Quantum state tomography by continuous measurement and compressed sensing

 

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A. Smith¹, C. A. Riofrio², B. E. Anderson¹, H. Sosa-Martinez¹, I. H. Deutsch², P.S. Jessen¹

1. Center for Quantum Information and Control, College of Optical Sciences and Department of Physics, University of Arizona, Tucson, Arizona 85721, USA
2. Center for Quantum Information and Control, Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA
 

The need to perform quantum state tomography on ever-larger systems has spurred a search for methods that yield good estimates from incomplete data. We study the performance of compressed sensing (CS) and least squares (LS) estimators in a fast protocol based on continuous measurement on an ensemble of cesium atomic spins. They both efficiently reconstruct nearly pure states in the 16-dimensional ground manifold, reaching average fidelities F_CS = 0.92 and F_LS = 0.88 using similar amounts of incomplete data. Surprisingly, the main advantage of CS in our protocol is an increased robustness to experimental imperfections.
 

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Three-axis measurement and cancellation of background magnetic fields to less than 50 μG in a cold atom experiment

 

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A. Smith¹, B.E. Anderson¹, S. Chaudhury², P.S. Jessen^1,2

1. Center for Quantum Information and Control (CQuIC) and Department of Physics, University of Arizona, Tucson, AZ 85721, USA
2. Center for Quantum Information and Control (CQuIC) and College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA
 

Many experiments involving cold and ultracold atomic gases require very precise control of magnetic fields that couple to and drive the atomic spins. Examples include quantum control of atomic spins, quantum control and quantum simulation in optical lattices, and studies of spinor Bose condensates. This makes accurate cancellation of the (generally time dependent) background magnetic field a critical factor in such experiments. We describe a technique that uses the atomic spins themselves to measure dc and ac components of the background field independently along three orthogonal axes, with a resolution of a few tens of ?G in a bandwidth of ~1 kHz. Once measured, the background field can be cancelled with three pairs of compensating coils driven by arbitrary waveform generators. In our laboratory, the magnetic field environment is sufficiently stable for the procedure to reduce the field along each axis to less than ~50 ?G rms, corresponding to a suppression of the ac part by about one order of magnitude. This suggests that our approach can provide access to a new low-field regime in cold atom experiments.
 

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Strongly Enhanced Spin Squeezing via Quantum Control

 

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Collin M. Trail, Poul S. Jessen, and Ivan H. Deutsch

Center for Quantum Information and Control (CQuIC) and Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico, USA, CQuIC and College of Optical Sciences and Department of Physics, University of Arizona, Tuscon, Arizona, USA
 

We describe a new approach to spin squeezing based on a double-pass Faraday interaction between an optical probe and an optically dense atomic sample. A quantum eraser is used to remove residual spin- probe entanglement, thereby realizing a single-axis twisting unitary map on the collective spin. This interaction can be phase matched, resulting in exponential enhancement of squeezing as a function of optical density for times short compared to the decoherence time. In practice the scaling and peak squeezing depends on decoherence, technical loss, and noise. Including these imperfections, our model indicates that ~10 dB of squeezing should be achievable with laboratory parameters.
 

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Quantum signatures of chaos in a kicked top

 

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Chaudhury, S., Smith, A., Anderson, B. E., Ghose, S., Jessen, P. S.

Center for Quantum Information and Control (CQuIC) and Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico, USA, CQuIC and College of Optical Sciences and Department of Physics, University of Arizona, Tuscon, Arizona, USA
 

Chaotic behaviour is ubiquitous and plays an important part in most fields of science. In classical physics, chaos is characterized by hypersensitivity of the time evolution of a system to initial conditions. Quantum mechanics does not permit a similar definition owing in part to the uncertainty principle, and in part to the Schrodinger equation, which preserves the overlap between quantum states. This fundamental disconnect poses a challenge to quantum classical correspondence, and has motivated a long-standing search for quantum signatures of classical chaos. Here we present the experimental realization of a common paradigm for quantum chaos, the quantum kicked top and the observation directly in quantum phase space of dynamics that have a chaotic classical counterpart. Our system is based on the combined electronic and nuclear spin of a single atom and is therefore deep in the quantum regime; nevertheless, we find good correspondence between the quantum dynamics and classical phase space structures. Because chaos is inherently a dynamical phenomenon, special significance attaches to dynamical signatures such as sensitivity to perturbation, or the generation of entropy and entanglement, for which only indirect evidence has been available. We observe clear differences in the sensitivity to perturbation in chaotic versus regular, non-chaotic regimes, and present experimental evidence for dynamical entanglement as a signature of chaos.
 

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Coherent control of atomic transport in spinor optical lattices

 

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Brian E. Mischuck, Poul S. Jessen, Ivan H. Deutsch
 

Coherent transport of atoms trapped in an optical lattice can be controlled by microwave-induced spin flips that correlate with site-to-site hopping. We study the controllability of homogeneous one-dimensional systems of noninteracting atoms in the absence of site addressability. Given these restrictions, we construct a deterministic protocol to map an initially localized Wannier state to a wave packet that that is coherently distributed over n sites. This is extended to analytic solutions for arbitrary unitary maps given homogenous systems and in the presence of time-dependent uni- form forces. Such control is important for applications in quantum information processing such as quantum computing and quantum simulations of condensed matter phenomena.
 

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Constructing general unitary maps from state preparations

 

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Merkel, Seth T., Brennen, Gavin, Jessen, Poul S., Deutsch, Ivan H.
 

We present an efficient algorithm for generating unitary maps on a d-dimensional Hilbert space from a time-dependent Hamiltonian through a combination of stochastic searches and geometric construction. The protocol is based on the eigendecomposition of the map. A unitary matrix can be implemented by sequentially mapping each eigenvector to a fiducial state, imprinting the eigenphase on that state, and mapping it back to the eigenvector. This requires the design of only d state-to-state maps generated by control wave forms that are efficiently found by a gradient search with computational resources that scale polynomially in d. In contrast, the complexity of a stochastic search for a single wave form that simultaneously acts as desired on all eigenvectors scales exponentially in d. We extend this construction to design maps on an n-dimensional subspace of the Hilbert space using only n stochastic searches. Additionally, we show how these techniques can be used to control atomic spins in the ground-electronic hyperfine manifold of alkali metal atoms in order to implement general qudit logic gates as well to perform a simple form of error correction on an embedded qubit.
 

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Quantum control and measurement of atomic spins in polarization spectroscopy

 

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Ivan H. Deutsch, Poul S. Jessen
 

Quantum control and measurement are two sides of the same coin. To affect a dynamical map, well-designed time-dependent control fields must be applied to the system of interest. To read out the quantum state, information about the system must be transferred to a probe field. We study a particular example of this dual action in the context of quantum control and measurement of atomic spins through the light-shift interaction with an off-resonant optical probe. By introducing an irreducible tensor decomposition, we identify the coupling of the Stokes vector of the light field with moments of the atomic spin state. This shows how polarization spectroscopy can be used for continuous weak measurement of atomic observables that evolve as a function of time. Simultaneously, the state-dependent light shift induced by the probe field can drive nonlinear dynamics of the spin, and can be used to generate arbitrary unitary transformations on the atoms. We revisit the derivation of the master equation in order to give a unified description of spin dynamics in the presence of both nonlinear dynamics and photon scattering. Based on this formalism, we review applications to quantum control, including the design of state-to-state mappings, and quantum-state reconstruction via continuous weak measurement on a dynamically controlled ensemble.
 

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Microwave Control of Atomic Motion in Optical Lattices

 

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Leonid Forster, Michal Karski, Jai-Min Choi, Andreas Steffen, Wolfgang Alt, Dieter Meschede, Artur Widera, Enrique Montano, Jae Hoon Lee, Worawarong Rakreungdet, Poul S. Jessen
 

We control the quantum mechanical motion of neutral atoms in an optical lattice by driving microwave transitions between spin states whose trapping potentials are spatially offset. Control of this offset with nanometer precision allows for adjustment of the coupling strength between different motional states, analogous to an adjustable effective Lamb-Dicke factor. This is used both for efficient one-dimensional sideband cooling of individual atoms to a vibrational ground state population of 97 percent, and to drive coherent Rabi oscillation between arbitrary pairs of vibrational states. We further show that microwaves can drive well resolved transitions between motional states in maximally offset, shallow lattices, and thus in principle allow for coherent control of long range quantum transport.
 

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Accurate microwave control and real-time diagnostics of neutral-atom qubits

 

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Rakreungdet, Worawarong, Lee, Jae Hoon, Lee, Kim Fook, Mischuck, Brian E., Montano, Enrique, Jessen, Poul S.
 

We demonstrate accurate single-qubit control in an ensemble of atomic qubits trapped in an optical lattice. The qubits are driven with microwave radiation, and their dynamics tracked by optical probe polarimetry. Real-time diagnostics is crucial to minimize systematic errors and optimize the performance of single-qubit gates, leading to fidelities of 0.99 for single-qubit rotations. We show that increased robustness to large, deliberately introduced errors can be achieved through the use of composite rotations. However, during normal operation the combination of very small intrinsic errors and additional decoherence during the longer pulse sequences precludes any significant performance gain in our current experiment.
 

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

 

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Merkel, Seth T., Jessen, Poul S., Deutsch, Ivan H.
 

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 with its d=16 dimensional Hilbert space of magnetic sublevels in the 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 SU2, 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.
 

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