Algorithmic Cooling in a Momentum State Quantum Computer

We describe a quantum computer based upon the coherent manipulation of two-level atoms between discrete one-dimensional momentum states. Combinations of short laser pulses with kinetic energy dependent free phase evolution can perform the logical invert, exchange, controlled-NOT, and Hadamard operations on any qubits in the binary representation of the momentum state, as well as conditional phase inversion. These allow a binary right rotation, which halves the momentum distribution in a single coherent process. Fields for the coherent control of atomic momenta may thus be designed as quantum algorithms.

Figure 1

(a) Fractional $\pi$ pulses tuned to the two-level atom couple adjacent momentum states, which (b) we label in units of $\hslash k$. This number, written in binary, gives the qubits of the momentum state quantum computer. (c) Two $\pi /2$ pulses act as the beam splitters of an atomic interferometer; the relative phase between the paths determines whether the pulses add or subtract, and hence whether the electronic state is inverted.

Figure 2

Bloch-vector representation of the first stage [Eq. (2)] of the right rotation. The first pulse rotates the four ground states into the horizontal plane; free evolution distributes these around the vertical axis according to their momenta; the second pulse then returns two to eigenstates, leaving the others as superpositions. Aside from phase corrections, the full right rotation takes 18 $\pi /2$ pulses and 26 $\pi$ pulses.

Figure 3

Cooling via the right-rotation operation, shown here applied to the lowest three qubits. The initial distribution across ground states $\{0,2,4,6\}$ is transferred (bold arrows) to the lowest momentum states $\{0,1,2,3\}$; subsequent spontaneous emission leaves population in states $\{0,2,4\}$. The width of the momentum distribution may thus be reduced by a factor of nearly 2 in a single coherent step.

Figure 4

Simulated evolution of the momentum distribution, shown after one, two, four, and eight cycles of the three-qubit cooling algorithm. The initial probability density is unity. While the process is ideal only for even, integral momenta, significant cooling remains apparent and, after a few cycles, the distribution is less than a single photon impulse wide. Spontaneous emission scrambles the exact momenta. There is little migration between adjacent, $8\hslash k$-wide, capture ranges.