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Chemical potential and variable number of particles control the quantum state: Quantum oscillators as a showcase

Benedikt M. Reible, Ana Djurdjevac, Luigi Delle Site

TL;DR

The paper develops an open quantum-system framework with a variable particle number, showing that the effective Hamiltonian $H_n^\mathrm{eff} = H_n - \mu \mathcal{N}$ imposes a mandatory condition on the energy spectrum, governed by the chemical potential $\mu$. By applying this to ensembles of quantum harmonic oscillators, translational-vibrational gases, and a chain of coupled oscillators, it derives explicit positivity conditions and $n$-dependent spectral restrictions for bosons and fermions, revealing how $\mu$ can be used to engineer accessible quantum states. The approach extends familiar results from the ideal Bose gas to more complex vibrational and transport settings, highlighting practical routes for spectral control in quantum technologies and potentially informing studies of entanglement and heat/mass transport in molecular systems. Overall, the work provides a rigorous basis for externally manipulating system spectra via chemical potential in open quantum systems with variable particle number, with broad implications for quantum devices and materials science.

Abstract

Despite their simplicity, quantum harmonic oscillators are ubiquitous in the modeling of physical systems. They are able to capture universal properties that serve as reference for the more complex systems found in nature. In this spirit, we apply a model of a Hamiltonian for open quantum systems in equilibrium with a particle reservoir to ensembles of quantum oscillators. By treating (i) a dilute gas of vibrating particles and (ii) a chain of coupled oscillators as showcases, we demonstrate that the property of varying number of particles leads to a mandatory condition on the energy of the system. In particular, the chemical potential plays the role of a parameter of control that can externally manipulate the spectrum of a system and the corresponding accessible quantum states.

Chemical potential and variable number of particles control the quantum state: Quantum oscillators as a showcase

TL;DR

The paper develops an open quantum-system framework with a variable particle number, showing that the effective Hamiltonian imposes a mandatory condition on the energy spectrum, governed by the chemical potential . By applying this to ensembles of quantum harmonic oscillators, translational-vibrational gases, and a chain of coupled oscillators, it derives explicit positivity conditions and -dependent spectral restrictions for bosons and fermions, revealing how can be used to engineer accessible quantum states. The approach extends familiar results from the ideal Bose gas to more complex vibrational and transport settings, highlighting practical routes for spectral control in quantum technologies and potentially informing studies of entanglement and heat/mass transport in molecular systems. Overall, the work provides a rigorous basis for externally manipulating system spectra via chemical potential in open quantum systems with variable particle number, with broad implications for quantum devices and materials science.

Abstract

Despite their simplicity, quantum harmonic oscillators are ubiquitous in the modeling of physical systems. They are able to capture universal properties that serve as reference for the more complex systems found in nature. In this spirit, we apply a model of a Hamiltonian for open quantum systems in equilibrium with a particle reservoir to ensembles of quantum oscillators. By treating (i) a dilute gas of vibrating particles and (ii) a chain of coupled oscillators as showcases, we demonstrate that the property of varying number of particles leads to a mandatory condition on the energy of the system. In particular, the chemical potential plays the role of a parameter of control that can externally manipulate the spectrum of a system and the corresponding accessible quantum states.
Paper Structure (14 sections, 3 theorems, 68 equations)

This paper contains 14 sections, 3 theorems, 68 equations.

Table of Contents

  1. introduction
  2. Effective Hamiltonian for an open quantum system
  3. Quantum harmonic oscillators
  4. Single harmonic oscillator
  5. Extension to many oscillators
  6. Second quantization formalism
  7. Ensembles of quantum harmonic oscillators in a particle reservoir
  8. Condition on the purely vibrational energy spectrum
  9. Interpretation of the condition for fermions
  10. Ideal quantum gas of vibrating particles
  11. Linear chain of coupled harmonic oscillators
  12. Discussion and Conclusion
  13. The ideal Bose gas
  14. Proof of convergence of the effective energy

Key Result

Lemma B.2

The following double series converges: $\sum_{k \in \mathbb{Z}} \sum_{q \in \mathbb{N}_0} \left[\varepsilon_k + \hbar \omega \left(q + \frac{1}{2}\right)\right] n_{k, q}$.

Theorems & Definitions (7)

  • Remark B.1
  • Lemma B.2
  • proof
  • Lemma B.3
  • proof
  • Corollary B.4
  • proof