Coherent cooling: a momentum state quantum computer

1. Coherent cooling: a momentum state quantum computer Danny Segal Tim Freegarde Dipartimento di Fisica, Università di Trento, 38050 Povo, Italy Quantum Optics & Laser Science, Imperial College, London SW7 2BZ, UK Quantum States & Qubits Manipulation of Individual Qubits • 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 • interaction with 1-D optical field couples ladder of momentum states • short pulse, broad bandwidth interaction is Doppler insensitive • states alternate between ground and excited electronic levels • Qubits form binary representation of state momentum Bloch vector representation of first element Quantum Operations Notation • The first element is an inter-ferometer which inverts, mixes or leaves unchanged the state, depending upon the momentum • Two such elements are combined to yield a selective state swap • A sequence of four swaps forms the EX(2,1) qubit exchange Interaction with laser field Matrices p-pulse (inversion) • W+,- couple pairs of states, according to the photon direction • interactions rotate Bloch vector through angle 2a about axis in horizontal plane • angle 2a corresponds to fraction of Rabi cycle performed • phase f defines azimuthal angle of • rotation axis • 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: p/2-pulse (beamsplitter) 2p-pulse (identity) Coherent Cooling Free evolution • F and G describe free evolution according to Schrödinger's equation • F accounts for the phase evolution due to the electronic energy • G accounts for the phase evolution due to the kinetic energy • separation possible using technique borrowed from interferometric cooling Atomic Interferometry f • The initial ground-state distribution is transferred to the lowest momentum states • Subsequent spontaneous emission leaves population in the lowest momentum ground states • Repeating the process further narrows the momentum distribution • Although the process applies perfectly only to even, integral momenta, significant cooling remains apparent • Two p/2 pulses act as beamsplitters for an atomic interferometer • The relative phase between the two paths determines whether the p/2 pulses add or subtract, and hence whether or not the electronic state is inverted • Combining the EX(2,1) and EX(1,0) qubit exchanges produces a right-rotation of the binary number • If Q0 is initially zero, the right-rotation corresponds to division by 2 QUANTUM COMPUTING G(t) • 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 See especially M Weitz, B C Young, S Chu, Phys Rev Lett 73 (19) 2563 (1994) Quantum Computer Instruction Set COHERENT COOLING Bloch vectors • 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 pure state FUTURE ALGORITHMS mixture • Grover-type search for cold states • More complex entanglement (>2 states) pure state Simulated evolution of the momentum state distribution, shown after 1, 2, 4 and 8 cycles of the 3-qubit coherent cooling algorithm. radiative interaction free evolution