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arXiv:2310.13631 · Physical Review Research

Fluctuating parametric drive of coupled classical oscillators can simulate dissipative qubits

Lorenzo Bernazzani, Guido Burkard

Published Mar 14, 2024• 2 citations
arXivHTMLPDF

Abstract

We investigate a system composed of two coupled oscillators subject to stochastic fluctuations in its internal parameters. In particular, we answer the question whether the well-known classical analogy of the quantum dynamics of two-level systems (TLSs), i.e., qubits, provided by two coupled oscillators can be extended to simulate the dynamics of dissipative quantum systems. In the context of nanomechanics, the analogy in the dissipation-free case has already been tested in multiple experimental setups, e.g., doubly clamped or cantilever string resonators and optically levitated particles. A well-known result of this classical analogy is that the relaxation and decoherence times of the analog quantum system must be equal, i.e., T1=T2 , in contrast to the general case of quantum TLSs. We show that this fundamentally quantum feature, i.e., T1≠T2 , can be implemented as well in the aforementioned classical systems by adding stochastic fluctuations in their internal parameters. Moreover, we show that these stochastic contributions can be engineered in the control apparatus of those systems, discussing, in particular, the application of this theory to levitated nanoparticles and to nanostring resonators. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.013284

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We investigate a system composed of two coupled oscillators subject to stochastic fluctuations in its internal parameters. In particular, we answer the question whether the well-known classical analogy of the quantum dynamics of two-level systems (TLSs), i.e., qubits, provided by two coupled oscillators can be extended to simulate the dynamics of dissipative quantum systems. In the context of nanomechanics, the analogy in the dissipation-free case has already been tested in multiple experimental setups, e.g., doubly clamped or cantilever string resonators and optically levitated particles. A well-known result of this classical analogy is that the relaxation and decoherence times of the analog quantum system must be equal, i.e., T1=T2 , in contrast to the general case of quantum TLSs. We show that this fundamentally quantum feature, i.e., T1≠T2 , can be implemented as well in the aforementioned classical systems by adding stochastic fluctuations in their internal parameters. Moreover, we show that these stochastic contributions can be engineered in the control apparatus of those systems, discussing, in particular, the application of this theory to levitated nanoparticles and to nanostring resonators. Published by the American Physical Society 2024
Decoherence is an essential mechanism that defines the boundary between classical and quantum behaviours, while imposing technological bounds for quantum devices. Little is known about quantum coherence of mechanical systems, as opposed to electromagnetic degrees of freedom. But decoherence can also be thought of in a purely classical context, as the loss of phase coherence in the classical phase space. Indeed the bridge between quantum and classical physics is under intense investigation, using, in particular, classical nanomechanical analogues of quantum phenomena. In the present work, by separating pure dephasing from dissipation, we quantitatively model the classical decoherence of a mechanical resonator: through the experimental control of frequency fluctuations, we engineer artificial dephasing. Building on the fruitful analogy introduced between spins/quantum bits and nanomechanical modes, we report on the methods available to define pure dephasing in these systems, while demonstrating the intrinsic almost-ideal properties of silicon nitride beams. These experimental and theoretical results, at the boundary between classical nanomechanics and quantum information fields, are prerequisite in the understanding of decoherence processes in mechanical devices, both classical and quantum. Introduction and background. Frequency noise in nano-mechanical systems (NEMS) is a topic intensely studied these last years, for both fundamental and practical reasons (references [ 13 – 23 ] of the paper): the frequency fluctuations can have an intrinsic origin linked to essential material properties, but they also limit the resolution of actual devices used as sensors. Recently, the Delft group proposed to analyze the effects of this noise on the dynamics of NEMS within a framework analogous to one of Nuclear Magnetic Resonance (NMR), or quantum bits (2014 Nat. Commun. 5 5819 ). Main results. Building on the analogy, in this paper we engineer frequency noise to study its impact on nano-resonators. We present a model experiment, with the full theory enabling the fit of data without free parameters. The effect of frequency noise is analogous to 'pure dephasing' in quantum bits, even though the device is a classical harmonic oscillator. As such, the analogy is very informative and interrogates the crossover between classical and quantum physics; it enters the class of 'classical analogues of quantum phenomena' studied in recent model nano-mechanical experiments (e.g. reference [ 10 ]). Wider implications. This is a broad topic that is of interest to many areas of science, far beyond NEMS and NMR. For the focused audience, we present the methods to quantitatively describe this 'classical pure dephasing' of NEMS, which can be used e.g. with carbon-based systems. As a conclusion, our results are a new tool for the understanding of classical decoherence in nano-resonators, and a pre-requisite for future experiments on quantum NEMS.
The model of a two-level system (TLS) coupled to a harmonic oscillator has found a widespread application in physics and chemistry. In this thesis we focus on the field of circuit quantum electrodynamics where the TLS stands for a quantum bit (qubit), which is the basic building unit of a potential quantum computer, and the oscillator represents a transmission-line resonator, which can be used, e.g., to store information contained on the qubit or for communication between several qubits. Furthermore, the oscillator can model a superconducting quantum interference device (SQUID) determining the state of the qubit. In first experimental realizations the coupling between the qubit and the oscillator was small compared to characteristic frequencies of the two devices. In such a situation the Jaynes-Cummings model provides a realistic and completely analytical description of the energy spectrum and the dynamics of the setup. However, it relies on a rotating-wave approximation (RWA) which neglects counter-rotating terms in the full qubit-oscillator Hamiltonian. These counter-rotating terms become important under increase of the coupling strength, as it has been achieved in recent experiments. We present two analytical approaches to take these additional contributions into account: The first one treats the qubit-oscillator Hamiltonian perturbatively in the coupling strength and predicts a frequency shift in its energy spectrum, the Bloch-Siegert shift. The second one considers the qubit's tunneling splitting as small parameter and thus treats the problem to all orders in the coupling allowing to enter the ultrastrong coupling regime, where the coupling strength becomes comparable to the qubit's and/or oscillator's frequency. For both cases we give a thorough analysis of the qubit's dynamics. In order to perform operations the qubit is usually probed by an external driving force. We model this situation by coupling the TLS to a classical oscillator and examine the resulting time dependent Hamiltonian using a combination of Floquet theory and Van Vleck perturbation theory. Thus, we provide an analysis of the qubit's energy spectrum and dynamics which is nonperturbative in the driving amplitude. We investigate effects like coherent destruction of tunneling and driving-inducing tunneling oscillations. By coupling the driven TLS to a quantized oscillator we give for the first time an analytical treatment of the qubit-oscillator system being simultaneously exposed to both ultrastrong coupling and extreme driving. We observe interesting phenomena in the dynamics like the suppression of specific frequencies under a variation of the coupling strength. To mimic environmental influences on the qubit we apply the Caldeira-Leggett master equation approach to the driven TLS and to the qubit-oscillator system and provide for both cases analytical expressions for the relaxation and dephasing rates.
The center-of-mass motion of a single optically levitated nanoparticle resembles three uncoupled harmonic oscillators. We show how a suitable modulation of the optical trapping potential can give rise to a coupling between two of these oscillators, such that their dynamics are governed by a classical equation of motion that resembles the Schrödinger equation for a two-level system. Based on experimental data, we illustrate the dynamics of this parametrically coupled system both in the frequency and in the time domain. We discuss the limitations and differences of the mechanical analogue in comparison to a true quantum mechanical system.
Coupled nanomechanical resonators have recently attracted much attention for both fundamental studies in physics and broad applications in high-precession detection or sensing. By studying the Landau-Zener transitions and Rabi oscillation of two coupled resonators, it has been shown that such a two-mode system acts as a classical two-level system bearing the analog to a quantum mechanical two-level one. Here we construct a Stückelberg interferometer with two coupled cantilevers by driving the system through the avoided crossing twice. By measuring the non-adiabatic phase acquired at the Landau-Zener transition, we unveil an in-depth analog between the two-mode and quantum two-level systems. Our study opens up new opportunities for constructing interferometers with classical devices.
Two-level systems (TLSs) are the major source of dephasing of spin qubits in numerous quantum computing platforms. In spite of much effort, it has been difficult to substantially mitigate the effects of this noise or, in many cases, to fully understand its physical origin. We propose a method to make progress on both of these issues. When an oscillating field is applied to a TLS, stochastic resonance can occur and the noise spectrum is moved to higher frequencies. This shift in the TLS noise spectrum will increase the dephasing times of the qubits that they influence. Furthermore, the details of this effect depend on the physical properties of the noise sources. Thus one can use qubit spectroscopy to investigate their physical properties, specifically the extent to which the TLSs themselves possess quantum coherence. We find that it should be possible to determine the dephasing rate and the energy-level separation of the TLS themselves in this way.
Two-level systems (TLS) are widely recognized as a dominant source of energy relaxation in superconducting qubits, limiting coherence times and device performance. While TLS-induced dissipation is often modeled as independent channels, interacting TLS can form finite many-body environments whose dynamical role in qubit relaxation remains insufficiently understood. Building on the information-erasure-based framework for irreversible thermalization developed in our previous work, we present a microscopic model and numerical simulations of a superconducting qubit coupled to a finite interacting many-body TLS environment as a concrete physical realization of this mechanism. We show that effective irreversible energy relaxation can emerge from intrinsic many-body dynamics without invoking external baths or phenomenological dissipation. The introduction of information erasure drives a transition from coherent oscillatory energy exchange to nonoscillatory exponential relaxation of the qubit. The relaxation times exhibit the scaling T1,T2∝Jq,τ−2T_1, T_2 \propto J_{q,τ}^{-2}T1​,T2​∝Jq,τ−2​ with respect to the qubit-TLS coupling strength, consistent with Fermi's golden rule and Lindblad-type descriptions. Importantly, both exponential decay and its scaling behavior arise directly from microscopic many-body dynamics. These results provide quantitative insight into TLS-induced decoherence mechanisms and establish a microscopic link between information erasure, many-body thermalization, and irreversible relaxation in superconducting quantum systems.