum computer‘

• Qubits form the binary representation of the momentum state

• Operations are combinations of laser pulses with kinetic energy dependent free phase evolution

• The logical invert, exchange, XOR and Walsh-Hadamard operations can be performed on any qubits, as well as conditional phase inversion

• These allow a binary right-rotation, which halves the width of the ground state momentum distribution in a single coherent process

• The problem of field design for the coherent control of atomic momenta may thus be tackled using techniques from quantum information processing

Bloch vectorsg

e

mixture

pure state

pure state

radiative interaction

free evolution

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• Candidate ‘toy’ system• Size scales with number of states, so number

of qubits limited• Practical implementation using stimulated

Raman transitions between hyperfine levels• Extension to 2-D for parallel computing

QUANTUM COMPUTING

COHERENT COOLING

• Offers maximum narrowing of momentum distribution within coherent process

• Imperfect application nonetheless cools non-integer momenta

• Complex optical pulse sequences related to ‘coherent control’ fields

FUTURE ALGORITHMS

• Grover-type search for cold states• More complex entanglement (>2 states)

cos i e-isin 0 0 0 0 0 0

i eisin cos 0 0 0 0 0 0

0 0 cos i e-isin 0 0 0 0

0 0 i eisin cos 0 0 0 0

0 0 0 0 cos i e-isin 0 0

0 0 0 0 i eisin cos 0 0

0 0 0 0 0 0 cos i e-isin0 0 0 0 0 0 i eisin cos

p0 + 4k

p0 + 3k

p0 + 2k

p0 + k

p0

p0 - k

p0 - 2k

p0 - 3k