Abstract
Symmetry plays the central role in the structure of quantum states of bipartite (or many-body) fermionic systems. Typically, symmetry leads to the phenomenon of quantum coherence and correlations (entanglement) inherent to quantum systems only. In the present work, we study the role of symmetry (i.e., quantum correlations) in invasive quantum measurements. We consider the influence of a direct or indirect measurement process on a composite quantum system. We derive explicit analytical expressions for the case of two quantum spins positioned on both sides of the quantum cantilever. The spins are coupled indirectly to each others via their interaction with a magnetic tip deposited on the cantilever. Two types of quantum witnesses can be considered, which quantify the invasiveness of a measurement on the systems’ quantum states: (i) A local quantum witness stands for the consequence on the quantum spin states of a measurement done on the cantilever, meaning we first perform a measurement on the cantilever, and subsequently a measurement on a spin. (ii) The non-local quantum witness signifies the response of one spin if a measurement is done on the other spin. In both cases the disturbance must involve the cantilever. However, in the first case, the spin-cantilever interaction is linear in the coupling constant Ω , where as in the second case, the spin-spin interaction is quadratic in Ω . For both cases, we find and discuss analytical results for the witness.
Highlights
Since the foundational development of quantum mechanics, the measurement process and the wave function collapse have been under discussion [1–15]
The modulation of the witness strength amplitude is inherently a quantum effect related to the entanglement between the cavity field and NV centers
For two effective quantum spins coupled to each other indirectly via the quantum oscillations of a cantilever, we derived an explicit expression for the local quantum witness W L that measures the invasiveness of a measurement done on the cantilever on the quantum states and on a subsequent measurement done on spins
Summary
Since the foundational development of quantum mechanics, the measurement process and the wave function collapse have been under discussion [1–15]. We consider two nitrogen vacancy (NV) centers described by the density operator ≡ ρ AB interacting via quantum modes of a nanomechanical oscillator f The paradigmatic model of NAMR consists of the spin of the NV center interacting with a magnetic tip. Centers A and B interact through a magnetic tip positioned at the end of nanomechanical oscillators (cantilever). Ω is the coupling constant of the NV spin with the magnetic tip, ↠, â are the phononic creation and annihilation operators describing the cantilever oscillations with the frequency ωf , and ω0 is the frequency of the NV spin. We consider the analytical solution in the resonant and highly off-resonant cases These two situations are distinguished by the detuning of the cantilever’s oscillation with regard to the frequency of the NV spins.
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