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Quantum Electronics and Laser Science Conference

Application of quantum control theory to manipulate the Zeeman states of an atom

Daniel J. Heinzen, G. Xu

Published Jan 20, 2003• 1 citation
Publisher

Abstract

Summary form only given. We present experimental results that demonstrate a high degree of control over the spin state of an atom. The experiment is carried out on a sample of about 10/sup 7/ laser-cooled /sup 133/Cs atoms in their F=4 ground state hyperfine level released from a magneto-optical trap. Our method is to apply a static magnetic field B/sub 0/, a time-dependent transverse magnetic field B/sub x/(t), and a linearly polarized off-resonance laser field to the atoms. The Hamiltonian is H=-/spl mu//spl middot/B(t)+ M/sup 2//spl omega//sub E/|M><M|, where B(t)=B/sub 0/z+B/sub x/(t)x. B/sub 0/ induces a precession of the spins at the Larmor frequency /spl omega//sub L/=/spl mu//sub B/B/4/spl planck/. The second term in H arises from the laser field light shift; together with the static field B/sub 0/ it yields a finite anharmonic ladder of quantum states, labeled by the spin projection quantum number M=-F,...,F. Because of the anharmonicity, B/sub x/(t) induces motions of the spin that are not simply rotations of the initial state. Rather, through the application of quantum control theory, it is possible to find a pulse B/sub x/(t) that produces time evolution connecting arbitrary spin states.

DOI
10.1109/qels.1999.807491

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Summary form only given. We present experimental results that demonstrate a high degree of control over the spin state of an atom. The experiment is carried out on a sample of about 10/sup 7/ laser-cooled /sup 133/Cs atoms in their F=4 ground state hyperfine level released from a magneto-optical trap. Our method is to apply a static magnetic field B/sub 0/, a time-dependent transverse magnetic field B/sub x/(t), and a linearly polarized off-resonance laser field to the atoms. The Hamiltonian is H=-/spl mu//spl middot/B(t)+ M/sup 2//spl omega//sub E/|M><M|, where B(t)=B/sub 0/z+B/sub x/(t)x. B/sub 0/ induces a precession of the spins at the Larmor frequency /spl omega//sub L/=/spl mu//sub B/B/4/spl planck/. The second term in H arises from the laser field light shift; together with the static field B/sub 0/ it yields a finite anharmonic ladder of quantum states, labeled by the spin projection quantum number M=-F,...,F. Because of the anharmonicity, B/sub x/(t) induces motions of the spin that are not simply rotations of the initial state. Rather, through the application of quantum control theory, it is possible to find a pulse B/sub x/(t) that produces time evolution connecting arbitrary spin states.
We experimentally investigate the coherence properties of a qubit stored in the Zeeman substates of the 52S1/2,F=1{5}^{\phantom{\rule{0.16em}{0ex}}2}{S}_{1/2},\phantom{\rule{0.222222em}{0ex}}F=152S1/2​,F=1 hyperfine ground level of a single optically trapped 87{}^{87}87Rb atom. Larmor precession of a single atomic spin-1 system is observed by preparing the atom in a defined initial spin state and then measuring the resulting state after a programmable period of free evolution. Additionally, by performing quantum-state tomography, maximum knowledge about the spin coherence is gathered. By using an active magnetic field stabilization and without application of a magnetic guiding field, we achieve transverse and longitudinal dephasing times of ${T}_{2}^{*}=75\ensuremath{-}150\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{s}$ and T1>0.5ms,{T}_{1}>0.5\phantom{\rule{0.28em}{0ex}}\mathrm{ms},T1​>0.5ms, respectively. We derive the light-shift distribution of a single atom in the approximately harmonic potential of a dipole trap and show that the measured atomic spin coherence is limited mainly by residual position- and state-dependent effects in the optical trapping potential. The improved understanding enables longer coherence times, an important prerequisite for future applications in long-distance quantum communication and computation with atoms in optical lattices, or for a loophole-free test of Bell's inequality.
Ultracold atom experiments use a gas of neutral atoms with temperatures less than 100 µK above absolute zero and offer unmatched experimental control of quantum states and coherence, which has allowed ultracold atom-based measurements to be some of the most precise to date. While ultracold atom experiments can control almost all atomic degrees of freedom, spin-dependent trapping and spatial manipulation has remained difficult if not inaccessible. We are developing a method of spin-dependent trapping and spatial manipulation for ultracold neutral atoms using the AC Zeeman force produced by a microwave magnetic near-field gradient generated by an atom chip. We measure the AC Zeeman force on ultracold rubidium atoms by observing its effect on the motion of atoms in free-fall and on those confined in a trap. We have studied the force as a function of microwave frequency detuning from a hyperfine transition at 6.8 GHz at several magnetic field strengths and have observed its characteristic bipolar and resonant features predicted by two-level dressed atom theory. We find that the force is several times the strength of gravity in our setup, and that it can be targeted to a specific hyperfine transition while leaving other hyperfine states and transitions relatively unaffected. We find that our measurements are reasonably consistent with parameter-free theoretical predictions.
Optical manipulation of spin coherence in rubidium atoms is studied. The effect of off-resonant and circularly polarized light on optically induced magnetization is investigated. The change in precession frequency caused by the light-shift effect is verified. Absorption-free phase control of spin precession and pure spin rotation about an arbitrary axis are demonstrated. A theory of precession frequency shift that includes the effect of absorption is considered by using the density matrix and the experimental results are in agreement with the predictions of the theory. Thus, we show that it is possible to carry out off-resonant control of spin coherence and all-optical manipulation of spins.
We demonstrate quantum control of a large spin angular momentum associated with the F=3 hyperfine ground state of 133Cs. Time-dependent magnetic fields and a static tensor light shift are used to implement near-optimal controls and map a fiducial state to a broad range of target states, with yields in the range 0.8-0.9. Squeezed states are produced also by an adiabatic scheme that is more robust against errors. Universal control facilitates the encoding and manipulation of qubits and qudits in atomic ground states and may lead to the improvement of some precision measurements.
About 2/spl times/10/sup 7/ neutral strontium atoms have been loaded into a magneto-optical trap (MOT) that uses the narrow (/spl Gamma/=2/spl pi/ 7 kHz) intercombination line at 689 nm as the cycling transition. The atoms are second-stage laser cooled to /spl sim/10 /spl mu/K from an initial 5 mK atomic sample (/spl sim/5/spl times/10/sup 7/ atoms), generated in a vapor cell MOT that employs the strong (/spl Gamma/=2/spl pi/ 32 MHz) cycling transition at 461 nm. To demonstrate the low temperatures of these atoms, optical Ramsey spectroscopy is performed upon the /sup 1/S/sub 0/-/sup 3/P/sub 1/ transition using only two excitation pulses; the time dependence of the fringe contrast indicates an RMS velocity of 3 cm/s for the 689 nm MOT-cooled sample.
Summary form only given. The AC-Stark shift of a set of interfering laser beams together with the Zeeman interaction with external magnetic fields provide flexible mechanisms for coherently controlling atomic wave packets. When detuned far from atomic resonance, these optical lattices can trap cold atoms with minimal dissipation and thus provide an ideal environment for observing quantum coherent phenomena. We consider a one dimensional lattice of double wells produced by counterpropagating linearly polarized laser beams and a static magnetic field transverse to the laser wave vectors.