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Quantum Acoustics: A Unified Quantum Theory of Phonons Analogous to Quantum Optics

shifa liu

Published Dec 31, 2025• 0 citations
Publisher

Abstract

We develop quantum acoustics as an independent quantum discipline. The elementary bosonic quantum is the phonon, obtained by canonical quantization of the elastic displacement field of a medium. The central replacement is c → cs, ε0 → ρ0, μ0 → 1/K, ℏ → ℏ, while the Planck constant is not replaced. Phonons obey canonical commutation relations, Bose–Einstein statistics with vanishing chemical potential, and energy quantization E = ℏω. The optical–acoustic dictionary is exact at the level of canonical quantization, mode algebra, Fock, coherent, squeezed, thermal and entangled states, and at the level of linear interference in the long-wavelength, single-branch, dissipationless limit. Multi-branch, dispersive, dissipative and finite-lifetime effects are handled by the generalized dictionary stated in Chapter 56. Quantum acoustics contains a macroscopic quantum branch, but it is not identical to macro- scopic quantum mechanics. The manuscript covers classical elasticity, variational principles, canonical quantization, phonon field algebra, propagators, Green functions, wave–particle duality, Bose–Einstein statistics, blackbody radiation, Debye heat capacity, acoustic quantum constants, acoustic Bekenstein–Hawking entropy, phonon thermodynamics, phonon BEC, phonon lasers, superfluidity, Fock, coherent, squeezed, thermal, entangled and non-classical states, beam splitters, Mach–Zehnder interferometers, Fabry–Perot cavities, Hong–Ou–Mandel interference, second-order coherence, phonon qubits, single- and two-qubit gates, quantum algorithms, quantum error correction and surface-code thresholds, quantum computing architecture, quantum networks and repeaters, quantum simulation, quantum sensing, topological phononic crystals, the SSH phonon model, two-dimensional topological phononic insulators, three-dimensional phonon Weyl semimetals, non-Hermitian acoustics and the phonon skin effect, non-Bloch band theory, topological defect classification, acoustic metrics, acoustic horizons, acoustic Hawking radiation, acoustic black- hole entropy and the information paradox, acoustic cosmology, phonon–graviton mixing, gravitational-wave coupling, the phonon analogy of dark matter and dark energy, surface acoustic wave resonators, optomechanical systems, superconducting qubit–phonon hybrid systems, Bose–Einstein condensates, phononic crystals, acoustic metamaterials, gravitational-wave detectors, the complete optical–acoustic dictionary, the generalized dictionary for multi-branch, dispersive, dissipative and finite-lifetime phonons, and strong-field QED vacuum and media as an analogue of multi-refringent, dispersive and dissipative media. Acoustic Hawking radiation has been observed in Bose–Einstein condensates by Steinhauer (2016). All definitions, derivations, mathematical details, experimental proposals and comparison factors are presented with explicit assumptions and complete proofs.

DOI
10.5281/zenodo.23647668

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This Tech–Science text of article dedicated to the QUANTUM ACOUSTIC Sensu Stricto says about phonons, the sound quazi–particles elements, which are very interesting in nano, micro, makro scales of nature. This New Concept of QUANTUM ACOUSTIC PHYSICS OR QUANTUM PHYSICS OF SOUND IS ILLUSTRATED ON EMPIRIO–CRITICISM PHILOSOPHY OF EXPERIMENTAL PHYSICS OF MORAVIAN BORN PROFESSOR ERNST MACH, the text operated with subjects like MACH’S NUMBER, MACH’S CONE, MACH’S PRINCIPLE, MACH’S WAVEMACHINE. The Highlights is focused on Author’s Sketches in problematic connected with AERODYNAMICS AND AERONAUTICS.

•Dec 1, 2018•0 citations
We develop quantum acoustics as an independent quantum discipline. The elementary bosonic quantum is the phonon, obtained by canonical quantization of the elastic displacement field of a medium. The central replacement is c → cs, ε0 → ρ0, μ0 → 1/K, ℏ → ℏ, while the Planck constant is not replaced. Phonons obey canonical commutation relations, Bose–Einstein statistics with vanishing chemical potential, and energy quantization E = ℏω. The optical–acoustic dictionary is exact at the level of canonical quantization, mode algebra, Fock, coherent, squeezed, thermal and entangled states, and at the level of linear interference in the long-wavelength, single-branch, dissipationless limit. Multi-branch, dispersive, dissipative and finite-lifetime effects are handled by the generalized dictionary stated in Chapter 56. Quantum acoustics contains a macroscopic quantum branch, but it is not identical to macro- scopic quantum mechanics. The manuscript covers classical elasticity, variational principles, canonical quantization, phonon field algebra, propagators, Green functions, wave–particle duality, Bose–Einstein statistics, blackbody radiation, Debye heat capacity, acoustic quantum constants, acoustic Bekenstein–Hawking entropy, phonon thermodynamics, phonon BEC, phonon lasers, superfluidity, Fock, coherent, squeezed, thermal, entangled and non-classical states, beam splitters, Mach–Zehnder interferometers, Fabry–Perot cavities, Hong–Ou–Mandel interference, second-order coherence, phonon qubits, single- and two-qubit gates, quantum algorithms, quantum error correction and surface-code thresholds, quantum computing architecture, quantum networks and repeaters, quantum simulation, quantum sensing, topological phononic crystals, the SSH phonon model, two-dimensional topological phononic insulators, three-dimensional phonon Weyl semimetals, non-Hermitian acoustics and the phonon skin effect, non-Bloch band theory, topological defect classification, acoustic metrics, acoustic horizons, acoustic Hawking radiation, acoustic black- hole entropy and the information paradox, acoustic cosmology, phonon–graviton mixing, gravitational-wave coupling, the phonon analogy of dark matter and dark energy, surface acoustic wave resonators, optomechanical systems, superconducting qubit–phonon hybrid systems, Bose–Einstein condensates, phononic crystals, acoustic metamaterials, gravitational-wave detectors, the complete optical–acoustic dictionary, the generalized dictionary for multi-branch, dispersive, dissipative and finite-lifetime phonons, and strong-field QED vacuum and media as an analogue of multi-refringent, dispersive and dissipative media. Acoustic Hawking radiation has been observed in Bose–Einstein condensates by Steinhauer (2016). All definitions, derivations, mathematical details, experimental proposals and comparison factors are presented with explicit assumptions and complete proofs.
In a quantum fluid such as a condensed Bose gas, elementary excitations such as phonons behave as quantized fields with dispersion relations and non-linearities that bear some resemblance to those of the electromagnetic field. The very low temperatures of these gases means that in some cases quantum fluctuations are important. Several remarkable phenomena in quantum electrodynamics have found acoustic analogs, and I will discuss two of them. The first is the dynamical Casimir effect, a prediction that the vacuum can generate real particles when boundary conditions are suddenly changed. That is, photon pairs can be spontaneously generated in an empty cavity with nonuniformly accelerating boundaries. A second is Hawking radiation. Here again photon pairs are generated at the horizon of a black hole. I will discuss how certain aspects of both these effects can be mimicked in a Bose Einstein condensate, and outline the state of the experimental art in this domain.
Due to its high coherence in transmission over a large distance in the ambient environment, the quantum optical system has been a prevailing platform for long-distance quantum communication, which was recently realized over a continental distance with a low earth orbit satellite and ground stations [102, 70]. However, the pure quantum optical system has so far shown weak interactions between photon and matter, which makes it inefficient in carrying out deterministic quantum gates for quantum repeater based scalable quantum network and quantum computing. On the other hand, superconducting quantum systems operating in the microwave domain with Josephson junction transmon qubits have proven to be capable of efficient deterministic quantum operations on quantum states [86, 87, 66]. Nevertheless, such architecture is prone to errors and decoherence due to cross-talk between microwave elements in a large-scale superconducting quantum circuit. Furthermore, superconducting systems, in general, also have large footprint (100s um) elements (resonators and superconducting quantum bits) [92, 60] that limit the ability to scale up a superconducting quantum system. Moreover, microwave quantum circuits require cooling to around 10 mK, making it unsuitable for communicating quantum information outside a dilution refrigerator (DF). Micro- and nano- acoustic elements have been extensively used in conventional integrated information processing systems due to their compactness and high coherence [97]. Acoustic systems in quantum engineering also have the advantage of being a platform for universal couplings between various quantum systems including spins, optical photons, and superconducting circuits. As it will be discussed in this thesis, elements critical to scalable optical quantum network and superconducting quantum circuit can be constructed relying on the cavity optomechanics and piezoelectric interactions. Optomechanical interaction is concerned with the light pressure coupling of cavity mechanical deformation to a strong optical field. This interaction has allowed the close to mechanical ground state cooling of mechanical resonators using laser and the ultra-sensitive displacement measurement that led to the detection of gravitational waves in the LIGO collaboration [125, 25]. Optomechanical crystals (OMCs) are lithographically patterned devices which contain a periodic structure that host bandgaps for both optical band electromagnetic waves and microwave band acoustic waves. A properly engineered defect in the crystal can confine and localize acoustic and electromagnetic modes of similar wavelengths into a small mode volume [17, 20, 21]. A strong optomechanical coupling, which can be achieved between such strongly confined co-localized optical and acoustic modes, can be used in engineering the quantum state of mechanical motion to realize useful quantum devices such as a high-coherence quantum memory [74] and an optomechanical high efficiency optical isolator for unidirectionally connecting distant optical cavities via an acoustic bus [37]. To strongly couple the mechanical degree of freedom with a superconducting quantum circuit, various methods can be used, ranging from electromechanic coupling (electric coupling to a mechanically compliant capacitor), magnetomechanical coupling (magnetic coupling to a vibrating SQUID loop), and piezoelectric coupling. The recent advent of quantum acoustics [23, 8, 9] was realized with the strong piezoelectric coupling between a superconducting transmon qubit and a high-coherence mechanical resonator. The engineered strong piezoacoustic coupling provides the possibility to carry out deterministic ultra-high fidelity two-qubit quantum gates on non-classical mechanical quantum states [52]. This ability together with the recent demonstration of ultra-long phonon lifetime mechanical resonators show the possibility of integrating the ultra-high quality mechanical resonator as a compact quantum memory element and even a new ultra-compact (10s um) quantum bit architecture for scalable superconducting quantum circuits. Furthermore, the strong piezoelectric coupling that can transduce quantum state in a superconducting circuit into mechanical wave also makes it possible to efficiently transduce a quantum state between a superconducting quantum circuit and a telecommunication band optical channel via a mechanical waveguide connected to an optomechanical crystal cavity.
Acoustic analogues of black holes (dumb holes) are generated when a supersonic fluid flow entrains sound waves and forms a trapped region from which sound cannot escape. The surface of no return, the acoustic horizon, is qualitatively very similar to the event horizon of a general relativity black hole. In particular Hawking radiation (a thermal bath of phonons with temperature proportional to the "surface gravity") is expected to occur. In this note we consider quasi-one-dimensional supersonic flow of a Bose–Einstein condensate (BEC) in a Laval nozzle (converging-diverging nozzle), with a view to finding which experimental settings could magnify this effect and provide an observable signal. We discuss constraints and problems for our model and identify the issues that should be addressed in the near future in order to set up an experiment. In particular we identify an experimentally plausible configuration with a Hawking temperature of order 70 n K; to be contrasted with a condensation temperature of the order of 90 n K.
The monogamy of quantum entanglement, applied by Almheiri-Marolf-Polchinski-Sully (AMPS) to black holes, obstructs a smooth horizon vacuum after the Page time. We transcribe this argument to Hawking-like phonon radiation from a sonic horizon in the Unruh acoustic metric. An exact purity identity shows that post-Page-time unitarity forces the entanglement between an outgoing phonon and its interior partner to vanish, selecting a non-Hadamard (Boulware-like) phonon state, which we define as an acoustic firewall. Its renormalized stress tensor differs from the smooth state by a constant, negative near-horizon flux, and the thermal-atmosphere energy density it removes, measured by a static calorimeter, grows as (δr)−2(δr)^{-2}(δr)−2 in the radial coordinate toward the horizon (singular in the free-fall frame), cut off at the healing length. The construction is kinematic and does not resolve the information paradox; it yields one concrete, falsifiable prediction: a differential phonon-calorimetry signal R(δr)=∣ΔE∣/E(0)→(ℓκ/δr)2\mathcal{R}(δr)=|Δ\mathcal{E}|/\mathcal{E}^{(0)}\to(\ell_κ/δr)^{2}R(δr)=∣ΔE∣/E(0)→(ℓκ​/δr)2, present only after the analogue Page time in a Bose-Einstein condensate.
This Tech–Science text of article dedicated to the QUANTUM ACOUSTIC Sensu Stricto says about phonons, the sound quazi–particles elements, which are very interesting in nano, micro, makro scales of nature. This New Concept of QUANTUM ACOUSTIC PHYSICS OR QUANTUM PHYSICS OF SOUND IS ILLUSTRATED ON EMPIRIO–CRITICISM PHILOSOPHY OF EXPERIMENTAL PHYSICS OF MORAVIAN BORN PROFESSOR ERNST MACH, the text operated with subjects like MACH’S NUMBER, MACH’S CONE, MACH’S PRINCIPLE, MACH’S WAVEMACHINE. The Highlights is focused on Author’s Sketches in problematic connected with AERODYNAMICS AND AERONAUTICS.