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arXiv:0804.2920 · Physical Review A

Quantum control of the hyperfine-coupled electron and nuclear spins in alkali-metal atoms

Seth Merkel, Poul Jessen, Ivan Deutsch

Published Aug 12, 2008• 40 citations
arXivHTMLPDF

Abstract

We study quantum control of the full hyperfine manifold in the ground-electronic state of alkali-metal atoms based on applied radio frequency and microwave fields. Such interactions should allow essentially decoherence-free dynamics and the application of techniques for robust control developed for NMR spectroscopy. We establish the conditions under which the system is controllable in the sense that one can generate an arbitrary unitary map on the system. We apply this to the case of $^{133}\mathrm{Cs}$ with its $d=16$ dimensional Hilbert space of magnetic sublevels in the $6{S}_{1∕2}$ state, and design control wave forms that generate an arbitrary target state from an initial fiducial state. We develop a generalized Wigner function representation for this space consisting of the direct sum of two irreducible representations of SU(2), allowing us to visualize these states. The performance of different control scenarios is evaluated based on the ability to generate a high-fidelity operation in an allotted time with the available resources. We find good operating points commensurate with modest laboratory requirements.

DOI
10.1103/physreva.78.023404

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In this dissertation I analyze Hamiltonian control of $d$-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.

arXiv:0906.4790•Jun 25, 2009•3 citations
We study quantum control of the full hyperfine manifold in the ground-electronic state of alkali-metal atoms based on applied radio frequency and microwave fields. Such interactions should allow essentially decoherence-free dynamics and the application of techniques for robust control developed for NMR spectroscopy. We establish the conditions under which the system is controllable in the sense that one can generate an arbitrary unitary map on the system. We apply this to the case of 133Cs^{133}\mathrm{Cs}133Cs with its d=16d=16d=16 dimensional Hilbert space of magnetic sublevels in the 6S1∕26{S}_{1∕2}6S1∕2​ state, and design control wave forms that generate an arbitrary target state from an initial fiducial state. We develop a generalized Wigner function representation for this space consisting of the direct sum of two irreducible representations of SU(2), allowing us to visualize these states. The performance of different control scenarios is evaluated based on the ability to generate a high-fidelity operation in an allotted time with the available resources. We find good operating points commensurate with modest laboratory requirements.
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.
We implement arbitrary maps between pure states in the 16-dimensional Hilbert space associated with the ground electronic manifold of ^{133}Cs. This is accomplished by driving atoms with phase modulated radio-frequency and microwave fields, using modulation waveforms found via numerical optimization and designed to work robustly in the presence of imperfections. We evaluate the performance of a sample of randomly chosen state maps by randomized benchmarking, obtaining an average fidelity >99%. Our protocol advances state-of-the-art quantum control and has immediate applications in quantum metrology and tomography.
Cold atomic spins are a great platform for developing and testing control and measurement techniques. This thesis presents experimental investigations into quantum control and measurement using laser cooled cesium atoms. On the control side, we present an experimental realization of a protocol to achieve full controllability of the entire hyperfine ground manifold of cesium. In particular, we demonstrate the ability to map between arbitrary states with fidelity greater than 0.99, using a combination of static, radio frequency, and microwave magnetic fields. On the measurement side, we present an experimental realization of quantum state tomography. The tomography protocol begins by measuring expectation values of an informationally complete set of observables using a weak optical probe in combination with dynamical control. The measurement record is processed using two different state estimation algorithms, allowing us to estimate a quantum state with fidelity greater than 0.9.
We study the ability to implement unitary maps on states of the I=9/2I=9/2I=9/2 nuclear spin in 87Sr^{87}\mathrm{Sr}87Sr, a d=10d=10d=10 dimensional (qudecimal) Hilbert space, using quantum optimal control. Through a combination of nuclear spin resonance and a tensor ac Stark shift, by solely modulating the phase of a radio-frequency magnetic field, the system is quantum controllable. Alkaline-earth-metal atoms, such as 87Sr^{87}\mathrm{Sr}87Sr, have a very favorable figure of merit for such control due to narrow intercombination lines and the large hyperfine splitting in the excited states. We numerically study the quantum speed limit, optimal parameters, and the fidelity of arbitrary state preparation and full SU(10) maps, including the presence of decoherence due to optical pumping induced by the light-shifting laser. We also study the use of robust control to mitigate some dephasing due to inhomogeneities in the light shift. We find that with an rf Rabi frequency of ${\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ and 0.5% inhomogeneity in the the light shift we can prepare an arbitrary Haar-random state in a time $T=4.5\ensuremath{\pi}/{\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ with average fidelity $\ensuremath{\langle}{\mathcal{F}}_{\ensuremath{\psi}}\ensuremath{\rangle}=0.9992$, and an arbitrary Haar-random SU(10) map in a time $T=24\ensuremath{\pi}/{\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ with average fidelity $\ensuremath{\langle}{\mathcal{F}}_{U}\ensuremath{\rangle}=0.9923$.
Quantum state reconstruction based on weak continuous measurement has the advantage of being fast, accurate and almost non-perturbative. In this work we present a pedagogical review of the protocol proposed by Silberfarb et al (2005 Phys. Rev. Lett. 95 030402), whereby an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner so as to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time through a maximum-likelihood estimate. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sublevels of a large-spin alkali atom, 133 Cs. We detail two different protocols for control. Using magnetic interactions and a quadratic ac Stark shift, we can reconstruct a chosen hyperfine manifold F , e.g. the seven-dimensional F = 3 manifold in the electronic ground state of Cs. We review the procedure as implemented in experiments (Smith et al 2006 Phys. Rev. Lett. 97 180403). We extend the protocol to the more ambitious case of reconstruction of states in the full 16-dimensional electronic ground subspace ( F = 3⊕ F = 4), controlled by microwaves and radio-frequency (RF) magnetic fields. We give detailed derivations of all physical interactions, approximations, numerical methods and fitting procedures, tailored to the realistic experimental setting. For the case of light-shift and magnetic control, reconstruction fidelities of ∼0.95 have been achieved, limited primarily by inhomogeneities in the light-shift. For the case of microwave/RF-control we simulate fidelity >0.97, limited primarily by signal-to-noise.
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.