Coherent cooling: a momentum state quantum computer Tim Freegarde Dipartimento di Fisica, Università di Trento, Povo, ItalyQuantum Optics & Laser Science, Imperial College, London SW7 2BZ, UK Danny Segal 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 Quantum States & Qubits Quantum Operations W +,- couple pairs of states, according to the photon direction interactions rotate Bloch vector through angle 2  about axis in horizontal plane angle 2  corresponds to fraction of Rabi cycle performed phase  defines azimuthal angle of rotation axis 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 Interaction with laser field Free evolution Matrices Atomic Interferometr y See especially M Weitz, B C Young, S Chu, Phys Rev Lett 73 (19) 2563 (1994) Non-zero elements cluster around leading diagonal m i,j and m i+2n,j+2n differ only through momentum dependence Matrices therefore summarized as 4x4 elements: G(t)  Quantum Computer Instruction Set levelnamedescriptionsequence 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 Q 0 W + (  /4,  /4). F(  ) W + ( ,0) 2 qubit EX(1,0) exchange Q 1, Q 0 F(  /2). W - (  /4,  ). G(  /4). W - (  /4,  /4). F(5  /4) XOR(1,0) CNOT Q 1, Q 0 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 Q 1, Q 0 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 vectors mixture pure state radiative interactionfree evolution Manipulation of Individual Qubits Notatio n  -pulse (inversi on)  -pulse (beamsp litter )  -pulse (identit y) Coherent Cooling Simulated evolution of the momentum state distribution, shown after 1, 2, 4 and 8 cycles of the 3-qubit coherent cooling algorithm. 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 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 Bloch vector representation of first element Combining the EX(2,1) and EX(1,0) qubit exchanges produces a right-rotation of the binary number If Q 0 is initially zero, the right-rotation corresponds to division by 2 Two  /2 pulses act as beamsplitters for an atomic interferometer The relative phase between the two paths determines whether the  /2 pulses add or subtract, and hence whether or not the electronic state is inverted 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 -i  sin  i e i  sin  cos  i e -i  sin  i e i  sin  cos  i e -i  sin  i e i  sin  cos  i e -i  sin  i e i  sin  cos   p  k   p  k   p  k   p 0 +  k  p0p0  p 0 -  k   p  k   p  k 