The cavity can be driven by optical or microwave radiation depending on the size of the system. This includes some of the following important studies on light squeezing [6,7], non-destructive quantum detection of light intensity [8] and mechanical oscillator feedback cooling [9].
Cavity optomechanics: Basic theory
The optomechanical Hamiltonian
HOM =ωc(q)a†a+ωmb†b, (1.2) is the cavity field annihilation operator and the mechanical mirror annihilation bis operator. is the optomechanical coupling strength to a photon. Then, to get rid of the obvious time dependence of the Hamiltonian of the motion, we switch to a frame rotating with the laser frequency ωl and write the full Hamiltonian (1.5) in the following way:
Optomechanical equations of motion
Outline of the thesis
Also, in this work we are particularly interested in the following quantum correlations: quantum entanglement, quantum disagreement and quantum synchronizations. In the context of continuously variable (CV) quantum information, Gaussian states are critical.
Two-mode Gaussian states
- Entanglement in two-mode Gaussian states
- Gaussian quantum discord
- Brief overview
- System dynamics
- Optomechanical entanglement
- Quantum mutual information and Gaussian quantum discord . 23
On the other hand, the dynamics of the quantum fluctuations is linearized in the limit |as| 1 and written as:. We note that, unlike the standard optomechanical system [22], the stability of the system now depends on the strength of the mirror-qubit coupling, and one must have η < ωm. 3.3(b) shows the same logarithmic negativity EN as a function of the dimensionless effective optomechanical coupling G/ωm, for different mirror-qubit coupling strengths.
Thus, the bipartite entanglement in the presence of the mirror-qubit coupling becomes more robust against thermal phonon fluctuations. 3.5(a) and 3.5(b), we draw the quantum mutual information IM, respectively. 2.16) and the Gaussian quantum mismatch DG (2.18) as a function of the normalized detuning ∆/ωm, for different mirror-qubit coupling strengths.
Enhancing quantum correlations in an optomechanical system via
- Brief overview
- System and steady-state behavior
- Optomechanical entanglement
- Gaussian quantum discord
- Conclusion
3.7(a), we plot the steady-state amplitudes of the optical and mechanical modes together with the stability region of the system (with gck = 10−3g0), as a function of the driving force. 3.8 we plot EN as a function of the normalized cross-Kerr coupling strength gck/g0, for a fixed driving force and optical tuning. Kerr coupling on a generic optomechanical system, we can significantly improve the degree of steady-state optomechanical entanglement.
Here too we find the improvement of the steady-state optomechanism with increasing cross-Kerr coupling strength. Furthermore, we have shown that in the presence of the cross-Kerr nonlinearity the generated entanglement becomes more robust against the.
Model and dynamics
4.1), the first and second terms correspond to the Hamiltonian of the driven cavity and the mechanical oscillators, respectively, where ∆0 = ωc − ωl is the optical detuning. The third term describes the optomechanical interaction between the cavity field and the mechanical oscillators, while the fourth term refers to the bilinear coupling between the two oscillators. Moreover, the system dynamics are inevitably subjected to the fluctuation-dissipation processes that affect both the cavity field and the mechanical oscillators.
The equation of motion corresponding to classical mean values is given by the following set of nonlinear differential equations:. Due to the above linearized dynamics and the Gaussian nature of zero-mean quantum noises, the quantum fluctuations in the stable regime evolve into an asymptotic Gaussian state that is fully characterized by its 6×6 correlation matrix given by: .
Results and discussions
The dependence of the stationary mechanical entanglement on the oscillator temperature is displayed in Fig. On the contrary, the “−” mode exhibits both the localization and delocalization phenomena depending on the strength of the mechanical coupling. Therefore, we can conclude that the asymptotic nature of the mechanical entanglement is directly related to the instability in the "−" mode.
We can see that the points corresponding to the above-mentioned coupling strengths are in the stable, on the border and in the unstable zone of the “-” mode, respectively. This well justifies our previously obtained entanglement dynamics, which correspond to the different sets of mechanical coupling strengths.
![Figure 4.2: Entanglement dynamics of the two coupled mechanical oscillators for λ 0 /ω m = [0, 0.005] (from bottom to top) at T = 0](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10556793.0/69.892.185.732.168.433/figure-entanglement-dynamics-coupled-mechanical-oscillators-λ-005.webp)
Conclusion
We consider binary and ternary mechanical PT symmetric architectures, realized over an optomechanical platform. We show that a significant delay in ESD can be achieved in the PT symmetric binary system by pushing the system towards an extraordinary point. While the search for PT symmetric devices is ongoing, it appears that one can easily implement such notions by carefully assigning gain and loss to an optical system.
Therefore, any PT symmetric device whose dynamics is governed by an intrinsic quantum mechanical equation of motion will provide a better insight into this theory. These quantum PT symmetric devices facilitate us to explore many intrinsic quantum properties, such as critical phenomena [171], entanglement [172], chiral population transfer decoherence dynamics [175], and information recovery and criticality [176].
Cavity optomechanics based architecture to realize the gain (loss) in
On the other hand, the dynamics of the quantum fluctuations are given by the linearized QLEs (valid in the. To do this, we first assume that the cavity is resonant with the Stokes sideband of the driving laser, ∆ = −ωm, and call the rotating wave approximation (RWA) (which is justified in the limit of ωm {G, κ, γ}) to obtain: Here one must note the inclusion of the following two terms: Γ = 4Gκ2 which quantifies the quantity. of optomechanically induced gain in the mechanical resonator, and, the cavity induced noise term√.
On the other hand, when the cavity decouples with the Anti-Stoke sideband, i.e., ∆ =ωm, one ends up with an interaction of the form, HI = -G a†b+ab†. Then, after an elimination of the cavity field, one eventually obtains a transition from|ni to|n−1i, with the same damping rate Γ.
Entanglement in PT symmetric binary systems
Here, following RWA, one can easily conclude that the interaction Hamiltonian is of the form, HI = -G a†b†+ab. In the following, we find that we can have such solutions only when the system remains in the symmetric PT phase. We further note that due to the above linearized dynamics and the zero-mean Gaussian nature of the quantum noises, the system retains its Gaussian characteristics.
5.3(a) we first show the time evolution of the quantum entanglement (2.14) in the absence of any noise. It can be seen that when the system is in the PT-symmetric phase, the entanglement oscillates periodically.
Entanglement in PT symmetric ternary systems
Here b1, b2, b3 (b†1, b†2, b†3) refer to the destruction (creation) operators of the gain, neutral and lossy resonators respectively, while the other parameters remain unchanged with the previous descriptions. It is observed that, with the inclusion of quantum noise, the tripartite state quickly suffers a sudden death of (true tripartite) entanglement, followed by a fully tripartite, inseparable state. However, it is noteworthy to note that at EP3 the entanglement of such three-party state losses proceeds more slowly than at any other point in the continuous PT-symmetric phase.
However, it is worth noting that the available bipartite (tripartite) entanglement achieved in the immediate vicinity of the EPs (EP3) is quite robust. It may also be useful to note that although we assume that the frequencies of both mechanical oscillators and their coupling rates with the cavity field are exactly the same, we find that small deviations from these conditions do not change the overall conclusion of the study. the results obtained.
Conclusion
Spontaneous synchronization, in particular with the quantum domain, has been found to have a very strong connection with primordial quantum correlations. Only recently, synchronization of optomechanical arrays has been achieved, using seven such micro-disc oscillators sharing a common optical field. However, it should be noted that measuring quantum synchronization is still quite challenging and has been provided in Refs.
Moreover, numerous attempts have been made to link the onset of quantum synchronization and the generation of quantum correlations. Notably, mutual information as a pure information-theoretic measure of quantum synchronization was proposed in Ref.
Model and dynamics
Such an interaction could be simulated by a Hamiltonian of the form Hc =−λ(a†1a2+a1a†2), where λ is any arbitrary coupling strength. Taking into account the dissipative effects, we can write the following quantum Langevin equations (in a frame rotating atωl):. where ∆0j is the optical input tuning, κj and γj are the optical and mechanical damping rates, respectively, and ainj and binj are the corresponding input bath operators. 1i−1 defines the average thermal occupation number at a temperature T. We now expand these operators O(t) as sums of classical expectation values hO(t)i plus quantum fluctuation operators δO(t), i.e. we write O( t) = hO(t)i+ 8O(t).
Since these solutions also obtain large coherent amplitudes, we can safely linearize the quantum Langevin equations for the fluctuation operators δO(t). We then have the following set of linearized quantum Langevin equations for the two-way configuration.
Quantum synchronization measure
Results and discussions
This suggests a possible connection between the onset of quantum synchronization and the generation of quantum correlation. Now, following these observations, a natural question that arises is whether the Gaussian quantum discordance is a sufficient order parameter to map the essential properties of quantum (phase) synchronization. 6.3(b)), as a function of the normalized coupling strength λ/κ and the frequency detuning between these two mechanical oscillators δ/ωmL.
It is observed that when the two optomechanical cells interact in a bidirectional manner, both synchronization and quantum discord exhibit a tongue-like pattern, which is the quantum analogue of the classical "Arnold tongue". 6.4(a)), we find that the degree of quantum synchronization hSpi is not maximal at the resonant state, rather it peaks around finite frequency tunings, depending on the strength of the transmission.
Conclusion
We found that in the presence of mirror-qubit coupling, the system becomes unstable with a lower optomechanical coupling strength, which leads to a significant increase of the steady-state entanglement, as well as the quantum Gaussian discordance, near the instability threshold . . We found that depending on the strength of the mechanical coupling, either stationary or dynamic mechanical entanglement behavior can be observed, which is also remarkably robust to oscillator temperature. Moreover, we have shown that this entanglement dynamics is strongly related to the stability of normal modes.
Here we provide a brief discussion on the generation of the additional second-order term, introduced in the system Hamiltonian. Here µq is the interaction strength between the mechanical mirror and auxiliary qubit, and q be the dimensionless position operator of the mechanical mirror.
Matheu’s Equation
