1. Introduction

The reduction of quantum noise is one of the important issues in quantum optics, laser physics, and nonlinear optics. Quantum noise is harmful for the high-precision experiments and thus ones seek many ways to suppress the quantum noise. Earlier, Caves et al. [1] first noted the possibilities of manipulating quantum fluctuations with the aim of precision measurement. When the quantum noise of one quadrature of an optical field is reduced below the standard quantum limit at the expense of the increasing noise in the conjugate quadrature, we can say that the fields are prepared in the squeezed states [2,3]. The bi- or multimode squeezed states are most closely related to the entangled states, which are indispensable resources in quantum information and quantum communication networks [4–7]. Many schemes for generation of the squeezed and entangled states have been proposed based on either the coherent evolutions [8–16] or the reservoir engineering [17–23]. The essential physics is due to the two-photon processes, which occur typically in a correlated emission laser or in four-wave mixing [16, 24, 25]. It is meaningful to seek other systems for realizing the noise squeezing of fields.

It has become aware that the bichromatic or ploychromatic interactions constitute an important field in quantum optics, laser physics, and nonlinear optics. Bichromatic excitation on one common atomic transition can lead to the modulation frequency dependent nonlinearities. Thus many novel effects occur while they are absent for monochromatic interactions with atoms. The spontaneous emission, absorption and dispersion spectra become comblike and dependent on the modulation frequency [26–44]. Spontaneous-emission-mediated emissive multiphoton transitions (the usual processes of stimulated emission and absorption are reversible) as irreversible processes can give rise to lasing without population inversion [45, 46]. Two-photon absorption is dramatically suppressed when a pair of bichromatic fields drive a three-level atomic system [47]. Experimental and theoretical investigation of cavity-modified dynamics of two-level atoms driven by a strong bichromatic field also shows that the width of atomic resonances may become smaller than the cavity linewidth [48]. Additionally, selective elimination of the spectral lines is obtained by varying the amplitudes and phases of the trichromatic components [49]. For a bichromatically excited two-level atom, there exist fluorescent two-photon processes, which are responsible for the possible noise squeezing in the central fluorescence spectrum [50, 51]. However, the above work has focused on the dynamics of atoms and there has been little work on noise squeezing of fields under bichromatic excitation. Once two fields of different frequencies and considerable amplitudes are simultaneously coupled to one common atomic transition, there appear the cross nonlinearities and both of the self and cross nonlinearities depend strongly on the modulation frequency. In order to study the effects of the modulation frequency dependent nonlinearities, one usually has to employ the harmonic expansions. In particular, the expanded terms of so many high orders prevent one from identifying the direct mechanisms for the quantum correlations.

Here we present an efficient way without the harmonic expansions and show clearly that the Bogoliubov mode dissipation causes the noise squeezing of fields in bichromatic interactions with atoms. The atom-field nonlinearities are transferred to the dressed atoms for proper cavity-detunings, and the Bogoliubov mode dissipative interactions are separated from the system dynamics. We first consider a fundamental model in which an ensemble of two-level atoms interact with the two cavity fields of different frequencies and considerable amplitudes. It is shown that the Bogoliubov mode dissipative interactions lead to the two-photon processes of two involved fields and thus to the two-mode squeezing. Then an ensemble of three-level Λ atoms is considered for cascade bichromatic interactions. It is found that dissipative interactions of the Bogoliubov-like modes with the dressed atoms lead to a quadrilateral of the two-photon processes of four involved fields and thus to the four-mode squeezing. Our work can be extended to other coherent systems and facilitate the analysis of the noise squeezing of fields in frequency dependent nonlinear interactions.

Particularly, in our system the bichromatic fields are not in Λ-configuration interaction with three-level atoms as in electromagnetic induced transparency and/or Raman processes but are directly coupled to the two-level atomic system, where the nonlinearities occur and provide the ability to suppress the quantum noise. When the driving field couples to the two-level atoms, the most remarkable feature of the atomic dynamics is Rabi resonance. We use driving fields of the same frequencies and tune the cavity fields resonant with Rabi sidebands. The Rabi resonances give rise to the collective dissipation and to squeezing. Using cascade bichromatic excitations in a Raman system, we also can obtain the four-mode squeezing.

The remaining part of the present paper is organized as follows. In section 2, we describe the bichromatic interactions with two-level atoms in cavities, derive the dressed atom-field interactions, and present the Bogoliubov mode dissipation effects on the field correlations. In section 3, we extend to the cascade bichromatic interactions in three-level atoms for the four-mode squeezing. Finally, the conclusion is given in section 4.

2. Bichromatic interactions with two-level atoms

We consider an ensemble of N two-level atoms at the intersection of two optical cavities, each of which is pumped by one external field, as shown in Fig. 1(a). The two cavity modes are simultaneously coupled to the common electronic dipole-allowed transition of the two-level atoms from the ground state |1〉 to the excited state |2〉, as shown in Fig. 1(b). The system under consideration is clearly different from those that are based on the wave mixing interactions. For the latter, the cavities are driven only by the vacuum reservoir but not by the external classical fields and the atoms are excited only by one monochromaic field [17–19]. In the present system, in sharp contrast, the two fields have different frequencies and considerable amplitudes, and the atoms are bichromatically excited. The master equation for density operator ρ of the atom-field system is written in the dipole approximation and in an appropriate rotating frame as [2,3]

 

Fig. 1 (a) An ensemble of two-level atoms is placed at the intersection of two cavities, into which two external coherent fields are input respectively. (b) The two cavity fields (annihilation operators a_1,2) of different frequencies are coupled to the common transition of the atoms.

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For the driven atom-cavity system, the high order nonlinearities can be seen from the fields equations through adiabatic elimination of the atomic variables under the symmetrical cavity detunings Δ_c1 = −Δ_c2 = Δ_c, and symmetrical parameters g_1,2 = g, and κ_1,2 = κ. For equal input intensity $I_{\text{in}}=\frac{4{\left|g{\epsilon }_{1,2}\right|}^2}{{\kappa }^2{\gamma }^2}$, we obtain the steady state equation for the equal field intensity $I=\frac{{\left|g〈a_{1,2}〉\right|}^2}{{\gamma }^2}$ as

2.1. Bogoliubov mode interactions with dressed atoms

First we make a unitary transformation [52] via $U_1={\prod }_{j=1}^2\text{exp}\left({\lambda }_j{a}_j^{†}-{\lambda }_j^{*}{a}_j\right)$, where ${\lambda }_j=\frac{{\epsilon }_j}{\kappa +i{\mathrm{\Delta }}_{cj}}$ (j = 1, 2), and linearize the cavity fields a_j = 〈a_j〉 + δa_j. By doing this we rewrite the Hamiltonian (2) as

Then we transform to a dressed states representation. We assume that the Rabi frequencies are real, equal and much stronger than the atomic and cavity decay rates Ω = |Ω| ≫ (γ, κ). By diagonalizing the Hamiltonian H_a, we obtain the dressed states that are expressed in terms of the bare atomic states as [53]

In the following calculations, while we give dependence of quantum correlations on the scaled atom-driving detunging $\frac{\mathrm{\Delta }}{\mathrm{\Omega }}$, the cavities are tuned to be resonant with the Rabi sidebands Δ_c1 = −Δ_c2 = d. All results we will obtain below hold when ||Δ_c1,2| − d| ≪ d. An increase in the cavity detunings will spoil the quantum correlations. If the deviations of the cavity detunings from Rabi resonances are comparable to or larger than the level spacing d, the two cavity fields are not in selective interactions with the higher and lower sidebands and the present treatment does no longer hold. Throughout the present work we assume that the above conditions for Rabi resonances are guaranteed.

Following the above treatment, we write the total free Hamiltonian H₀ of atom-field system as H₀ = H_a + H_c. By using the dressed atomic states, and making a further unitary transformation with U₂ = exp (−iH₀t/ħ) and taking a rotating-wave approximation, we rewrite the interaction Hamiltonian H_I as

 

Fig. 2 The individual cavity fields a_1,2 are both in resonant interactions with the dressed atoms. The two fields combine into the Bogoliubov mode b₁ for Δ > 0 (N₋ > N₊) or b₂ for Δ < 0 (N₊ > N₋).

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2.2. Dissipation and two-mode squeezing

Now we use the interactions of the Bogoliubov modes with the dressed atoms to analyze the field correlations and the responsible mechanism.

2.2.1. Dissipation and two-photon processes

We assume that the atomic variables decay much more rapidly than the cavity fields, i.e., γ ≫ κ. Then by tracing out the atomic variables, we derive the master equation for the reduced density operator of the cavity modes ρ̃ = Tr_a ρ as

When A − B ≫ κ, i.e., the absorption (i.e., the dissipation) by the atoms dominates [3] over the gain (i.e., the amplification). Since the Bogoliubov mode b₁ undergoes dissipation, and it evolves into thermal vacuum state. It should be noted that b₂ mode decouples from the atoms. In contrast, for the case of Δ < 0, the Bogoliubov mode b₂ undergoes dissipation and evolves into thermal vacuum state while b₁ mode decouples from the atoms. Expanding Eq. (15) in terms of a_1,2 we have the terms

2.2.2. Approximately analytic two-mode variances

To show clearly the effects of dissipation on the field correlations, we first give the approximately analytic description. For this purpose we assume that dissipation by the dressed atoms dominates over the vacuum reservoir and neglect temporarily the cavity losses. We define the quadrature operators $\delta x_o=\frac{1}{\sqrt{2}}(o+o^{†})$ and $\delta p_o=\frac{-i}{\sqrt{2}}(o-o^{†})$, o = a_1,2, b_1,2. By using the Bogoliubov transformation relations, we can express the two-mode quadrature operators in terms of the Bogoliubov modes as

Due to the dissipation by the atoms, the b₁ mode evolves into its thermal vacuum state. From Eqs. (14) and (15) we can obtain nonvanishing correlation $〈b_1^{†}{b}_1〉=\frac{N_{+}}{N_{-}-N_{+}}$ for Δ > 0. Similarly we have the nonvanishing correlation $〈b_2^{†}{b}_2〉=\frac{N_{-}}{N_{+}-N_{-}}$ for Δ < 0. Thus we have the quantum fluctuations $〈:{\left(\delta x_{b_j}\right)}^2:〉=\frac{1}{2}〈b_j^2+{b_j^{†}}^2+2b_j^{†}{b}_j〉=〈b_j^{†}{b}_j〉$ for Δ > 0 (j = 1) and Δ < 0 (j = 2). Finally the two-mode correlations are obtained as

 

Fig. 3 Approximately analytic variance 〈(δX)²〉 versus the scaled detuning $\frac{\mathrm{\Delta }}{\mathrm{\Omega }}$. Within the shadow at $\frac{\mathrm{\Delta }}{\mathrm{\Omega }}=0$, the conditions for approximately analytic solution are not well met.

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2.2.3. Numerical verification and discussion

In order to include the cavity losses and to calculate the quantum correlations, we can apply a standard Langevin approach by means of the generalized P representation [54–56]. The individual operators are arranged in the normal ordering $a_1^{†}$, $a_2^{†}$, a₂, a₁, and the correspondences between the c numbers and the operators are defined as α_j ↔ a_j, ${\alpha }_j^{*}\leftrightarrow a_j^{†}$ (j = 1, 2). The c number Langevin equations for the individual modes are derived from Eqs. (14) and (15) as

In Fig. 4, we show the numerical results for different rates of cavity losses. It is seen that the squeezing exists almost in the entire region except at Δ = 0. This is in a good agreement with the analytic solution. Near the point Δ = 0, the populations trend to be equal, N₊ ≈ N₋. The absorption term proportional to A is no longer dominant over the amplification term proportional to B. Therefore, the Bogoliubov mode dissipation by the atoms is greatly degraded and can’t suppress the cavity loss. As Δ approaches zero, the squeezing reduces and disappears finally at Δ = 0. This shows that the squeezing and entanglement don’t occur when the dispersive nonlinearity vanishes completely at Δ = 0. The best squeezing arrives at 50%.

 

Fig. 4 The variance 〈(δX)²〉 versus the scaled detuning $\frac{\mathrm{\Delta }}{\mathrm{\Omega }}$ for different rates of cavity losses κ = 0.2γ (red dashed), κ = 0.1γ (black dotted), and κ = 0.01γ (blue solid). We have used the other parameter $g\sqrt{N}=10\gamma$.

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Finally, we should note the essentially different ways between the monochromatic and bichromatic interactions with atoms for establishing the two-photon processes, which are responsible for the squeezing. For the present case, the two-photon processes are established by using two fields simultaneously to a single atomic transition. In sharp contrast, for the monochromatic case, the two-photon processes are created by using two cascaded atomic transitions (e.g., the three-level atoms in Λ configurations), each of which is coupled to one monochromatic field. For the bichromatic case on a single transition, although only either of two Bogoliubov modes mediates interactions and the best squeezing is limited to 50%, the parameter range for the squeezing is the almost entire region except Δ = 0. For the cascaded transitions with monochromatic field, although two Bogoliubov modes mediate interactions and the squeezing can be enhanced for especial parameters, the parameter range for squeezing is greatly limited [20]. Combining the above two cases we can use the three-level atoms for cascade bichromatic interactions and for four-mode squeezing, as will be shown in the following section.

3. Cascade bichromatic interactions with three-level atoms

In our setup, four optical cavities intersect at a common spot, where an ensemble of N three-level Λ-type atoms is placed, as shown in Fig. 5(a). The atom has two metastable states |1〉 and |2〉 and one excited state |3〉. The two cavity fields (the annihilation operators a_j and a_j+2, the creation operators $a_j^{†}$ and $a_{j+2}^{†}$, and the frequencies ω_cj and ω_cj+2) are driven by the classical fields of the same frequency ω_j, and coupled to the electronic dipole-allowed transition |j〉 − |3〉 (j = 1, 2), as shown in Fig. 5(b). The master equation for density operator ρ of the atom-field system is written in the dipole approximation and in an appropriate rotating frame as [2, 3] $\dot{\rho }=-\frac{i}{\hslash }[H,\rho ]+ℒ\rho$, where the Hamiltonian of the system is written as

 

Fig. 5 (a) Four optical cavities intersect at a common spot, where an atomic ensemble of three-level atoms is placed. (b) The interaction of the four cavity fields with the atoms in Λ configuration.

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For the bistable or multistable system, the high order nonlinearities can be seen from the fields equations through adiabatic elimination of the atomic variables under the symmetrical detunings Δ₁ = −Δ₂ = −Δ and Δ_cj = −Δ_cj+2 = Δ_c (j = 1, 2), and equal parameters g_1–4 = g, γ_1,2 = γ, and κ_1–4 = κ. For equal input intensity $I_{\mathit{in}}=\frac{{\left|g{\epsilon }_l\right|}^2}{{\kappa }^2{\gamma }^2}$, and the collective field intensity $I=\frac{{\left|g〈a_l〉\right|}^2}{{\gamma }^2}$ (l = 1 – 4) satisfies the nonlinear equation I_in = I[(1 + 𝒜)² + (Δ̄_c + 𝒟)²], where 𝒜 and 𝒟 represent respectively the nonlinear absorption and dispersion by the atoms and depend strongly on the cavity field intensity I through the relations

3.1. Bogoliubov-like four-mode interactions with dressed atoms

In order to extract the mechanism hidden behind the nonlinearities, we merge the nonlinearities into the dressed atomic states and analyze the dressed atom-photon interactions as follows. First, we make a unitary transformation [52] via $U_1={\prod }_{j=1}^4\text{exp}\left({\lambda }_j{a}_j^{†}-{\lambda }_j^{*}{a}_j\right)$, where ${\lambda }_j=\frac{{\epsilon }_j}{\kappa +i{\mathrm{\Delta }}_{c_j}}$ (j = 1 – 4). Then by linearizing the cavity fields a_j = 〈a_j〉 + δa_j, we rewrite the Hamiltonian as H = H_a + H_c + H_I, where

We transfer the common phase of Ω_j + Ω_j+2 to the atomic operators and are left with ${\overline{\mathrm{\Omega }}}_j=\left|{\mathrm{\Omega }}_j+{\mathrm{\Omega }}_{j+2}{e}^{i\left({\phi }_{j+2}-{\phi }_j\right)}\right|=\frac{\mathrm{\Omega }}{2}\left|1+e^{i\mathrm{\Phi }}\right|$, where we have assmued that Rabi frequencies have their phases φ_j+2 − φ_j = Φ, j = 1, 2. We also assume that the amplitudes of Rabi frequencies are equal and much stronger than the atomic and cavity decay rates, Ω̄_j ≫ (γ, κ). By diagonalizing the Hamiltonian H_a, we obtain the dressed states, which are expressed in terms of the bare atomic states as [53]

Here we focus on the off-resonance case, Δ ≠ 0. As in the case of two-level atoms, we also assume that the conditions for Rabi resonances are guaranteed, Δ_c1 = −Δ_c2 = d. The total free Hamiltonian H₀ of atom-field system is written as H₀ = H_a + H_c. At the same time we write the interaction Hamiltonian H_I in terms of the dressed atomic states, and make a further unitary transformation with U₂ = exp (−iH₀t/ħ) and a rotating-wave approximation. Then we obtain the interaction Hamiltonian

 

Fig. 6 The dressed atomic transitions for N₀ > N_±. The left blue and red lines with arrows mean respectively the annihilation of the a_1,2 modes and the creation of the a_3,4 modes and constitute the collective b₁ mode. The right red and blue lines with arrows show the annihilation of the a_3,4 modes and the creation of the a_1,2 modes and constitute the collective b₂ mode.

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We define four Bogoliubov-like modes for the parameter regime (i) $\frac{\left|\mathrm{\Delta }\right|}{\overline{\mathrm{\Omega }}}<1$ as follows,

 

Fig. 7 (a) Interactions of the collective b_1,2 modes with the dressed atoms via the transitions (i) from |0〉 to |±〉 (N₀ > N_±) (ii) from |±〉 to |0〉 (N_± > N₀). (b) The interactions between any two individual fields. All four fields are in a quadrilateral loop of two-photon interactions and in the X lines of two quantum-beat interactions. Any three fields are in a triangle of two two-photon interactions and one quantum-beat interaction.

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3.2. Dissipation and four-mode squeezing

In what follows we use the above Hamiltonian to discuss the four-mode correlations. We first show a quadrilateral loop of two-photon processes, then give the apprroximately analytic solution, and finally present our numerical results.

3.2.1. Dissipation and a quadrilateral of two-photon processes

We assume that the atomic variables decay much more rapidly than the cavity fields, i.e., γ ≫ κ. Then by tracing out the atomic variables, we derive the master equation for the reduced density operator of the cavity modes ρ̃ = Tr_a ρ as $\dot{\tilde{\rho }}={𝒦}_a\tilde{\rho }+{ℒ}_c\tilde{\rho }$, where ${𝒦}_a\tilde{\rho }=-\frac{i}{\hslash }{\text{Tr}}_a\left[H_I,\rho \right]$ describes the atom-field interaction. Following the standard technique [2,3], we derive the atomic contribution part for (i) $\frac{\left|\mathrm{\Delta }\right|}{\overline{\mathrm{\Omega }}}<1$ as

When A − B ≫ (κ, γ_c), i.e., the absorption by the atoms dominates [3], and thus the collective modes b_1,2 undergo dissipation and reduce to the thermal vacuum states. This is the very effect of the dissipation by the dressed atoms. Expanding the dominant terms (those proportional to A and B), we can find two types of interactions between the different individual modes a_k,l. One type is the two-photon interactions as in the squeezed vacuum reservoir case [2,3] and described by the terms

The loop interactions are the most remarkable feature of the multimode atom-field interactions. All of these interactions happen either in the absorption of collective modes (described by the A term in Eq. (32)) or in the amplification of the collective modes (described by the B term) [2, 3]. When the absorption dominates over the amplification, the four-mode system will evolve into a multimode steady state. As will be shown below, the dissipation based two-photon processes will determine the existence of four-mode squeezing.

3.2.2. Approximately analytic four-mode variances

In order to show the dissipation effects on the multimode correlations, we rewrite the multimode annihilation operator b₁ in terms of collective annihilation and creation operators as

To show this clearly, we first consider the dominant terms (the A and B terms in Eq. (32)) and neglect temporarily those minor terms (the cavity relaxations and the coherence transfer). Under such conditions, we can give the approximately analytic expression of the four-mode quantum correlations. We introduce the quadrature operators $\delta x_o=\frac{1}{\sqrt{2}}(o+o^{†})$ and $\delta p_o=\frac{-i}{\sqrt{2}}(o-o^{†})$, o = a_1–4. By using the reversible transformation from Eq. (30), we can express the individual modes a_1–4 (or b_1–4) in terms of collective modes b_1–4. On the basis of the above two-collective-mode Bogolubov transform in Eq. (38) we focus on the four-mode operator

If the variance satisfies the relation 〈(δX)²〉 < 1, the multimode squeezing exists. We take Φ = 0 and plot the variance 〈(δX)²〉 versus the scaled detuning $\frac{\mathrm{\Delta }}{\mathrm{\Omega }}$ in Fig. 8. The four-mode squeezing exists in the almost entire regime except at the specific points, i.e., $\frac{\left|\mathrm{\Delta }\right|}{\mathrm{\Omega }}=(0,1,\sqrt{2})$. At $\frac{\left|\mathrm{\Delta }\right|}{\mathrm{\Omega }}=0$, the atoms are trapped in the dark state and the cavity fields are decoupled from the atoms. No squeezing occurs at the exact dark resonance. At $\frac{\left|\mathrm{\Delta }\right|}{\mathrm{\Omega }}=\sqrt{2}$, the two-collective-mode squeezing parameter is not well defined. This problem is avoided when we treat the individual modes, because this definition is no longer necessary. At $\frac{\left|\mathrm{\Delta }\right|}{\mathrm{\Omega }}=1$, the atoms are equally populated in three dressed states i.e., N₀ = N_±, the absorption is equal to the gain and no net dissipation occurs. In fact, about $\frac{\left|\mathrm{\Delta }\right|}{\mathrm{\Omega }}=1$ (as shown by shadows in Fig. 8), the condition A − B ≫ κ is not well met, i.e., the absorption by the atoms does no more play a dominant role. The approximately analytic solution does not hold about $\frac{\left|\mathrm{\Delta }\right|}{\mathrm{\Omega }}=1$. As will be shown in the following subsection, the best achievable squeezing is 50% and appears still near $\frac{\left|\mathrm{\Delta }\right|}{\mathrm{\Omega }}=1$. We should also point out that we can obtain the same variance for other collective modes

 

Fig. 8 Approximately analytic four-mode variance 〈(δX)²〉 versus the scaled detuning $\frac{\mathrm{\Delta }}{\mathrm{\Omega }}$ for a fixed phase Φ = 0. Four-mode squeezing occurs in almost entire region except at $\frac{\left|\mathrm{\Delta }\right|}{\mathrm{\Omega }}=0,1,\sqrt{2}$. Within two shadows at $\frac{\left|\mathrm{\Delta }\right|}{\mathrm{\Omega }}=1$, the conditions for approximately analytic solution are not well met.

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3.2.3. Numerical verification and discussion

Now we present a numerical calculation by including the vacuum damping and the coherence transfer. We can apply a standard Langevin approach by means of the generalized P representation [54–56]. The individual operators are arranged in the normal ordering $a_1^{†}$, $a_2^{†}$, $a_3^{†}$, $a_4^{†}$, a₄, a₃, a₂, a₁, and the correspondences between the c numbers and the operators are defined as α_j ↔ a_j, ${\alpha }_j^{*}\leftrightarrow a_j^{†}$ (j = 1 – 4). The c number Langevin equations for individual modes are derived as