Much of the physical world around us can be described in terms of harmonic oscillators in thermodynamic equilibrium. At the same time, the far from equilibrium behavior of oscillators is important in many aspects of modern physics. Here, we investigate a resonating system subject to a fundamental interplay between intrinsic nonlinearities and a combination of several driving forces. We have constructed a controllable and robust realization of such a system using a macroscopic doubly clamped string. We experimentally observe a hitherto unseen double hysteresis in both the amplitude and the phase of the resonator's response function and present a theoretical model that is in excellent agreement with the experiment. Our work provides a thorough understanding of the double-hysteretic response through a symmetry breaking of parametric phase states that elucidates the selection criteria governing transitions between stable solutions. Our study motivates applications ranging from ultrasensitive force detection to low-energy computing memory units. PACS numbers:Parametric excitation of resonators plays an important role in many areas of science and technology. In its bestknown form, parametric excitation describes the modulation of a resonator's natural frequency at twice the natural frequency itself [1][2][3][4]. In this case, energy is pumped into or out of the resonator depending on the phase of the modulation relative to the oscillation. This ubiquitous feature finds applications in a wide range of fields including signal amplification and noise squeezing [5][6][7][8][9][10][11][12][13] with contemporary proposals also including topological chiral amplifiers [14], generation of quantum entanglement [15,16], as well as mechanical logic operations with the so-called parametron [17][18][19][20].The last decade has seen remarkable progress in the fabrication and control of nanomechanical resonators which serve as an ideal platform for harnessing parametric excitations [21][22][23]. As the resonators scale down, they attain unprecedented sensitivity towards minute masses, forces and magnetic moments [24][25][26]. At the same time, they enter a regime where nonlinearities become a defining characteristic that offers new functionality for parametrical detectors [22,23,[27][28][29]. Indeed, for sufficiently strong parametric driving, the effective damping of the linear resonator becomes negative and the oscillation amplitude is stabilized by nonlinearities [30].The negative effective damping regime of the parametric resonator is particularly interesting because it features two stable oscillation solutions [4]. These solutions, which we term 'parametric phase states' for the rest of this paper, are a result of the double periodicity of the parametric excitation. They are degenerate in amplitude, but phase shifted by π, and they are fascinating because they allow for the study of broken timetranslation symmetry and activated interstate switching in both classical and quantum systems [31][32][33]. Recently, it was shown that a...
We propose a novel method for linear detection of weak forces using parametrically driven nonlinear resonators. The method is based on a peculiar feature in the response of the resonator to a near resonant periodic external force. This feature stems from a complex interplay between the parametric drive, external force and nonlinearities. For weak parametric drive, the response exhibits the standard Duffing-like single jump hysteresis. For stronger drive amplitudes, we find a qualitatively new double jump hysteresis which arises from stable solutions generated by the cubic Duffing nonlinearity. The additional jump exists only if the external force is present and the frequency at which it occurs depends linearly on the amplitude of the external force, permitting a straightforward ultrasensitive detection of weak forces. With state-of-the-art nanomechanical resonators, our scheme should permit force detection in the atto-newton range.
We introduce and describe the multiconfigurational time-depenent Hartree for indistinguishable particles (MCTDH-X) software, which is hosted, documented, and distributed at http://ultracold.org. This powerful tool allows the investigation of ground state properties with time-independent Hamiltonians, and dynamics of interacting quantum many-body systems in different spatial dimensions. The MCTDH-X software is a set of programs and scripts to compute, analyze, and visualize solutions for the time-dependent and time-independent many-body Schrödinger equation for indistinguishable quantum particles. As the MCTDH-X software represents a general solver for the Schrödinger equation, it is applicable to a wide range of problems in the fields of atomic, optical, molecular physics as well as condensed matter systems. In particular, it can be used to study light-matter interactions, correlated dynamics of electrons in solid states, as well as some aspects related to quantum information and computing. The MCTDH-X software solves a set of non-linear coupled working equations based on the application of the variational principle to the Schrödinger equation. These equations are obtained by using an ansatz for the many-body wavefunction that is a time-dependent expansion in a set of time-dependent, fully symmetrized bosonic (X=B) or fully anti-symmetrized fermionic (X=F) many-body basis states. It is the time-dependence of the basis set, that enables MCTDH-X to deal with quantum dynamics at a superior accuracy as compared to, for instance, exact diagonalization approaches with a static basis, where the number of basis states necessary to capture the dynamics of the wavefunction typically grows rapidly with time.Herein, we give an introduction to the MCTDH-X software via an easy-to-follow tutorial with a focus on accessibility.The illustrated exemplary problems are hosted at http://ultracold.org/tutorial and consider the physics of a few interacting bosons or fermions in a double-well potential. We explore computationally the position-space and momentum-space density, the one-body reduced density matrix, Glauber correlation functions, phases, (dynamical) phase transitions as well as the imaging of the quantum systems. Although a few particles in a double well potential represent a minimal model system, we are able to demonstrate a rich variety of phenomena with it. We use the double well to illustrate the fermionization of bosonic particles, the crystallization of fermionic particles, characteristics of the superfluid and Mott-insulator quantum phases in Hubbard models, and even dynamical phase transitions. We provide a complete set of input files and scripts to redo all computations in this paper at http://ultracold.org/data/tutorial_input_files.zip, accompanied by tutorial videos at https://www.youtube.com/playlist?list=PLJIFUqmSeGBKxmLcCuk6dpILnni_uIFGu. Our tutorial should guide the potential users to apply the MCTDH-X software also to more complex systems. arXiv:1911.00525v2 [cond-mat.quant-gas] 16 Jan 2020The time-dependen...
We investigate harmonically-trapped, laser-pumped bosons with infinite-range interactions induced by a dissipative high-finesse red-detuned optical cavity with numerical and analytical methods. We obtain multiple cavity and atomic observables as well as the full phase diagram of the system using the multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X) approach. Besides the transition from an unorganized normal phase to a superradiant phase where atoms self-organize, we focus on an in-depth investigation of the self-organized superfluid to self-organized Mott insulator phase transition in the superradiant phase as a function of the cavity-atom coupling. The numerical results are substantiated by an analytical study of an effective Bose-Hubbard model. We numerically analyze cavity fluctuations and emergent strong correlations between atoms in the many-body state across the Mott transition via the atomic density distributions and Glauber correlation functions. Unexpectedly, the weak harmonic trap leads to features like a lattice switching between the two symmetry-broken Z2 configurations of the untrapped system and a reentrance of superfluidity in the Mott insulating phase. Our analytical considerations quantitatively explain the numerically observed correlation features. arXiv:1811.09634v1 [cond-mat.quant-gas]
We numerically obtain the full time-evolution of a parametrically-driven dissipative Bose-Einstein condensate in an optical cavity and investigate the implications of driving for the phase diagram. Beyond the normal and superradiant phases, a third nonequilibrium phase emerges as a manybody parametric resonance. This dynamical normal phase switches between two symmetry-broken superradiant configurations. The switching implies a breakdown of the system's mapping to the Dicke model. Unlike the other phases, the dynamical normal phase shows features of nonintegrability and thermalization.
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