Peter A. Ivanov

With coupled detuned oscillators, either individual or collective oscillations are observable. The latter is used in quantum information processing in linear Paul traps. Here, we study the transition from collective radial modes at stronger axial confinements into individual radial oscillations at low axial confinements in a tapered Paul trap. The eigenmodes are experimentally studied in detail in the transition regime and compared with theoretical predictions. The features studied will enable investigation of modified heat transport phenomena and defect formation in trapped ions.

We propose an adiabatic Ramsey interferometry technique for detecting weak nonlinearities with trapped ions. The method relies on using the quantum Rabi model as a probe, which is sensitive to nonlinear symmetry-breaking perturbations. We show that the couplings which arise either from anharmonic terms of the trapping potential or due to higher order terms in the Coulomb interaction expansion can be efficiently estimated by measuring the spin state probabilities alone. We show that the spin signal is amplified by the mean-phonon excitations, which results in the estimation precision reaching the super-Heisenberg limit. Notably, achieving such high-precision estimation does not require specific entangled state preparation and can be reached even for initial thermal motion state. Furthermore, we show that the super-Heisenberg scaling can be observed even in the presence of weak spin-dephasing.

We propose the realization of a mechanically squeezed Kerr oscillator with a single ion in a tapered trap. We show that the motion coupling between the axial and radial modes caused by the trap geometry leads to Kerr nonlinearity of the radial mode with magnitude controlled by the trap frequencies. This allows the realization of non-Gaussian quantum gates, which play a significant role in the universal set of continuous variable quantum gates. Furthermore, we show that, because of the nonlinearity of the ion trap, applying an off-resonant time-varying electric field along the trap axis causes a motion squeezing of the radial mode. Finally, we discuss the motion mode frequency spectrum of an ion crystal in a tapered trap. We show that the frequency gap between the motion modes increases with trap nonlinearity, which benefits the realization of faster quantum gates.

We propose a spin-motion state for high-precision quantum metrology with super-Heisenberg scaling of the parameter estimation uncertainty using a trapped ion system. Such a highly entangled state can be created using the Tavis-Cummings Hamiltonian which describes the interaction between a collective spin system and a single vibrational mode. Our method relies on an adiabatic evolution in which the initial motional squeezing is adiabatically transferred into collective spin squeezing. In the weak squeezing regime, we show that the adiabatic evolution creates a spin-squeezed state, which reduces the quantum projective noise to a sub-shot noise limit. For strong bosonic squeezing we find that the quantum Fisher information follows a super-Heisenberg scaling law $\propto N^{5/2}$ in terms of the number of ions $N$. Furthermore, we discuss the spin squeezing parameter which quantifies the phase sensitivity enhancement in Ramsey spectroscopic measurements and show that it also exhibits a super-Heisenberg scaling with $N$. Our work enables the development of high-precision quantum metrology based on entangled spin-boson states that lead to faster scaling of the parameter estimation uncertainty with the number of spins.

We propose a laser-free method for creation of a phonon two-mode squeezed state and a beam-splitter transformation, using time-varying electric fields and non-linear couplings between the normal modes in a linear ion crystal. Such non-linear Coulomb-mediated interactions between the collective vibrational modes arise under specific trap-frequency conditions in an ion trap. We study the quantum metrological capability for parameter estimation of the two quantum states and show that a Heisenberg limit of precision can be achieved when the initial state with $n$ phonons evolves under the action of the beam-splitter transformation. Furthermore, we show that the phonon non-linearity and the spin-dependent force can be used for creation of a three-qubit Fredkin gate.

I propose an efficient method for measuring non-linear coupling between the collective axial breathing mode and the radial rocking mode induced by the mutual Coulomb repulsion in linear ion crystal. The quantum sensing technique is based on the laser induced coupling between one of the vibrational modes and the internal ion's spin states which allows to estimate the non-linear coupling either by measuring the phonon probability distribution or directly be observing the Ramsey-type oscillations of the ion spin states. I show that due to the presence of non-linear phonon coupling the off-resonance interaction between the ion spin states and the axial breathing mode leads to spin-dependent phonon squeezing of the radial rocking mode. Thus the non-linear coupling can be estimated by measuring population distribution of the motional squeezed state. Furthermore, I show that the off-resonance interaction between the spin and the radial rocking mode creates a spin-dependent beam splitter operation between the two vibrational modes. Thus, the parameter estimation can be carried out by detecting the ion spin populations. Finally, I show that the measurement uncertainty precision can reach the Heisenberg limit by using an entangled states between the two collective modes.

We propose an adiabatic method for optimal phonon temperature estimation using trapped ions which can be operated beyond the Lamb-Dicke regime. The quantum sensing technique relies on a time-dependent red-sideband transition of phonon modes, described by the non-linear Jaynes-Cummings model in general. A unique feature of our sensing technique is that the relevant information of the phonon thermal distributions can be transferred to the collective spin-degree of freedom. We show that each of the thermal state probabilities is adiabatically mapped onto the respective collective spin-excitation configuration and thus the temperature estimation is carried out simply by performing a spin-dependent laser fluorescence measurement at the end of the adiabatic transition. We characterize the temperature uncertainty in terms of the Fisher information and show that the state projection measurement saturates the fundamental quantum Cramér-Rao bound for quantum oscillator at thermal equilibrium.

We propose a method for the realization of the two-qubit quantum Fourier transform (QFT) using a Hamiltonian which possesses the circulant symmetry. Importantly, the eigenvectors of the circulant matrices are the Fourier modes and do not depend on the magnitude of the Hamiltonian elements as long as the circulant symmetry is preserved. The QFT implementation relies on the adiabatic transition from each of the spin product states to the respective quantum Fourier superposition states. We show that in ion traps one can obtain a Hamiltonian with the circulant symmetry by tuning the spin-spin interaction between the trapped ions. We present numerical results which demonstrate that very high fidelity can be obtained with realistic experimental resources. We also describe how the gate can be accelerated by using a "shortcut-to-adiabaticity" field.

We propose a scheme for detecting time-varying weak forces using quantum probe consisting of single spin and quantum oscillator under the effect of collective dissipation. We study the force estimation in the steady-state regime where the information of the force is extracted by measuring observable of the oscillator such as quadrature and mean phonon excitation. We quantify the force sensitivity in terms of quantum Fisher information and show that it diverges approaching the critical spin-boson coupling making the system sensitive to very small force perturbation. We show that close to the critical coupling the measurement of the oscillator quadrature is optimal in a sense that saturates the fundamental Cramer-Rao bound. Furthermore, we study the force estimation in the presence of phonon squeezing and show that it can significantly improve the sensitivity reaching minimal detectable force of order of xN (1${\rm xN=10^{-27}{\rm N}}$).

The quadrupole interaction between the Rydberg electronic states of a Rydberg ion and the radio frequency electric field of the ion trap is analyzed. Such a coupling is negligible for the lowest energy levels of a trapped ion but it is important for a trapped Rydberg ion due to its large electric dipole moment. This coupling cannot be neglected by the standard rotating-wave approximation because it is comparable to the frequency of the trapping electric field. We investigate the effect of the quadrupole coupling by performing a suitable effective representation of the Hamiltonian. For a single ion we show that in this effective picture the quadrupole interaction is replaced by rescaled laser intensities and additional Stark shifts of the Rydberg levels. Hence this detrimental quadrupole coupling can be efficiently compensated by an appropriate increase of the Rabi frequencies. Moreover, we consider the strong dipole-dipole interaction between a pair of Rydberg ions in the presence of the quadrupole coupling. In the effective representation we observe reducing of the dipole-dipole coupling as well as additional spin-spin interaction.

We introduce the estimation protocol for detecting the temperature of the transverse vibrational modes of linear ion crystal. We show that thanks to the laser induced laser coupling between the vibrational modes and the collective spin states the estimation of the temperature is carried out by set of measurements of the spin populations. We show that temperature estimation protocol using single ion as a quantum probe is optimal in a sense that the set of state projective measurement saturates the fundamental Cramer-Rao bound. We find a plateau of the maximal temperature sensitivity using ion chain as a quantum probe. Moreover, we show that the non-classical part of the quantum Fisher information could leads to enhancement of the temperature sensitivity compared to the single ion case.

We introduce a quantum sensing protocol for detecting the parameters characterizing the phase space displacement by using a single trapped ion as a quantum probe. We show that thanks to the laser-induced coupling between the ion's internal states and the motion mode the estimation of the two conjugated parameters describing the displacement can be efficiently performed by a set of measurements of the atomic state populations. Furthermore, we introduce a three-parameter protocol capable to detect the magnitude, the transverse direction and the phase of the displacement. We characterize the uncertainty of the two- and three-parameter problems in terms of the Fisher information and show that state projective measurement saturates the fundamental quantum Cramer-Rao bound.

We introduce quantum sensing protocol for detection spatially varying fields by using two coupled harmonic oscillators as a quantum probe. We discuss a physical implementation of the sensing technique with two trapped ions coupled via Coulomb mediated phonon hopping. Our method relies on using the coupling between the localized ion oscillations and the internal states of the trapped ions which allows to measure spatially varying electric and magnetic fields. First we discuss an adiabatic sensing technique which is capable to detect a very small force difference simply by measuring the ion spin population. We also show that the adiabatic method can be used for detection magnetic field gradient which is independent of the magnetic offset. Second, we show that the strong spin phonon coupling could leads to improve sensitivity to force as well as to phase estimations. We quantify the sensitivity in terms of quantum Fisher information and show that it diverges by approaching the critical coupling.

We introduce quantum sensing schemes for measuring very weak forces with a single trapped ion. They use the spin-motional coupling induced by the laser-ion interaction to transfer the relevant force information to the spin-degree of freedom. Therefore, the force estimation is carried out simply by observing the Ramsey-type oscillations of the ion spin states. Three quantum probes are considered, which are represented by systems obeying the Jaynes-Cummings, quantum Rabi (in 1D) and Jahn-Teller (in 2D) models. By using dynamical decoupling schemes in the Jaynes-Cummings and Jahn-Teller models, our force sensing protocols can be made robust to the spin dephasing caused by the thermal and magnetic field fluctuations. In the quantum-Rabi probe, the residual spin-phonon coupling vanishes, which makes this sensing protocol naturally robust to thermally-induced spin dephasing. We show that the proposed techniques can be used to sense the axial and transverse components of the force with a sensitivity beyond the yN $/\sqrt{\text{Hz}}$ range, i.e. in the xN$ /\sqrt{\text{Hz}}$ (xennonewton, $10^{-27}$). The Jahn-Teller protocol, in particular, can be used to implement a two-channel vector spectrum analyzer for measuring ultra-low voltages.

We propose a quantum sensing scheme for measuring weak forces based on a symmetry-breaking adiabatic transition in the quantum Rabi model. We show that the system described by the Rabi Hamiltonian can serve as a sensor for extremely weak forces with sensitivity beyond the yN $/\sqrt{\text{Hz}}$ range. We propose an implementation of this sensing protocol using a single trapped ion. A major advantage of our scheme is that the force detection is performed by projective measurement of the population of the spin states at the end of the transition, instead of the far slower phonon number measurement used hitherto.

We present a system for the simulation of Heisenberg models with spins $s=\frac{1}{2}$ and $s=1$ with a linear crystal of trapped ions. We show that the laser-ion interaction induces a Jaynes-Cummings-Hubbard interaction between the atomic V-type level structure and the two phonon species. In the strong-coupling regime the collective atom and phonon excitations become localized at each lattice site and form an effective spin system with varying length. We show that the quantum-mechanical superexchange interaction caused by the second-order phonon hopping processes creates a Heisenberg-type coupling between the individual spins. Trapped ions allow to control the superexchange interactions by adjusting the trapping frequencies, the laser intensity, and the detuning.

The electric dipole moments of various neutral elementary particles, such as neutron, neutrinos, certain hypothetical dark matter particles and others, are predicted to exist by the standard model of high energy physics and various extensions of it. However, the predicted values are beyond the present experimental capabilities. We propose to simulate and emulate the electric dipole moment of neutral relativistic particles and the ensuing effects in the presence of electrostatic field by emulation of an extended Dirac equation in ion traps.

We study theoretically the collective E$\otimes$e Jahn-Teller-Dicke distortion in a system of trapped ions. We focus in the limit of infinite range interactions in which an ensemble of effective spins interacts with two collective vibrational modes with U(1) symmetric couplings. Our model is exactly solvable in the thermodynamical limit and it is amenable to be solved by exact numerical diagonalization for a moderate number of ions. We show that trapped ions are ideally suited to study the emergence of spontaneous symmetry breaking of a continuous symmetry and magnetic structural phase transition in a mesoscopic system.

We propose a geometric phase gate in a decoherence-free subspace with trapped ions. The quantum information is encoded in the Zeeman sublevels of the ground-state and two physical qubits to make up one logical qubit with ultra long coherence time. Single- and two-qubit operations together with the transport and splitting of linear ion crystals allow for a robust and decoherence-free scalable quantum processor. For the ease of the phase gate realization we employ one Raman laser field on four ions simultaneously, i.e. no tight focus for addressing. The decoherence-free subspace is left neither during gate operations nor during the transport of quantum information.

We propose an ion trap implementation of Grover's quantum search algorithm for an unstructured database of arbitrary length N. The experimental implementation is appealingly simple because the linear ion trap allows for a straightforward construction, in a single interaction step and without a multitude of Hadamard transforms, of the reflection operator, which is the engine of the Grover algorithm. Consequently, a dramatic reduction in the number of the required physical steps takes place, to just O(\sqrt{N}), the same as the number of the mathematical steps. The proposed setup allows for demonstration of both the original (probabilistic) Grover search and its deterministic variation, and is remarkably robust to imperfections in the register initialization.