Publications
2018



(2018). Phaselocking between different partialwaves in atomion spinexchange collisions. Physical Review Letters. 121:173402.


(2018). Direct observation of atomion nonequilibrium sympathetic cooling. Physical Review Letters. 121:053402.

(2018). New Methods for Testing Lorentz Invariance with Atomic Systems. Physical Review Letters. 120:103202.




(2018). Experimental apparatus for overlapping a groundstate cooled ion with ultracold atoms. Journal of Modern Optics. 65:501.

(2018). Probing new light forcemediators by isotope shift spectroscopy. Physical Review Letters. 120:091801. Abstract[All authors]
We explore a method to probe new long and intermediaterange interactions using precision atomic isotope shift spectroscopy. We develop a formalism to interpret linear King plots as bounds on new physics with minimal theory inputs. We focus only on bounding the new physics contributions that can be calculated independently of the standard model nuclear effects. We apply our method to existing Ca+ data and project its sensitivity to conjectured new bosons with spinindependent couplings to the electron and the neutron using narrow transitions in other atoms and ions, specifically, Sr and Yb. Future measurements are expected to improve the relative precision by 5 orders of magnitude, and they can potentially lead to an unprecedented sensitivity for bosons within the 0.3 to 10 MeV mass range.
2017

(2017). Fast Dynamical Decoupling of the MolmerSorensen Entangling Gate. Physical Review Letters. 119:220505. Abstract
Engineering entanglement between quantum systems often involves coupling through a bosonic mediator, which should be disentangled from the systems at the operation's end. The quality of such an operation is generally limited by environmental and control noise. One of the prime techniques for suppressing noise is by dynamical decoupling, where one actively applies pulses at a rate that is faster than the typical time scale of the noise. However, for bosonmediated gates, current dynamical decoupling schemes require executing the pulses only when the boson and the quantum systems are disentangled. This restriction implies an increase of the gate time by a factor of root N, with N being the number of pulses applied. Here we propose and realize a method that enables dynamical decoupling in a bosonmediated system where the pulses can be applied while spinboson entanglement persists, resulting in an increase in time that is at most a factor of pi/2, independently of the number of pulses applied. We experimentally demonstrate the robustness of our entangling gate with fast dynamical decoupling to sigma(z) noise using ions in a Paul trap.

(2017). Probing atomic Higgslike forces at the precision frontier. Physical Review D. 96:(9)093001. Abstract
We propose a novel approach to probe new fundamental interactions using isotope shift spectroscopy in atomic clock transitions. As a concrete toy example we focus on the Higgs boson couplings to the building blocks of matter: the electron and the up and down quarks. We show that the attractive Higgs force between nuclei and their bound electrons, which is poorly constrained, might induce effects that are larger than the current experimental sensitivities. More generically, we discuss how new interactions between the electron and the neutrons, mediated via light new degrees of freedom, may lead to measurable nonlinearities in a King plot comparison between isotope shifts of two different transitions. Given stateoftheart accuracy in frequency comparison, isotope shifts have the potential to be measured with subHz accuracy, thus potentially enabling the improvement of current limits on new fundamental interactions. A candidate atomic system for this measurement requires two different clock transitions and four zero nuclear spin isotopes. We identify several systems that satisfy this requirement and also briefly discuss existing measurements. We consider the size of the effect related to the Higgs force and the requirements for it to produce an observable signal.

(2017). Observation of Optomechanical Strain in a Cold Atomic Cloud. Physical Review Letters. 119:163201. Abstract
We report the observation of optomechanical strain applied to thermal and quantum degenerate Rb87 atomic clouds when illuminated by an intense, far detuned homogeneous laser beam. In this regime the atomic cloud acts as a lens that focuses the laser beam. As a backaction, the atoms experience a force opposite to the beam deflection, which depends on the atomic cloud density profile. We experimentally demonstrate the basic features of this force, distinguishing it from the wellestablished scattering and dipole forces. The observed strain saturates, ultimately limiting the momentum impulse that can be transferred to the atoms. This optomechanical force may effectively induce interparticle interactions, which can be optically tuned.

(2017). Singleshot energy measurement of a single atom and the direct reconstruction of its energy distribution. Physical Review A. 96:(2)020701(R). Abstract
An ensemble of atoms in a steady state, whether or not in thermal equilibrium, has a welldefined energy distribution. Since the energy of single atoms within the ensemble cannot be individually measured, energy distributions are typically inferred from statistical averages. Here, we show how to measure the energy of a single atom in a single experimental realization (single shot). The energy distribution of the atom over many experimental realizations can thus be readily and directly obtained. We apply this method to a single ion trapped in a linear Paul trap for which the energy measurement in a single shot is applicable from 10 K x k(B) and above. Our energy measurement agrees within 5% to a different thermometry method which requires extensive averaging. Apart from the total energy, we also show that the motion of the ion in different trap modes can be distinguished. We believe that this method will have profound implications on singleparticle chemistry and collision experiments.

(2017). Quantum lockin force sensing using optical clock Doppler velocimetry. Nature Communications. 8:14157. Abstract
Force sensors are at the heart of different technologies such as atomic force microscopy or inertial sensing. These sensors often rely on the measurement of the displacement amplitude of mechanical oscillators under applied force. The best sensitivity is typically achieved when the force is alternating at the mechanical resonance frequency of the oscillator, thus increasing its response by the mechanical quality factor. The measurement of lowfrequency forces, that are below resonance, is a more difficult task as the resulting oscillation amplitudes are significantly lower. Here we use a singletrapped Sr88(+) ion as a force sensor. The ion is electrically driven at a frequency much lower than the trap resonance frequency. We measure small amplitude of motion by measuring the periodic Doppler shift of an atomic optical clock transition, enhanced using the quantum lockin technique. We report frequency force detection sensitivity as low as 2.8 x 10(20) NHz(1/2).

(2017). Doppler cooling thermometry of a multilevel ion in the presence of micromotion. Physical Review A. 96:012519.
2016

(2016). Dynamics of a GroundState Cooled Ion Colliding with Ultracold Atoms. Physical Review Letters. 117:243401. Abstract
Ultracold atomion mixtures are gaining increasing interest due to their potential applications in ultracold and statecontrolled chemistry, quantum computing, and manybody physics. Here, we studied the dynamics of a single groundstate cooled ion during few, to many, Langevin (spiraling) collisions with ultracold atoms. We measured the ion's energy distribution and observed a clear deviation from the MaxwellBoltzmann distribution, characterized by an exponential tail, to a powerlaw distribution best described by a Tsallis function. Unlike previous experiments, the energy scale of atomion interactions is not determined by either the atomic cloud temperature or the ion's trap residual excessmicromotion energy. Instead, it is determined by the force the atom exerts on the ion during a collision which is then amplified by the trap dynamics. This effect is intrinsic to ion Paul traps and sets the lower bound of atomion steadystate interaction energy in these systems. Despite the fact that our system is eventually driven out of the ultracold regime, we are capable of studying quantum effects by limiting the interaction to the first collision when the ion is initialized in the ground state of the trap.

(2016). Atomic Quadrupole Moment Measurement Using Dynamic Decoupling. Physical Review Letters. 116:(14)140801. Abstract
We present a method that uses dynamic decoupling of a multilevel quantum probe to distinguish small frequency shifts that depend on m(j)(2), where m(j)(2) is the angular momentum of level vertical bar j > along the quantization axis, from large noisy shifts that are linear in m(j), such as those due to magnetic field noise. Using this method we measured the electricquadrupole moment of the 4D(5/2) level in Sr88(+) to be 2.973(0.033)(+0.026)ea(0)(2). Our measurement improves the uncertainty of this value by an order of magnitude and thus helps mitigate an important systematic uncertainty in Sr88(+) based optical atomic clocks and verifies complicated manybody quantum calculations.
2015

(2015). Universal gateset for trappedion qubits using a narrow linewidth diode laser. New Journal of Physics. 17:113060. Abstract
We report on the implementation of a high fidelity universal gateset on optical qubits based on trapped Sr88(+) ions for the purpose of quantum information processing. All coherent operations were performed using a narrow linewidth diode laser. We employed a masterslave configuration for the laser, where an ultra low expansion glass FabryPerot cavity is used as a stable reference as well as a spectral filter. We characterized the laser spectrum using the ions with a modified Ramsey sequence which eliminated the affect of the magnetic field noise. We demonstrated high fidelity single qubit gates with individual addressing, based on inhomogeneous micromotion, on a twoion chain as well as the MolmerSorensen twoqubit entangling gate.

(2015). Measurement of the SpinDipolar Part of the Tensor Polarizability of Rb87. Physical Review Letters. 115:(18)183001. Abstract
We report on the measurement of the contribution of the magneticdipole hyperfine interaction to the tensor polarizaility of the electronic ground state in Rb87. This contribution was isolated by measuring the differential shift of the clock transition frequency in Rb87 atoms that were optically trapped in the focus of an intense CO2 laser beam. By comparing to previous tensor polarizability measurements in Rb87, the contribution of the interaction with the nuclear electricquadrupole moment was isolated as well. Our measurement will enable better estimation of blackbody shifts in Rb atomic clocks. The methods reported here are applicable for future spectroscopic studies of atoms and molecules under strong quasistatic fields.

(2015). Constraints on Exotic DipoleDipole Couplings between Electrons at the Micrometer Scale. Physical Review Letters. 115:(8)081801. Abstract
New constraints on exotic dipoledipole interactions between electrons at the micrometer scale are established, based on a recent measurement of the magnetic interaction between two trapped Sr88(+) ions. For light bosons (mass
2014

(2014). Cooperative Lamb Shift in a Mesoscopic Atomic Array. Physical Review Letters. 113:(19)193002. Abstract
According to quantum electrodynamics, the exchange of virtual photons in a system of identical quantum emitters causes a shift of its energy levels. Such shifts, known as cooperative Lamb shifts, have been studied mostly in the nearfield regime. However, the resonant electromagnetic interaction persists also at large distances, providing coherent coupling between distant atoms. Here, we report a direct spectroscopic observation of the cooperative Lamb shift of an optical electricdipole transition in an array of Sr+ ions suspended in a Paul trap at interion separations much larger than the resonance wavelength. By controlling the precise positions of the ions, we studied the farfield resonant coupling in chains of up to eight ions, extending to a length of 40 mu m. This method provides a novel tool for experimental exploration of cooperative emission phenomena in extended mesoscopic atomic arrays.

(2014). Quantum process tomography of a MolmerSorensen interaction. Physical Review A. 90:(1)010103(R). Abstract
We present a simple tomographic protocol, for twoqubit systems, that relies on a single discriminatory transition and no direct spatially selective imaging. This scheme exploits excess micromotion in the trap to realize all operations required to prepare all input states and analyze all output states. We demonstrate a twoqubit entangling gate with a Bell state production fidelity of 0.981(6), and apply the above protocol to perform the first quantum process tomography of a MolmerSorensen entangling gate. We characterize its chiprocess matrix, the simplest for an entanglement gate on a separablestates basis, and observe that our dominant source of error is accurately modeled by a quantum depolarization channel.

(2014). Measurement of the magnetic interaction between two bound electrons of two separate ions. Nature. 510:(7505)376380. Abstract
Electrons have an intrinsic, indivisible, magnetic dipole aligned with their internal angular momentum (spin). The magnetic interaction between two electronic spins can therefore impose a change in their orientation. Similar dipolar magnetic interactions exist between other spin systems and have been studied experimentally. Examples include the interaction between an electron and its nucleus and the interaction between several multielectron spin complexes(15). The challenge in observing such interactions for two electrons is twofold. First, at the atomic scale, where the coupling is relatively large, it is often dominated by the much larger Coulomb exchange counterpart(1). Second, on scales that are substantially larger than the atomic, the magnetic coupling is very weak and can be well below the ambient magnetic noise. Here we report the measurement of the magnetic interaction between the two groundstate spin1/2 valence electrons of two Sr88(+) ions, cotrapped in an electric Paul trap. We varied the ion separation, d, between 2.18 and 2.76 micrometres and measured the electrons' weak, millihertzscale, magnetic interaction as a function of distance, in the presence of magnetic noise that was six orders of magnitude larger than the magnetic fields the electrons apply on each other. The cooperative spin dynamics was kept coherent for 15 seconds, during which spin entanglement was generated, as verified by a negative measured value of 0.16 for the swap entanglement witness. The sensitivity necessary for this measurement was provided by restricting the spin evolution to a decoherencefree subspace that is immune to collective magnetic field noise. Our measurements show a d(3.0(4)) distance dependence for the coupling, consistent with the inversecube law.
2013

(2013). Addressing TwoLevel Systems Variably Coupled to an Oscillating Field. Physical Review Letters. 111:(7)073001. Abstract
We propose a simple method to spectrally resolve an array of identical twolevel systems coupled to an inhomogeneous oscillating field. The addressing protocol uses a dressing field with a spatially dependent coupling to the atoms. We validate this scheme experimentally by realizing singlespin addressing of a linear chain of trapped ions that are separated by similar to 3 mu m, dressed by a laser field that is resonant with the micromotion sideband of a narrow optical transition.

(2013). Emergence of a Measurement Basis in AtomPhoton Scattering. Science. 339:(6124)11871191. Abstract
After measurement, a wavefunction is postulated to collapse on a predetermined set of statesthe measurement basis. Using quantum process tomography, we show how a measurement basis emerges in the evolution of the electronic spin of a single trapped atomic ion after spontaneous photon scattering and detection. This basis is determined by the excitation laser polarization and the direction along which the photon was detected. Quantum tomography of the combined spinphoton state reveals that although photon scattering entangles all superpositions of the measurementbasis states with the scattered photon polarization, the measurementbasis states themselves remain classically correlated with it. Our findings shed light on the process of quantum measurement in atomphoton interactions.

(2013). Nonlinear SingleSpin Spectrum Analyzer. Physical Review Letters. 110:(11)110503. Abstract
Qubits have been used as linear spectrum analyzers of their environments. Here we solve the problem of nonlinear spectral analysis, required for discrete noise induced by a strongly coupled environment. Our nonperturbative analytical model shows a nonlinear signal dependence on noise power, resulting in a spectral resolution beyond the Fourier limit as well as frequency mixing. We develop a noise characterization scheme adapted to this nonlinearity. We then apply it using a single trapped ion as a sensitive probe of strong, nonGaussian, discrete magnetic field noise. Finally, we experimentally compared the performance of equidistant vs Uhrig modulation schemes for spectral analysis. DOI: 10.1103/PhysRevLett.110.110503

2012

(2012). Reversal of PhotonScattering Errors in Atomic Qubits. Physical Review Letters. 109:(10)103601. Abstract
Spontaneous photon scattering by an atomic qubit is a notable example of environmentinduced error and is a fundamental limit to the fidelity of quantum operations. In the scattering process, the qubit loses its distinctive and coherent character owing to its entanglement with the photon. Using a single trapped ion, we show that by utilizing the information carried by the photon, we are able to coherently reverse this process and correct for the scattering error. We further used quantum process tomography to characterize the photonscattering error and its correction scheme and demonstrate a correction fidelity greater than 85% whenever a photon was measured.

(2012). Quantum control of Sr88(+) in a miniature linear Paul trap. Applied Physics BLasers And Optics. 107:(4)11671174. Abstract
We report on the construction and characterization of an apparatus for quantum information experiments using Sr88(+) ions. A miniature linear radiofrequency (rf) Paul trap was designed and built. Trap frequencies above 1 MHz in all directions are obtained with 50 V on the trap endcaps and less than 1 W of rf power. We encode a quantum bit (qubit) in the two spin states of the S (1/2) electronic groundstate of the ion. We constructed all the necessary laser sources for laser cooling and full coherent manipulation of the ions' external and internal states. Oscillating magnetic fields are used for coherent spin rotations. Highfidelity readout as well as a coherence time of 2.5 ms are demonstrated. Following resolved sideband cooling the average axial vibrational quanta of a single trapped ion is and a heating rate of is measured.
2011

(2011). Highfidelity state detection and tomography of a singleion Zeeman qubit. New Journal of Physics. 13:073027. Abstract
We demonstrate highfidelity Zeeman qubit state detection in a single trapped (88)Sr(+) ion. Qubit readout is performed by shelving one of the qubit states to a metastable level using a narrow linewidth diode laser at 674 nm, followed by stateselective fluorescence detection. The average fidelity reached for the readout of the qubit state is 0.9989(1). We then measure the fidelity of state tomography, averaged over all possible singlequbit states, which is 0.9979(2). We also fully characterize the detection process using quantum process tomography. This readout fidelity is compatible with recent estimates of the detection error threshold required for faulttolerant computation, whereas highfidelity state tomography opens the way for highprecision quantum process tomography.

(2011). Singleion quantum lockin amplifier. Nature. 473:(7345)6165. Abstract
Quantum metrology(1) uses tools from quantum information science to improve measurement signaltonoise ratios. The challenge is to increase sensitivity while reducing susceptibility to noise, tasks that are often in conflict. Lockin measurement is a detection scheme designed to overcome this difficulty by spectrally separating signal from noise. Here we report on the implementation of a quantum analogue to the classical lockin amplifier. All the lockin operationsmodulation, detection and mixingare performed through the application of noncommuting quantum operators to the electronic spin state of a single, trapped Sr(+) ion. We significantly increase its sensitivity to external fields while extending phase coherence by three orders of magnitude, to more than one second. Using this technique, we measure frequency shifts with a sensitivity of 0.42 Hz Hz(1/2) (corresponding to a magnetic field measurement sensitivity of 15 pT Hz(1/2)), obtaining an uncertainty of less than 10 mHz (350 fT) after 3,720 seconds of averaging. These sensitivities are limited by quantum projection noise and improve on other singlespin probe technologies(2,3) by two orders of magnitude. Our reported sensitivity is sufficient for the measurement of parity nonconservation(4), as well as the detection of the magnetic field of a single electronic spin one micrometre from an ion detector with nanometre resolution. As a first application, we perform light shift spectroscopy of a narrow optical quadrupole transition. Finally, we emphasize that the quantum lockin technique is generic and can potentially enhance the sensitivity of any quantum sensor.

(2011). The trappedion qubit tool box. Contemporary Physics. 52:(6)531550. Abstract
In this tutorial we review the basic building blocks of Quantum Information Processing with cold trapped atomic ions. We mainly focus on methods to implement singlequbit rotations and twoqubit entangling gates, which form a universal set of quantum gates. Different ion qubit choices and their respective gate implementations are described.
2010

(2010). Singleion nonlinear mechanical oscillator. Physical Review A. 82:(6) Abstract
We study the steadystate motion of a single trapped ion oscillator driven to the nonlinear regime. Damping is achieved via Doppler laser cooling. The ion motion is found to be well described by the Duffing oscillator model with an additional nonlinear damping term. We demonstrate here the unique ability of tuning both the linear as well as the nonlinear damping coefficients by controlling the lasercooling parameters. Our observations pave the way for the investigation of nonlinear dynamics on the quantumtoclassical interface as well as mechanical noise squeezing in lasercooling dynamics.

(2010). Decoherence due to Elastic Rayleigh Scattering. Physical Review Letters. 105:(20)200401. Abstract
We present theoretical and experimental studies of the decoherence of hyperfine groundstate superpositions due to elastic Rayleigh scattering of light off resonant with higher lying excited states. We demonstrate that under appropriate conditions, elastic Rayleigh scattering can be the dominant source of decoherence, contrary to previous discussions in the literature. We show that the elasticscattering decoherence rate of a twolevel system is given by the square of the difference between the elasticscattering amplitudes for the two levels, and that for certain detunings of the light, the amplitudes can interfere constructively even when the elasticscattering rates from the two levels are equal. We confirm this prediction through calculations and measurements of the total decoherence rate for a superposition of the valence electron spin levels in the ground state of (9)Be(+) in a 4.5 T magnetic field.
2009

(2009). Qubit coherent control and entanglement with squeezed light fields. Physical Review A. 80:(3)033803. Abstract
We study the use of squeezed light for qubit coherent control and compare it with the coherentstate control field case. We calculate the entanglement between a short pulse of resonant squeezed light and a twolevel atom in free space and the resulting operation error. We find that the squeezing phase, the phase of the light field, and the atomic superposition phase all determine whether atompulse mode entanglement and the gate error are enhanced or suppressed. When averaged over all possible qubit initial states, the gate error would not decrease by a practically useful amount and would in fact increase in most cases. However, the enhanced entanglement may be of use in quantum communication schemes. We discuss the possibility of measuring the increased gate error as a signature of the enhancement of entanglement by squeezing.

(2009). Entangled mechanical oscillators. Nature. 459:(7247)683U84. Abstract
Hallmarks of quantum mechanics include superposition and entanglement. In the context of large complex systems, these features should lead to situations as envisaged in the 'Schrodinger's cat'(1) thought experiment (where the cat exists in a superposition of alive and dead states entangled with a radioactive nucleus). Such situations are not observed in nature. This may be simply due to our inability to sufficiently isolate the system of interest from the surrounding environment(2,3) a technical limitation. Another possibility is some asyetundiscovered mechanism that prevents the formation of macroscopic entangled states(4). Such a limitation might depend on the number of elementary constituents in the system(5) or on the types of degrees of freedom that are entangled. Tests of the latter possibility have been made with photons, atoms and condensed matter devices(6,7). One system ubiquitous to nature where entanglement has not been previously demonstrated consists of distinct mechanical oscillators. Here we demonstrate deterministic entanglement of separated mechanical oscillators, consisting of the vibrational states of two pairs of atomic ions held in different locations. We also demonstrate entanglement of the internal states of an atomic ion with a distant mechanical oscillator. These results show quantum entanglement in a degree of freedom that pervades the classical world. Such experiments may lead to the generation of entangled states of largerscale mechanical oscillators(810), and offer possibilities for testing nonlocality with mesoscopic systems(11). In addition, the control developed here is an important ingredient for scalingup quantum information processing with trapped atomic ions(1214).
2008

(2008). Randomized benchmarking of quantum gates. Physical Review A. 77:(1) 012307. Abstract[All authors]
A key requirement for scalable quantum computing is that elementary quantum gates can be implemented with sufficiently low error. One method for determining the error behavior of a gate implementation is to perform process tomography. However, standard process tomography is limited by errors in state preparation, measurement and onequbit gates. It suffers from inefficient scaling with number of qubits and does not detect adverse errorcompounding when gates are composed in long sequences. An additional problem is due to the fact that desirable error probabilities for scalable quantum computing are of the order of 0.0001 or lower. Experimentally proving such low errors is challenging. We describe a randomized benchmarking method that yields estimates of the computationally relevant errors without relying on accurate state preparation and measurement. Since it involves long sequences of randomly chosen gates, it also verifies that error behavior is stable when used in long computations. We implemented randomized benchmarking on trapped atomic ion qubits, establishing a onequbit error probability per randomized pi/2 pulse of 0.00482(17) in a particular experiment. We expect this error probability to be readily improved with straightforward technical modifications.
2007

(2007). Fluorescence during Doppler cooling of a single trapped atom. Physical Review A. 76:(5)053416. Abstract[All authors]
We investigate the temporal dynamics of Doppler cooling of an initially hot single trapped atom in the weakbinding regime using a semiclassical approach. We develop an analytical model for the simplest case of a single vibrational mode for a harmonic trap, and show how this model allows us to estimate the initial energy of the trapped particle by observing the timedependent fluorescence during the cooling process. The experimental implementation of this temperature measurement provides a way to measure atom heating rates by observing the temperature rise in the absence of cooling. This method is technically relatively simple compared to conventional sideband detection methods, and the two methods are in reasonable agreement. We also discuss the effects of rf micromotion, relevant for a trapped atomic ion, and the effect of coupling between the vibrational modes on the cooling dynamics.

(2007). Simplified motional heating rate measurements of trapped ions. Physical Review A. 76:(3)033411. Abstract[All authors]
We have measured motional heating rates of trapped atomic ions, a factor that can influence multiion quantum logic gate fidelities. Two simplified techniques were developed for this purpose: one relies on Raman sideband detection implemented with a single laser source, while the second is even simpler and is based on timeresolved fluorescence detection during Doppler recooling. We applied these methods to determine heating rates in a microfrabricated surfaceelectrode trap made of gold on fused quartz, which traps ions 40 mu m above its surface. Heating rates obtained from the two techniques were found to be in reasonable agreement. In addition, the trap gives rise to a heating rate of 300 +/ 30 s(1) for a motional frequency of 5.25 MHz, substantially below the trend observed in other traps.

(2007). Errors in trappedion quantum gates due to spontaneous photon scattering. Physical Review A. 75:(4)042329. Abstract[All authors]
We analyze the error in trappedion, hyperfine qubit, quantum gates due to spontaneous scattering of photons from the gate laser beams. We investigate singlequbit rotations that are based on stimulated Raman transitions and twoqubit entangling phase gates that are based on spindependent optical dipole forces. This error is compared between different ion species currently being investigated as possible quantuminformation carriers. For both gate types we show that with attainable laser powers the scattering error can be reduced to below current estimates of the faulttolerance error threshold.