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arXiv:0706.2450 · Physical Review Letters

Quantum Control of the Hyperfine Spin of a Cs Atom Ensemble

Souma Chaudhury, Seth Merkel, Tobias Herr, Andrew Silberfarb, Ivan Deutsch, Poul Jessen

Published Oct 16, 2007• 138 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.99.163002

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arXiv:0906.4790•Jun 25, 2009•3 citations
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.
Laboratory techniques to manipulate and observe ultracold atoms make these an attractive platform for testing new ideas in quantum control and measurement. Over the last decade we have revisited the tensor interaction between light fields and multilevel atoms, and have developed a theoretical framework suited for applications in quantum control and measurement (see [1] for a review). One important finding is that the combined action of a light shift and magnetic field on an atomic ground state can be used to implement a non-linear Hamiltonian for a hyperfine spin, and that its action can lead to nonlinear spin dynamics such as wavepacket collapse and revival. Using concepts from classical control theory it is straightforward to show that this Hamiltonian is sufficiently general for full control of an arbitrarily large spin. On this foundation we have developed a new protocol for quantum state reconstruction, based on continuous weak measurement of a spin observable during carefully designed coherent evolution [2]. We have further used our tools for control and state reconstruction to implement and verify protocols for optimal control of Cs hyperfine spns (Fig. 1), showing that an initial fiducial state can be transformed into any desired target state with a fidelity in the 80-90% range [3].
Quantum control of many body atomic spins is often pursued in the context of an atom-light quantum interface, where a quantized light field acts as a "quantum bus" that can be used to entangle distant atoms. One key challenge is to improve the coherence of the atom-light interface and the amount of atom-light entanglement it can generate, given the constraints of working with multilevel atoms and optical fields in a 3D geometry. We have explored new ways to achieve this, through rigorous optimization of the spatial geometry, and through control of the internal atomic state. Our basic setup consists of a quantized probe beam passing through an atom cloud held in a dipole trap, first generating spin-probe entanglement through the Faraday interaction, and then using backaction from a measurement of the probe polarization to squeeze the collective atomic spin. The relevant figure of merit is the metrologically useful spin squeezing determined by the enhancement in the resolution of rotations of the collective spin, relative to the commonly used spin coherent state. With an optimized free-space geometry, and by using a 2-color probe scheme to suppress tensor light shifts, we achieve 3(2) dB of metrologically useful spin squeezing. We can further increase atom-light coupling by implementing internal state control to prepare spin states with larger initial projection noise relative to the spin coherent state. Under the right conditions this increase in projection noise can lead to stronger measurement backaction and increased atom-atom entanglement. With further internal state control the increased atom-atom entanglement can then be mapped to a basis where it corresponds to improved squeezing of, e.g., the physical spin-angular momentum or the collective atomic clock pseudospin. In practice, controlling the collective spin of N ~ 10⁶ atoms in this fashion is an extraordinarily difficult challenge because errors in the control of individual atoms tend to be highly correlated. By employing precise internal state control, we have prepared and detected projection noise limited "cat" states (which have initial projection noise that is larger by a factor of 2f = 8 for Cs relative to the spin coherent state) and estimate that we can generate up to 6.0(5) dB of metrologically useful spin squeezing, demonstrating the advantage of using the internal atomic structure as a resource for ensemble control.
Laser-cooled atoms offer an excellent platform for testing new ideas of quantum control and measurement. I will discuss experiments where we use light and magnetic fields to drive and monitor non-trivial quantum dynamics of a large spin-angular momentum associated with an atomic hyperfine ground state. We can design Hamiltonians to generate arbitrary spin states and perform a full quantum state reconstruction of the results. We have implemented and verified time optimal controls to generate a broad variety of spin states, including spin-squeezed states useful for metrology. Yields achieved are of the range 0.8-0.9.We present a first experimental demonstration of the quantum kicked top, a popular paradigm for quantum and classical chaos. We make `movies' of the evolving quantum state which provides a direct observation of phase space dynamics of this system. The spin dynamics seen in the experiment includes dynamical tunneling between regular islands, rapid spreading of states throughout the chaotic sea, and surprisingly robust signatures of classical phase space structures. Our data show differences between regular and chaotic dynamics in the sensitivity to perturbations of the quantum kicked top Hamiltonian and in the average electron-nuclear spin entanglement during the first 40 kicks. The difference, while clear, is modest due to the small size of the spin.
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 study the ability to control ddd-dimensional quantum systems (qudits) encoded in the hyperfine spin of alkali-metal atoms through the application of radio- and microwave-frequency magnetic fields in the presence of inhomogeneities in amplitude and detuning. Such a capability is essential to the design of robust pulses that mitigate the effects of experimental uncertainty and also for application to tomographic addressing of particular members of an extended ensemble. We study the problem of preparing an arbitrary state in the Hilbert space from an initial fiducial state. We prove that inhomogeneous control of qudit ensembles is possible based on a semianalytic protocol that synthesizes the target through a sequence of alternating rf and microwave-driven SU(2) rotations in overlapping irreducible subspaces. Several examples of robust control are studied, and the semianalytic protocol is compared to a brute-force, full numerical search. For small inhomogeneities, <<<11%1, both approaches achieve average fidelities greater than 0.99, but the brute-force approach performs superiorly, reaching high fidelities in shorter times and capable of handling inhomogeneities well beyond experimental uncertainty. The full numerical search is also applied to tomographic addressing whereby two different nonclassical states of the spin are produced in two halves of the ensemble.
In this dissertation I analyze Hamiltonian control of ddd-dimensional quantum systems as realized in alkali atomic spins. Alkali atoms provide an ideal platform for studies of quantum control due to the extreme precision with which the control fields are characterized as well as their isolation from their environment. In chapter 2, I review some background material on open-loop quantum control theory. Chapter 3 provides a derivation of the Hamiltonians arising from electromagnetic fields that we use to control our alkali atomic spins. In chapter 4, I develop an algorithm for state preparation, that is mapping a fiducial state to some arbitrary target state, and show numerical and experimental implementations for making arbitrary superpositions of hyperfine states in $^{133}Cs. Finally, chapter 5 presents a protocol for generating full unitary maps efficiently by utilizing the ability to construct state mappings.