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Optically trapped atoms in arrays of optical tweezers have emerged as a powerful platform for quantum information processing given the recent demonstrations of high-fidelity quantum logic gates and on-demand reconfigurable geometry. Both in gate operations and atomic transport, additional errors will occur due to leakage out of the computation space, atomic motional heating, or loss of an atom out of a trap completely. In this work, we address these error channels in a unified manner through laser fluorescence that can detect and cool the atom without disturbing the quantum information encoded therein. As only the electrons in the atom couple directly to the laser field, such quantum nondemolition (QND) processes are made possible by encoding quantum information in the nuclear spin of alkaline earth-like atoms and avoiding the effects of the hyperfine interaction which couples it to the electrons. By detuning a fluorescence laser off-resonantly from the $\mathrm{^1S_0} \rightarrow \mathrm{^1P_1}$ transition, far compared to the (small) hyperfine spitting, optical pumping between nuclear states falls off rapidly with detuning, scaling as $~1/\Delta^4$. In contrast, Rayleigh scattering falls off as $~1/\Delta^2$. We also consider a resonant leakage detection protocol off the $^1\mathrm{P}_1$ line. This is achieved by disabling the hyperfine coupling via a strong AC stark effect and canceling the residual lightshifts via dressing. The same scheme can be used to recool the atoms towards the vibrational ground state for the quantum information encoded in the ground state of alkaline earth atoms while preserving the coherence. These advances could significantly improve the prospect of neutral atoms for fault-tolerant quantum computation., Comment: Comments and suggestions are Welcome!

Leakage out of the computational subspace is a major limitation of current state-of-the-art neutral-atom quantum computers and a significant challenge for scalable systems. In a quantum processor with cesium atoms, we demonstrate proof-of-principle circuit-based conversion of leakage errors to erasure errors via Leakage Detection Units (LDUs), which non-destructively map information about the presence or absence of the qubit onto the state of an ancilla. With a standard LDU circuit, we successfully convert leakage errors to erasure errors for all major leakage pathways while preserving the quantum information in the case that no leakage occurred. We benchmark the performance of the LDU using a three-outcome low-loss state detection method and also explore the advantages of three-outcome measurements for LDUs. We find that the LDU detects atom-loss errors with ~93.4% accuracy, limited by technical imperfections of our apparatus. We further compile and execute a SWAP LDU, wherein the roles of the original data atom and ancilla atom are exchanged under the action of the LDU, providing 'free refilling' of atoms in the case of leakage errors. This circuit-based leakage-to-erasure error conversion is a critical component of a neutral-atom quantum processor where the quantum information may significantly outlive the lifetime of any individual atom in the quantum register., Comment: 14 pages, 8 figures, published in PRX Quantum

We show how the decoherence that occurs in an entangling atomic spin-light interface can be simply modeled as the dynamics of a bosonic mode. Although one seeks to control the collective spin of the atomic system in the permutationally invariant (symmetric) subspace, diffuse scattering and optical pumping are local, making an exact description of the many-body state intractable. To overcome this issue we develop a generalized Holstein-Primakoff approximation for collective states which is valid when decoherence is uniform across a large atomic ensemble. In different applications the dynamics is conveniently treated as a Wigner function evolving according to a thermalizing diffusion equation, or by a Fokker-Planck equation for a bosonic mode decaying in a zero temperature reservoir. We use our formalism to study the combined effect of Hamiltonian evolution, local and collective decoherence, and measurement backaction in preparing nonclassical spin states for application in quantum metrology., Comment: 22 pages, 6 figures

We construct a fault-tolerant quantum error-correcting protocol based on a qubit encoded in a large spin qudit using a spin-cat code, analogous to the continuous variable cat encoding. With this, we can correct the dominant error sources, namely processes that can be expressed as error operators that are linear or quadratic in the components of angular momentum. Such codes tailored to dominant error sources {can} exhibit superior thresholds and lower resource overheads when compared to those designed for unstructured noise models. To preserve the dominant errors during gate operations, we identify a suitable universal gate set. A key component is the CNOT gate that preserves the rank of spherical tensor operators. Categorizing the dominant errors as phase and amplitude errors, we demonstrate how phase errors, analogous to phase-flip errors for qubits, can be effectively corrected. Furthermore, we propose a measurement-free error correction scheme to address amplitude errors without relying on syndrome measurements. Through an in-depth analysis of logical CNOT gate errors, we establish that the fault-tolerant threshold for error correction in the spin-cat encoding surpasses that of standard qubit-based encodings. We consider a specific implementation based on neutral-atom quantum computing, with qudits encoded in the nuclear spin of $^{87}$Sr, and show how to generate the universal gate set, including the rank-preserving CNOT gate, using quantum control and the Rydberg blockade. These findings pave the way for encoding a qubit in a large spin with the potential to achieve fault tolerance, high threshold, and reduced resource overhead in quantum information processing., Comment: Published in PRX Quantum

We study the tractability of classically simulating critical phenomena in the quench dynamics of one-dimensional transverse field Ising models (TFIMs) using highly truncated matrix product states (MPS). We focus on two paradigmatic examples: a dynamical quantum phase transition (DQPT) that occurs in nonintegrable long-range TFIMs, and the infinite-time correlation length of the integrable nearest-neighbor TFIM when quenched to the critical point. For the DQPT, we show that the order parameters can be efficiently simulated with surprisingly heavy truncation of the MPS bond dimension. This can be used to reliably extract critical properties of the phase transition, including critical exponents, even when the full many-body state is not simulated with high fidelity. The long-time correlation length near the critical point is more sensitive to the full many-body state fidelity, and generally requires a large bond dimension MPS. Nonetheless, we find that this can still be efficiently simulated with strongly truncated MPS because it can be extracted from the short-time behavior of the dynamics where entanglement is low. Our results demonstrate that while accurate calculation of the full many-body state (microstate) is typically intractable due to the volume-law growth of entanglement, a precise specification of an exact microstate may not be required when simulating phases of matter of many-body systems (macrostates). We also study the tractability of simulation using truncated MPS based on quantum chaos and equilibration in the models. We find a counterintuitive inverse relationship, whereby local expectation values are most easily approximated for chaotic systems whose exact many-body state is most intractable.

We study the generation of two-qudit entangling quantum logic gates using two techniques in quantum optimal control. We take advantage of both continuous, Lie-algebraic control and digital, Lie-group control. In both cases, the key is access to a time-dependent Hamiltonian which can generate an arbitrary unitary matrix in the group SU($d^2$). We find efficient protocols for creating high-fidelity entangling gates. As a test of our theory, we study the case of qudits robustly encoded in nuclear spins of alkaline earth atoms and manipulated with magnetic and optical fields, with entangling interactions arising from the well-known Rydberg blockade. We applied this in a case study based on a $d=10$ dimensional qudit encoded in the $I=9/2$ nuclear spin in $^{87}$Sr, controlled through a combination of nuclear spin-resonance, a tensor AC-Stark shift, and Rydberg dressing, which allows us to generate an arbitrary symmetric entangling two-qudit gate such as CPhase. Our techniques can be used to implement qudit entangling gates for any $2\le d \le10$ encoded in the nuclear spin. We also studied how decoherence due to the finite lifetime of the Rydberg states affects the creation of the CPhase gate and found, through numerical optimization, a fidelity of $0.9985$, $0.9980$, $0.9942$, and $0.9800$ for $d=2$, $d=3$, $d=5$, and $d=7$ respectively. This provides a powerful platform to explore the various applications of quantum information processing of qudits including metrological enhancement with qudits, quantum simulation, universal quantum computation, and quantum error correction., Comment: Included more details and figures

We revisit well-known protocols in quantum metrology using collective spins and propose a unifying picture for optimal state preparation based on a semiclassical description in phase space. We show how this framework allows for quantitative predictions of the timescales required to prepare various metrologically useful states, and that these predictions remain accurate even for moderate system sizes, surprisingly far from the classical limit. Furthermore, this framework allows us to build a geometric picture that relates optimal (exponentially fast) entangled probe preparation to the existence of separatrices connecting saddle points in phase space. We illustrate our results with the paradigmatic examples of the two-axis counter-twisting and twisting-and-turning Hamiltonians, where we provide analytical expressions for all the relevant optimal time scales. Finally, we propose a generalization of these models to include $p$-body collective interaction (or $p$-order twisting), beyond the usual case of $p=2$. Using our geometric framework, we prove a no-go theorem for the local optimality of these models for $p>2$., Comment: 15 pages, 6 figures, 9 pages appendix

We revisit the implementation of a two-qubit entangling gate, the M{\o}lmer-S{\o}rensen gate, using the adiabatic Rydberg dressing paradigm for neutral atoms as studied in [Phys. Rev. A 101, 030301 (R) (2020)]. We study the implementation of rapid adiabatic passage using a two-photon transition, which does not require the use of an ultra-violet laser, and can be implemented using only amplitude modulation of one field with all laser frequencies fixed. We find that entangling gate fidelities, comparable to the one-photon excitation, are achievable with the two-photon excitation. Moreover, we address how the adiabatic dressing protocol can be used to implement entangling gates outside the regime of a perfect Rydberg blockade. We show that using adiabatic dressing we can achieve scaling of gate fidelity set by the fundamental limits to entanglement generated by the Rydberg interactions while simultaneously retaining a limited population in the doubly-excited Rydberg state. This allows for fast high fidelity gates for atoms separated beyond the blockade radius.

Neutral atoms are building blocks of ground-up quantum many-body systems. Well-controlled and high-fidelity entangling gates are an essential component for realizing complex neutral atom architectures for quantum computing, quantum simulation, and measurement with precision better than the standard quantum limit. In this Letter we report the realization of a M{\o}lmer-S{\o}rensen unitary between two neutral atoms, based on adiabatic single-photon dressing to Rydberg levels. We show that this technique is highly robust to noise sources and experimental imperfections that have limited the fidelity of other approaches to neutral atom gates., Comment: 7 pages, 4 figures

We study the behavior of errors in the quantum simulation of spin systems with long-range multi-body interactions resulting from the Trotter-Suzuki decomposition of the time-evolution operator. We identify a regime where the Floquet operator underlying the Trotter decomposition undergoes sharp changes even for small variations in the simulation step size. This results in a time evolution operator that is very different from the dynamics generated by the targeted Hamiltonian, which leads to a proliferation of errors in the quantum simulation. These regions of sharp change in the Floquet operator, referred to as structural instability regions, appear typically at intermediate Trotter step sizes and in the weakly-interacting regime, and are thus complementary to recently revealed quantum chaotic regimes of the Trotterized evolution (Sieberer et al., npj Quantum Information 5, 1 (2019)). We characterize these structural instability regimes in $p$-spin models, transverse-field Ising models with all-to-all $p$-body interactions, and analytically predict their occurrence based on unitary perturbation theory. We further show that the effective Hamiltonian associated with the Trotter decomposition of the unitary time-evolution operator, when the Trotter-step size is chosen to be in the structural instability region, is very different from the target Hamiltonian, which explains the large errors that can occur in the simulation in the regions of instability. These results have implications for the reliability of near-term gate-based quantum simulators, and reveal an important interplay between errors and the physical properties of the system being simulated., Comment: 20 pages, 4 figures

We study the ability to implement unitary maps on states of the $I=9/2$ nuclear spin in \textsuperscript{87}Sr, a $d=10$ 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 atoms, such as \textsuperscript{87}Sr, 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 $\Omega_\text{rf}$ and 0.5\% inhomogeneity in the the light shift we can prepare an arbitrary Haar-random state in a time $T={4.5}\pi/\Omega_\text{rf}$ with average fidelity $\langle \mathcal{F}_\psi \rangle =0.9992$, and an arbitrary Haar-random SU(10) map in a time $T=24\pi/\Omega_\text{rf}$ with average fidelity $\langle \mathcal{F}_U \rangle = 0.9923$.

We study how chaos, introduced by a weak perturbation, affects the reliability of the output of analog quantum simulation. As a toy model, we consider the Lipkin-Meshkov-Glick (LMG) model. Inspired by the semiclassical behavior of the order parameter in the thermodynamic limit, we propose a protocol to measure the quantum phase transition in the ground state and the dynamical quantum phase transition associated with quench dynamics. We show that the presence of a small time-dependent perturbation can render the dynamics of the system chaotic. We then show that the estimates of the critical points of these quantum phase transitions, obtained from the quantum simulation of its dynamics, are robust to the presence of this chaotic perturbation, while other aspects of the system, such as the mean magnetization are fragile, and therefore cannot be reliably extracted from this simulator. This can be understood in terms of the simulated quantities that depend on the global structure of phase space vs. those that depend on local trajectories., Comment: 13 pages, 7 figures

We introduce kicked $p$-spin models describing a family of transverse Ising-like models for an ensemble of spin-$1/2$ particles with all-to-all $p$-body interaction terms occurring periodically in time as delta-kicks. This is the natural generalization of the well-studied quantum kicked top ($p$=2)[Haake, Ku\'s, and Scharf, Z. Phys. B 65, 381 (1987)]. We fully characterize the classical nonlinear dynamics of these models, including the transition to global Hamiltonian chaos. The classical analysis allows us to build a classification for this family of models, distinguishing between $p=2$ and $p>2$, and between models with odd and even $p$'s. Quantum chaos in these models is characterized in both kinematic and dynamic signatures. For the latter we show numerically that the growth rate of the out-of-time-order correlator is dictated by the classical Lyapunov exponent. Finally, we argue that the classification of these models constructed in the classical system applies to the quantum system as well., Comment: 14 pages, 6 figures, and appendices. Close to the published version

The second quantum revolution has been built on a foundation of fundamental research at the intersection of physics and information science, giving rise to the discipline we now call Quantum Information Science (QIS). The quest for new knowledge and understanding drove the development of new experimental tools and rigorous theory, which defined the roadmap for second-wave quantum technologies, including quantum computers, quantum-enhanced sensors, and communication systems. As technology has matured, the race to develop and commercialize near-term applications has accelerated. In the current regime of Noisy Intermediate Scale Quantum (NISQ) devices, the continued necessity of basic research is manifest. Under what conditions can we truly harness quantum complexity and what are its implications for potential useful applications? These questions remain largely unanswered, and as the QIS industry ramps up, a continuous feedback between basic science and technology is essential. In this Perspective I review how curiosity-driven research led to radical new technologies and why the quest for basic understanding is essential for further progress.

Quantum simulators are widely seen as one of the most promising near-term applications of quantum technologies. However, it remains unclear to what extent a noisy device can output reliable results in the presence of unavoidable imperfections. Here we propose a framework to characterize the performance of quantum simulators by linking the robustness of measured quantum expectation values to the spectral properties of the output observable, which in turn can be associated with its macroscopic or microscopic character. We show that, under general assumptions and on average over all states, imperfect devices are able to reproduce the dynamics of macroscopic observables accurately, while the relative error in the expectation value of microscopic observables is much larger on average. We experimentally demonstrate the universality of these features in a state-of-the-art quantum simulator and show that the predicted behavior is generic for a highly accurate device, without assuming any knowledge about the nature of the imperfections., Comment: 16 pages including appendices, 13 figures. Content previously in Supplementary Material has been added to appendices

We study the application of a new method for simulating nonlinear dynamics of many-body spin systems using quantum measurement and feedback [Mu\~noz-Arias et al., Phys. Rev. Lett. 124, 110503 (2020)] to a broad class of many-body models known as $p$-spin Hamiltonians, which describe Ising-like models on a completely connected graph with $p$-body interactions. The method simulates the desired mean field dynamics in the thermodynamic limit by combining nonprojective measurements of a component of the collective spin with a global rotation conditioned on the measurement outcome. We apply this protocol to simulate the dynamics of the $p$-spin Hamiltonians and demonstrate how different aspects of criticality in the mean-field regime are readily accessible with our protocol. We study applications including properties of dynamical phase transitions and the emergence of spontaneous symmetry breaking in the adiabatic dynamics of the collective spin for different values of the parameter $p$. We also demonstrate how this method can be employed to study the quantum-to-classical transition in the dynamics continuously as a function of system size., Comment: 16 pages, 7 figures

The Rydberg blockade mechanism is now routinely considered for entangling qubits encoded in clock states of neutral atoms. Challenges towards implementing entangling gates with high fidelity include errors due to thermal motion of atoms, laser amplitude inhomogeneities, and imperfect Rydberg blockade. We show that adiabatic rapid passage by Rydberg dressing provides a mechanism for implementing two-qubit entangling gates by accumulating phases that are robust to these imperfections. We find that the typical error in implementing a two-qubit gate, such as the controlled phase gate, is dominated by errors in the single atom light shift, and that this can be easily corrected using adiabatic dressing interleaved with a simple spin echo sequence. This results in a two-qubit M{\o}lmer-S{\o}renson gate. A gate fidelity $\sim 0.995$ is achieveable with modest experimental parameters and a path to higher fidelities is possible for Rydberg states in atoms with a stronger blockade, longer lifetimes, and larger Rabi frequencies.

Analog quantum simulation is widely considered a step on the path to fault tolerant quantum computation. If based on current noisy hardware, the accuracy of an analog simulator will degrade after just a few time steps, especially when simulating complex systems that are likely to exhibit quantum chaos. Here we describe a small, highly accurate quantum simulator and its use to run high fidelity simulations of three different model Hamiltonians for $>100$ time steps. While not scalable to exponentially large Hilbert spaces, this platform provides the accuracy and programmability required for systematic exploration of the interplay between dynamics, imperfections, and accuracy in quantum simulation., Comment: Published version. 10 pages, 5 figures, including Supplemental Material

We study a method to simulate quantum many-body dynamics of spin ensembles using measurement-based feedback. By performing a weak collective measurement on a large ensemble of two-level quantum systems and applying global rotations conditioned on the measurement outcome, one can simulate the dynamics of a mean-field quantum kicked top, a standard paradigm of quantum chaos. We analytically show that there exists a regime in which individual quantum trajectories adequately recover the classical limit, and show the transition between noisy quantum dynamics to full deterministic chaos described by classical Lyapunov exponents. We also analyze the effects of decoherence, and show that the proposed scheme represents a robust method to explore the emergence of chaos from complex quantum dynamics in a realistic experimental platform based on an atom-light interface., Comment: 6 pages, 4 figures and supplementary material

Squeezing of collective atomic spins has been shown to improve the sensitivity of atomic clocks and magnetometers to levels significantly below the standard quantum limit. In most cases the requisite atom-atom entanglement has been generated by dispersive interaction with a quantized probe field, or by state dependent collisions in a quantum gas. Such experiments typically use complex multilevel atoms like Rb or Cs, with the relevant interactions designed so atoms behave like pseudo-spin-$1/2$ particles. We demonstrate the viability of spin squeezing for collective spins composed of the physical angular momenta of $\sim 10^6$ Cs atoms, each in an internal spin-4 hyperfine state. A peak metrological squeezing of $\gtrsim -5$dB was generated by quantum backaction from a dispersive quantum nondemolition (QND) measurement, implemented using a two-color optical probe that minimizes tensor light shifts without sacrificing measurement strength. Other significant developments include the successful application of composite pulse techniques for accurate dynamical control of the collective spin, enabled by broadband suppression of background magnetic fields inside a state-of-the-art magnetic shield. The absence of classical noise has allowed us to compare the observed quantum projection noise and squeezing to a theoretical model that properly accounts for both the relevant atomic physics and the spatial mode of the collective spin, finding good quantitative agreement and thereby validating its use in other contexts. Thus, our work sets the stage for experiments on quantum feedback, deterministic squeezing, closed-loop magnetometry, and new types of quantum simulation based on continuous QND measurement and feedback., Comment: 14 pages, 4 figures

We study the sampling complexity of a probability distribution associated with an ensemble ofidentical noninteracting bosons undergoing a quantum random walk on a one-dimensional lattice.With uniform nearest-neighbor hopping we show that one can efficiently sample the distribution fortimes logarithmic in the size of the system, while for longer times there is no known efficient samplingalgorithm. With time-dependent hopping and optimal control, we design the time evolution toapproximate an arbitrary Haar-random unitary map analogous to that designed for photons in alinear optical network. This approach highlights a route to generating quantum complexity byoptimal control only of a single-body unitary matrix. We study this in the context of two potentialexperimental realizations: a spinor optical lattice of ultracold atoms and a quantum gas microscope., Comment: 10+3 pages, 6 figures: Published version

The spin-coherent-state positive-operator-valued-measure (POVM) is a fundamental measurement in quantum science, with applications including tomography, metrology, teleportation, benchmarking, and measurement of Husimi phase space probabilities. We prove that this POVM is achieved by collectively measuring the spin projection of an ensemble of qubits weakly and isotropically. We apply this in the context of optimal tomography of pure qubits. We show numerically that through a sequence of weak measurements of random directions of the collective spin component, sampled discretely or in a continuous measurement with random controls, one can approach the optimal bound., Comment: 5 pages, 2 figures

We study the enhancement of cooperativity in the atom-light interface near a nanophotonic waveguide for application to quantum nondemolition (QND) measurement of atomic spins. Here the cooperativity per atom is determined by the ratio between the measurement strength and the decoherence rate. Counterintuitively, we find that by placing the atoms at an azimuthal position where the guided probe mode has the lowest intensity, we increase the cooperativity. This arises because the QND measurement strength depends on the interference between the probe and scattered light guided into an orthogonal polarization mode, while the decoherence rate depends on the local intensity of the probe. Thus, by proper choice of geometry, the ratio of good to bad scattering can be strongly enhanced for highly anisotropic modes. We apply this to study spin squeezing resulting from QND measurement of spin projection noise via the Faraday effect in two nanophotonic geometries, a cylindrical nanofiber and a square waveguide. We find that, with about 2500 atoms and using realistic experimental parameters, $ \sim 6.3 $ and $ \sim 13 $ dB of squeezing can be achieved on the nanofiber and square waveguide, respectively., Comment: 8 pages + 2 pages appendix with 7 figures. To appear on Phys. Rev. A

We observe the nonlinearity of the Jaynes-Cummings (JC) ladder in the Autler-Townes spectroscopy of the hyperfine ground states for a Rydberg-dressed two-atom system. Here the role of the two-level system in the JC model is played by the presence or absence of a collective Rydberg excitation, and the bosonic mode manifests as the number $n$ of single atom spin flips, symmetrically distributed between the atoms. We measure the normal-mode splitting and $\sqrt{n}$ nonlinearity as a function of detuning and Rabi frequency, thereby experimentally establishing the isomorphism with the JC model., Comment: 5 pages, 4 figures

Symmetric ensembles of neutral atoms interacting via the Rydberg blockade are well-described by the Jaynes-Cummings Hamiltonian. We use this framework to study the problem of generating arbitrary superpositions of Dicke states of hyperfine qubits in such ensembles. The combination of the symmetric Rydberg blockade and microwaves that drive the qubits with a time-dependent phase is sufficient to make these ensembles completely controllable, in the sense that one can generate an arbitrary unitary transformation on the system. We apply this to the problem of state mapping. With currently feasible parameters, it is possible to generate arbitrary symmetric states of ~ 10 hypefine qubits with high fidelity in ~ 1 microsecond, assuming fast microwave phase switching times. To reduce the requirements on phase switching, we propose a "dressed ground control" scheme, in which the control task is simplified by restricting the system's dynamics to the dressed ground subspace., Comment: 7 pages (including 2 of supplemental material), 3 figures

We consider the problem of quantum-state tomography under the assumption that the state is pure, and more generally that its rank is bounded by a given value $r$. In this scenario two notions of informationally complete measurements emerge: rank-$r$ complete measurements and rank-$r$ strictly-complete measurements. Whereas in the first notion, a rank-$r$ state is uniquely identified from within the set of rank-$r$ states, in the second notion the same state is uniquely identified from within the set of all physical states, of any rank. We argue, therefore, that strictly-complete measurements are compatible with convex optimization, and we prove that they allow robust quantum state estimation in the presence of experimental noise. We also show that rank-$r$ strictly-complete measurements are as efficient as rank-$r$ complete measurements. We construct examples of strictly-complete measurements and give a complete description of their structure in the context of matrix completion. Moreover, we numerically show that a few random bases form such measurements. We demonstrate the efficiency-robustness property for different strictly-complete measurements with numerical experiments. We thus conclude that only strictly-complete measurements are useful for practical tomography., Comment: This is a combined version of arXiv:1510.02736 and arXiv:1511.01433

We study the strong coupling between photons and atoms that can be achieved in an optical nanofiber geometry when the interaction is dispersive. While the Purcell enhancement factor for spontaneous emission into the guided mode does not reach the strong-coupling regime for individual atoms, one can obtain high cooperativity for ensembles of a few thousand atoms due to the tight confinement of the guided modes and constructive interference over the entire chain of trapped atoms. We calculate the dyadic Green's function, which determines the scattering of light by atoms in the presence of the fiber, and thus the phase shift and polarization rotation induced on the guided light by the trapped atoms. The Green's function is related to a full Heisenberg-Langevin treatment of the dispersive response of the quantized field to tensor polarizable atoms. We apply our formalism to quantum nondemolition (QND) measurement of the atoms via polarimetry. We study shot-noise-limited detection of atom number for atoms in a completely mixed spin state and the squeezing of projection noise for atoms in clock states. Compared with squeezing of atomic ensembles in free space, we capitalize on unique features that arise in the nanofiber geometry including anisotropy of both the intensity and polarization of the guided modes. We use a first principles stochastic master equation to model the squeezing as function of time in the presence of decoherence due to optical pumping. We find a peak metrological squeezing of ~5 dB is achievable with current technology for ~2500 atoms trapped 180 nm from the surface of a nanofiber with radius a=225 nm., Comment: To be appeared on PRA

We explore quantum signatures of classical chaos by studying the rate of information gain in quantum tomography. The tomographic record consists of a time series of expectation values of a Hermitian operator evolving under application of the Floquet operator of a quantum map that possesses (or lacks) time reversal symmetry. We find that the rate of information gain, and hence the fidelity of quantum state reconstruction, depends on the symmetry class of the quantum map involved. Moreover, we find an increase in information gain and hence higher reconstruction fidelities when the Floquet maps employed increase in chaoticity. We make predictions for the information gain and show that these results are well described by random matrix theory in the fully chaotic regime. We derive analytical expressions for bounds on information gain using random matrix theory for different class of maps and show that these bounds are realized by fully chaotic quantum systems., Comment: 19 pages, Invited review for Pramana J of physics. arXiv admin note: substantial text overlap with arXiv:1212.4572

Characterizing complex quantum systems is a vital task in quantum information science. Quantum tomography, the standard tool used for this purpose, uses a well-designed measurement record to reconstruct quantum states and processes. It is, however, notoriously inefficient. Recently, the classical signal reconstruction technique known as "compressed sensing" has been ported to quantum information science to overcome this challenge: accurate tomography can be achieved with substantially fewer measurement settings, thereby greatly enhancing the efficiency of quantum tomography. Here we show that compressed sensing tomography of quantum systems is essentially guaranteed by a special property of quantum mechanics itself---that the mathematical objects that describe the system in quantum mechanics are matrices with nonnegative eigenvalues. This result has an impact on the way quantum tomography is understood and implemented. In particular, it implies that the information obtained about a quantum system through compressed sensing methods exhibits a new sense of "informational completeness." This has important consequences on the efficiency of data taking for quantum tomography, and enables us to construct informationally complete measurements that are robust to noise and modeling errors. Moreover, our result shows that one can expand the numerical tool-box used in quantum tomography and employ highly efficient algorithms developed to handle large dimensional matrices on a large dimensional Hilbert space. While we mainly present our results in the context of quantum tomography, they apply to the general case of positive semidefinite matrix recovery., Comment: 9 + 5 pages, 3 figures; v3 published version

We study a scheme for implementing a controlled-Z (CZ) gate between two neutral-atom qubits based on the Rydberg blockade mechanism in a manner that is robust to errors caused by atomic motion. By employing adiabatic dressing of the ground electronic state, we can protect the gate from decoherence due to random phase errors that typically arise because of atomic thermal motion. In addition, the adiabatic protocol allows for a Doppler-free configuration that involves counterpropagating lasers in a $\sigma_+/\sigma_-$ orthogonal polarization geometry that further reduces motional errors due to Doppler shifts. The residual motional error is dominated by dipole-dipole forces acting on doubly-excited Rydberg atoms when the blockade is imperfect. For reasonable parameters, with qubits encoded into the clock states of $^{133}$Cs, we predict that our protocol could produce a CZ gate in $<10$ $\mu$s with error probability on the order of $10^{-3}$., Comment: 9 pages, 5 figures

We study quantum tomography based on a stochastic continuous-time measurement record obtained from a probe field collectively interacting with an ensemble of identically prepared systems. In comparison to previous studies, we consider here the case in which the measurement-induced backaction has a nonnegligible effect on the dynamical evolution of the ensemble. We formulate a maximum likelihood estimate for the initial quantum state given only a single instance of the continuous diffusive measurement record. We apply our estimator to the simplest problem -- state tomography of a single pure qubit, which, during the course of the measurement, is also subjected to dynamical control. We identify a regime where the many-body system is well approximated at all times by a separable pure spin coherent state, whose Bloch vector undergoes a conditional stochastic evolution. We simulate the results of our estimator and show that we can achieve close to the upper bound of fidelity set by the optimal POVM. This estimate is compared to, and significantly outperforms, an equivalent estimator that ignores measurement backaction., Comment: 10 pages, 5 epic figures

We study quantum process tomography given the prior information that the map is a unitary or close to a unitary process. We show that a unitary map on a $d$-level system is completely characterized by a minimal set of $d^2{+}d$ elements associated with a collection of POVMs, in contrast to the $d^4{-}d^2$ elements required for a general completely positive trace-preserving map. To achieve this lower bound, one must probe the map with a particular set of $d$ pure states. We further compare the performance of different compressed sensing algorithms used to reconstruct a near-unitary process from such data. We find that when we have accurate prior information, an appropriate compressed sensing method reduces the required data needed for high-fidelity estimation, and different estimators applied to the same data are sensitive to different types of noise. Compressed sensing techniques can therefore be used both as indicators of error models and to validate the use of the prior assumptions., Comment: 11 pages, 3 figures

We study the three-dimensional nature of the quantum interface between an ensemble of cold, trapped atomic spins and a paraxial laser beam, coupled through a dispersive interaction. To achieve strong entanglement between the collective atomic spin and the photons, one must match the spatial mode of the collective radiation of the ensemble with the mode of the laser beam while minimizing the effects of decoherence due to optical pumping. For ensembles coupling to a probe field that varies over the extent of the cloud, the set of atoms that indistinguishably radiates into a desired mode of the field defines an inhomogeneous spin wave. Strong coupling of a spin wave to the probe mode is not characterized by a single parameter, the optical density, but by a collection of different effective atom numbers that characterize the coherence and decoherence of the system. To model the dynamics of the system, we develop a full stochastic master equation, including coherent collective scattering into paraxial modes, decoherence by local inhomogeneous diffuse scattering, and backaction due to continuous measurement of the light entangled with the spin waves. This formalism is used to study the squeezing of a spin wave via continuous quantum nondemolition (QND) measurement. We find that the greatest squeezing occurs in parameter regimes where spatial inhomogeneities are significant, far from the limit in which the interface is well approximated by a one-dimensional, homogeneous model., Comment: 24 pages, 7 figures

Ultracold atoms in optical lattices are an important platform for quantum information science, lending itself naturally to quantum simulation of many-body physics and providing a possible path towards a scalable quantum computer. To realize its full potential, atoms at individual lattice sites must be accessible to quantum control and measurement. This challenge has so far been met with a combination of high-resolution microscopes and resonance addressing that have enabled both site-resolved imaging and spin-flips. Here we show that methods borrowed from the field of inhomogeneous control can greatly increase the performance of resonance addressing in optical lattices, allowing us to target arbitrary single-qubit gates on desired sites, with minimal crosstalk to neighboring sites and greatly improved robustness against uncertainty in the lattice position. We further demonstrate the simultaneous implementation of different gates at adjacent sites with a single global control waveform. Coherence is verified through two-pulse Ramsey interrogation, and randomized benchmarking is used to measure an average gate fidelity of ~95%. Our control-based approach to reduce crosstalk and increase robustness is broadly applicable in optical lattices irrespective of geometry, and may be useful also on other platforms for quantum information processing, such as ion traps and nitrogen-vacancy centers in diamond., Comment: Originally submitted version. Longer version with some substantive edits to appear in Nature Communications

We find quantum signatures of classical chaos in various metrics of information gain in quantum tomography. We employ a quantum state estimator based on weak collective measurements of an ensemble of identically prepared systems. The tomographic measurement record consists of a sequence of expectation values of a Hermitian operator that evolves under repeated application of the Floquet map of the quantum kicked top. We find an increase in information gain and hence higher fidelities in the reconstruction algorithm when the chaoticity parameter map increases. The results are well predicted by random matrix theory.

We study an architecture for implementing adiabatic quantum computation with trapped neutral atoms. Ground state atoms are dressed by laser fields in a manner conditional on the Rydberg blockade mechanism, thereby providing the requisite entangling interactions. As a benchmark we study the performance of a Quadratic Unconstrained Binary Optimization (QUBO) problem whose solution is found in the ground state spin configuration of an Ising-like model. We model a realistic architecture, including details of the atomic implementation, with qubits encoded into the clock states of 133Cs, effective B-fields implemented through stimulated Raman transitions, and atom-atom coupling achieved by excitation to the 100P3/2 Rydberg level. Including the fundamental effects of photon scattering, we find the fidelity of two-qubit implementation to be on the order of 0.99, with higher fidelities possible with improved laser sources., Comment: 5 pages, 3 figures

Unitary control of qudits can improve the collective spin squeezing of an atomic ensemble. Preparing the atoms in a state with large quantum fluctuations in magnetization strengthens the entangling Faraday interaction. The resulting increase in interatomic entanglement can be converted into metrologically useful spin squeezing. Further control can squeeze the internal atomic spin without compromising entanglement, providing an overall multiplicative factor in the collective squeezing. We model the effects of optical pumping and study the tradeoffs between enhanced entanglement and decoherence. For realistic parameters we see improvements of ~10 dB., Comment: 5 pages, 2 figures

We study the ability to control d-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 semi-analytic 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 semi-analytic protocol is compared to a brute force, full numerical search. For small inhomogeneities, < 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 study the quantum phases of a three-color Hubbard model that arises in the dynamics of the p-band orbitals of spinless fermions in an optical lattice. Strong, color-dependent interactions are induced by an optical Feshbach resonance. Starting from the microscopic scattering properties of ultracold atoms, we derive the orbital exchange constants at 1/3 filling on the cubic optical lattice. Using this, we compute the phase diagram in a Gutzwiller ansatz. We find novel phases with 'axial orbital order' in which pz and px + ipy (or px - ipy) orbitals alternate., Comment: 4+epsilon pages, 3 figures. Similar to version published in PRA(R)

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., PRL 95 030402 (2005), 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, 133Cs. 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 7-dimensional F=3 manifold in the electronic-ground state of Cs. We review the procedure as implemented in experiments (Smith et al., PRL 97 180403 (2006)). We extend the protocol to the more ambitious case of reconstruction of states in the full 16-dimensional electronic-ground subspace (F=3 \oplus F=4), controlled by microwaves and radio-frequency 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 \sim 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.

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. In practice the scaling and peak squeezing depends on decoherence, technical loss, and noise. A simplified model indicates ~10 dB of squeezing should be achievable with current laboratory parameters., Comment: 4 pages, 2 figures.

We study the possibility of performing quantum state reconstruction from a measurement record that is obtained as a sequence of expectation values of a Hermitian operator evolving under repeated application of a single random unitary map, U_0. We show that while this single-parameter orbit in operator space is not informationally complete, it can be used to yield surprisingly high-fidelity reconstruction. For a d-dimensional Hilbert space with the initial observable in su(d), the measurement record lacks information about a matrix subspace of dimension > d-2 out of the total dimension d^2-1. We determine the conditions on U_0 such that the bound is saturated, and show they are achieved by almost all pseudorandom unitary matrices. When we further impose the constraint that the physical density matrix must be positive, we obtain even higher fidelity than that predicted from the missing subspace. With prior knowledge that the state is pure, the reconstruction will be perfect (in the limit of vanishing noise) and for arbitrary mixed states, the fidelity is over 0.96, even for small d, and reaching F > 0.99 for d > 9. We also study the implementation of this protocol based on the relationship between random matrices and quantum chaos. We show that the Floquet operator of the quantum kicked top provides a means of generating the required type of measurement record, with implications on the relationship between quantum chaos and information gain., Comment: 8 pages, 4 figures

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.

Nuclear spin exchange occurs in ultracold collisions of fermionic alkaline-earth-like atoms due to a difference between s- and p-wave phase shifts. We study the use of an optical Feshbach resonance, excited on the ${}^1S_0 \to {}^3P_1$ intercombination line of ${}^{171}$Yb, to affect a large modification of the s-wave scattering phase shift, and thereby optically mediate nuclear exchange forces. We perform a full multichannel calculation of the photoassociation resonances and wave functions and from these calculate the real and imaginary parts of the scattering length. As a figure of merit of this interaction, we estimate the fidelity to implement a $\sqrt{SWAP}$ entangling quantum logic gate for two atoms trapped in the same well of an optical lattice. For moderate parameters one can achieve a gate fidelity of $\sim95% $ in a time of $\sim 50 \mu$s., Comment: 5 pages, 1 figure

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 uniform forces. Such control is important for applications in quantum information processing such as quantum computing and quantum simulations of condensed matter phenomena., Comment: 20 pages, 3 figures

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 eigen-decomposition 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 waveforms 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 waveform 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 atoms in order to implement general qudit logic gates as well to perform a simple form of error correction on an embedded qubit., Comment: 9 pages, 3 figures

We study quantum control of the full hyperfine manifold in the ground-electronic state of alkali 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 on the system. We apply this to the case of $^{133}$Cs with its $d=16$ dimensional Hilbert space of magnetic sublevels in the $6S_{1/2}$ state, and design control waveforms 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 representation of SU(2), allowing us to visualize these states. The performance of different control scenarios is evaluated based on the ability to generate high-fidelity operation in an allotted time with the available resources. We find good operating points commensurate with modest laboratory requirements., Comment: 14 pages, 7 figures; corrected typos