Iwo Bialynicki-Birula, Tomasz Sowinski
We study the influence of the nonlinearity in the Schrodinger equation on the motion of quantum particles in a harmonic trap. In order to obtain exact analytic solutions, we have chosen the logarithmic nonlinearity. The unexpected result of our study is the existence in the presence of nonlinearity of two or even three coexisting Gaussian solutions.
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A logarithmic Schrödinger equation with time-dependent coupling to the non-linearity is presented as a model of collisional decoherence of the wavefunction of a quantum particle in position-space. The particular mathematical form of the logarithmic Schrödinger equation has been shown to follow from conditional wave theory, but the validity of the logarithmic Schrödinger equation has not yet been investigated numerically for general initial conditions. Using an operator-splitting approach, we solve the non-linear equation of motion for the wavefunction numerically and compare it it to the solut…
Favored schemes for trapped-ion quantum logic gates use bichromatic laser fields to couple internal qubit states with external motion through a "spin-dependent force." We introduce a new degree of freedom in this coupling that reduces its sensitivity to phase decoherence. We demonstrate bichromatic spin-dependent forces on a single trapped $^{111}$Cd$^+$ ion, and show that phase coherence of the resulting "Schrödinger-cat" states of motion depends critically upon the spectral arrangement of the optical fields. This applies directly to the operation of entangling gates on multiple ions.
We investigate the feasibility of observing matter-wave interference of a micron-sized charged particle by putting it into a quantum superposition of states with a distinguishable separation. In the proposed method, an atomic ion is confined in a linear Paul trap along with the massive charged particle so that we can make use of the extensive toolbox of experimental techniques developed to control quantum states of trapped ions, and to manipulate their motions with high fidelity operations. This approach provides a stringent test of the predictions of dynamical reduction models of delocalised …
We summarize efforts at NIST to implement quantum computation using trapped ions, based on a scheme proposed by J.I. Cirac and P. Zoller (Innsbruck University). The use of quantum logic to create entangled states, which can maximize the quantum-limited signal-to-noise ratio in spectroscopy, is discussed.
The studied model describes a particle that obeys a one-dimensional nonlinear Schrödinger equation in the potential of a double-well. Transitions between the two lowest self-trapped states of this system under the influence of the external time-dependent perturbation are studied in the two-mode approximation. If the perturbation dependence on time is harmonic with the frequency $ω$, then transitions between the states become possible if the amplitude of the perturbation $F$ exceeds some threshold value $F_c(ω)$; above the threshold motion of the system becomes chaotic. If the perturbation is a…
Linear square-well Schrödinger equation endowed with a singular logarithmic spike in the origin is studied. The study is methodical, motivated by the problem of non-gausson states $ψ_n(x)$, $n \neq 0$ generated by nonlinear Schrödinger equations. Once the state-dependent self-interaction term is chosen logarithmic, $\sim -g\,\ln[ψ^*_n(x)ψ_n(x)]$, the nonlinear model develops the puzzling logarithmic (i.e., weakly singular) repulsive barriers near the nodal zeros of $ψ_n(x)$ at $n \neq 0$. In our linearized approach the weak-coupling regime is shown reliably described by the routine Rayleigh-Sc…
We study the effect of a logarithmic nonlinearity in the Schrödinger equation (SE) on the dynamics of a freely expanding Bose-Einstein condensate (BEC). The logarithmic nonlinearity was one of the first proposed nonlinear extensions to the SE which emphasized the conservation of important physical properties of the linear theory, e.g.: the separability of noninteracting states. Using this separability, we incorporate it into the description of a BEC obeying a logarithmic Gross-Pittaevskii equation. We investigate the dynamics of such BECs using variational and numerical methods and find that, …
A proof is given that if the nonlinear Schrodinger wave function is constrained to have support over only a finite volume in configuration space, then the total energy is bounded from below for either sign of the logarithmic term in the Hamiltonian. It is concluded that the usual assumption about the sign of the logarithmic term made by Bialynicki-Birula and Mycielski is not the only possibility, and that a sensible theory can be made with the opposite sign as well.
We carry out probabilistic phase-space sampling of mesoscopic Schrödinger cat quantum states, demonstrating multipartite Bell violations for up to 60 qubits. We use states similar to those generated in photonic and ion-trap experiments. These results show that mesoscopic quantum superpositions are directly accessible to probabilistic sampling, and we analyze the properties of sampling errors. We also demonstrate dynamical simulation of super-decoherence in ion traps. Our computer simulations can be either exponentially faster or slower than experiment, depending on the correlations measured.
Contributions to the study of time dependent oscillators in Paul traps. Semiclassical approach
本研究针对保罗阱中振荡四极场束缚离子的量子动力学，通过两种经典方法推导其运动方程等价于Hill方程，引入绝热不变量确定系统态，证实量子态为Fock态、薛定谔方程解为准能态，并拓展至多体系统半经典研究，给出传播子。
We examine the time evolution of a two level ion interacting with a light field in harmonic oscillator trap and in a trap with anharmonicities. The anharmonicities of the trap are quantified in terms of the deformation parameter $τ$ characterizing the q-analog of the harmonic oscillator trap. Initially the ion is prepared in a Schrödinger cat state. The entanglement of the center of mass motional states and the internal degrees of freedom of the ion results in characteristic collapse and revival pattern. We calculate numerically the population inversion I(t), quasi-probabilities $Q(t),$ and pa…
The Schrödinger-Newton equation has gained attention in the recent past as a nonlinear modification of the Schrödinger equation due to a gravitational self-interaction. Such a modification is expected from a fundamentally semi-classical theory of gravity, and can therefore be considered a test case for the necessity of the quantisation of the gravitational field. Here we provide a thorough study of the effects of the Schrödinger-Newton equation for a micron-sized sphere trapped in a harmonic oscillator potential. We discuss both the effect on the energy eigenstates and the dynamical behaviour …
No experimental evidence exists, to date, whether or not the gravitational field must be quantised. Theoretical arguments in favour of quantisation are inconclusive. The most straightforward alternative to quantum gravity, a coupling between classical gravity and quantum matter according to the semi-classical Einstein equations, yields a nonlinear modification of the Schrödinger equation. Here, effects of this so-called Schrödinger-Newton equation are discussed, which allow for technologically feasible experimental tests.
It is shown that the energy levels of the one-dimensional nonlinear Schrodinger, or Gross-Pitaevskii, equation with the homogeneous trap potential $x^{2p}$, $p\geq 1$, obey an approximate scaling law and as a consequence the energy increases approximately linearly with the quantum number. Moreover, for a quadratic trap, $p=1$, the rate of increase of energy with the quantum number is independent of the nonlinearity: this prediction is confirmed with numerical calculations. It is also shown that the energy levels computed using a variational approximation do not satisfy this scaling law.
Condensation phenomena of ions in an electrostatic logarithmic trap
本文研究静电对数阱中离子气体在库仑相互作用下的低温热力学，发现离子在径向局域态的坍缩导致玻色子和费米子在特定温度下发生从非简并到强简并的突变转变，并分析了压力可观测性条件。
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