1. Introduction

Second-order squeezing and entanglement of optical fields occurs as the Bogoliubov modes evolve towards the vacuum states due to two-photon processes [1–6]. Such evolutions happen through two fundamentally different kinds of induced interactions with the materials plus the external driving fields. One kind is parametric processes [1–12], while the other kind is engineered dissipative processes [13–20]. While the former was well known so early [1–6], the latter, which is usually called reservoir engineering, has been revealed much later [13–18]. The essential difference between them lies in that the former is coherent while the other is incoherent. Their common conditions are the dominance of the induced interactions over the environmental damping.

Higher-order squeezed and entangled states are important resources for universal quantum computation and entanglement distillation because it can improve the efficiency of quantum information tasks [21–30]. Not only by second-order moments as for second-order squeezing [1–6], and but also by higher-order moments, one defines higher-order squeezing when quantum fluctuations in a quadrature to higher orders are reduced below what are for the vacuum states [31,32]. For example, the fourth-order squeezing and entanglement of two optical fields involves fourth-order moments of the annihilation and creation operators $a_{1,2}$ and $a_{1,2}^{\dagger }$ such as $\langle a_{1}a_{2} a_{1}a_{2}\rangle$, $\langle a_{1}^{\dagger }a_{2}^{\dagger } a_{1}a_{2}\rangle$. Genuine non-Gaussian entanglement also requires criteria involving higher-order moments [27–30]. For non-Gaussian two-mode states, the second-order criteria are only sufficient and may fail to reveal entanglement. So far, the parametric interactions have been examined for higher-order squeezing [31–48] and higher-order nonclassical photon statistics [49–54].

In principle, the appearance or not of second-order squeezing does not mean the same for higher-order squeezing, because higher-order squeezing does not necessarily happen with second-order squeezing, but instead it is determined by different criteria involving higher-order moments. There has been much inspiring experimental progress in the higher-order nonclassicality. For example, higher-order photon correlations are measured with multimode pulsed quantum states [55,56]; twin beams generated in parametric down-conversion are most often used for multimode photon correlations [57,58]; and multiphoton detectors are developed for measurements of photon distribution [59,60].

To our knowledge, however, there has been no report so far that shows any relation of the reservoir engineering to higher-order squeezing. Our motivation is to fill in gaps in this aspect. As for the reservoir engineering, the dissipation rates and the squeezing parameters are, more often than not, oppositely dependent on the system features. For this purpose we exemplify the reservoir engineering based on coherent population trapping (CPT). CPT is one of the most remarkable coherent effects in the near-resonant systems and has widely been investigated theoretically and experimentally [61–70]. A great number of optical phenomena are surprisingly modified or various new phenomena are ceaselessly established [71–78]. Recent progresses have demonstrated the experimental steps towards the CPT-based quantum control [79–83]. To our greatest interest, the CPT-based systems have recently been shown to support both the parametric [19] and dissipative interactions [20] for second-order squeezing in different parameter regimes. The former originates from the two-photon resonances between the second adjacent dark states, while the latter is based on the collective one-photon resonances between the adjacent dark and bright states.

In view of the absence of any relation between the reservoir engineering and higher-order squeezing and in view of the importance of the CPT systems for quantum coherent control, we push the CPT-based reservoir engineering towards higher-order squeezing. It is shown that the CPT-based atomic reservoir makes the Bogoliubov modes evolve almost completely to the vacuum states and thus yields fourth-order two-mode squeezing of an almost perfect degree ($90\%\sim 100\%$). The advantages lie in the strong strengths for the atom-field interactions, the giant nonlinearities, and the robustness against spontaneous emission because of the dark resonances between the atomic ground states. As a by-product, we make a parallel comparison with the two-level reservoir and show that the two-level atoms only supports the squeezing of less than $75\%$.

We should emphasize the essential difference of the present work from the previous one [45]. The previous scheme used the parametric processes for CPT fields themselves, while the present scheme is to use the dissipative processes for the Bogoliubov modes of the fields that differ from the CPT feilds. The former is coherent while the latter is incoherent. The purpose of the present work is to push both the dissipative mechanism and CPT to the higher-order squeezing. Although it seems that the fields are in parametric interactions with the dressed CPT atoms as in Eq. (21), actually it is not the case. Induced by the CPT atoms is the dissipation of the Bogoliubov field modes as in Eq. (23). Our focus is put on the dissipation effects of the CPT atoms. As the secondary difference, the previous scheme related the higher-order fluctuations approximately to the second-order quadrature correlations. This approximation holds only when the fields are strong and the fluctuations are small. In contrast, the present cavity fields, different from the CPT fields, operate below threshold and thus are weak. All fourth-order quadrature moments are included into the quantum correlations to be considered

In the remaining part of the present paper is organized as follows. Given in Sec. 2 are the criteria for the forth-order squeezing and the comprehensive mechanism for the reservoir engineering. In Sec. 3 we describe the CPT-based scheme and reveal its essential features. In Sec. 4 a parallel comparison is made with the two-level reservoir. Finally, the conclusion is given in Sec. 5.

2. Conditions for fourth-order squeezing

What is the most important is the creation of the two-photon processes and the reduction of spontaneous emission in a realistic atomic or molecular system. With this in mind, before presenting the CPT-based scheme, we first see the general conditions for higher-order squeezing in a case. Our analysis is based on a representative master equation [2,3,5] (also see below)

2.1 Criteria for fourth-order squeezing

We expect to consider two kinds of the fourth-order two-mode squeezing. The first kind occurs when the fourth-order moments guarantee the inequality

For a quadrature, $n$-th order squeezing involves $n$th order moments of the annihilation and creation operators. The usual squeezing is in second order. However, an intensity operator is itself the second order, and the lowest order for antibunching (or squeezing) requires fourth order moments of the annihilation and creation operators. Without confusion, it is more reasonable and more convenient to define the order consistently in terms of the highest order that is required for the criterion.

Usually there is a difference between the measures of nonclassicality and the witnesses of various kinds of nonclassicality [84,85]. Some operational quantities might be more convenient in testing the nonclassicality of specific states generated in experiments. However, here we are only involved in the reduction of the quantum noise, the fluctuations relative to the corresponding coherent states are sufficient to describe the degree of noise reduction.

For clearness, we define second-order moments

2.2 Bogoliubov modes for a comprehensive mechanism

Bogoliubov modes help us understand the conditions for a good higher-order squeezing. More often than not, as will be seen later, the annihilation and creation operators $a_{k}$ and $a_{l}^{\dagger }$ ($k\neq l$) appear in pairs in the Hamiltonian for the interactions of the dressed atoms with the cavity fields. They combine into the Bogoliubov modes in the standard form [2–4]

Substitution of Eq. (10) into Eq. (1) gives us the dominant part as [2–4]

 

Fig. 1. Dissipation of the Bogoliubov modes, $b_{1,2}$, substituting for the original fields $a_{1,2}$, as a comprehensive mechanism for the reservoir engineering, for which the two-photon processes are generally concealed in a strongly coupled atomic or molecular system and are not seen intuitively.

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3. CPT-based scheme

The CPT-based scheme is made up of near-resonantly dressed three-level atoms in $\Lambda$ configuration. As shown in Fig. 2 (a), the bare atom has two ground or metastable states $|1,2\rangle$ and one excited state $|3\rangle$. Two wings in $\Lambda$ configuration are the dipole-allowed transitions $|1, 2\rangle \leftrightarrow |3\rangle$, to which two external dressing fields of frequencies $\omega _{1,2}$ are coupled with Rabi frequencies $\Omega _{1,2}$, respectively. It has been aware that CPT happens when the dressing fields are on two-photon resonance between the ground states [61,67–70]. We focus on the near-resonant case, where the CPT effect maintains a crucial role in manipulating the atom-field interactions. Two quantized fields $a_{1,2}$ are generated from two Rabi sidebands of the two different transitions, respectively.

 

Fig. 2. The CPT-based scheme. (a) Two external coherent fields $\Omega_{1,2}$, while trapping the atoms in the coherent superposition of the ground states, induce the two-photon transitions with respect to the two quantized fields $a_{1,2}$ between the ground states. (b) The two quantized fields $a_{1,2}$ behave as a pair of Bogoliubov modes, which are on the Rabi resonances with the atoms in terms of dressed states.

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Our focus is on the creation of the fourth-order squeezing via the two-photon processes, which are induced by the dressing fields and are rooted deeply in the near-resonant coupled system. The dressed atomic states [see Fig. 2 (b)] and the Bogoliubov modes are most suitable for the description of the essential mechanism. In order to understand more conveniently the physical mechanism we present first its effect on the higher-order correlations before presenting a lengthy analysis. Fig. 3 and Fig. 4 (with the parameters given later) show almost perfect fourth-order squeezing ($90\%\sim 100\%$) at a special location. The cascaded transitions of the three-level atoms can be in respective interactions with microwave and optical fields, in addition to being resonant with either microwave or optical fields. Our analysis is performed in three steps: the dressed atomic states, the dissipation in terms of the Bogoliubov modes, and the fourth-order squeezing.

 

Fig. 3. The fourth-order moment $\Delta X_{a}^{4}$ $(=\Delta P_{a}^{4})$ versus the normalized detuning $\Delta/\Omega$ for resonantly dressed three-level atoms as an engineered reservoir. The parameters are chosen as $C=20$ (dotted), $C=500$ (dashed) and $C=5000$ (solid), and $\kappa_{1,2}=0.1\gamma$. The optimal squeezing approaches $100\%$ below what is for the minimal uncertainty state at $\Delta/\Omega\rightarrow-1/\sqrt{6}\approx-0.408.$

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Fig. 4. The normalized variance $\Delta V^{2}$ versus the normalized detuning $\Delta/\Omega$ for resonantly dressed three-level atoms as an engineered reservoir. The parameters are the same as in Fig. 3. The optimal squeezing is about $90\%$ below what is for the minimal uncertainty state at $\Delta/\Omega\rightarrow-1/\sqrt{6}\approx-0.408.$

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3.1 Dressed atomic states

The master equation for the atom-dressing-field density operator $\rho$ of the atom-field is derived in the dipole approximation and in an appropriate rotating frame as [3]

It is most convenient to transfer the dressing field induced nonlinearity to the atoms. That is, we now turn to a picture of the dressed atomic states. For the clarity of presentation of physical mechanism, we take a special kind of parameters: $\Delta _{1}=-\Delta _{2}=-\Delta$ and $\Omega _{1,2}=\Omega$ (real). After diagonalization of $H_{0}$, we obtain three dressed states $|0,\pm \rangle$, which have equally spaced eigenvalues $\lambda _{0,\pm }=0,\hbar \bar {\Omega }$, and are written in terms of bare states as [86]

3.2 Dissipation in terms of Bogoulibov modes

Now we focus on the interaction of the CPT-based atomic reservoir with two quantized fields. The Hamiltonian for the interaction

The interaction of the quantized fields with the CPT atoms is shown in Fig. 2 (b). The dressing field induced nonlinearity merged in the dressed atoms is reflected in three aspects. (i) The creation and annihilation of quantized fields is accompanied with the spin-flips $\sigma _{\pm 0}$ in terms of the dressed atomic states. The dressed state population differences will play a crucial role in the quantum correlations. (ii) The atom-field interaction strengths are merged with the nonlinearities, which are seen from $-\frac {1}{2}g\cos ^{2}\theta$ and $\frac {1}{2}g\sin \theta (1-\sin \theta )$. (iii) The two cavity fields coupled to respective bare-state transitions are now pulled into a collective or simultaneous interactions with a common dressed-state transition. The collective modes are just the same as the Bogoliubov modes in Eq. (10). In other words, the Bogoliubov modes substitute for the original fields.

In our interest is the symmetric forms of the induced parametric interactions. This symmetry gives us the convenience for the use of the Bogoliubov modes. Now we are in a position to analyze the dissipation of the Bogoliubov modes substituting for the original fields. In terms of the definition in Eq. (10) we have hyperbolic tangent of the squeezing parameter $\tanh r=\frac {1+\sin \theta }{|\sin \theta |}$ for $\frac {\Delta }{\Omega }<-\frac {1}{\sqrt {6}}$ and $\tanh r=\frac {|\sin \theta |}{1+\sin \theta }$ for $\frac {\Delta }{\Omega }>-\frac {1}{\sqrt {6}}$. By using the Bogoliubov modes we rewrite the interaction Hamiltonian in Eq. (21) for $-\frac {1}{\sqrt {6}}<\frac {\Delta }{\Omega }<0$ and $0<\frac {\Delta }{\Omega }<\frac {1}{2}$ as

We assume that the atomic variables decay much more rapidly than the cavity fields. Following the standard techniques as in [3], we derive the master equation of motion for the reduced density operator $\varrho =\textrm {Tr}_{\textrm {atom}}\rho$ as

Equation (23) shows that the present engineered reservoir supports two dissipative channels respectively for two Bogoliubov modes. The mechanism is shown pictorially in Fig. 1. The original modes $a_{1,2}$ combine as the Bogoliubov modes $b_{1,2}$ and undergo absorption by the dressed atoms. In principle, large $\sinh (2r)$ for good squeezing needs to match large net dissipation rate $(A-B)\sinh (2r)\gg B$. As shown in Fig. 5, there are three different cases as follows for the compatibility of the squeezing parameter $\sinh (2r)$ with the net dissipation rate $A-B$.

 

Fig. 5. The squeezing parameter $\sinh(2r)$ (dotted), the net dissipation rate $(A-B)/\kappa C$ (dashed), and the compatibility parameter $R/\kappa C$ (solid line with a breaking point at $\Delta/\Omega \rightarrow -1/\sqrt{6}\approx -0.408$) for the CPT-based atoms and the chosen cavity field detunings $\delta_{1,2}=\pm\bar{\Omega}$. Good compatibility of $\sinh(2r)$ with $A-B$ appears in the shaded regime. For $\Delta/\Omega\approx -0.408$ ($x\rightarrow0^{+}$), we have $\sinh(2r)\rightarrow1/x$ and $A-B\rightarrow x\kappa C$, which leads to $R\propto \kappa C$ even comparable to the maximal value. When the cavity fields are tuned to $\delta_{1,2}=\mp\bar{\Omega}$, the related regime of $\Delta/\Omega$ is turned to the symmetric side with respect to the vertical axis.

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(1) Here is the first part of our main results. As $\frac {\Delta }{\Omega }\rightarrow -\frac {1}{\sqrt {6}}$ (i.e., $\sin \theta \rightarrow -\frac {1}{2}$, the shaded regime), the squeezing parameter is dependent on a nonlinear function

(2) As $\frac {|\Delta |}{\Omega }\rightarrow 0^{-}$, we have $\sinh (2r) \rightarrow 0$ and $R\rightarrow 0$ even when $A-B$ takes its maximal value.

(3) For intermediate value of the normalized detuning: $\sinh (2r)$ is so small although $A-B$ is large, and so the obtainable squeezing is weak although $R$ takes a large value.

In addition, if the cavity fields are tuned to their respective symmetric Rabi sidebands, $\delta _{1,2}=\mp \bar {\Omega }$, the above normalized detuning $\frac {\Delta }{\Omega }\rightarrow -\frac {1}{\sqrt {6}}$ for good compatibility is correspondingly changed to its symmetric location, $\frac {\Delta }{\Omega }\rightarrow \frac {1}{\sqrt {6}}$.

3.3 Almost perfect fourth-order squeezing

In this subsection we verify the prediction as above by calculating the higher-order correlations. We rewrite Eq. (23) in terms of the original $a_{1,2}$ modes in an explicit form

At $\frac {\Delta }{\Omega }\rightarrow -\frac {1}{\sqrt {6}}$, there happen almost $100\%$ squeezing for $X_{a}$ and $P_{a}$ and about $90\%$ squeezing for the sum operator $V$. The accessibility of the three conditions (i)-(iii) in Eqs. (13)–(15) that have to be satisfied by the nonlinear function is present as follows.

(1) The first condition necessary for the optimal higher-order squeezing is a large value of the squeezing parameter: $\sinh (2r)\gg 1$. For the present case, $\sinh (2r)$ takes the large values only as the normalized detuning approaches a limiting value: $\frac {\Delta }{\Omega }\rightarrow -\frac {1}{\sqrt {6}}$ (i.e., $\sin \theta \rightarrow -\frac {1}{2}$). In this case, we have $\sinh (2r) \propto \frac {1}{x}\rightarrow \infty$, where $x=1+2\sin \theta >0$ is a nonlinear function for the present three-level atoms. The best squeezing is reached when the squeezing parameter $\sinh (2r)$ takes its value as large as possible and when the conditions for the net dissipation rates are well satisfied. Beyond this regime, the small $\sinh (2r)$ represents weak squeezing even although the other conditions are well met.

(2) The second condition is dominance of the engineered dissipation over the vacuum dissipation, $A \gg \kappa \cosh ^{2}r$. In terms of the Boboliubov modes, the cavity dissipation rates become $\kappa \cosh ^{2}r$. In order to overcome the vacuum dissipation we have to make the artificial reservoir dominate over it. When this condition is met, the Bogoliubov modes can evolve close to their vacuum state. In order to satisfy this condition, we need overcome two-fold effects. One is the rise in the squeezing parameter $r$ and the other is the fall in the effective interaction strength $g_{b}\propto g \sqrt {x}$. This can be reached by increasing the density of atoms involved in the interaction.

(3) The third condition is a considerable compatibility parameter as given in Eq. (14): $R=(A-B)\sinh (2r)\gg B$. Note that the dressed population differences do not tend to zero, $N_{0}-N_{\pm }\rightarrow \frac {4}{19}N\neq 0$ for $\frac {|\Delta |}{\Omega }\rightarrow -\frac {1}{\sqrt {6}}$. Thus we have the engineered net dissipation $A-B\propto x\kappa C$. Usually, for the near-resonantly dressed atoms, the net dissipation rates and the squeezing parameter are not compatibly large. One tends towards zero while the other is large because of the opposite effects of the nonlinearities. However, it does not matter. The important thing is to guarantee that the compatibility parameter keeps a considerable value, $R\propto \kappa C$. It is indeed the case for the CPT-based three-level atoms, as analyzed in the above subsection. Under this condition, even if $A-B\rightarrow 0$, higher-order squeezing is well preserved as much as possible. The above three conditions are essential for the almost ideal squeezing for the present scheme.

3.4 Experimental aspects

Proposed as follows is a possible experimental scheme for the above CPT-based reservoir engineering for high-order squeezing. On one hand, cold atoms behaves as a new kind of media for quantum optics, as it has been demonstrated recently. Both field squeezing and spin squeezing have been observed using cold atoms in a magneto-optical trap (MOT) [87–94]. The use of ultracold atomic gases has advantages to allow high densities of the atoms and the high intensities of driving fields, and to reduce the Doppler shifts and the collision dephasing rate of the ground state transition. The atoms (for example, alkali $^{87}$Rb) are trapped by using a standard six-beams $\sigma ^{+}/\sigma ^{-}$ MOT configuration [95] on the $|5^{2}S_{1/2}, F=2\rangle -|5^{2}P_{3/2}, F=3\rangle$ transition (D2 line at 780 nm). A repumping beams is tuned on the $|5^{2}S_{1/2}, F=1\rangle -|5^{2}P_{3/2}, F=2\rangle$ transition. On the other hand, the CPT effect can be realized most widely in atoms, molecules, and semiconductors, so long as a $\Lambda$ atom-field interaction configuration with stable lower-lying states can be isolated out [61,62,67–70]. It is well aware that both CPT and electromagnetically induced transparency (EIT) have the same origin, which is just the dark resonance. EIT and the related nonlinear and quantum effects have already been observed in ultracold atoms by a great number of investigation groups (see references in [67–70] for examples). The MOT is built in a large ultrahigh vacuum chamber, inside which a doubly resonant optical cavity is set up around the cold-atom cloud [87,88]. The D1 transition hyperfine structure (795 nm) is used for the CPT-controlled quantum control. The Zeeman sublevels $|5^{2}S_{1/2}, F=1, m=\pm 1\rangle$ correspond respectively to the lower-lying states $|1\rangle$ and $|2\rangle$, and the sublevel $|5^{2}P_{1/2}, F=1, m=0\rangle$ serves as the excited state $|3\rangle$. Correspondingly, the $\sigma ^{+}$ and $\sigma ^{-}$ circularly polarized beams are used for the CPT effect and the CPT-based reservoir engineering.

The parameters of the proposed system can be estimated as follows [87–94]. A waist of $w\sim 35$ $\mu$m and a homogeneous laser beam of width $d\sim 50$ $\mu\textrm{m}$ give us an interaction volume of $10^{-7}\, \textrm{cm}^{3}$. The coupling constant of the cavity fields with the atoms is about $g\sim 2\pi \times 0.45$ MHz. The atomic and cavity damping rates are $\gamma \sim 2\pi \times 5.3$ MHz and $\kappa \sim 2\pi \times 2.5$ MHz, respectively. The Rabi frequency of the CPT field $\Omega \sim 2\pi \times (15\sim 30)$ MHz (a power smaller than $5.5~\mu$W) is enough to guarantee that the quantum fields are well separated from the adjacent fluorescence spectral lines. For a cooperativity parameter of $C=500$, a required number of atoms is $3.3\times 10^{4}$, which corresponds to a density of atomic density. $3.3\times 10^{11} / \, \textrm{cm}^{3}$ The density is small enough to prevent coherence losses due to collisions at temperature of $\mu$K. The above is just a rather conservative set of parameters, for a chosen experimental setup, from a wide range of possibilities.

In addition, since the cascaded transitions of the three-level atoms can be used in the microwave and/or optical regimes, the corresponding mechanism is suitable for higher-order two-mode squeezing of microwave and optical fields simultaneously, in addition to either microwave only or optical fields only. As an example, we can use hydrogen atom, for which $|1S\rangle$ and $|2S\rangle$ act as the lower-lying states $|1\rangle$ and $|2\rangle$, and $|3P\rangle$ is used as the excited state $|3\rangle$. While the $|1\rangle -|2\rangle$ transition is in the microwave regime, the $|2\rangle -|3\rangle$ transition locates in the optical regime.

4. Two-level atoms as a comparison

In this section, we show that the usual two-level atomic reservoir has an essential difference in the nonlinear dependence from the above CPT-based three-level atomic reservoir. Shown in Fig. 6 (a) is the interactions of the dressing and quantized fields with one common transition. The dressing field of frequency $\omega _{0}$ dresses the atom with Rabi frequency $\Omega$, and two cavity fields $a_{1,2}$ of frequencies $\nu _{1,2}$ are generated from two Rabi sidebands of the dressed atoms.

 

Fig. 6. Interactions in the two-level atoms. (a) One external coherent field $\Omega$ creates the two-photon processes for the quantized fields $a_{1,2}$ between the ground and excited states. (b) The quantized fields $a_{1,2}$, which act as either of the Bogoliubov modes, are on Rabi resonances with the atoms in terms of dressed states.

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The two-photon processes by the near-resonant dressing field necessarily involve the excitation of a large fraction of the atoms, which is essentially different from the CPT-based scheme. Shown in Fig. 6 (b) is the dressed-state description, for analysis below, of the quantized fields with the dressed atoms. One ensemble of two-level atoms supports only one channel for either of Bogoliubov modes and two ensembles are needed as a composite reservoir to set up two dissipative channels. Whether for a single ensemble or two composite ensembles, the best achievable fourth-order squeezing is in a limited degree. Similarly, in order to present more conveniently the physical mechanism we first show the numerical results before analysis. It is shown in Fig. 7 (for one ensemble with the parameters given later) that the best squeezing is $75\%$ for the two-mode quadratures $X_{a}=x_{a_{1}}-x_{a_{2}}$ and $P_{a}=p_{a_{1}}+p_{a_{2}}$. The analysis is now described as follows.

 

Fig. 7. The fourth-order moment $\Delta X_{a}^{4}$ $(=\Delta P_{a}^{4})$ versus the normalized detuning $\Delta/\Omega$ for the two-level atomic reservoir. The parameters used are $C=20$ (dotted), $C=500$ (dashed) and $C=5000$ (solid), and $\kappa_{1,2}=0.1\gamma$. The optimal squeezing is about $75\%$ below what is for the vacuum state.

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4.1 Dressed atomic states

The master equation for the density operator $\rho$ of the atom-field takes the same form as in Eq. (16), where the Hamiltonian for the interaction of the atoms with the dressing field is substituted into

In order to show clearly the nonlinear effects, we can merge the nonlinearities into the atoms, then we have the dressed atoms. After diagonalization of $H_{0}$ we obtain the dressed states [86]

4.2 Dissipation in terms of Bogoliubov modes

Added to $H_{0}$ is the interaction Hamiltonian

In order to describe the mechanisms more clearly, we can define the Bogoliubov modes as in Eq. (10) for the cavity fields: $b_{1}=a_{1}\cosh r-a^{\dagger }_{2}\sinh r$, $b_{2}=a_{2}\cosh r-a^{\dagger }_{1}\sinh r$. When $g_{1,2}=g$ we have the hyperbolic tangent of the squeezing parameter for $b_{1,2}$ modes as $\tanh r=\tan ^{2}\theta$ for $\Delta >0$ and $\tanh r=\cot ^{2}\theta$ for $\Delta <0$. We rewrite the interaction Hamiltonian in Eq. (36) as

The near-resonantly dressed two-level atoms constitute the engineered dissipation reservoir for the Bogoliubov field modes. Under the adiabatic condition $(\gamma \gg \kappa _{1,2})$ we derive the master equation for the reduced density operator $\varrho$ of the cavity fields as

 

Fig. 8. The normalized variance $\Delta V^{2}$ versus the normalized detuning $\Delta/\Omega$ for the two-level atomic reservoir. The parameters are the same as in Fig. 7. The optimal squeezing is about $50\%$ below what is for the vacuum state.

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(1) As $\Delta \rightarrow 0$ (i.e., $\cos \theta \rightarrow \sin \theta$), the squeezing parameter depends on a nonlinear function

(2) For $\frac {|\Delta |}{\Omega }\rightarrow \infty$ we have $\sinh (2r) \rightarrow 0$. Squeezing tends towards zero even for $A-B$ takes its maximal value.

(3) In the intermediate regime, $\sinh (2r)$ and $A-B$ are relatively compatibly large. In this case, only squeezing defined by intermediate value of $r$ is preserved through the dominance of the engineered dissipation over the vacuum dissipation. Therefore, achievable squeezing will be limited to a certain degree.

4.3 Fourth-order squeezing in a limited degree

Now we proceed to the numerical verification of the above analysis. The master equation of the reduced density operator $\varrho$ for the original fields $a_{1,2}$ is derived in the same way as for Eq. (1) with rates $(\mathcal {A}_{1}, \mathcal {B}_{1})=\frac {2}{\Gamma }g^{2}_{1}N_{\mp }\cos ^{4}\theta$, $(\mathcal {A}_{2}, \mathcal {B}_{2})=\frac {2}{\Gamma }g^{2}_{2}N_{\pm }\sin ^{4}\theta$, and $(\alpha , \beta )=\frac {1}{4\Gamma }g_{1}g_{2}N_{\mp }\sin ^{2}(2\theta )$. We assume $\kappa _{1,2}=0.1\gamma$ and $g_{1,2}=g$. It is enough for us to compare the fourth-order squeezing of $X_{a}$ and $P_{a}$. Plotted in Fig. 7 is $\Delta X_{a}^{4}$ $(=\Delta P_{a}^{4})$ versus the normalized detuning $\Delta /\Omega$ for $C=20$ (dotted), $C=500$ (dashed) and $C=5000$ (solid), and $\kappa _{1,2}=0.1\gamma$. The main features are summarized as follows.

(1) The best achievable squeezing is limited to $75\%$ for the two-mode quadratures $X_{a}$ and $P_{a}$. This occurs in the intermediate regime of the normalized detuning $\Delta /\Omega$. The higher-order squeezing is achievable even in the presence of one dissipation channel. For $\Delta >0$ or $\Delta <0$, the original $a_{1,2}$ modes combine as $b_{1}$ or $b_{2}$ to mediate the interaction, but the other Bogoliubov mode is decoupled from the atoms. It is understandable that the engineered dissipation of one Bogoliubov mode only yields squeezing to a limited degree. As a comparison, we should also note that for the two-level atomic system, the best second-order squeezing for $X_{a}$ and $P_{a}$ is about $50\%$ [4]. This shows that a larger fractional reduction of the higher moments is possible than of the second moments.

(2) The parameter $R$ for the compatibility of the squeezing parameter with the net dissipation rates tends towards zero as $\sinh (2r)$ rises to a large value. This is because two multiplying factors tend to be vanishing. One is the strength for the interactions of the Bogoliubov modes with the dressed atoms $g_{b}=g\sqrt {x}$ ($x=|\cos (2\theta )|$) and the other is the dressed population difference $|N_{+}-N_{-}|\rightarrow 2xN$. At the same time we have $\sinh (2r)\propto \frac {1}{x}$. For $\frac {\Delta }{\Omega }\rightarrow 0$, the simultaneous disappearance of $|g_{b}|^{2}$ and $|N_{+}-N_{-}|$ determines the disappearance of $R$: $R\propto x\kappa C \rightarrow 0$. This is in sharp contrast to the CPT-based three-level atoms, in which the dressed population differences tend to be a nonvanishing value.

(3) The regime of $\Delta /\Omega$ for the intermediate squeezing is much wider for the two-level atoms than for the three-level atoms. This is because for the two-level atoms, there exists a much wider regime where the squeezing parameter and the net dissipation rates are compatibly large, although the squeezing parameter is intermediately large. In addition, there exists a symmetry in the regimes for higher-order squeezing. In contrast, no such symmetry

4.4 Two two-level atomic ensembles

One can expect that two two-level atomic ensembles as a composite reserovir enhance higher-order squeezing. Beyond our expectation, however, there is little enhancement. The dependence of the fourth-order moment $\Delta X_{a}^{4}$ $(=\Delta P_{a}^{4})$ on various parameters is shown in Fig. 9 for the parameters given later. It is shown that there remains still the optimal squeezing of about $50\%\sim 75\%$. Physically, this originates from the unchanged fact that net dissipation rate falls more rapidly ($A_{l}-B_{l}\propto x^{2}N\rightarrow 0$) than the squeezing parameter rises ($\sinh (2r)\propto 1/x\rightarrow \infty$). This is analyzed as follows.

 

Fig. 9. The fourth-order moment $\Delta X_{a}^{4}$ $(=\Delta P_{a}^{4})$ for two composite two-level atomic reservoir versus the normalized detuning $\Delta/\Omega$ for different cooperativity parameter $C$. The other parameters are given in the text.

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Two single-ended folded cavities cross at two crossing sections, in which are placed two atomic ensembles of different but close resonance frequencies. One common coherent field of frequency $\omega _{0}$ drives the $l$th atomic ensembles with Rabi frequency $\Omega _{l}$ and detunings $\Delta _{l}$, respectively, and induces the simultaneous Rabi resonances with quantized cavity fields. Following the same technique, we obtain the atom-field interaction Hamiltonian

As a special case, $\Delta _{1}=-\Delta _{2}$, $\cos ^{2}\theta _{1}=\sin \theta _{2}=\cos \theta$, $g_{kl}=g$, $\varphi =0$, we rewrite the interaction Hamiltonian in Eq. (43) as

Then we turn to the general case, including the asymmetry of the dissipation channels and the phase dependence. The reduced master equation $\varrho$ is in exactly the same form as Eq. (1) except for the substitutions of rates ($\mathcal {A}_{1,2}, \mathcal {B}_{1,2},\alpha , \beta$): $(\mathcal {A}_{1}, \mathcal {B}_{1})=\frac {2}{\Gamma _{1}}g^{2}_{11}N^{(1)}_{\mp }\cos ^{4}\theta _{1}+\frac {2}{\Gamma _{2}}g^{2}_{12}N^{(2)}_{\mp }\cos ^{4}\theta _{2}$, $(\mathcal {A}_{2}, \mathcal {B}_{2})=\frac {2}{\Gamma _{1}}g^{2}_{21}N^{(1)}_{\pm }\sin ^{4}\theta _{1}+\frac {2}{\Gamma _{2}}g^{2}_{22}N^{(2)}_{\pm }\sin ^{4}\theta _{2}$, and $(\alpha , \beta )=\frac {1}{4\Gamma _{1}}g_{11}g_{22}N^{(1)}_{\mp }\sin ^{2}(2\theta _{1})+\frac {1}{4\Gamma _{2}}g_{12}g_{22} \\ N^{(2)}_{\mp }\sin ^{2}(2\theta _{2})e^{-i\varphi }$ where $\Gamma _{l}=\frac {\gamma _{l}}{2}+\frac {\gamma _{l}}{4}\sin ^{2}(2\theta _{l})$, $\gamma _{l}$ is the rate of spontaneous emission for the $l$th atomic ensemble. Plotted in Fig. 9 are the normalized moments $\Delta X_{a}^{4}$ $(=\Delta P_{a}^{4})$ for different cooperativity parameter $C$. There is little difference between the single-channel case and the two-channel case for the two-level atomic reservoirs. The best squeezing remains $75\%$ for $X_{a}$ and $P_{a}$. The parameters used in Fig. 9 are $C=20$ (dotted), $C=500$ (dashed), $C=5000$ (solid), $\zeta =-1$, $\varphi =0$, and $\kappa =0.1\gamma$. As for the case of one ensemble of dressed two-level atoms, since the net dissipation rates fall more rapidly than the squeezing parameter rises, as shown in Fig. 7, the optimal squeezing is generally confined to an intermediate degree.

5. Conclusion

In conclusion, we have presented a CPT-based scheme for higher-order squeezing of two optical fields. This scheme is robust against the spontaneous emission and experimentally more accessible in a realistic atomic or molecular system because the intrinsic two-photon processes are induced by the dressing fields between the atomic ground states. In a comprehensive way, we owe the effects of the two-photon processes to the dissipation of the Bogoliubov modes, as the equivalent substitutes for two original fields, towards their vacuum states. The CPT-induced nonlinearity supports an almost ideal dissipation (i.e., the compatibility of the large dissipation rates with the large squeezing parameter), and results in almost ideal higher-order squeezing ($90\%\sim 100\%$). As a by-product, there is only a limited degree of the fourth-order squeezing in two-level atoms ($50\%\sim 75\%$). The characteristic features of the two-representative systems are comparatively summarized as follows.

As the common feature of these two representative reservoirs, there is an opposite change in the squeezing parameter and the net dissipation rate as a system-dependent nonlinear function $x$ tends towards zero. While their hyperbolic sine function of the squeezing parameter $r$ depends on $x$ in a reversely proportional form: $\sinh (2r)\propto 1/x\rightarrow \infty$ as $x\rightarrow 0^{+}$, the net dissipation rate $\Delta A=A-B$ tends towards zero: $\Delta A\rightarrow 0$. After the nonlinearities induced by dressing fields are merged into the atom-field interactions, the atoms act as an engineered reservoir, and the Bogoliubov modes are established from the original fields through a squeezing parameter and substitute for the original fields as the two equivalent modes. The squeezing parameter stands for the potentially possible squeezing. Whether the possible squeezing is reached is determined by the net dissipation rates. The good squeezing requires that the large squeezing parameter is accompanied with the dominant net dissipation rates over both the dissipation rates due to the vacuum and the excitation rates due to the engineered reservoir.

The essential difference of these two representative reservoirs lies in their own different ways that the net dissipation rate tends towards zero as $x\rightarrow 0$. The CPT-based atomic reservoir has an $x$-linear dependence on the nonlinear function for two dissipative channels: $\Delta A\propto x\kappa C$ as $x\rightarrow 0$. In sharp contrast, the two-level atomic reservoir displays an $x$-square dependence only for one dissipative channel: $\Delta A\propto x^{2}\kappa C$. Good squeezing requires a dominance of a compatibility parameter $R=\Delta A\sinh (2r)$ over the accompanying excitation rate. This condition is well satisfied for the CPT-based reservoir ($R\propto \kappa C$) but not so well for the two-level reservoir ($R\propto x\kappa C \rightarrow 0$). It is the essential difference that the fourth-order two-mode squeezing is reached $90\%\sim 100\%$ for CPT-based reservoir but is limited to $50\%\sim 75\%$ for the two-level reservoir.

As a further origin, the above essential difference is due to the different dependence of the dressed population difference on the system-dependent nonlinear function $x$. Commonly, the net dissipation rates depend on both the effective coupling strengths $g_{b}\propto g\sqrt {x}$ ($g$ is the atom-field coupling coefficient) and the dressed population difference $\Delta \tilde {N}$. The most remarkable difference lies in that, the CPT-based reservoir has an $x$-independent population difference $\Delta \tilde {N}\rightarrow \frac {4}{19} N$ as $x\rightarrow 0$ ($N$ is the number of atoms), but the two-level reservoir has an additional $x$-dependent population difference $\Delta \tilde {N}\sim 2xN\rightarrow 0$. As a consequence, the CPT-based reservoir has an $x$-linear net dissipation rate, which supports the considerable compatibility parameter. The two-level reservoir has an $x$-square dependent net dissipation rate, which gives rise to a vanishing compatibility parameter.