arXiv subjects

R. Blümel

Publications and source records attributed to R. Blümel.

At least 19 recordsLinked to original sources

arXiv2017

Explicit, analytical radio-frequency heating formulas for spherically symmetric nonneutral plasmas in a Paul trap

We present explicit, analytical heating formulas that predict the heating rates of spherical, nonneutral plasmas stored in a Paul trap as a function of cloud size $S$, particle number $N$, and Paul-trap control parameter $q$ in the low-temperature regime close to the cloud $\rightarrow$ crystal phase transition. We find excellent agreement between our analytical heating formulas and detailed, time-dependent molecular-dynamics simulations of the trapped plasmas. We also present the results of our numerical solutions of a temperature-dependent mean-field equation, which are consistent with our numerical simulations and our analytical results. This is the first time that analytical heating formulas are presented that predict heating rates with reasonable accuracy, uniformly for all $S$, $N$, and $q$.

physics.plasm-ph↗

arXiv2017

Symmetry boosts quantum computer performance

Frequently, subroutines in quantum computers have the structure $\mathcal{F}\mathcal{U}\mathcal{F}^{-1}$, where $\mathcal{F}$ is some unitary transform and $\mathcal{U}$ is performing a quantum computation. In this paper we suggest that if, in analogy to spin echoes, $\mathcal{F}$ and $\mathcal{F}^{-1}$ can be implemented symmetrically such that $\mathcal{F}$ and $\mathcal{F}^{-1}$ have the same hardware errors, a symmetry boost in the fidelity of the combined $\mathcal{F}\mathcal{U}\mathcal{F}^{-1}$ quantum operation results. Running the complete gate--by--gate implemented Shor algorithm, we show that the fidelity boost can be as large as a factor 10. Corroborating and extending our numerical results, we present analytical scaling calculations that show that a symmetry boost persists in the practically interesting case of a large number of qubits. Our analytical calculations predict a minimum boost factor of about 3, valid for all qubit numbers, which includes the boost factor 10 observed in our low-qubit-number simulations. While we find and document this symmetry boost here in the case of Shor's algorithm, we suggest that other quantum algorithms might profit from similar symmetry-based performance boosts whenever $\mathcal{F}\mathcal{U}\mathcal{F}^{-1}$ sub-units of the corresponding quantum algorithm can be identified.

quant-ph↗

arXiv2017

Loading a linear Paul trap to saturation from a magneto-optical trap

We present experimental measurements of the steady-state ion number in a linear Paul trap (LPT) as a function of the ion-loading rate. These measurements, taken with (a) constant Paul trap stability parameter $q$, (b) constant radio-frequency (rf) amplitude, or (c) constant rf frequency, show nonlinear behavior. At the loading rates achieved in this experiment, a plot of the steady-state ion number as a function of loading rate has two regions: a monotonic rise (region I) followed by a plateau (region II). Also described are simulations and analytical theory which match the experimental results. Region I is caused by rf heating and is fundamentally due to the time dependence of the rf Paul-trap forces. We show that the time-independent pseudopotential, frequently used in the analytical investigation of trapping experiments, cannot explain region I, but explains the plateau in region II and can be used to predict the steady-state ion number in that region. An important feature of our experimental LPT is the existence of a radial cut-off $\hat R_{\rm cut}$ that limits the ion capacity of our LPT and features prominently in the analytical and numerical analysis of our LPT-loading results. We explain the dynamical origin of $\hat R_{\rm cut}$ and relate it to the chaos border of the fractal of non-escaping trajectories in our LPT. We also present an improved model of LPT ion-loading as a function of time.

physics.atom-ph↗

arXiv2016

Ion Crystal Metamorphoses in a Paul trap

The standard second-order pseudo-oscillator potential used in many analytical investigations of the properties of ions stored in a Paul trap has serious limitations. In this paper we show that ion-crystal configurations exhibited by 2, 3, and 4 simultaneously stored ions in a Paul trap are not predicted by the standard pseudo-oscillator potential, but are all captured qualitatively and quantitatively by an extended pseudopotential derived in this paper. The power of our extended pseudopotential extends in particular to the prediction of the border lines between different crystal configurations (morphologies) in the Paul trap's $a$, $q$ stability diagram. In the three- and four-ion cases, several of the ion-crystal structures predicted by our improved pseudopotential have never been observed experimentally before. We present them here as a challenge for experiments.

physics.atom-ph↗

arXiv2016

Infrared refractive index dispersion of PMMA spheres from synchrotron extinction spectra

We performed high-resolution Fourier-transform infrared (FTIR) spectroscopy of a polymethyl methacrylate (PMMA) sphere of unknown size in the Mie scattering region. Apart from a slow, oscillatory structure (wiggles), which is due to an interference effect, the measured FTIR extinction spectrum exhibits a ripple structure, which is due to electromagnetic resonances. We fully characterize the underlying electromagnetic mode structure of the spectrum by assigning two mode numbers to each of the ripples in the measured spectrum. We show that analyzing the ripple structure in the spectrum in the wavenumber region from about $3000\,$cm$^{-1}$ to $8000\,$cm$^{-1}$ allows us to both determine the unknown radius of the sphere and the PMMA index of refraction, which shows a strong frequency dependence in this near-infrared spectral region. While in this paper we focus on examining a PMMA sphere as an example, our method of determining the refractive index and its dispersion from synchrotron infrared extinction spectra is generally applicable for the determination of the index of refraction of any transparent substance that can be shaped into micron-sized spheres.

physics.optics↗

arXiv2015

Universal non-monotonic structure in the saturation curves of MOT-loaded Na$^+$ ions stored in an ion-neutral hybrid trap: Prediction and observation

We predict that the steady-state ion number $N_s$ for radio-frequency (rf) traps, loaded at a rate of $λ$ particles per unit time, shows universal non-monotonic behavior as a function of loading rate $λ$. The shape of $N_s(λ)$, characterized by four dynamical regions, is universal in the sense that it is predicted to manifest itself in all rf traps independently of the details of their construction. For $λ\ll$ 1 particles / rf cycle (Region I), as expected, $N_s(λ)$ increases monotonically with $λ$. However, contrary to intuition, at intermediate $λ\sim 1$ particles / rf cycle (Region II), $N_s(λ)$ reaches a maximum, followed by a minimum of $N_s(λ)$ (Region III). For $λ\gg 1$ particles / rf cycle (Region IV), $N_s(λ)$ again rises monotonically. In Region IV numerical simulations, analytical calculations, and experiments show $N_s(λ)\sim λ^{2/3}$. We confirm this prediction experimentally with MOT-loaded Na$^+$ ions stored in a hybrid ion-neutral trap.

physics.atom-ph↗

arXiv2015

Critical exponents for the cloud-crystal phase transition of charged particles in a Paul Trap

It is well known that charged particles stored in a Paul trap, one of the most versatile tools in atomic and molecular physics, may undergo a phase transition from a disordered cloud state to a geometrically well-ordered crystalline state (the Wigner crystal). In this paper we show that the average lifetime $\barτ_m$ of the metastable cloud state preceding the cloud $\rightarrow$ crystal phase transition follows a powerlaw, $\barτ_m \sim (γ-γ_c)^{-β}$, $γ>γ_c$, where $γ_c$ is the critical value of the damping constant $γ$ at which the cloud $\rightarrow$ crystal phase transition occurs. The critical exponent $β$ depends on the trap control parameter $q$, but is independent of the number of particles $N$ stored in the trap and the trap control parameter $a$, which determines the shape (oblate, prolate, or spherical) of the cloud. For $q=0.15,0.20$, and $0.25$, we find $β=1.20\pm 0.03$, $β=1.61\pm 0.09$, and $β=2.38\pm 0.12$, respectively. In addition we find that for given $a$ and $q$, the critical value $γ_c$ of the damping scales approximately like $γ_c=C \ln [ \ln (N)] + D$ as a function of $N$, where $C$ and $D$ are constants. Beyond their relevance for Wigner crystallization of nonneutral plasmas in Paul traps and mini storage rings, we conjecture that our results are also of relevance for the field of crystalline beams.

physics.comp-ph↗

arXiv2014

Measurement of low-energy Na^+ -- Na total collision rate in an ion--neutral hybrid trap

We present measurements of the total elastic and resonant charge-exchange ion-atom collision rate coefficient $k_\mathrm{ia}$ of cold sodium (\ce{Na}) with optically-dark low energy \ce{Na+} ions in a hybrid ion-neutral trap. To determine $k_\mathrm{ia}$, we measured the trap loading and loss from both a \ce{Na} magneto-optical trap (MOT) and a linear radio frequency quadrupole Paul trap. We found the total rate coefficient to be $7.4 \pm 1.9 \times 10^{-8}$ cm$^3$/s for the type I \ce{Na} MOT immersed within an $\approx 140$ K ion cloud and $1.10 \pm 0.25 \times 10^{-7}$ cm$^3$/s for the type II \ce{Na} MOT within an $\approx 1070$ K ion cloud. Our measurements show excellent agreement with previously reported theoretical fully quantal \textit{ab initio} calculations. In the process of determining the total rate coefficient, we demonstrate that a MOT can be used to probe an optically dark ion cloud's spatial distribution within a hybrid trap.

physics.atom-ph↗

arXiv2013

Scaling laws for Shor's algorithm with a banded quantum Fourier transform

We investigate the performance of a streamlined version of Shor's algorithm in which the quantum Fourier transform is replaced by a banded version that for each qubit retains only coupling to its $b$ nearest neighbors. Defining the performance $P(n,b)$ of the $n$-qubit algorithm for bandwidth $b$ as the ratio of the success rates of Shor's algorithm equipped with the banded and the full bandwidth ($b=n-1$) versions of the quantum Fourier transform, our numerical simulations show that $P(n,b) \approx \exp[-φ_{max}^2 (n,b)/100]$ for $n < n_t(b)$ (non-exponential regime) and $P(n,b) \approx 2^{-ξ_b (n-8)}$ for $n>n_t(b)$ (exponential regime), where $n_{t}(b)$, the location of the transition, is approximately given by $n_{t}(b)\approx b+5.9 + \sqrt{7.7(b+2)-47}$ for $b\gtrsim 8$, $φ_{max} (n,b) = 2π[2^{-b-1} (n-b-2) + 2^{-n}]$, and $ξ_b\approx 1.1 \times 2^{-2b}$. Analytically we obtain $P(n,b) \approx \exp[-φ_{max}^2 (n,b)/64]$ for $n n_t(b)$, where $ξ_{b}^{(a)} \approx \frac{π^2}{12 \ln(2)} \times 2^{-2b} \approx 1.19 \times 2^{-2b}$. Thus, our analytical results predict the $φ_{max}^2$ scaling ($n n_t$) of the data perfectly. In addition, in the large-$n$ regime, the prefactor in $ξ_b^{(a)}$ is close to the results of our numerical simulations and, in the low-$n$ regime, the numerical scaling factor in our analytical result is within a factor 2 of its numerical value. As an example we show that $b=8$ is sufficient for factoring RSA-2048 with a 95% success rate.

quant-ph↗

arXiv2011

Conceptually new mechanism for trapping neutral, polar particles

It is shown that a superposition of static and rapidly oscillating electric {\it monopole} (source) fields is capable of trapping particles with a permanent electric dipole moment. Thus, the new trapping mechanism differs fundamentally from saddle-point traps that use static and oscillating higher-multipole fields. An analytical stability analysis together with detailed molecular dynamics trajectory calculations prove that the trap is stable. Thin rods of barium titanate (BaTiO$_3$) provide an illustrative example for the working principle of the new trap. The effects of gravity are considered. The existence of a bifurcation regime is predicted. A particular strength of the new trap is that it also works for zero orbital angular momentum with respect to the field-generating electrodes.

physics.atom-ph↗

arXiv2004

Explicit Spectral formulae for scaling quantum graphs

We present an exact analytical solution of the spectral problem of quasi one-dimensional scaling quantum graphs. Strongly stochastic in the classical limit, these systems are frequently employed as models of quantum chaos. We show that despite their classical stochasticity all scaling quantum graphs are explicitly solvable in the form $E_n=f(n)$, where $n$ is the sequence number of the energy level of the quantum graph and $f$ is a known function, which depends only on the physical and geometrical properties of the quantum graph. Our method of solution motivates a new classification scheme for quantum graphs: we show that each quantum graph can be uniquely assigned an integer $m$ reflecting its level of complexity. We show that a taut string with piecewise constant mass density provides an experimentally realizable analogue system of scaling quantum graphs.

quant-ph↗

arXiv2003

Solution of scaling quantum networks

We show that all scaling quantum graphs are explicitly integrable, i.e. any one of their spectral eigenvalues $E_n$ is computable analytically, explicitly, and individually for any given $n$. This is surprising, since quantum graphs are excellent models of quantum chaos [see, e.g., T. Kottos and H. Schanz, Physica E {\bf 9}, 523 (2001)].

quant-ph↗

arXiv2003

Explicit, analytical solution of scaling quantum graphs

Based on earlier work on regular quantum graphs we show that a large class of scaling quantum graphs with arbitrary topology are explicitly analytically solvable. This is surprising since quantum graphs are excellent models of quantum chaos and quantum chaotic systems are not usually explicitly analytically solvable.

quant-ph↗

arXiv2002

Integrability in 1D Quantum Chaos

Explicit, exact periodic orbit expansions for individual eigenvalues exist for a subclass of quantum networks called regular quantum graphs. We prove that all linear chain graphs have a regular regime.

quant-ph↗

arXiv2002

Mathematical Foundations of Regular Quantum Graphs

We define a class of quantum systems called regular quantum graphs. Although their dynamics is chaotic in the classical limit with positive topological entropy, the spectrum of regular quantum graphs is explicitly computable analytically and exactly, state by state, by means of periodic orbit expansions. We prove analytically that the periodic orbit series exist and converge to the correct spectral eigenvalues. We investigate the convergence properties of the periodic orbit series and prove rigorously that both conditionally convergent and absolutely convergent cases can be found. We compare the periodic orbit expansion technique with Lagrange's inversion formula. While both methods work and yield exact results, the periodic orbit expansion technique has conceptual value since all the terms in the expansion have direct physical meaning and higher order corrections are obtained according to physically obvious rules. In addition our periodic orbit expansions provide explicit analytical solutions for many classic text-book examples of quantum mechanics that previously could only be solved using graphical or numerical techniques.

quant-ph↗

arXiv2001

Spectra of regular quantum graphs

We consider a class of simple quasi one-dimensional classically non-integrable systems which capture the essence of the periodic orbit structure of general hyperbolic nonintegrable dynamical systems. Their behavior is simple enough to allow a detailed investigation of both classical and quantum regimes. Despite their classical chaoticity, these systems exhibit a ``nonintegrable analog'' of the Einstein-Brillouin-Keller quantization formula which provides their spectra explicitly, state by state, by means of convergent periodic orbit expansions.

quant-ph↗

arXiv2001

Exact, convergent periodic-orbit expansions of individual energy eigenvalues of regular quantum graphs

We present exact, explicit, convergent periodic-orbit expansions for individual energy levels of regular quantum graphs. One simple application is the energy levels of a particle in a piecewise constant potential. Since the classical ray trajectories (including ray splitting) in such systems are strongly chaotic, this result provides the first explicit quantization of a classically chaotic system.

quant-ph↗

arXiv2001

One-dimensional quantum chaos: Explicitly solvable cases

We present quantum graphs with remarkably regular spectral characteristics. We call them {\it regular quantum graphs}. Although regular quantum graphs are strongly chaotic in the classical limit, their quantum spectra are explicitly solvable in terms of periodic orbits. We present analytical solutions for the spectrum of regular quantum graphs in the form of explicit and exact periodic orbit expansions for each individual energy level.

quant-ph↗