The theory of measurement with white and colored noise operators is discussed, utilizing the Collision Model representation of weak continuous measurement. While the former embodies an ideal measurement, the latter admits correlations between pointer states at different times, as it embodies a quantum delay line. The response of this apparatus is shown to be equivalent to a sort of comb filter. We apply these measurements to a simplified scheme for the process of musical sensing. While the measurement with white noise leads to a monotone rise in the certainty of a specific sound, measuring with colored noise introduces several patterns in the output, which we term as the quantum color of the sound and propose several sonifications of the outcome. Much like how feedback delay networks in digital acoustic models create the richness of simulated spaces, our quantum sensor introduces temporal correlations to sound perception through coherent feedback mechanism. Under the speculation that the human ear is sensitive to such temporal correlations, measurement with colored noise could be used to charge a sound signal with rhythmic meaning.
The theory of measurement with white and colored noise operators is discussed, utilizing the Collision Model representation of weak continuous measurement. While the former embodies an ideal measurement, the latter admits correlations between pointer states at different times, as it embodies a quantum delay line. The response of this apparatus is shown to be equivalent to a sort of comb filter. We apply these measurements to a simplified scheme for the process of musical sensing. While the measurement with white noise leads to a monotone rise in the certainty of a specific sound, measuring with colored noise introduces several patterns in the output, which we term as the quantum color of the sound and propose several sonifications of the outcome. Much like how feedback delay networks in digital acoustic models create the richness of simulated spaces, our quantum sensor introduces temporal correlations to sound perception through coherent feedback mechanism. Under the speculation that the human ear is sensitive to such temporal correlations, measurement with colored noise could be used to charge a sound signal with rhythmic meaning.
We explore an unconventional bridge between quantum mechanical density matrices and sound by mapping elements of the density matrix and their phases to auditory signals, thus introducing a framework for Open Quantum Sonification. Employing the eigenstates of the Hamiltonian operator as a basis, each quantum state contributes a frequency proportional to its energy level. The off-diagonal terms, which encode coherence and phase relationships between energy levels, are rendered as binaural signals presented separately to the left and right ears. We illustrate this method within the context of open quantum system dynamics governed by Lindblad equations, presenting first an example of quantum Brownian motion of a particle in a thermal bath, and second, a recoherence process induced by boundary driving that results in spin-helix states. This document serves as a companion to the corresponding audio visual simulations of these models available on the YouTube channel Open Quantum Sonification with the Python Codes on GitHub. The auditory analogy presented here provides an intuitive and experiential means of describing quantum phenomena such as tunnelling, thermalisation, decoherence, and recoherence.
In the last chapter, we dealt with the wide class of physical operations that preserve the Gaussian character of the initial state, treating them as discrete transformations. Here, we describe what could loosely be considered the continuous-time version of such transformations. This chapter is intended to provide the reader with the tools to describe a wide range of notable quantum dynamics, with a very broad applicative scope. After a note on the action of linear and quadratic Hamiltonians, whose description follows directly from the theory of Gaussian states presented in Chapter 3 but was not explicitly clarified so far, we shall address systems linearly coupled to a reservoir of modes that are delta-correlated in time (a situation commonly referred to as ‘white noise’), and thus give rise to Markovian dynamics which we shall refer to as ‘diffusive’ for reasons that will be made clear in due time. Our story will commence with a derivation of the Gaussian dynamics resulting from such an environmental coupling, in the form of dynamical equations for the first and second statistical moments. Besides their interest per se , the latter will be instrumental in obtaining an unconventional derivation of the master equations that govern general, not necessarily Gaussian, quantum states under the same conditions. The corresponding quantum Langevin equations for the canonical operators in the Heisenberg picture will also be derived. Next, we will turn our attention to the study of the system's dynamics that result from the measurement of the environment. This is a topic with a longstanding tradition in quantum mechanics, grounded in the formalism of quantum stochastic calculus that emerged within the mathematical physics community as early as the seventies, and then re-introduced as the quantum trajectories approach in quantum optics almost twenty years later. The ‘weak’ POVMs resulting from observing a weakly coupled environment (as opposed to ‘strong’, projective measurements), carried out continuously in time (a prescription customarily subsumed by the term ‘monitoring’), leads to a stochastic dynamics for the system, where the randomness of the measurement outcomes is compactly and most elegantly represented by infinitesimal stochastic increments, giving rise to continuous but not differentiable quantum dynamics. Existing approaches to this subject tend to be quite impervious to the uninitiated. In the quantum optics literature, this is largely due to the technicalities involved in incorporating the discontinuous update of the quantum state prescribed by the von Neumann postulate. More mathematically oriented treatments have instead typically preferred an algebraic approach where the usual difficulty of describing measurements in the Heisenberg picture is compounded by the adoption of a rather abstract hierarchy of filters, in part inspired by classical control theory, to describe different possible updates of the quantum state. This state of affairs is further complicated by the added difficulties that come with the necessary introduction of some form of quantum stochastic calculus. Our approach to monitoring will therefore take a different route. Like in our derivation of diffusion and master equations, we shall keep restricting to linear couplings with a white noise environment, forfeiting full generality This situation is well modelled by the so called input-output formalism, whose physical origin will be discussed in detail in Chapter 9 , which approximates reliably, among other instances, the interaction between internal resonating modes and electromagnetic radiation leaking in and out of an optical cavity. First, we shall introduce stochastic Wiener increments and the Itô rule, which will arise naturally from the identification of the continuum field's modes subject to monitoring. We shall deliberately keep our arsenal of stochastic methods to these bare essentials, sacrificing to some extent both depth and generality, but still providing the reader with a powerful and seminal demonstration of the formalism. We shall then restrict to Gaussian general-dyne detections and apply the exact formalism developed in the previous chapter to the continuous filtering of the system's state conditioned by such measurements' outcomes. Next, as in our derivation of the unconditional master equations, we shall use the Gaussian dynamics as a key to unlock the full stochastic master equations governing general states. Hence, we will build on the quantum theory of filtering in the Gaussian regime to derive dynamical equations for linear feedback control and also apply our theoretical tools to the enhanced production of squeezing. Finally, we will turn to the closely related paradigm of coherent feedback, where environmental modes do not undergo measurements but are fed back to interact coherently with the system, see in detail how it modifies Gaussian diffusive dynamics and assess its potential for control by reconsidering the production of squeezing assisted by interferometric coherent feedback loops.
Abstract Quantum technologies based on photons rely on correlated pairs generated through nonlinear optics. Hence, understanding the impact of disturbances is of paramount importance for the development of this innovative field. Here we focus on the quantum spectroscopy as one of the most promising quantum technique showing a realistic perspective for a future employment. In this field, the most interesting disturbance is represented by the frequency noise. We present an experiment aimed at testing and characterizing the resilience against different levels of white and colored noise and we include simulations to generalize our findings. With this work we demonstrate that the spectroscopical properties of both the target and the noise, as well as the strength of the quantum correlations, play a crucial role and could have e huge impact on the performances, especially in terms of achievable spectral resolution. Remarkably, the presence of a specific region where the technology is always robust against any kind of noise represents an advantage allowing to deal with optimal measurements and fully exploit the capabilities of this technology.
Quantum sensing has been receiving researcher’s attention these years due to its ultrahigh sensitivity and precision. However, the bandwidth of the sensors may be low, thus limiting the scope of their practical applications. The low-bandwidth problem is conquered by feedback control methods, which are widely utilized in classic control fields. Based on a quantum harmonic oscillator model operating near the resonant point, the bandwidth and sensitivity of the quantum sensor are analyzed. The results give two important conclusions: (a) the bandwidth and sensitivity are two incompatible performance parameters of the sensor, so there must be a trade-off between bandwidth and sensitivity in practical applications; (b) the quantum white noise affects the signal to be detected in a non-white form due to the feedback control.
A quantum mechanical approach to the psychoacoustic effect of Shepard tones has been explored. Melodic intervals are represented by qubits. The phenomenon of bistable perception of intervals and their acoustic collapse has been considered and described in the language of quantum mechanics. It is shown that the perception of intervals can be understood as a result of measuring the standard basis vector in different bases rotated in a plane of the chroma circle. For comparison, the Stern-Gerlach and photon polarization experiments have been cited to show an analogy. A quantum analog of the Shepard interval source, with a photonic random number generator, has been proposed. The topic of quantum entanglement transferred to the field of sound and psychoacoustics is also discussed. Based on quantum-like calculations, it has been shown how disturbances in time coherence between the playing of different music band members could affect the modification of the melody.
Quantum sensors represent a revolutionary technology with the potential to surpass the limits of classical sensors across a wide range of applications, including precision measurement, navigation, and medical diagnostics. However, the realization of this potential is significantly hindered by the pervasive presence of noise. This work addresses the critical need for accurate noise modeling and effective suppression techniques within quantum sensors. We introduce a novel noise model that incorporates both thermal and coherent noise sources, explicitly accounting for the quantum mechanical nature of these sensors. This model is then utilized to design and analyze various noise suppression strategies, including feedback control algorithms and advanced filtering methods. The theoretical framework presented here provides a foundation for improved sensor design and operation, ultimately driving advancements in quantum sensing technology. The core claim of this work is that precise noise modeling and effective suppression techniques are paramount for enhancing the precision of quantum sensors. This is achieved through the establishment of a new noise model and the subsequent design of suppression techniques such as feedback control and filtering. The novelty lies in the detailed and accurate representation of noise sources within the quantum sensor system.