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arXiv:1212.3077 · Physical Review A

Effects of an exceptional point on the dynamics of a single particle in a time-dependent harmonic trap

Raam Uzdin, Emanuele G. Dalla Torre, Ronnie Kosloff, Nimrod Moiseyev

Published Aug 12, 2013• 21 citations
arXivHTMLPDF

Abstract

The time evolution of a single particle in a harmonic trap with time-dependent frequency $\ensuremath{\omega}(t)$ has been well studied. Nevertheless, here we show that when the harmonic trap is opened (or closed) as a function of time while keeping the adiabatic parameter $\ensuremath{\mu}=[d\ensuremath{\omega}(t)/dt]/{\ensuremath{\omega}}^{2}(t)$ fixed, a sharp transition from an oscillatory to a monotonic exponential dynamics occurs at $\ensuremath{\mu}=2$. At this transition point, the time evolution has an exceptional point (EP) at all instants. This situation, where an EP of a time-dependent Hermitian Hamiltonian is obtained at any given time, is very different from other known cases. In the present case, we show that the order of the EP depends on the set of observables used to describe the dynamics. Our finding is relevant to the dynamics of a single ion in a magnetic, optical, or rf trap, and of diluted gases of ultracold atoms in optical traps.

DOI
10.1103/physreva.88.022505

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The time evolution of a single particle in a harmonic trap with time-dependent frequency \ensuremathω(t)\ensuremath{\omega}(t)\ensuremathω(t) has been well studied. Nevertheless, here we show that when the harmonic trap is opened (or closed) as a function of time while keeping the adiabatic parameter \ensuremathμ=[d\ensuremathω(t)/dt]/\ensuremathω2(t)\ensuremath{\mu}=[d\ensuremath{\omega}(t)/dt]/{\ensuremath{\omega}}^{2}(t)\ensuremathμ=[d\ensuremathω(t)/dt]/\ensuremathω2(t) fixed, a sharp transition from an oscillatory to a monotonic exponential dynamics occurs at \ensuremathμ=2\ensuremath{\mu}=2\ensuremathμ=2. At this transition point, the time evolution has an exceptional point (EP) at all instants. This situation, where an EP of a time-dependent Hermitian Hamiltonian is obtained at any given time, is very different from other known cases. In the present case, we show that the order of the EP depends on the set of observables used to describe the dynamics. Our finding is relevant to the dynamics of a single ion in a magnetic, optical, or rf trap, and of diluted gases of ultracold atoms in optical traps.
It has been shown already that when the harmonic trap is opened (or closed) as a function of time while keeping the adiabatic parameter $\ensuremath{\mu}=[d\ensuremath{\omega}(t)/dt]/{\ensuremath{\omega}}^{2}(t)$ fixed, a sharp transition from an oscillatory to a monotonic exponential dynamics occurs at $\ensuremath{\mu}=2$ [Uzdin, Dalla Torre, Kosloff, and Moiseyev, Phys. Rev. A 88, 022505 (2013)]. Here we show that by using time-dependent linear coordinate transformation the time-dependent Schr\"odinger equation with Hermitian time-dependent Hamiltonian is transformed into a time-dependent Schr\"odinger equation with a time-independent harmonic oscillator with the dimensionless frequency of $\sqrt{1\ensuremath{-}{(\ensuremath{\mu}/2)}^{2}}$. At $\ensuremath{\mu}=2$ a transition to a non-Hermitian Hamiltonian is obtained as the potential well is transformed to a parabolic potential barrier. While in a harmonic trap noninteracting particles have classical periodic motions, they are pushed apart exponentially in time as the potential well is suddenly transformed into a parabolic potential barrier in the new variable representation.
Resonances associated with complex-frequency poles are ubiquitous across physics and can arise in any open system, ranging from subwavelength particles and cavities to biological structures. When two such resonances coalesce, they form exceptional points (EPs), non-Hermitian singularities known to produce unusual spectral and dynamical behavior. However, the dynamics of the response of such resonances and exceptional points to complex frequency drive remained largely unexplored. Here, we experimentally observe the temporal response of complex-frequency resonances and theoretically study this for exceptional points. We unveil a universal transient phenomenon of open cavities driven at complex frequencies: the system's initial response grows linearly, with enhanced growth at exceptional points (EPs), even though the system is passive and the excitation decays. Closed-form theory for general resonators, extended to higher-order modes, predicts efficient power transfer with ttt and t2t^2t2 scaling for complex single poles and exceptional points (EPs), respectively, at all times. We demonstrate these effects in subwavelength optical scatterers and experimentally in an electrical circuit analogue, with excellent agreement, and explore configurations that capture EP-enhanced growth.
In this contribution to the memorial issue of Göran Lindblad, we investigate the periodically driven Lindblad equation for a two-level system. We analyze the system using both adiabatic diagonalization and numerical simulations of the time-evolution, as well as Floquet theory. Adiabatic diagonalization reveals the presence of exceptional points in the system, which depend on the system parameters. We show how the presence of these exceptional points affects the system evolution, leading to a rapid dephasing at these points and a staircase-like loss of coherence. This phenomenon can be experimentally observed by measuring, for example, the population inversion. We also observe that the presence of exceptional points seems to be related to which underlying Lie algebra the system supports. In the Floquet analysis, we map the time-dependent Liouvillian to a non-Hermitian Floquet Hamiltonian and analyze its spectrum. For weak decay rates, we find a Wannier-Stark ladder spectrum accompanied by corresponding Stark-localized eigenstates. For larger decay rates, the ladders begin to dissolve, and new, less localized states emerge. Additionally, their eigenvalues are exponentially sensitive to perturbations, similar to the skin effect found in certain non-Hermitian Hamiltonians.
Exceptional points (EPs) of a non-Hermitian Hamiltonian with parity-time-reversal (PT) symmetry have the potential to drastically enhance the capabilities of metrology and sensing through their power-law growing sensitivity to external perturbation. With the ability of generating and tuning dissipation in a single trapped ion system, we observe rich dynamics and detailed quantum phase transitions from the PT-symmetric phase to the symmetry-breaking phase. In this single qubit full quantum system, we develop a method to precisely determine the location of EP without any fitting parameter, and extract the eigenvalues in a unified way through all parameter regions. We can also obtain the full density matrix by quantum state tomography. Finally, we suggest from theoretical analysis that the periodically driving PT-symmetric non-Hermitian system can be used to measure the magnitude, frequency, and phase of time-dependent perturbation with EP enhancement.
We study interacting ultracold atoms in a three-dimensional (3D) harmonic trap with spin-selective dissipations, which can be effectively described by non-Hermitian parity-time (PT\mathcal{PT}PT) symmetric Hamiltonians. By exactly solving the non-Hermitian two-body problem of spin-1/2 (spin-1) bosons in a 3D harmonic trap, we find that the system can exhibit third-order (fifth-order) exceptional points (EPs) with ultrasensitive cube-root (fifth-root) spectral response due to interaction anisotropies in spin channels. We also present the general principle for the creation of high-order EPs and their spectral sensitivities with arbitrary particle number NNN and arbitrary spin sss. Generally, with spin-independent interactions, the EP order of bosons can be as high as 2Ns+12Ns+12Ns+1, and the spectral response around EP can be as sensitive as $\ensuremath{\sim}{\ensuremath{\epsilon}}^{1/(2ks+1)}$ under a kkk-body interaction anisotropy. Moreover, we propose to detect the ultrasensitive spectral response through the probability dynamics of certain state. These results suggest a convenient route towards more powerful sensor devices in spinor cold atomic systems.
We show how a single trapped ion may be used to test a variety of important physical models realized as time-dependent harmonic oscillators. The ion itself functions as its own motional detector through laser-induced electronic transitions. Alsing et al., [Phys. Rev. Lett. 94, 220401 (2005)] proposed that an exponentially decaying trap frequency could be used to simulate (thermal) Gibbons-Hawking radiation in an expanding universe, but the Hamiltonian used was incorrect. We apply our general solution to this experimental proposal, correcting the result for a single ion and showing that while the actual spectrum is different from the Gibbons-Hawking case, it nevertheless shares an important experimental signature with this result.