1. IntroductionSome quantum interference phenomena in the cavity optomechanical system (OMS), such as the optomechanically induced transparency (OMIT)[1,2] and the optomechanically induced absorption (OMIA),[3] have attracted a great deal of attention. These effects promote the developments of the subluminal (superluminal) propagation, optical storage, optical switches, and so on.[4–7] On the other hand, the appearance of the normal-mode splitting (NMS) as an unambiguous evidence for the strong coupling between quantum systems has been investigated. For instance, the optomechanical NMS for recognizing the strong coupling of a cavity mode to a mechanical mode was observed.[8] The optomechanical systems have developed from pure systems to hybrid systems,[9, 10, 11] and some remarkable phenomena were found in the hybrid optomechanical systems. For example, a new way for detecting entanglement was presented.[12] A tunable multi-channel inverse OMIT was investigated in the optomechanical system assisted with the Coulomb interaction.[13] A nonlinear optomechanical system with two movable mirrors, in which EIT can be controlled and manipulated, was theoretically studied.[14]
With the hybrid optomechanial systems, it is worth noting that a crowd of novel optical phenomena were found under the assistance of an atomic ensemble. For example, the bistable phenomenon in a Fabry–Perot (FP) cavity, assisted by a cold two-level atomic ensemble, was observed.[15] The properties of multipartite entanglement, the Einstein–Podolshy–Rosen (EPR) state, and the mechanical vibrator cooled to the ground state were demonstrated in the hybrid system containing an atomic ensemble.[16–18] It was proven that the radiation pressure between the cavity field and the movable mirror can be enhanced by adding atomic ensembles.[19] It was also shown that a large number of atoms can broaden the window of OMIT.[20,21] The above studies indicate that the atomic ensemble plays an important role in the researches of optomechanics.
We also know from the previous researches that the two-mode cavity system has a more flexible controllability and becomes a research hot spot.[22–25] For example, Agarwal and Huang studied the inverse EIT (IEIT) and showed that its effect is equivalent to the perfect coherent absorption.[25] In a double-cavity optomechanical system, the perfect coherent absorption, transmission, and synthesis were achieved.[26]
In this paper, we propose a system for achieving some adjustable quantum coherence effects, including the NMS, the OMIT, and the OMIA. The system consists of two tunnel-coupled optomechanical cavities, in which there are two atomic ensembles. The two cavities are also driven by some external fields. When the system works under different conditions, we can obtain the NMS, the OMIT, and the OMIA, respectively. These effects can be flexibly adjusted by the power of the coupling lasers and the coupling strength between the atomic ensembles and the cavity fields. This is different from the system proposed in Ref. [27], in which the system just achieves these coherent processes but cannot adjust them flexibly. Comparing with the system in Ref. [26], the high quality factor of the oscillator is not required in our system.
The organization of this paper is as follows. In Section 2, we propose the model and solve the quantum Langevin equations. In Section 3, we study the NMS in this system without any coupling fields. In Section 4, we discuss how to adjust the phenomena of OMIT and OMIA in this system by inputting one coupling field. Section 5 presents a brief summary.
2. Model and general formalismThe system studied in this work is shown schematically in Fig. 1. The system consists of two single-cavity optomechanical systems and two two-level atomic ensembles with
atoms. Each single-cavity optomechanical system is constructed with one fixed end mirror and one partially transmissive oscillating mirror with the effective mass m, the frequency
, and the decay rate
. Two identical optical cavities of length L, frequency
, and decay rate
are coupled via the photon tunneling and are driven by two coupling fields with the frequencies of
and
, respectively. At the same time, one probe field with the frequency
is injected into the left optical cavity to detect the transparency properties of the overall system. We define
,
and
as the input powers of the relevant fields and
=
,
, and
as the real field amplitudes. The Hamiltonian of the system can be written as
Here the first four terms respectively describe the free Hamiltonian of the two cavities and the two oscillating mirrors, in which
and
are the annihilation (creation) operators of the cavity fields and the mechanical modes, respectively. The fifth and the sixth terms are the free Hamiltonian of the two atom ensembles.
,
, and
are the Pauli matrices for each atom.
is the number of atoms in the
i-th atomic ensemble. We assume that all the atoms have the same excited (ground) state
. As a result, they have the same transition frequency
. The seventh and the eighth terms are the interaction Hamiltonian showing the radiation pressure interaction between the cavities and the oscillating mirrors with optomechanical coupling rate
. For simplicity, we assume
. The ninth term expresses the interaction between the two cavities, where
is the tunnel coupling rate, which is proportional to the transmittance of the two oscillating mirrors in the two cavities. The tenth and the eleventh terms give the atom–field interaction Hamiltonian describing the coupling between each cavity and the atomic ensemble in the cavity. Here,
is the atom–field coupling rate, where
μ is the electric dipole between the two levels,
V describes the cavity mode volume, and
is the permittivity of the vacuum. We restrain each atomic ensemble in a thin layer whose size is much smaller than the wavelength of the cavity field in the cavity axial direction, so all the atoms have the same coupling strength
with the cavity field. The last three terms describe the interaction of the two coupling fields with the cavity fields and that of the probe field with the left cavity field. It is necessary to point out that there is no direct interaction between the atoms and the mirrors, the interaction between them indirectly relies on the cavity fields.
As all the atoms have the same transition frequency
, we can define the collective quasi-spin operators of the two atomic ensembles
and
to simplify the form of the Hamiltonian. In the large-
and low-excitation conditions, the collective quasi-spin operators satisfy the bosonic commutation relation[21]
. Then we have
and
, and the Hamiltonian can be simplified as
where
and
are the effective coupling strengths between the atomic ensembles and the cavity fields. We can see that the coupling coefficient
can be enhanced by increasing the atomic number
in each ensemble. In the interaction picture with respect to
, the Hamiltonian can be rewritten as
where
,
,
,
,
, and
are the detunings. Without loss of generality, we assume that the frequencies of the two coupling fields are the same, i.e.,
, then
,
, and
. By using Hamiltonian (
3) and these conditions, we obtain the following Heisenberg–Langevin equations:
in which we have included the noise and damping terms. Here
,
, and
are the thermal noise on the
i-th oscillating mirror, the input quantum vacuum noise operator in the
i-th cavity, and the quantum noise operator of the
i-th atomic ensemble, respectively, and have zero mean value, i.e.,
. Moreover, the input quantum vacuum noise operators of all cavities and atomic ensembles satisfy the nonvanishing correlation functions
and
.
[29] The thermal noise operators of the oscillating mirrors obey the temperature-dependent correlation function
[30]
where
is the Boltzmann constant, and
T is the temperature of the oscillating mirrors. Each operator can be expressed as the sum of its steady value and its small fluctuation, i.e.,
. By solving the equation of motion for the fluctuation
, we can understand the dynamical behaviors of the system.
First we can obtain the steady solutions of Eq. (4) as follows:
where
and
give the effective detunings between the cavity modes and the coupling fields. Since
are typically very small compared to
, we can linearize this problem, and the linearized quantum Langevin equation can be obtained as
Then we make transformations
,
,
, and assume
, i.e., the detuning between the cavity field and the coupling field is the same as that between the atomic ensemble and the coupling field. We drive the mechanical modes with red-detuned coupling fields, i.e.,
. Under these conditions, equation~(
6) can be simplified to
where
is the detuning of the probe field from the cavity resonance frequency. Now we begin to derive an expression for the transmission coefficient of the system. For this reason, we use the input–output relation
[29]
and make use of the ansatz
where
and
are the components of
oscillating at
and
in the original frame. From Eqs. (
8)–(
10), we have
By taking mean values in the set of equations (
7) and using Eq. (
9), we can obtain
By substituting Eq. (
12) into Eq. (
11), we obtain the expression for the transmission coefficient
There are three kinds of direct interactions in the system: the coupling between the two cavities, the interaction between the cavities and the oscillators, and the interaction of the cavities with the atomic ensembles. In the following, we will discuss the impacts of these three kinds of interactions on the output spectrum. In this paper, we choose realistic parameters in a recent experiment:
[17] the wavelength of the control fields
nm,
mm,
ng,
Hz,
Hz,
Hz,
Hz (the quality factor
). In addition, we have considered the situation in which the atoms are resonant with the cavity, i.e.,
, this is corresponding to
. This assumption has also been used by the well-known work.
[17]
3. Adjustable normal-mode splittingIn this section, we focus on the impact of the tunnel coupling between the two cavities on the output spectrum, i.e., we consider the situation in which
. In this case, the transmission coefficient (13) can be simplified as
Based on Eq. (
14), we plot the transmission coefficient
T as a function of the detuning
in Fig.
2(a). It can be seen that when the tunnel coupling between the two cavities is weak, there appears a single resonance around
in the probe output spectrum. If the tunnel coupling
is tuned into the strong regime, i.e.,
, the resonance splits into two, i.e., there appears the normal-mode splitting, and the separation between the two resonances is increased with the tunnel coupling
. From Eq. (
14), we can obtain the frequencies of the two resonance peaks as
and their separation
We plot the normalized separation
versus the normalized tunnel coupling
in Fig.
2(b). It can be seen that the separation of the split window is approximately linearly increased with increasing tunnel coupling when the tunnel coupling
is switched into the strong regime, i.e.,
. So we can control
to achieve and adjust the behavior of the NMS.
4. Adjustable optomechanically induced transparency and optomechanically induced absorptionIn what follows, we examine the nature of the output probe spectrum when we input the coupling field into the system. We find that the optomechanically induced transparency (OMIT) and the optomechanically induced absorption (OMIA) can be achieved in this system, and their windows can be adjusted by controlling the power of the coupling field and the interaction of the cavity field with the atomic ensemble. What is more, in the optomechanical system with a single cavity, the window of the OMIT is not perfect at a higher decay
(low quality factor) of the oscillating mirror. However, we will show that it is still perfect in our system even if a larger mechanical decay
exists. To see this clearly, we choose a large mechanical decay
HZ and a low quality factor
, which are based on Ref. [28], to complete the numerical simulation.
4.1. Adjustable OMIT in the case of
In this subsection, we study the transparency spectra in the situation of
, i.e., there is no right atomic ensemble and no right coupling field. The transparency spectra are plotted in Figs. 3(a) and 3(b), respectively. Figure 3(a) shows the transparency spectrum with the power
of the left coupling field as the parameter. It can be seen that there appears the normal-mode splitting when
and
(the coupling between the two cavities is in the strong coupling regime), this is in agreement with the discussion in Section 3. When
, i.e., when we input the left coupling field, there appears the optomechanically induced transparency, and the transparency window near
becomes higher and wider with increasing
. Figure 3(b) shows the transparency spectrum with the coupling strength
between the left atomic ensemble and the left cavity field as the parameter. We can see that the transparency window near
becomes higher and wider with increasing
. It should be pointed out that we can obtain a perfect transparency
when
is large enough. These features can be explained as follows.
According to the previous researches, the OMIT window can be adjusted via the effective optomechanical coupling strength
in the left cavity, which is equivalent to the Rabi frequency in the atomic electromagnetically induced transparency.[2,3] By using our previous assumption
in the present situation (
), we can obtain from Eq. (5) the effective optomechanical coupling strength between the left cavity and the mechanical mode, which is proportional to
, and
with
where
and
are the effective decay and the effective detuning, respectively. We can see from Eq. (
17) that the effective optomechanical coupling strength
is increased with increasing power
of the left coupling field (
). We also find that our result is the same as that in Ref. [
27] if there is no atomic ensemble, i.e., if
. Now because of the existence of the atomic ensemble, the effective decay
adds a term
and the effective detuning
reduces a term
, their overall effect is that the effective optomechanical coupling is increased when the system is in the sideband resolved limit, i.e.,
.
[25] This broadens and heightens the window of OMIT.
It should be pointed out that, in our system, even if the oscillating mirror has a relatively larger decay rate (
kHz), the perfect transparency window can also be obtained by reasonably controlling the power of the coupling field and the interaction strength between the left cavity and the atomic ensemble, as shown in Figs. 3(a) and 3(b). This is better than that in Ref. [27], in which the perfect transparency window only appears for a relatively smaller decay rate (
Hz).
4.2. Adjustable OMIA in the case of
=0In this subsection, we study the transparency spectra in the situation of
, i.e., there is no left atomic ensemble and no left coupling field. The transparency spectra are plotted in Figs. 3(c) and 3(d), respectively. Figure 3(c) shows the transparency spectrum with the power
of the right coupling field as the parameter. It can be seen that there appears the normal-mode splitting when
and
(the coupling between the two cavities is in the strong coupling regime), this is in agreement with the discussion in Section 3. When
, i.e., when we input the right coupling field, instead of a resonance peak near
as in Fig. 3(a), there appears a resonance valley, this is the so-called optomechanically induced absorption. The absorption window near
becomes deeper and wider with increasing
. Figure 3(d) shows the transparency spectrum with the coupling strength
between the right atomic ensemble and the right cavity field as the parameter. We can see that the absorption window near
becomes deeper and wider with increasing
. It should be pointed out that one can obtain a full absorption
when
is large enough. These features can be explained as follows.
It was shown that the width and the depth of the OMIA can be adjusted by changing the effective optomechanical coupling
in the right cavity.[3,27] For the present situation (
), we can obtain
with
It can be seen from Eq. (
19) that
is proportional to the power
(
) of the right coupling field. Equation (
20) tells us that when adding the right atomic ensemble, the effective detuning
subtracts an extra term
, the effective decay
adds an extra term
, and these lead to that the effective optomechanical coupling in the right cavity is enhanced in the sideband resolved limit. Thus we can broaden and deepen the windows of OMIA by increasing the power
of the right coupling field and the interaction strength
between the right cavity and the right atomic ensemble.
We have also examined the situation in which both the two atomic ensembles and the two coupling fields exist, but not found any more interesting features, so we do not report the situation here.
