4. The Stationary Entanglement of the LG-Cavity Field with the Rotating End Mirror in the Presence of the Nonlinear CK Interaction and the OPA
In the following, we analyze the impacts of the cavity detuning , the parametric gain G and phase of the OPA, the strength of the nonlinear CK interaction, the Gaussian laser power ℘, the topological charge l of the cavity field, the mass m of the rotating end mirror, and the ambient temperature T on the stationary entanglement of the cavity field with the rotating end mirror.
The parameters we use in the numerical calculations are similar to those in Ref. [
11], which studies the entanglement of an intracavity LG field with a rotating end mirror: the wavelength of the incident Gaussian laser is
nm, the cavity length is
mm, the cavity field decays at a rate of
MHz, the rotating end mirror has the radius
m, the resonance frequency
MHz, the mechanical quality factor
, and the damping rate
Hz. Thus, the studied system is operating in the resolved-sideband limit
.
Figure 2 plots the logarithmic negativity
against the normalized cavity detuning
for different gains
G of the OPA when
ng,
mW,
,
,
g, and
K. When the driving phase
of the OPA is 0 (
Figure 2a), the system stability conditions require
for
,
,
, respectively, and the corresponding maximum logarithmic negativity
values just before the unstable regime are 0.250, 0.266, 0.297, respectively. When the driving phase
of the OPA is set to 0.2
(
Figure 2b), the system stability conditions for
,
,
restrict the normalized cavity detuning to the range of
, respectively, the corresponding maximum logarithmic negativity
just before the unstable regime is 0.252, 0.316, 0.237, respectively. In the absence of the OPA (
), the system stability conditions require
, and we observe that the logarithmic negativity
increases from 0 to a peak value of about 0.05 and then gradually decreases to 0 as the cavity detuning
increases. Thus, the maximum entanglement in the presence of the OPA far exceeds that in the absence of the OPA (
), indicating that the nonlinear gain
G of the OPA significantly enhances the steady-state entanglement. For a fixed phase
of the OPA, as the gain
G of the OPA increases, the regime where the entanglement occurs shifts towards larger cavity detunings
, and the amount of entanglement decreases with increasing cavity detuning
. In
Figure 2b, when
, it is noted that the maximum entanglement does not occur at the highest gain
G of the OPA. This shows the influence of the driving phase
of the OPA on the entanglement, which is discussed further in the following.
Figure 3 plots the logarithmic negativity
against the normalized gain
of the OPA for different strengths
of the nonlinear CK interaction when
ng,
mW,
,
,
, and
K. When
(
Figure 3a), for
,
,
,
, when the gain
G of the OPA satisfies
0.156, 0.132, 0.108, 0.085, respectively, the system remains stable. For
,
,
,
, as the gain
G of the OPA increases, the logarithmic negativity
increases and reaches the maximum values of about 0.263, 0.302, 0.293, and 0.300, respectively, just before entering the unstable regime. In
Figure 3b (
, for
,
,
,
, the stability condition imposes the limitation
0.206, 0.181, 0.155, 0.129, respectively. The maximum logarithmic negativity
values before approaching the unstable regime are about 0.273, 0.305, 0.279, and 0.292, respectively. This demonstrates that the gain
G of the OPA can effectively enhance the steady-state entanglement between the cavity field and the rotating end mirror. This is due to the fact that increasing the gain
G of the OPA leads to a larger photon number
in the cavity field and a larger phonon number
in the rotating end mirror, as shown in
Figure 4, which plots the number
of the photons in the cavity field and the number
of the phonons in the rotating end mirror at the steady state against the normalized gain
of the OPA when
ng,
mW,
,
,
, and
. A larger photon number
in the cavity field increases the coupling between the cavity field and the rotating end mirror. Meanwhile, a larger photon number
in the cavity field and a larger phonon number
in the rotating end mirror give rise to a stronger CK coupling between the cavity field and the rotating end mirror. Moreover, in
Figure 3, for fixed values of
and
G, the larger the CK coupling strength
, the larger the entanglement, but, for a fixed value of
, with increasing the CK coupling strength
, the stable regime becomes narrower. Comparing
Figure 3a with
Figure 3b, it is evident that changing the driving phase
of the OPA significantly alters the regime of steady-state entanglement.
Figure 5 plots the logarithmic negativity
against the phase
of the OPA for some different gains
G of the OPA when
ng,
mW,
,
,
, and
K. Without the OPA (
), for
, when the phase
of the OPA is changed from 0 to
, the values of the logarithmic negativity
remain unchanged, which are about 0.011, 0.019, respectively. Next we look at the case with the OPA. When
(
Figure 5a), for
, the system is stable when the phase
of the OPA satisfies
,
, and
, respectively. When
(
Figure 5b), for
, the system is stable when the phase
of the OPA satisfies
,
, and
, respectively. Thus, for a fixed strength
of the nonlinear CK interaction, increasing the gain
G of the OPA makes the system stable in a narrower range of phases
. And when the gain
G of the OPA is fixed, for
, the system is stable in a narrower range of phases
compared to that for
. When
(
Figure 5a), for
, the logarithmic negativity
takes its maximum values of about 0.320, 0.254, and 0.171 at
, respectively. When
(
Figure 5b), for
, the logarithmic negativity
takes its maximum value of about 0.274, 0.190, and 0.249 at
, respectively. Hence, for a fixed value of the CK coupling strength
, it is noted that the presence of the OPA can enhance the entanglement since the logarithmic negativity
in the presence of the OPA can be larger than that in the absence of the OPA, but the maximum entanglement does not increase with an increasing gain
G of the OPA, and the maximum entanglement happens at a phase
of the OPA at which the system is close to the unstable regime.
Figure 6 depicts the logarithmic negativity
versus the normalized strength
of the nonlinear CK interaction for different gains
G of the OPA when
ng,
,
,
mW,
,
, and
K. When
(
Figure 6a), for different OPA nonlinear gains
, the system stability requires
1.48, 1.03, 0.63, 0.312. As the strength
of the nonlinear CK interaction increases, the logarithmic negativity
for different
G values rapidly rises from 0, reaching a maximum value of about 0.299, 0.265, 0.213, 0.104, respectively, just before the system approaches instability. Similarly, in
Figure 6b (
), for
,
,
, the system stability conditions require
, 1.20, 0.82, 0.44, respectively. As the strength
of the nonlinear CK interaction increases, the maximum logarithmic negativity
values before approaching instability are about 0.299, 0.270, 0.235, 0.106, respectively. Thus, for a fixed phase
of the OPA, with an increasing gain
G of the OPA, the instability is triggered at a weaker strength
of the nonlinear CK interaction. For a given nonzero gain
G of the OPA, it is noted that the instability for
is triggered at a stronger strength
of the nonlinear CK interaction compared to the case for
. For a fixed phase
and gain
G of the OPA, it is possible to improve the entanglement by increasing the strength
of the nonlinear CK interaction. Additionally, as the gain
G of the OPA increases, the entanglement appears at a weaker strength
of the nonlinear CK interaction and reaches the maximum value at a weaker strength
of the nonlinear CK interaction, but the maximum entanglement decreases.
Figure 7 plots the logarithmic negativity
against the laser power
℘ for different gains
G of the OPA when
ng,
,
,
,
, and
K. When
(
Figure 7a), for
,
,
,
, the stability conditions require
1.42, 1.03, 0.68, 0.39 mW, and the corresponding driving power thresholds for generating entanglement are 0.36, 0.76, 0.59, 0.37 mW, the maximum logarithmic negativity
values near the unstable regime are 0.326, 0.298, 0.236, and 0.098, respectively. Similarly, when
(
Figure 7b), the stability conditions require
1.42, 1.17, 0.85, 0.52 mW, respectively, and the corresponding driving power thresholds for generating entanglement are 0.36, 0.87, 0.73, 0.48 mW, respectively; then, the maximum logarithmic negativity
values near the unstable regime are about 0.326, 0.259, 0.293, 0.168, respectively. For a fixed phase
of the OPA, from Equation (
4), it is noted that increasing the laser power
℘ can enhance the photon number
within the cavity; thus, it is possible to improve the steady-state entanglement by increasing the laser power
℘. Additionally, as the gain
G of the OPA increases, the entire entanglement regime shifts towards lower laser powers, with the threshold power appearing at lower values. This indicates that it is possible to achieve stronger steady-state entanglement with lower laser powers.
Figure 8 plots the logarithmic negativity
against the topological charge
l of the cavity field for different gains
G of the OPA when
ng,
,
mW,
,
, and
K. When
(
Figure 8a), for
, the entanglement occurs only when the topological charge
l is not less than the threshold values
, 12, 12, 11, respectively, the stable regime is found to be
20, 17, 15, 13, and the maximum logarithmic negativity
close to the unstable regime is about 0.189, 0.122, 0.170, 0.292, respectively. When
(
Figure 8b), for
,
,
,
, the entanglement appears only when the topological charge
l is not less than the threshold value
, 13, 14, 13, respectively, and the stable regime is found to be
, 19, 17, 15, respectively, the maximum logarithmic negativity
close to the unstable regime is about 0.188, 0.261, 0.195, 0.196, respectively. Notably, when the topological charge
l is less than the threshold value
, the optorotational coupling is not strong enough to generate the entanglement. In addition, the exchange of orbital angular momentum between photons and helical phase elements is fundamental for generating entanglement between the cavity and mechanical modes. Thus, it is possible that an increase in the topological charge
l facilitates the enhancement of entanglement. It is found that a higher gain
G of the OPA can result in the maximum entanglement occurring at a smaller topological charge
l, with a more rapid increase in entanglement and a narrowing stable regime. For a fixed nonzero gain
G of the OPA, the system for
becomes unstable at a larger topological charge
l compared to the case for
.
Figure 9 shows the logarithmic negativity
against the mass
m of the rotating end mirror for different gains
G of the OPA when
mW,
,
,
,
, and
K. When
(
Figure 9a), for
,
,
,
, the system’s stability conditions require
, 72, 96, 134 ng, respectively, and the logarithmic negativity
reaches the maximum values of about 0.281, 0.222, 0.251, 0.258 near the unstable regime, respectively. When
(
Figure 9b), for
,
,
,
, the system’s stability conditions require
, 62, 76, 98 ng, respectively, and the logarithmic negativity
reaches the maximum values of about 0.281, 0.281, 0.251, 0.245 near the unstable regime, respectively. For fixed values of
and
G, when the mass
m of the rotating end mirror increases, the entanglement decreases. When the phase
of the OPA is fixed, for a larger gain
G of the OPA, the system becomes stable at a larger mass
m of the rotating end mirror. For a fixed nonzero gain
G of the OPA, the system for
becomes stable at a smaller mass
m of the rotating end mirror in comparison with the case for
. When the phase
of the OPA is fixed, a larger gain
G of the OPA can generate the entanglement between the optical mode and a rotating end mirror with a larger mass
m. Even at around 150 ng (
Figure 9a), the entanglement can still be achieved, providing valuable insights for experimental realization.
Figure 10 plots the logarithmic negativity
against the ambient temperature
T for different gains
G of the OPA when
ng,
,
mW,
,
, and
. For
, the system is stable when
G is less than 0.109
, 0.156
, respectively. Thus, the system for
becomes unstable at a larger gain
G of the OPA in contrast to the case for
. When
(
Figure 10a), for
, the logarithmic negativity
takes its maximum values of about 0.012, 0.110, 0.172, and 0.295 at
K, respectively, and the logarithmic negativity
is zero when the ambient temperature
T is not less than 1.8 K, 8.3 K, 13.8 K, and 27.4 K, respectively. When
(
Figure 10b), for
, the logarithmic negativity
takes its maximum value of about 0.012, 0.034, 0.198, and 0.281 at
K, respectively, and the logarithmic negativity
is zero when the ambient temperature
T is not less than 1.8 K, 1.9 K, 13.1 K, and 21.8 K, respectively. When the phase
and gain
G of the OPA are fixed, as the ambient temperature
T increases, the entanglement decreases and eventually vanishes, which can be attributed to the detrimental effect of the thermal noise of the environment on the entanglement. For a fixed phase
of the OPA, an increase in the gain
G of the OPA results in a larger entanglement and allows the entanglement to persist at higher temperatures. This indicates that the gain
G of the OPA not only enhances the steady-state entanglement, but also improves the system’s ability to withstand decoherence in a thermal environment.
In our previous work [
20], it has been shown that the entanglement between the cavity field and the rotating end mirror with mass 100 ng can be improved by using the nonlinear CK interaction, and the entanglement
can be obtained with the nonlinear CK interaction
, which is much larger than the entanglement
(
Figure 5a) between the cavity field and the rotating end mirror with mass 100 ng in the presence of the nonlinear CK interaction
but in the absence of the OPA in this work. The reason is that a large topological charge
of the cavity field is used in Ref. [
20] and a small topological charge
of the cavity field are used in this work. Furthermore, it has been shown in this work that a larger entanglement
(
Figure 5a) between the cavity field and the rotating end mirror can be obtained under the combined action of the nonlinear CK interaction
and the OPA
.
Finally, we discuss the possibility of the experimental feasibility of this proposal. The optorotational entanglement can be verified by measuring the logarithmic negativity
, which requires one to measure the 10 independent elements of the correlation matrix
V [
27]. The quadratures of the LG-cavity mode can be measured by homodyne detecting the cavity output field [
27]. And the quadratures of the rotating end mirror can be measured by homodyning the output field from the adjacent cavity formed by the rotating end mirror and a third fixed spiral phase element [
27]. On the other hand, the LG-cavity optorotational system has not been realized experimentally yet [
12]. However, it has been demonstrated experimentally that an LG laser beam with a topological charge of
can be achieved by using spiral-phase mirrors [
30], that the torsional frequency of a nanomechanical resonator can be up to 8.4 MHz [
31], and that a micromechanical resonator with mass 25 ng, radius 15
m, and mechanical quality factor
can be cooled to 135 mK [
32]. Thus, with the rapid development of fabrication technologies, it is possible to experimentally realize the optorotational entanglement in the LG-cavity optorotational system with the rotating mirror having mass 100 ng, radius 10
m, frequency
MHz, and mechanical quality factor
at temperature
K.