S. Martínez-Garaot

Hamiltonian engineering via invariants and dynamical algebra

We use the dynamical algebra of a quantum system and its dynamical invariants to inverse engineer feasible Hamiltonians for implementing shortcuts to adiabaticity. These are speeded up processes that end up with the same populations than slow, adiabatic ones. As application examples we design families of shortcut Hamiltonians that drive two and a three-level systems between initial and final configurations imposing physically motivated constraints on the terms (generators) allowed in the Hamiltonian.

Shortcuts to adiabaticity for non-Hermitian systems

Adiabatic processes driven by non-Hermitian, time-dependent Hamiltonians may be sped up by generalizing inverse engineering techniques based on counter-diabatic (transitionless driving) algorithms or on dynamical invariants. We work out the basic theory and examples described by two-level Hamiltonians: the acceleration of rapid adiabatic passage with a decaying excited level and of the dynamics of a classical particle on an expanding harmonic oscillator.

Compact and high conversion efficiency mode-sorting asymmetric Y junction using shortcuts to adiabaticity.

We propose a compact and high conversion efficiency asymmetric Y junction mode multiplexer/demultiplexer for applications in on-chip mode-division multiplexing. Traditionally, mode sorting is achieved by adiabatically separating the arms of a Y junction. We shorten the device length using invariant-based inverse engineering and achieve better conversion efficiency than the adiabatic device.

Vibrational mode multiplexing of ultracold atoms

Sending multiple messages on qubits encoded in different vibrational modes of cold atoms or ions along a transmission waveguide requires us to merge first and then separate the modes at input and output ends. Similarly, different qubits can be stored in the modes of a trap and be separated later. We design the fast splitting of a harmonic trap into an asymmetric double well so that the initial ground vibrational state becomes the ground state of one of two final wells, and the initial first excited state becomes the ground state of the other well. This might be done adiabatically by slowly deforming the trap. We speed up the process by inverse engineering a double-function trap using dynamical invariants. The separation (demultiplexing) followed by an inversion of the asymmetric bias and then by the reverse process (multiplexing) provides a population inversion protocol based solely on trap reshaping.

Fast quasiadiabatic dynamics

We work out the theory and applications of a fast quasiadiabatic approach to speed up slow adiabatic manipulations of quantum systems by driving a control parameter as near to the adiabatic limit as possible over the entire protocol duration. We find characteristic time scales, such as the minimal time to achieve fidelity 1, and the optimality of the approach within the iterative superadiabatic sequence. Specifically, we show that the population inversion in a two-level system, the splitting and cotunneling of two-interacting bosons, and the stirring of a Tonks-Girardeau gas on a ring to achieve mesoscopic superpositions of many-body rotating and nonrotating states can be significantly speeded up.

Fast expansions and compressions of trapped-ion chains

We investigate the dynamics under diabatic expansions/compressions of linear ion chains.Combining a dynamical normal-mode harmonic approximation with the invariant-based inverse-engineering technique, we design protocols that minimize the final motional excitation of the ions. This can substantially reduce the transition time between high and low trap-frequency operations, potentially contributing to the development of scalable quantum information processing.

Fast phase gates with trapped ions

We implement faster-than-adiabatic two-qubit phase gates using smooth state-dependent forces. The forces are designed to leave no final motional excitation, independently of the initial motional state in the harmonic, small-oscillations limit. They are simple, explicit functions of time and the desired logical phase of the gate, and are based on quadratic invariants of motion and Lewis-Riesenfeld phases of the normal modes.

Fast driving between arbitrary states of a quantum particle by trap deformation

By performing a slow adiabatic change between two traps of a quantum particle, it is possible to transform an eigenstate of the original trap into the corresponding eigenstate of the final trap. If no level crossings are involved, the process can be made faster than adiabatic by setting first the interpolated evolution of the wave function from its initial to its final form and inferring from this evolution the trap deformation. We find a simple and compact formula which gives the trap shape at any time for any interpolation scheme. It is applicable even in complicated scenarios where there is no adiabatic process for the desired state transformation, e.g., if the state changes its topological properties. We illustrate its use for the expansion of a harmonic trap, for the transformation of a harmonic trap into a linear trap and into an arbitrary number of traps of a periodic structure. Finally, we study the creation of a node exemplified by the passage from the ground state to the first excited state of a harmonic oscillator.

Fast separation of two trapped ions

We design fast protocols to separate or recombine two ions in a segmented Paul trap. By inverse engineering the time evolution of the trapping potential composed of a harmonic and a quartic term, it is possible to perform these processes in a few microseconds without final excitation. These times are much shorter than the ones reported so far experimentally. The design is based on dynamical invariants and dynamical normal modes. Anharmonicities beyond the harmonic approximation at potential minima are taken into account perturbatively. The stability versus an unknown potential bias is also studied.

Fast atom transport and launching in a nonrigid trap

We study the shuttling of an atom in a trap with controllable position and frequency. Using invariant-based inverse engineering, protocols in which the trap is simultaneously displaced and expanded are proposed to speed up transport between stationary trap locations as well as launching processes with narrow final-velocity distributions. Depending on the physical constraints imposed, either simultaneous or sequential approaches may be faster. We consider first a perfectly harmonic trap, and then extend the treatment to generic traps. Finally, we apply this general framework to a double-well potential to separate different motional states with different launching velocities.