We discuss the application of quantum-mechanical supersymmetry to particle traps. The supersymmetric-partner wave functions may be used to describe a valence fermion in a trap system with an isotropic harmonic-oscillator potential. Interactions with the core are incorporated analytically. The close similarity of this approach to the application of supersymmetry in atomic systems is made explicit by means of a radial mapping between the two systems.
This paper begins with some background information and a summary of results in atomic supersymmetry. The connection between the supersymmetric Coulomb and oscillator problems in arbitrary dimensions is outlined. Next, I treat the issue of finding a description of supersymmetry-based quantum-defect theory in terms of oscillators. A model with an anharmonic term that yields analytical eigenfunctions is introduced to solve this problem in arbitrary dimensions. Finally, I show that geonium atoms (particles contained in a Penning trap) offer a realization of a multidimensional harmonic oscillator i…
A number of new L$\acute{e}$vi-Leblond type equations admitting four component spinor solutions have been proposed. The pair of linearized equations thus obtained in each case lead to Hamiltonians with characteristic features like L-S coupling and supersymmetry. The relevant momentum operators have often been understood in terms of Clifford algebraic bases producing Schr$\ddot{o}$dinger Hamiltonians with L-S coupling. As for example Hamiltonians representing Rashba effect or three dimensional harmonic oscillator have been constructed. The supersymmetric nature of one dimensional oscillator has…
Supersymmetry (SUSY) helps solve the hierarchy problem in high-energy physics and provides a natural groundwork for unifying gravity with other fundamental interactions. While being one of the most promising frameworks for theories beyond the Standard Model, its direct experimental evidence in nature still remains to be discovered. Here we report experimental realization of a supersymmetric quantum mechanics (SUSY QM) model, a reduction of the SUSY quantum field theory for studying its fundamental properties, using a trapped ion quantum simulator. We demonstrate the energy degeneracy caused by…
The supersymmetric quantum mechanical model based on higher-derivative supercharge operators possessing unbroken supersymmetry and discrete energies below the vacuum state energy is described. As an example harmonic oscillator potential is considered.
Rapid advances are being made toward optically cooling a single mode of a micro-mechanical system to its quantum ground state and observing quantum behavior at macroscopic scales. Reaching this regime in room-temperature environments requires a stringent condition on the mechanical quality factor $Q_m$ and frequency $f_m$, $Q_{m}f_{m}{\gtrsim}k_{B}T_{bath}/h$, which so far has been marginally satisfied only in a small number of systems. Here we propose and analyze a new class of systems that should enable unprecedented $Q_{m}f_m$ values. The technique is based upon using optical forces to "tra…
We formulate N-fold supersymmetry in quantum mechanical matrix models. As an example, we construct general two-by-two Hermitian matrix 2-fold supersymmetric quantum mechanical systems. We find that there are two inequivalent such systems, both of which are characterized by two arbitrary scalar functions, and one of which does not reduce to the scalar system. The obtained systems are all weakly quasi-solvable.
We show that the Ginzburg-Landau expansion of the grand potential for the Bogoliubov-de Gennes Hamiltonian is determined by the integrable nonlinear equations of the AKNS hierarchy, and that this provides the natural mathematical framework for a hidden nonlinear quantum mechanical supersymmetry underlying the dynamics.
Recently, we developed a method for calculating the lifetime of a particle inside a magnetic trap with respect to spin flips, as a first step in our efforts to understand the quantum-mechanics of magnetic traps. The 1D toy model that was used in this study was physically unrealistic because the magnetic field was not curl-free. Here, we study, both classically and quantum-mechanically, the problem of a neutral particle with spin S, mass m and magnetic moment mu, moving in 3D in an inhomogeneous magnetic field corresponding to traps of the Ioffe-Pritchard, `clover-leaf' and `baseball' type. Def…
We propose a technique aimed at cooling a harmonically oscillating mirror to its quantum mechanical ground state starting from room temperature. Our method, which involves the two-sided irradiation of the vibrating mirror inside an optical cavity, combines several advantages over the two-mirror arrangements being used currently. For comparable parameters the three-mirror configuration provides a stiffer trap for the oscillating mirror. Furthermore it prevents bistability from limiting the use of higher laser powers for mirror trapping, and also partially does so for mirror cooling. Lastly, it …
We investigate theoretically the mechanical effects of light on atoms trapped by an external potential, whose dipole transition couples to the mode of an optical resonator and is driven by a laser. We derive an analytical expression for the quantum center-of-mass dynamics, which is valid in presence of a tight external potential. This equation has broad validity and allows for a transparent interpretation of the individual scattering processes leading to cooling. We show that the dynamics are a competition of the mechanical effects of the cavity and of the laser photons, which may mutually int…
We theoretically study cooling of flexural modes of a mechanical oscillator by Bose-Einstein Condensate (BEC) atoms (Rb87) trapped in a magnetic trap. The mechanical oscillator with a tiny magnet attached on one of its free ends produces an oscillating magnetic field. When its oscillating frequency matches certain hyperfine Zeeman energy of Rb87 atoms, the trapped BEC atoms are coupled out of the magnetic trap by the mechanical oscillator, flying away from the trap with stolen energy from the mechanical oscillator. Thus the mode temperature of the mechanical oscillator is reduced. The mode tem…
More than one hundred years ago Heinrich Hertz succeeded in transmitting signals over a few meters to a receiving antenna using an electromagnetic oscillator and thus proving the electromagnetic theory developed by James C. Maxwell[1]. Since then, technology has developed, and today a variety of oscillators is available at the quantum mechanical level. For quantized electromagnetic oscillations atoms in cavities can be used to couple electric fields[2, 3]. For mechanical oscillators realized, for example, with cantilevers[4, 5] or vibrational modes of trapped atoms[6] or ions[7, 8], a quantum …
The harmonic oscillator is one of the simplest physical systems but also one of the most fundamental. It is ubiquitous in nature, often serving as an approximation for a more complicated system or as a building block in larger models. Realizations of harmonic oscillators in the quantum regime include electromagnetic fields in a cavity [1-3] and the mechanical modes of a trapped atom [4] or macroscopic solid [5]. Quantized interaction between two motional modes of an individual trapped ion has been achieved by coupling through optical fields [6], and entangled motion of two ions in separate loc…
High-gain resonant nonlinear Raman scattering on trapped cold atoms within a high-fineness ring optical cavity is simply explained under a nonlinear opto-mechanical mechanism, and a proposal using it to detect frequency of micro-trap on atom chip is presented. The enhancement of scattering spectrum is due to a coherent Raman conversion between two different cavity modes mediated by collective vibrations of atoms through nonlinear opto-mechanical couplings. The physical conditions of this technique are roughly estimated on Rubidium atoms, and a simple quantum analysis as well as a multi-body se…
We study an analogue of the equations describing TE and TM modes in a planar waveguide with an arbitrary continuous dependence of the electric permittivity and magnetic permeability on coordinates with the stationary Schrödinger equation. The effective potential energies involved in the Schrödinger equation for TE and TM modes are found. In general, the effective potential energies for TE and TM modes are different but in the limit of a weak dependence of the permittivity and permeability on coordinates they coincide. In the case when the product of a position-dependent permittivity and permea…
Phonon trapping has an immense impact in many areas of science and technology, from the antennas of interferometric gravitational wave detectors to chip-scale quantum micro- and nano-mechanical oscillators. It usually relies on the mechanical suspension--an approach, while isolating selected vibrational modes, leads to serious drawbacks for interrogation of the trapped phonons, including limited heat capacity and excess noises via measurements. To circumvent these constraints, we realize a new paradigm of phonon trapping using mechanical bound states in the continuum (BICs) with topological fe…
We investigate a new realization of extended quantum-mechanical supersymmetry. We first show that an $\mathcal{N}=2$ quantum-mechanical supersymmetry is hidden in the four-dimensional (4D) spectrum of the Kaluza-Klein decomposition for the higher dimensional Dirac field, that is, Kaluza-Klein mode functions of 4D right-handed spinors and 4D left-handed ones form $\mathcal{N}=2$ supermultiplets. In addition to $\mathcal{N}=2$ supersymmetry, we discover that an $\mathcal{N}$-extended supersymmetry ($\mathcal{N} = d+2\ (d+1)$ for $d=$ even (odd) extra dimensions) is further hidden in the 4D spect…
New types of irreducible second order Darboux transformations for the one dimensional Schroedinger equation are described. The main feature of such transformations is that the transformation functions have the eigenvalues grater then the ground state energy of the initial (or reference) Hamiltonian. When such a transformation is presented as a chain of two first order transformations, an intermediate potential is singular and therefore intermediate Hamiltonian can not be Hermitian while the final potential is regular and the final Hamiltonian is Hermitian. Second derivative supersymmetric quan…
We propose to realize quantized discrete kinks with cold trapped ions. We show that long-lived solitonlike configurations are manifested as deformations of the zigzag structure in the linear Paul trap, and are topologically protected in a circular trap with an odd number of ions. We study the quantum-mechanical time evolution of a high-frequency, gap separated internal mode of a static kink and find long coherence times when the system is cooled to the Doppler limit. The spectral properties of the internal modes make them ideally suited for manipulation using current technology. This suggests …
Complete description of the classical and quantum dynamics of a particle in an anisotropic, rotating, harmonic trap is given. The problem is studied in three dimensions and no restrictions on the geometry are imposed. In the generic case, for an arbitrary orientation of the rotation axis, there are two regions of instability with different characteristics. The analysis of the quantum-mechanical problem is made simple due to a direct connection between the classical mode vectors and the quantum-mechanical wave functions. This connection is obtained via the matrix Riccati equation that governs t…
In this work we present the simplest generic form of the propagator for the time-dependent quadratic Hamiltonian. We manifest the simplicity of our method by giving explicitly the propagators for a free particle in time-dependent electric field, forced harmonic oscillator and the Paul trap. Exact transition amplitudes and uncertainties are calculated analytically for the Paul trap and harmonic oscillator. The results show that near the instability regions very large quantum mechanical uncertainties are obtained as demonstrated in a special figure. The method is also applied to calculating the …
Arrays of neutral-atom qubits in optical tweezers are a promising platform for quantum computation. Despite experimental progress, a major roadblock for realizing neutral atom quantum computation is the qubit initialization. Here we propose that supersymmetry -- a theoretical framework developed in particle physics -- can be used for ultra-high fidelity initialization of neutral-atom qubits. We show that a single atom can be deterministically prepared in the vibrational ground state of an optical tweezer by adiabatically extracting all excited atoms to a supersymmetric auxiliary tweezer. The s…
We investigate the performance of different control techniques for ion transport in state-of-the-art segmented miniaturized ion traps. We employ numerical optimization of classical trajectories and quantum wavepacket propagation as well as analytical solutions derived from invariant based inverse engineering and geometric optimal control. We find that accurate shuttling can be performed with operation times below the trap oscillation period. The maximum speed is limited by the maximum acceleration that can be exerted on the ion. When using controls obtained from classical dynamics for wavepack…
Trapped ions are among the leading candidates for quantum computing technologies. Interfacing ion qubits in separate traps and interfacing ion qubits with superconducting qubits are two of the many challenges to scale up quantum computers. One approach to overcome both problems is to use a conducting wire to mediate the Coulomb interaction between ions in different traps, or between ions and superconducting qubits. To this end, a trapped charged particle inducing charge on a conductor has long been modeled as a system of equivalent lumped element electronic components. Careful consideration re…
Propagation and interference of quantum-mechanical particles comprise an important part of elementary processes in quantum physics, and their essence can be modeled using a quantum walk, a mathematical concept that describes the motion of a quantum-mechanical particle among discretized spatial regions. Here we report the observation of the quantum walks of a phonon, a vibrational quantum, in a trapped-ion crystal. By employing the capability of preparing and observing a localized wave packet of a phonon, the propagation of a single radial local phonon in a four-ion linear crystal is observed w…
We introduce the confluent version of the quantum-mechanical supersymmetry (SUSY) formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the transformed potentials are derived, and normalizability of the transformed solutions is established. Our findings extend and complement former results.
We propose an experimental configuration, within an ion trap, by which a quantum mechanical delta-kicked harmonic oscillator could be realized, and investigated. We show how to directly measure the sensitivity of the ion motion to small variations in the external parameters.
We study a 5d gravity theory with a warped metric and show that two N = 2 supersymmetric quantum-mechanical systems are hidden in the 4d spectrum. The supersymmetry can be regarded as a remnant of higher-dimensional general coordinate invariance and turns out to become a powerful tool to determine the physical 4d spectrum and the allowed boundary conditions. Possible extensions of the N = 2 supersymmetry are briefly discussed.
The concept of supersymmetry in a quantum mechanical system is extended, permitting the recognition of many more supersymmetric systems, including very familiar ones such as the free particle. Its spectrum is shown to be supersymmetric, with space-time symmetries used for the explicit construction. No fermionic or Grassmann variables need to be invoked. Our construction extends supersymmetry to continuous spectra. Most notably, while the free particle in one dimension has generally been regarded as having a doubly degenerate continuum throughout, the construction clarifies taht there is a sing…
I examine quantum mechanical Hamiltonians with partial supersymmetry, and explore two main applications. First, I analyze a theory with a logarithmic spectrum, and show how to use partial supersymmetry to reveal the underlying structure of this theory. This method reveals an intriguing equivalence between two formulations of this theory, one of which is one-dimensional, and the other of which is infinite-dimensional. Second, I demonstrate the use of partial supersymmetry as a tool to obtain the asymptotic energy levels in non-relativistic quantum mechanics in an exceptionally easy way. In the …
Coherent tripartite interactions among degrees of freedom of completely different nature are instrumental for quantum information and simulation technologies, but they are generally difficult to realize and remain largely unexplored. Here, we predict a tripartite coupling mechanism in a hybrid setup comprising a single NV center and a micromagnet. We propose to realize direct and strong tripartite interactions among single NV spins, magnons and phonons via modulating the relative motion between the NV center and the micromagnet. Specifically, by introducing a parametric drive (two-phonon drive…
Motivated by the duality of normalizable states and the presence of the quasi-parity quantum number q=+/-1 in PT symmetric (non-Hermitian) quantum mechanical potential models, the relation of PT symmetry and supersymmetry (SUSY) is studied. As an illustrative example the PT invariant version of the Scarf II potential is presented, and it is shown that the "bosonic" Hamiltonian has two different "fermionic" SUSY partner Hamiltonians (potentials) generated from the ground-state solutions with q=1 and q=-1. It is shown that the "fermionic" potentials cease to be PT invariant when the PT symmetry …
We study, both classically and quantum-mechanically, the problem of a neutral particle with spin S, mass m and magnetic moment mue, moving in two-dimensions in an inhomogeneous magnetic field given by B_x=B'*x; B_y=-B'*y; B_z=B; We identify K, the ratio between the precessional frequency of the particle and its vibration frequency, as the relevant parameter of the problem. Classicaly, we find that when the magnetic moment is antiparallel to B, the particle is trapped provided that K<sqrt{4/27}. We also find that viscous friction, be it translational or precessional, destabilizes the system. Qu…
This work presents some peculiarities of the near quantum ground state behaviour of curved (phonon trapping) Bulk Acoustic Wave (BAW) cavities when compared to a conventional mechanical resonator. The curved cavity system resolves the quandary of the conventional mechanical system where the Bose-Einstein distribution requires higher frequencies for lower quantum occupation factors contrary to the constraint of an inverse frequency dependence of the quantum fluctuations of displacement. We demonstrate how the non-trivial cavity geometry can lead to better phonon trapping, enhancing the variance…
We show that some simple well studied quantum mechanical systems without fermion (spin) degrees of freedom display, surprisingly, a hidden supersymmetry. The list includes the bound state Aharonov-Bohm, the Dirac delta and the Poschl-Teller potential problems, in which the unbroken and broken N=2 supersymmetry of linear and nonlinear (polynomial) forms is revealed.
An enduring challenge for contemporary physics is to experimentally observe and control quantum behavior in macroscopic systems. We show that a single trapped atomic ion could be used to probe the quantum nature of a mesoscopic mechanical oscillator precooled to 4K, and furthermore, to cool the oscillator with high efficiency to its quantum ground state. The proposed experiment could be performed using currently available technology.
We consider a possible route to ground state cooling of a levitated nanoparticle, magnetically trapped by a strong permanent magnet, using a combination of measurement and feedback. The trap frequency of this system is much lower than those involving trapped ions or nano-mechanical resonators. Minimisation of environmental heating is therefore challenging as it requires control of the system on a timescale comparable to the inverse of the trap frequency. We show that these traps are an excellent platform for performing optimal feedback control via real-time state estimation, for the preparatio…
I discuss the robustness of the pumped cavity dynamics against phase diffusion of the laser and conclude that opto-mechanical cooling has extreme sensitivity compared to laser cooling of atoms. Certain proposals of ground state opto-mechanical cooling by single cavity would require an unrealistic sharp laser linewidth or equivalently, a very low level of phase noise. A systematic way to cancel classical excess phase noise is the interferometric twin-cavity pumping, initiated for optically trapped macro-mirrors of future gravitational-wave detectors.
The non-commutativity of position and momentum observables is a hallmark feature of quantum physics. However this incompatibility does not extend to observables which are periodic in these base variables. Such modular-variable observables have been suggested as tools for fault-tolerant quantum computing and enhanced quantum sensing. Here we implement sequential measurements of modular variables in the oscillatory motion of a single trapped ion, using state-dependent displacements and a heralded non-destructive readout. We investigate the commutative nature of modular variable observables by de…
Supersymmetry is a technique that allows us to extract information about the states and spectra of quantum mechanical systems which may otherwise be unsolvable. In this paper we reconstruct Ioffe's set of states for the singular non-separable two-dimensional Morse potential using supersymmetry from a non-degenerate set of states constructed for the initial separable Morse Hamiltonian. We define generalised coherent states, compute their uncertainty relations, and we find that the singularity in the partner Hamiltonian significantly affects the localisation of the coherent state wavefunction.
We study a general problem of the translational/rotational/vibrational/electronic dynamics of a diatomic molecule exposed to an interaction with an arbitrary external electromagnetic field. The theory developed in this paper is relevant to a variety of specific applications. Such as, alignment or orientation of molecules by lasers, trapping of ultracold molecules in optical traps, molecular optics and interferometry, rovibrational spectroscopy of molecules in the presence of intense laser light, or generation of high order harmonics from molecules. Starting from the first quantum mechanical pr…
We investigate one-dimensional quantum mechanical systems which have type A N-fold supersymmetry with two different values of N simultaneously. We find that there are essentially four inequivalent models possessing the property, one is conformal, two of them are hyperbolic (trigonometric) including Rosen-Morse type, and the other is elliptic.
Quantum-mechanical principles can be used to process information (QIP). In one approach, linear arrays of trapped, laser cooled ion qubits (two-level quantum systems) are confined in segmented multi-zone electrode structures. The ion trap approach to QIP requires trapping and control of numerous ions in electrode structures with many trapping zones. I investigated microfabrication of structures to trap, transport and couple large numbers of ions. Using 24Mg+ I demonstrated loading and transport between zones in microtraps made of boron doped silicon. This thesis describes the fundamentals of i…
The interplay between supersymmetry and classical and quantum computation is discussed. First, it is shown that the problem of computing the Witten index of $\mathcal N \leq 2$ quantum mechanical systems is $\#P$-complete and therefore intractable. Then, the notions of supersymmetry in the space of qubits and supersymmetric quantum circuits are introduced and some of their properties discussed. In particular, it is shown that these define a nontrivial subclass of quantum algorithms with robustness properties typical of supersymmetric systems. Concrete examples, including the supersymmetric SYK…
We give the definition and explore the algebraic structure of a class of quantum symmetries, called topological symmetries, which are generalizations of supersymmetry in the sense that they involve topological invariants similar to the Witten index. A topological symmetry (TS) is specified by an integer n>1, which determines its grading properties, and an n-tuple of positive integers (m_1,m_2,...,m_n). We identify the algebras of supersymmetry, p=2 parasupersymmetry, and fractional supersymmetry of order n with those of the Z_2-graded TS of type (1,1), Z_2-graded TS of type (2,1), and Z_n-grad…
Levitated particles are a promising platform for precision sensing of external perturbations and probing the boundary between quantum and classical worlds. A critical obstacle for these applications is the difficulty of generating nonclassical states of motion which have not been realized so far. Here, we show that strong squeezing of the motion of a levitated particle below the vacuum level is feasible with available experimental parameters. Using suitable modulation of the trapping potential (which is impossible with clamped mechanical resonators) and coherent scattering of trapping photons …
We formulate the framework of N-fold supersymmetry in one-body quantum mechanical systems with position-dependent mass (PDM). We show that some of the significant properties in the constant-mass case such as the equivalence to weak quasi-solvability also hold in the PDM case. We develop a systematic algorithm for constructing an N-fold supersymmetric PDM system. We apply it to obtain type A N-fold supersymmetry in the case of PDM, which is characterized by the so-called type A monomial space. The complete classification and general form of effective potentials for type A N-fold supersymmetry i…
The stable operation of quantum computers will rely on error-correction, in which single quantum bits of information are stored redundantly in the Hilbert space of a larger system. Such encoded qubits are commonly based on arrays of many physical qubits, but can also be realized using a single higher-dimensional quantum system, such as a harmonic oscillator. A powerful encoding is formed from a periodically spaced superposition of position eigenstates. Various proposals have been made for realizing approximations to such states, but these have thus far remained out of reach. Here, we demonstra…
Force sensors are at the heart of different technologies such as atomic force microscopy or inertial sensing \cite{RMPforce2003, Rugar2004, YazdiIEEE}. These sensors often rely on the measurement of the displacement amplitude of mechanical oscillators under applied force. Examples for such mechanical oscillators include micro-fabricated cantilevers \cite{YazdiIEEE}, carbon nanotubes \cite{NanotubeForce} as well as single trapped ions \cite{Bollinger, Udem} . The best sensitivity is typically achieved when the force is alternating at the mechanical resonance frequency of the oscillator thus inc…
We review recent works on optomechanics of optically trapped microspheres and nanoparticles in vacuum, which provide an ideal system for studying macroscopic quantum mechanics and ultrasensitive force detection. An optically trapped particle in vacuum has an ultrahigh mechanical quality factor as it is well-isolated from the thermal environment. Its oscillation frequency can be tuned in real time by changing the power of the trapping laser. Furthermore, an optically trapped particle in vacuum may rotate freely, a unique property that does not exist in clamped mechanical oscillators. In this re…
Optically levitated mechanical sensors promise isolation from thermal noise far beyond what is possible using flexible materials alone. One way to access this potential is to apply a strong optical trap to a minimally supported mechanical element, thereby increasing its quality factor $Q_m$. Current schemes, however, require prohibitively high laser power ($\sim10$ W), and the $Q_m$ enhancement is ultimately limited to a factor of $\sim$ 50 by hybridization between the trapped mode and the dissipative modes of the supporting structure. Here we propose a levitation scheme taking full advantage …
We propose an experiment utilizing an array of cooled micro-cantilevers coupled to a sample of ultra-cold atoms trapped near a micro-fabricated surface. The cantilevers allow individual lattice site addressing for atomic state control and readout, and potentially may be useful in optical lattice quantum computation schemes. Assuming resonators can be cooled to their vibrational ground state, the implementation of a two-qubit controlled-NOT gate with atomic internal states and the motional states of the resonator is described. We also consider a protocol for entangling two or more cantilevers o…
We analyze the dynamics of single photon transport in a single-mode waveguide coupled to a micro-optical resonator using a fully quantum mechanical model. We examine the propagation of a single-photon Gaussian packet through the system under various coupling conditions. We review the theory of single photon transport phenomena as applied to the system and we develop a discussion on the numerical technique we used to solve for dynamical behavior of the quantized field. To demonstrate our method and to establish robust single photon results, we study the process of adiabatically lowering or rais…
