Inverse Mpemba Effect Demonstrated on a Single Trapped Ion Qubit.

The Mpemba effect is a counterintuitive phenomena in which a hot system reaches a cold temperature faster than a colder system, under otherwise identical conditions. Here, we propose a quantum analog of the Mpemba effect, on the simplest quantum system, a qubit. Specifically, we show it exhibits an inverse effect, in which a cold qubit reaches a hot temperature faster than a hot qubit. Furthermore, in our system a cold qubit can heat up exponentially faster, manifesting the strong version of the effect. This occurs only for sufficiently coherent systems, making this effect quantum mechanical, i.e., due to interference effects. We experimentally demonstrate our findings on a single ^{88}Sr^{+} trapped ion qubit. The existence of this anomalous relaxation effect in simple quantum systems reveals its fundamentality, and may have a role in designing and operating quantum information processing devices.

Robust Two-Qubit Gates for Trapped Ions Using Spin-Dependent Squeezing.

Entangling gates are an essential component of quantum computers. However, generating high-fidelity gates, in a scalable manner, remains a major challenge in all quantum information processing platforms. Accordingly, improving the fidelity and robustness of these gates has been a research focus in recent years. In trapped ions quantum computers, entangling gates are performed by driving the normal modes of motion of the ion chain, generating a spin-dependent force. Even though there has been significant progress in increasing the robustness and modularity of these gates, they are still sensitive to noise in the intensity of the driving field. Here we supplement the conventional spin-dependent displacement with spin-dependent squeezing, which creates a new interaction, that enables a gate that is robust to deviations in the amplitude of the driving field. We solve the general Hamiltonian and engineer its spectrum analytically. We also endow our gate with other, more conventional, robustness properties, making it resilient to many practical sources of noise and inaccuracies.

Precision Measurement of Atomic Isotope Shifts Using a Two-Isotope Entangled State.

Atomic isotope shifts (ISs) are the isotope-dependent energy differences between atomic electron energy levels. These shifts have an important role in atomic and nuclear physics, and have been recently suggested as unique probes of physics beyond the standard model under the condition that they are determined significantly more precisely than the current state of the art. In this Letter, we present a simple and robust method for measuring ISs by taking advantage of Hilbert subspaces that are insensitive to common-mode noise yet sensitive to the IS. Using this method we evaluate the IS of the 5S_{1/2}↔4D_{5/2} transition between ^{86}Sr^{+} and ^{88}Sr^{+} with a 1.6×10^{-11} relative uncertainty to be 570 264 063.435(5)(8) (statistical)(systematic) Hz. Furthermore, we detect a relative difference of 3.46(23)×10^{-8} between the orbital g factors of the electrons in the 4D_{5/2} level of the two isotopes. Our method is relatively easy to implement and is indifferent to element or isotope, paving the way for future tabletop searches for new physics, posing interesting prospects for testing quantum many-body calculations, and for the study of nuclear structure.

Collisions between cold molecules in a superconducting magnetic trap.

Collisions between cold molecules are essential for studying fundamental aspects of quantum chemistry, and may enable the formation of quantum degenerate molecular matter by evaporative cooling. However, collisions between trapped, naturally occurring molecules have not been directly observed so far owing to the low collision rates of dilute samples. Here we report the direct observation of collisions between cold trapped molecules, without the need for laser cooling. We magnetically capture molecular oxygen in an 800-millikelvin-deep superconducting trap and set bounds on the ratio between the elastic- and inelastic-scattering rates-the key parameter determining the feasibility of evaporative cooling. We further co-trap atoms and molecules and identify collisions between them, paving the way for studies of cold interspecies collisions in a magnetic trap.

Quadrupole Shift Cancellation Using Dynamic Decoupling.

We present a method that uses radio-frequency pulses to cancel the quadrupole shift in optical clock transitions. Quadrupole shifts are an inherent inhomogeneous broadening mechanism in trapped ion crystals and impose one of the limitations forcing current optical ion clocks to work with a single probe ion. Canceling this shift, at each interrogation cycle of the ion frequency, reduces the complexity in using N>1 ions in clocks, thus allowing for a reduction of the instability in the clock frequency by sqrt[N] according to the standard quantum limit. Our sequence relies on the tensorial nature of the quadrupole shift, and thus also cancels other tensorial shifts, such as the tensor ac stark shift. We experimentally demonstrate our sequence on three and seven ^{88}Sr^{+} ions trapped in a linear Paul trap, using correlation spectroscopy. We show a reduction of the quadrupole shift difference between ions to the ≈10 mHz level where other shifts, such as the relativistic second-order Doppler shift, are expected to limit our spectral resolution. In addition, we show that using radio-frequency dynamic decoupling we can also cancel the effect of first-order Zeeman shifts.

Robust Entanglement Gates for Trapped-Ion Qubits.

High-fidelity two-qubit entangling gates play an important role in many quantum information processing tasks and are a necessary building block for constructing a universal quantum computer. Such high-fidelity gates have been demonstrated on trapped-ion qubits; however, control errors and noise in gate parameters may still lead to reduced fidelity. Here we propose and demonstrate a general family of two-qubit entangling gates which are robust to different sources of noise and control errors. These gates generalize the renowned Mølmer-Sørensen gate by using multitone drives. We experimentally implemented several of the proposed gates on ^{88}Sr^{+} ions trapped in a linear Paul trap and verified their resilience.

Phase Locking between Different Partial Waves in Atom-Ion Spin-Exchange Collisions.

We present a joint experimental and theoretical study of spin dynamics of a single ^{88}Sr^{+} ion colliding with an ultracold cloud of Rb atoms in various hyperfine states. While spin exchange between the two species occurs after 9.1(6) Langevin collisions on average, spin relaxation of the Sr^{+} ion Zeeman qubit occurs after 48(7) Langevin collisions, which is significantly slower than in previously studied systems due to a small second-order spin-orbit coupling. Furthermore, a reduction of the endothermic spin-exchange rate is observed as the magnetic field is increased. Interestingly, we find that while the phases acquired when colliding on the spin singlet and triplet potentials vary largely between different partial waves, the singlet-triplet phase difference, which determines the spin-exchange cross section, remains locked to a single value over a wide range of partial waves, which leads to quantum interference effects.

Direct Observation of Atom-Ion Nonequilibrium Sympathetic Cooling.

Sympathetic cooling is the process of energy exchange between a system and a colder bath. We investigate this fundamental process in an atom-ion experiment where the system is composed of a single ion trapped in a radio-frequency Paul trap and prepared in a classical oscillatory motion with total energy of ∼200 K, and the bath is an ultracold cloud of atoms at µK temperature. We directly observe the sympathetic cooling dynamics with single-shot energy measurements during one to several collisions in two distinct regimes. In one, collisions predominantly cool the system with very efficient momentum transfer leading to cooling in only a few collisions. In the other, collisions can both cool and heat the system due to nonequilibrium dynamics in the presence of the ion trap's oscillating electric fields. While the bulk of our observations agree well with a molecular-dynamics simulation of hard-sphere (Langevin) collisions, a measurement of the scattering angle distribution reveals forward-scattering (glancing) collisions which are beyond the Langevin model. This work paves the way for further nonequilibrium and collision dynamics studies using the well-controlled atom-ion system.

Toward Heisenberg-Limited Rabi Spectroscopy.

The use of entangled states was shown to improve the fundamental limits of spectroscopy to beyond the standard-quantum limit. Here, rather than probing the free evolution of the phase of an entangled state with respect to a local oscillator, we probe the evolution of an initially separable two-atom register under an Ising spin Hamiltonian with a transverse field. The resulting correlated spin-rotation spectrum is twice as narrow as that of an uncorrelated rotation. We implement this ideally Heisenberg-limited Rabi spectroscopy scheme on the optical-clock electric-quadrupole transition of ^{88}Sr^{+} using a two-ion crystal. We further show that depending on the initial state, correlated rotation can occur in two orthogonal subspaces of the full Hilbert space, yielding entanglement-enhanced spectroscopy of either the average transition frequency of the two ions or their difference from the mean frequency. The use of correlated spin rotations can potentially lead to new paths for clock stability improvement.

Publisher Correction: Spin-controlled atom-ion chemistry.

The original version of this Article contained an error in the third sentence of the first paragraph of the 'Spin polarizing the Sr+ ion with ultracold atoms' section of the Results, which incorrectly read 'The Langevin collision rate is 1.' The correct version adds 'kHz' after '1.' The fifth sentence of this same paragraph originally read as "Although 87Rb has a I = 3/2 nuclear spin and a hyperfine-split ground-state manifold, 88Sr has no nuclear spin and a Zeeman split two-fold ground state", which is incorrect. The correct version states "88Sr+" instead of "88Sr". The first sentence of the fourth paragraph of this same section originally read as "As the collisional energies are on the mK energy scale, spin exchange between Sr+ and Rb prepared in the F = 1 state is allowed only as long as it does not require Rb to change its hyperfine state and climb the 330 m hyperfine energy gap", which is incorrect. The correct version states "330 mK" instead of "330 m".In the Discussion section, the text was originally incorrectly repeated.This has been corrected in both the PDF and HTML versions of the Article.

Spin-controlled atom-ion chemistry.

Quantum control of chemical reactions is an important goal in chemistry and physics. Ultracold chemical reactions are often controlled by preparing the reactants in specific quantum states. Here we demonstrate spin-controlled atom-ion inelastic (spin-exchange) processes and chemical (charge-exchange) reactions in an ultracold Rb-Sr+ mixture. The ion's spin state is controlled by the atomic hyperfine spin state via spin-exchange collisions, which polarize the ion's spin parallel to the atomic spin. We achieve ~ 90% spin polarization due to the absence of strong spin-relaxation channel. Charge-exchange collisions involving electron transfer are only allowed for (RbSr)+ colliding in the singlet manifold. Initializing the atoms in various spin states affects the overlap of the collision wave function with the singlet molecular manifold and therefore also the reaction rate. Our observations agree with theoretical predictions.

Fast Dynamical Decoupling of the Mølmer-Sørensen Entangling Gate.

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 boson-mediated 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 sqrt[N], with N being the number of pulses applied. Here we propose and realize a method that enables dynamical decoupling in a boson-mediated system where the pulses can be applied while spin-boson entanglement persists, resulting in an increase in time that is at most a factor of π/2, independently of the number of pulses applied. We experimentally demonstrate the robustness of our entangling gate with fast dynamical decoupling to σ_{z} noise using ions in a Paul trap.

Trapping of Molecular Oxygen together with Lithium Atoms.

We demonstrate simultaneous deceleration and trapping of a cold atomic and molecular mixture. This is the first step towards studies of cold atom-molecule collisions at low temperatures as well as application of sympathetic cooling. Both atoms and molecules are cooled in a supersonic expansion and are loaded into a moving magnetic trap that brings them to rest via the Zeeman interaction from an initial velocity of 375 m/s. We use a beam seeded with molecular oxygen, and entrain it with lithium atoms by laser ablation prior to deceleration. The deceleration ends with loading of the mixture into a static quadrupole trap, which is generated by two permanent magnets. We estimate 10^{9} trapped O_{2} molecules and 10^{5} Li atoms with background pressure limited lifetime on the order of 1 sec. With further improvements to lithium entrainment we expect that sympathetic cooling of molecules is within reach.

Molecular beam brightening by shock-wave suppression.

Supersonic beams are a prevalent source of cold molecules used in the study of chemical reactions, atom interferometry, gas-surface interactions, precision spectroscopy, molecular cooling, and more. The triumph of this method emanates from the high densities produced in relation to other methods; however, beam density remains fundamentally limited by interference with shock waves reflected from collimating surfaces. We show experimentally that this shock interaction can be reduced or even eliminated by cryocooling the interacting surface. An increase of nearly an order of magnitude in beam density was measured at the lowest surface temperature, with no further fundamental limitation reached. Visualization of the shock waves by plasma discharge and reproduction with direct simulation Monte Carlo calculations both indicate that the suppression of the shock structure is partially caused by lowering the momentum flux of reflected particles and significantly enhanced by the adsorption of particles to the surface. We observe that the scaling of beam density with source pressure is recovered, paving the way to order-of-magnitude brighter, cold molecular beams.

Dynamics of a Ground-State Cooled Ion Colliding with Ultracold Atoms.

Ultracold atom-ion mixtures are gaining increasing interest due to their potential applications in ultracold and state-controlled chemistry, quantum computing, and many-body physics. Here, we studied the dynamics of a single ground-state 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 Maxwell-Boltzmann distribution, characterized by an exponential tail, to a power-law distribution best described by a Tsallis function. Unlike previous experiments, the energy scale of atom-ion interactions is not determined by either the atomic cloud temperature or the ion's trap residual excess-micromotion 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 atom-ion steady-state 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.

Measurement of the magnetic interaction between two bound electrons of two separate ions.

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 multi-electron spin complexes. 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. 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 ground-state spin-1/2 valence electrons of two (88)Sr(+) ions, co-trapped in an electric Paul trap. We varied the ion separation, d, between 2.18 and 2.76 micrometres and measured the electrons' weak, millihertz-scale, 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 decoherence-free 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 inverse-cube law.

Addressing two-level systems variably coupled to an oscillating field.

We propose a simple method to spectrally resolve an array of identical two-level 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 single-spin addressing of a linear chain of trapped ions that are separated by ~3 µm, dressed by a laser field that is resonant with the micromotion sideband of a narrow optical transition.

Emergence of a measurement basis in atom-photon scattering.

After measurement, a wave-function is postulated to collapse on a predetermined set of states--the 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 spin-photon state reveals that although photon scattering entangles all superpositions of the measurement-basis states with the scattered photon polarization, the measurement-basis states themselves remain classically correlated with it. Our findings shed light on the process of quantum measurement in atom-photon interactions.

Nonlinear single-spin spectrum analyzer.

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, non-Gaussian, discrete magnetic field noise. Finally, we experimentally compared the performance of equidistant vs Uhrig modulation schemes for spectral analysis.

Single-ion quantum lock-in amplifier.

Quantum metrology uses tools from quantum information science to improve measurement signal-to-noise ratios. The challenge is to increase sensitivity while reducing susceptibility to noise, tasks that are often in conflict. Lock-in 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 lock-in amplifier. All the lock-in operations--modulation, detection and mixing--are performed through the application of non-commuting 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 single-spin probe technologies by two orders of magnitude. Our reported sensitivity is sufficient for the measurement of parity non-conservation, 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 lock-in technique is generic and can potentially enhance the sensitivity of any quantum sensor.