Quantum-beat based dissipation for spin squeezing and light entanglement

Chen Huang; Xiangming Hu; Yang Zhang; Lingchao Li; Shi Rao

doi:10.1364/OE.24.019094

1. Introduction

Squeezing and entanglement lie at the heart of quantum optics and quantum information because of their potential applications such as high-precision measurement [1, 2], quantum computing [3,4] and dense coding [5,6]. Once one of two orthonormal quadratures of a system has its fluctuations below the standard quantum limit, we call that the system is in its squeezed state [7]. Two systems are inseparable if the total variance of a pair of Einstein-Podolsky-Rosen-like operators is below the standard quantum limit [8, 9]. This is known as the continuous variable entanglement. Recently, the dissipation effects have been explored to prepare the squeezed and entangled states in different schemes. Rabi interactions with two successive beams of dressed atoms can act as dissipative processes and make two optical fields into their squeezed and entangled states [10]. Raman interactions mediated by two vacuum cavity fields and classical fields can be used as a dissipative reservoir to drive two atomic ensembles into their squeezed and entangled states [11–14]. In particular, it has been found that such dissipation mechanism, which is responsible for spin squeezing at steady state, hides deeply behind the coherence-induced nonlinearities in coherent population trapping (CPT) [15]. It is well known that CPT is one of the most remarkable coherence effects in the atom-photon interactions [16–19]. The atomic coherence is established between the two ground states and the long-lived coherence plays a decisive role in the quantum correlations. So far, the CPT coherence effects on the quantum correlations have been investigated intensively theoretically [20–23] and experimentally [24–27]. It seems that the long-lived coherence between the two ground states as in CPT case is indispensable for the spin squeezing.

In this article we suggest that quantum beat serves as a different alternative for the dissipation induced spin squeezing and light entanglement. In the quantum beat system [28–34], the atomic coherence is established between the excited states, and the long-lived coherence no longer appears. Typically, quantum beat is formed in a V-type three-level atom when it is resonantly coupled to two classical fields. In terms of the proper superposition states of the excited states, one knows that one of the superposition states is decoupled from the driving fields and becomes empty. Sometimes this is called the phenomenon of “coherent depopulation” [35], which is the very counterpart of CPT case. This shows that the coherence is created between two excited states. The coherence effects are essentially different between the CPT and quantum-beat cases. The absorption vanishes for the involved fields in the CPT case, while there are not spontaneous emission photons into the beat in the quantum-beat case although the atoms are excited [28–34]. The latter is the very essence of quantum noise reduction in the quantum beat lasers. So far, the related works for light entanglement in quantum beat have mainly focused on the vacuum fields or relatively weak fields which do not saturate the atoms. To our knowledge, however, little work has been done for the case in which the bright quantized fields near-resonantly excite the atoms and establish the quantum-beat based nonlinearities.

We reveal the existence of the same dissipation mechanism as in the CPT case behind the quantum-beat based nonlinearities and almost the same efficiency for the spin squeezing and light entanglement. Nearly ideal two-mode spin squeezing and light entanglement are reachable when we substitute the excited-state coherence for the long-lived ground-state coherence. It is a little surprising for us at the first glance. We will give a detailed analysis and a comprehensive comparison with the CPT case. Once the quantum beat is detuned from the resonance transitions, the coherence induced nonlinearities give rise to the dissipation of the Bogoliubov modes of the two cavity fields, and to the dissipation of the Bogoliubov modes of the two dressed atom spins. Two-mode squeezing and entanglement are obtainable either for two optical fields or for the two dressed atom spins. More importantly, the two-mode squeezing of dressed spins corresponds to the steady state squeezing of the excited-state spin. The present scheme is accessible experimentally based on the experimental progress in the quantum correlations [24–27] and in the quantum-beat laser [33,34].

The remaining part of this article is organized as follows. In Sec. 2, we first give a brief review of the quantum beat system. Then we give the master equation of the atom-field interacting system and show the interactions between dressed atoms and the cavity fields. In Sec. 3, we present the dissipation mechanism in our detuned quantum beat system. The two subsections present the physical analysis of the Bogoliubov mode dissipations for the cavity fields and for the dressed atom spins, respectively, under respective adiabatic conditions. In Sec. 4, we give a numerical verification of the quantum correlations. The first subsection gives the numerical calculation of two-mode squeezing and entanglement, and the second subsection shows the numerical verification of the excited-state spin squeezing. Finally, discussion and conclusion are given in Sec. 5.

2. Model and master equation

We now place an ensemble of N three-level atoms in V configuration, as shown in the inset of Fig. 1(a), at the intersection of two optical cavities. These optical cavities are pumped by their respective external fields. As depicted in Fig. 1, the two cavity fields are respectively coupled to the two electronic dipole-allowed transitions of the atoms from two excited states |1, 2〉 to the ground state |3〉. The master equation for the density operator ρ of the atom-field system is derived in the following form [36]

\dot{ρ} = - \frac{i}{ℏ} [H, ρ] + ℒ ρ,

where the system Hamiltonian reads as

H = ℏ \sum_{l = 1, 2} [Δ_{l} σ_{l l} + Δ_{c_{l}} a_{l}^{†} a_{l} + g_{l} (a_{l} σ_{l 3} + σ_{3 l} a_{l}^{†}) + i (ε_{l} a_{l}^{†} - ε_{l}^{*} a_{l})],

where

σ_{j k} = \sum_{μ = 1}^{N} σ_{j k}^{μ}

(

σ_{j k}^{μ} = | j_{μ} 〉 〈 k_{μ} |

; j, k = 1, 2, 3) are the collective projection operators for j = k and the spin-flip operators for j ≠ k. a_l and

a_{l}^{†}

(l = 1, 2) are annihilation and creation operators for the light fields. Δ_l = ω_3l − ω_l and Δ_{c_l} = ω_{c_l} − ω_l are, respectively, the detunings of atomic transition frequencies ω_3l and cavity resonance frequencies ω_{c_l} with respect to the driving field frequencies ω_l. g_l are the coupling strengths for the atom-cavity fields and ε_l are the classical driving field amplitudes. The damping term in the master equation takes the form

\begin{array}{l} ℒ ρ = & ℒ_{a} ρ + ℒ_{c} ρ, \\ ℒ_{a} ρ = & \sum_{l = 1, 2} \frac{γ_{l}}{2} (2 σ_{3 l} ρ σ_{l 3} - σ_{l 3} σ_{3 l} ρ - ρ σ_{l 3} σ_{3 l}), \\ ℒ_{c} ρ = & \sum_{l = 1, 2} κ_{l} (2 a_{l} ρ a_{l}^{†} - a_{l}^{†} a_{l} ρ - ρ a_{l}^{†} a_{l}), \end{array}

where ℒ_a ρ and ℒ_c ρ describe the atomic and cavity decays with rates γ_l and 2κ_l, respectively.

Fig. 1 Diagrammatic sketch of quantum beat system. (a) An ensemble of V-type atoms is coupled to two quantized fields a_1,2. Inset: The V-type atoms are placed at the intersection of two cavity fields which are driven by two classical fields ε_1,2, respectively. (b) As an equivalent form, the collective modes a_b,d are coupled to the two collective transitions |b, d〉 ↔ |3〉 respectively. No population is in the state |d〉 due to the spontaneous emission |d〉 → |3〉 and the absence of pumping field from |3〉 to |d〉.

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To obtain explicit expressions for the dressed atom-cavity interactions and analytical solution, we focus on the situation of symmetric parameters g_1,2 = g, γ_1,2 = γ and κ_1,2 = κ, in which the dressed sublevels are equally spaced and the cavity fields are resonant with the corresponding dressed transitions. It is convenient for us to make a unitary transformation [37] via $U_{1} = \prod_{l = 1, 2} exp (λ_{l} a_{l}^{†} - λ_{l}^{*} a_{l})$ with $λ_{l} = \frac{ε_{l}}{κ + i Δ_{c_{l}}}$ , and to linearize the cavity fields a_l = 〈a_l〉 + δa_l (l = 1, 2). After doing so, we decompose the Hamiltonian into three parts:

H = H_{a} + H_{c} + H_{I},

where

\begin{array}{l} H_{a} = & ℏ \sum_{l = 1, 2} (Δ_{l} σ_{l l} + Ω_{l} σ_{l 3} + Ω_{l}^{*} σ_{3 l}), \\ H_{c} = & ℏ \sum_{l = 1, 2} Δ_{c_{l}} δ a_{l}^{†} δ a_{l}, \\ H_{I} = & ℏ \sum_{l = 1, 2} g (δ a_{l} σ_{l 3} + σ_{3 l} δ a_{l}^{†}) . \end{array}

Here, H_a represents the interaction of the atoms with both the classical pump fields ε_l and the semiclassical part of the cavity fields 〈a_l〉 (with total Rabi frequency Ω_l = λ_l + g〈a_l〉). H_c represents the free Hamiltonian of the two fluctuating fields. H_I represents the interaction of the atoms with the fluctuating fields. In what follows we first recall the quantum correlation under resonance condition before going to the detuned quantum-beat case.

2.1. Quantum beat

The quantum coherence effect in quantum beat is more explicit in resonant system (Δ_l = Δ_{c_l} = 0). In that case, the second part of the Hamiltonian H_c vanishes. We introduce the superposition states of the atoms

\begin{array}{l} | b 〉 = & cos β | 1 〉 + sin β | 2 〉, \\ | d 〉 = & - sin β | 1 〉 + cos β | 2 〉, \end{array}

where Ω_1,2 have been assumed to be real and we have defined

tan β = \frac{Ω_{2}}{Ω_{1}}

. Substituting the superposition states Eq. (6) into Eq. (5), we rewrite the Hamiltonian H_a as

H_{a} = ℏ Ω (σ_{b 3} + σ_{3 b}),

where

Ω = \sqrt{Ω_{1}^{2} + Ω_{2}^{2}}

. From the Eq. (7), we notice that only the superposition state |b〉 is coupled to the driving fields and the other superposition state |d〉 no longer participates in the interaction, as shown in Fig. 1(b). Naturally, the spontaneous emission |d〉 → |3〉 empties the population on the superposition state |d〉 even when it is populated initially. Eventually, there will be no population at all on the superposition state |d〉 at steady state

〈 σ_{d d} 〉 = 0 .

This phenomenon is sometime known as coherence-induced depopulation [35]. Obviously, this is the very counterpart of CPT. The comparison of quantum beats with CPT will be given in Sec. 5. We stress that the spontaneous emission |d〉 → |3〉 vanishes because of 〈σ_dd〉 = 0 at steady state while the spontaneous emission |b〉 → |3〉 is kept. As a consequence, two spontaneous emissions |1, 2〉 → |3〉 superpose into one |b〉 → |3〉, which is the phenomenon of quantum beats [36]. For each atom, the coherence between the two excited states reaches its maximum value

\frac{〈 σ_{12} 〉}{N} = \frac{1}{4},

when

cos β = sin β = \frac{1}{\sqrt{2}}

and Ω ≫ γ are satisfied.

This coherence leads to both quantum beat signal and the quantum noise reduction. In terms of coherent superposition states as above, we define two orthonormal collective modes a_b,d as

\begin{array}{l} a_{b} = a_{1} cos β + a_{2} sin β, \\ a_{d} = - a_{1} sin β + a_{2} cos β, \end{array}

Here we call a_b the sum mode and a_d the difference mode. Hence, the Hamiltonian for the interaction of two fluctuating fields with the atoms is rewritten as

H_{I} = ℏ g (δ a_{b} σ_{b 3} + δ a_{d} σ_{d 3}) + H . c .,

where we have used H.c. for Hermitian conjugate of the terms before it. We note that the difference mode is only coupled to the superposition state |d〉. The transition from |3〉 to |d〉 corresponds to absorption of the difference mode. Since the superposition state |d〉 remains de-excited, the difference mode undergoes only absorption. This determines that the difference mode always stays in its vacuum state, 〈a_d〉 = 0. This also indicates that modes a_1,2 have the same phase φ. Essentially, two modes a_1,2 are pulled into a quantum beat. This leads to the absence of the noise from the spontaneous emission |d〉 → |3〉. This is the very essence of quantum beat laser [28–32]. The Hermitian operators corresponding to the relative amplitude r_a and relative phase ψ_a for the a_1,2 modes are defined as [32]

\begin{array}{l} r_{a} & \equiv x_{a_{1}} sin β - x_{a_{2}} cos β = - \sqrt{2} Re (a_{d} e^{i φ}), \\ ψ_{a} & \equiv p_{a_{1}} sin β - p_{a_{2}} cos β = - \sqrt{2} Im (a_{d} e^{i φ}), \end{array}

where

x_{a_{l}} = \frac{1}{\sqrt{2}} (a_{l} e^{i φ} + a_{l}^{†} e^{- i φ})

,

p_{a_{l}} = \frac{- i}{\sqrt{2}} (a_{l} e^{i φ} - a_{l}^{†} e^{- i φ})

, l = 1, 2. Since phase noise is reduced, the mode a_d remains in its vacuum state and the variances of the relative amplitude and the relative phase are at their vacuum levels

〈 {(δ r_{a})}^{2} 〉 = 〈 {(δ ψ_{a})}^{2} 〉 = \frac{1}{2} .

In the long time limit, we obtain the two quantum beat signals for the a_1,2 modes [32]

Re 〈 a_{1}^{†} a_{2} 〉 = \frac{1}{2} sin (2 β) cos [(ω_{c_{1}} - ω_{c_{2}}) t] 〈 a_{b}^{†} a_{b} 〉 \neq 0 .

2.2. Detuned quantum beat system

We now concentrate on the coherence and the nonlinearities of detuned quantum beat system. We introduce symmetrical detunings Δ₁ = −Δ₂ = Δ ≠ 0 and Δ_c₁ = −Δ_c₂ = Δ_c ≠ 0. Hence, different from the resonant case, the free part for the fluctuating fields H_c takes a non-vanishing form

H_{c} = ℏ Δ_{c} (a_{1}^{†} a_{1} - a_{2}^{†} a_{2}),

where we have dropped the symbol δ and do so from now on (i.e., by a_l we mean δa_l). By transferring the phase of Ω_l to the atomic operators, we have real values for Ω_l = |Ω_l|. We assume that the Rabi frequencies are equal and much stronger than the atomic and cavity decay rates Ω_l = Ω ≫ (γ_1,2, κ_1,2). After diagonalizing H_a, we obtain the dressed states that are expressed in terms of the bare atomic states as [38]

(\begin{matrix} | + 〉 \\ | 0 〉 \\ | - 〉 \end{matrix}) = (\begin{matrix} \frac{1 + sin θ}{2} & \frac{1 - sin θ}{2} & \frac{cos θ}{\sqrt{2}} \\ - \frac{cos θ}{\sqrt{2}} & \frac{cos θ}{\sqrt{2}} & sin θ \\ \frac{1 - sin θ}{2} & \frac{1 + sin θ}{2} & - \frac{cos θ}{\sqrt{2}} \end{matrix}) (\begin{matrix} | 1 〉 \\ | 2 〉 \\ | 3 〉 \end{matrix}),

where we have defined

cos θ = \frac{\sqrt{2} Ω}{Ω}

,

sin θ = \frac{Δ}{Ω}

and

\bar{Ω} = \sqrt{Δ^{2} + 2 Ω^{2}}

. These dressed states |0〉 and |±〉 have their eigenvalues E_0,± = 0, ±Ω̄, which are equally spaced. The free Hamiltonian H_a for the dressed atoms now becomes

H_{a} = ℏ \bar{Ω} (σ_{+ +} - σ_{- -}) .

Transforming the relaxation terms of the atoms to the dressed-state representation, we obtain the steady-state populations N_l = 〈σ_ll〉 (l = 0, ±) as

\begin{array}{l} N_{0} = \frac{N ({sin}^{4} θ + {sin}^{2} θ)}{3 {sin}^{4} θ - 3 {sin}^{2} θ + 2}, \\ N_{+} = N_{-} = \frac{1}{2} (N - N_{0}) . \end{array}

From Eq. (18), we obtain

\begin{array}{l} N_{\pm} > N_{0} for | \frac{Δ}{Ω} | < 1, \\ N_{0} > N_{\pm} for | \frac{Δ}{Ω} | > 1 . \end{array}

When Δ = 0, we have N₀ = 0, which is the same as Eq. (8). That means that the atoms are coherently depopulated [35].

Here, we focus on the case of non-vanishing detunings Δ₁ = −Δ₂ = Δ ≠ 0. In terms of the dressed atomic states, H_a and H_c constitute the total free Hamiltonian for the dressed atom-field system

\begin{array}{l} H_{0} & = H_{a} + H_{c} \\ = ℏ \bar{Ω} (σ_{+ +} - σ_{- -}) + ℏ Δ_{c} (a_{1}^{†} a_{1} - a_{2}^{†} a_{2}) . \end{array}

We tune the cavity fields resonant with the Rabi sidebands Δ_c = Ω̄. The case of Δ_c = −Ω̄ is treated in the same way, and the dependence of the quantum correlations on Δ/Ω is obtained by exchanging all figures below symmetrically with respect to the Δ = 0 axis. The dressed states are well separated from each other since Ω ≫ (γ_1,2, κ_1,2). We can make a unitary transformation with U₂ = exp(−iH₀t/ħ) and a rotating-wave approximation. After doing so, we obtain the interaction Hamiltonian

\begin{array}{l} H_{I} & = \frac{1}{2} ℏ g [a_{1} sin θ (1 + sin θ) + a_{2}^{†} {cos}^{2} θ] σ_{+ 0} \\ + \frac{1}{2} ℏ g [a_{2} sin θ (1 + sin θ) + a_{1}^{†} {cos}^{2} θ] σ_{- 0} + H . c . . \end{array}

We note that the operators a_1,2 represent the fluctuating parts of the cavity fields, and they do not change the population of the dressed atoms. The atoms have their commutation relations [σ_0±, σ_±0] = σ₀₀ − σ_±±. At steady state, we have vanishing spin modes in our system 〈σ_0±〉 = 〈σ_±0〉 = 0. We obtain δσ_0± = σ_0± and δσ_±0 = σ_±0. Therefore, we denote the fluctuations of the dressed state spins δσ_0± (δσ_±0) by σ_0± (σ_±0). The z quadratures of the two spin modes have the mean values N₀ − N_±. The nonlinearities created by the semiclassical parts Ω_1,2 are merged into the dressed atomic states |0, ±〉 and the parameters cos θ and sin θ. In terms of the effective atomic states and the parameters, we separate out the interactions between the fluctuating parts of the fields and the spins. The entire system is now looked upon as four interacting parts, two of which are the cavity fields a_1,2 and the other two are the dressed atom spins σ_±0. As shown by the Hamiltonian (21), each field is in simultaneous resonant interactions with two cascaded dressed transitions, as shown in Fig. 2. Or to say, each dressed atomic spin is simultaneously coupled to two cavity fields. That means that the cavity fields are in Bogoliubov-like collective interactions with each dressed spin or that the dressed spins are in Bogoliubov-like collective interactions with each cavity field. We will focus on the dissipation of the collective modes of a_1,2 by the dressed spins and on the dissipation of the collective modes of σ_±0 by the cavity fields. Once the collective modes are damped, the cavity fields or the dressed spins are pulled into two-mode squeezed and entangled states.

Fig. 2 Interactions between the fluctuating parts of the cavity fields and the dressed atomic spins. The two cavity modes are coupled to each dressed transition.

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3. Physical analysis of atom-photon interaction effects

Here we first present a physical analysis of the effects of the above atom-photon interactions on quantum correlations. We assume that the atom-photon interactions dominate over the environmental reservoir damping. It is reasonable for us to discard temporarily the damping terms due to the vacuum environment.

3.1. Bogoliubov field dissipation by the atoms

If the atomic spins undergo adiabatic evolutions (γ ≫ κ), that will lead to dissipation common for the two cavity fields. To describe the collective dissipation, we introduce the Bogoliubov modes for the cavity fields [40]

\begin{array}{l} b_{1} = a_{1} cosh r - e^{i ϕ} a_{2}^{†} sinh r, \\ b_{2} = a_{2} cosh r - e^{i ϕ} a_{1}^{†} sinh r, \end{array}

where we have defined the squeezing parameter

r = arctanh (\frac{| sin θ |}{1 + sin θ})

for

\frac{Δ}{Ω} < \sqrt{\frac{2}{3}}

and

r = arctanh (\frac{1 - sin θ}{sin θ})

, for

\frac{Δ}{Ω} > \sqrt{\frac{2}{3}}

, and ϕ = 0 for

\frac{Δ}{Ω} < 0

and ϕ = π for

\frac{Δ}{Ω} < 0

. Then the interaction Hamiltonian (21) is rewritten as

H_{I} = \sum_{l = 1, 2} ℏ \tilde{g} (b_{l} σ_{l}^{+} + σ_{l} b_{l}^{†}),

where the coupling strength

\tilde{g} = \frac{1}{2} g (1 + sin θ) \sqrt{| 1 - 2 sin θ |}

, and we have defined the atomic spins

\begin{array}{l} σ_{1, 2} = σ_{\mp 0} for \frac{Δ}{Ω} < \sqrt{\frac{2}{3}}, \\ σ_{1, 2} = σ_{0 \pm} for \frac{Δ}{Ω} > \sqrt{\frac{2}{3}} . \end{array}

As shown in Eq. (23), we notice that the behavior of the interactions between the Bogoliubov modes of cavity fields and the dressed atom spins relies on the atom-field normalized detuning $\frac{Δ}{Ω}$ . On the other hand, from Eqs. (19) and (24), we know that σ_1,2 ( $σ_{1, 2}^{+}$ ) are flip-down (flip-up) operators in the regions of $\frac{Δ}{Ω} = (- 1, 0)$ , (0, $\sqrt{\frac{2}{3}}$ ), (1, ∞). For the rest regions $\frac{Δ}{Ω} = (- \infty, - 1)$ , ( $\sqrt{\frac{2}{3}}$ , 1), σ_1,2 ( $σ_{1, 2}^{+}$ ) are flip-up (flip-down) operators. Thus, combining that and the Hamiltonian (23), we find that the absorption of b_1,2 modes is dominant over the amplification in the regions of $\frac{Δ}{Ω} = (- 1, 0)$ , (0, $\sqrt{\frac{2}{3}}$ ), (1, +∞). The amplification of b_1,2 modes dominates over the absorption in the rest regions $\frac{Δ}{Ω} = (- \infty, - 1)$ , ( $\sqrt{\frac{2}{3}}$ , 1). It should be noted that the squeezing and entanglement of the fields is possible only when the absorption (dissipation) of the Bogoliubov modes dominates. As the Bogoliubov modes dissipate to thermal vacuum state, the squeezing and entanglement of the two cavity fields a_1,2 emerges. From the discussion we conclude that the squeezing and entanglement may appear in the regions of $\frac{Δ}{Ω} = (- 1, 0)$ , (0, $\sqrt{\frac{2}{3}}$ ), (1, +∞).

In terms of the Bogoliubov modes, we obtain an effective physical mechanism of the atom-photon interactions. This reveals that the Bogoliubov modes dissipation can lead to the squeezing and entanglement of the two cavity fields in certain regions. Now we concentrate on the origin of the squeezing and entanglement and search for the two-photon-like processes in our system. If we use the definitions of the σ_1,2 as shown in Eq. (24) to rewrite the Hamiltonian (21), we obtain a loop of quadripartite interactions between the two cavity fields and two dressed atom spins in the regions of $\frac{Δ}{Ω} = (- 1, 0)$ , (0, $\sqrt{\frac{2}{3}}$ ), (1, +∞), as depicted in Fig. 3(a). We stress that σ_1,2 ( $σ_{1, 2}^{+}$ ) are flip-down (flip-up) operators in these areas. Thus, the dressed atom-cavity interactions described in Hamiltonian (23) are both in the parametric amplifier types and in the beam-splitter types. It is known that the beam-splitter interactions lead to the quantum state transfer. The parametric amplifier interactions which are sometime called two-photon-like process are responsible for squeezing and entanglement. The adiabatically evolving atomic spin σ₁ (σ₂) entangles itself with a₂ (a₁), and transfers immediately its state to a₁ (a₂). As a consequence, the synchronized transferring and squeezing processes lead to the two-mode squeezing and entanglement of the cavity modes a_1,2. This also explains why there are no squeezing and entanglement at $\frac{Δ}{Ω} = 0$ . At that occasion, the two squeezing interaction terms in Eq. (22) vanish (sinh r = 0) while the two transferring interaction terms are kept. Physically, this breaks the synchronizing of the squeezing and transferring interactions between the engineered reservoir and the constituents of Bogoliubov modes. Thus, the disappearance of the two-mode spin squeezing in the resonance case is understood.

Fig. 3 Dissipation and entanglement of the cavity fields a_1,2 by the dressed atom spins σ_1,2. (a) The two-mode squeezing and entanglement of two cavity fields a_1,2 are based on the synchronized transferring and squeezing interactions that are introduced by the engineered reservoirs σ_1,2. (b) The Bogoliubov field modes b_1,2 are dissipated by the dressed atom spins σ_1,2.

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In order to simplify this sophisticated four-body interactions, we can transform the four-body interaction loop into two much simpler separated two-body interactions by using the Bogoliubov modes b_1,2 [see Eq. (22)]. More importantly, Fig. 3(b) gives us an explicit picture of how the dissipation mechanism affects our system, that is, two Bogoliubov field modes b_1,2 are dissipated by the two dressed atom spins σ_1,2 respectively. Once the dissipation of the Bogoliubov modes dominates over the environmental dissipation, the two cavity fields evolve to their squeezing and entangled states.

3.2. Bogoliubov spin dissipation by the fields

Similarly, if the cavity fields evolve adiabatically (κ ≫ γ), they will induce dissipation common for the two dressed atom spins. To show this clearly, we use the definitions of σ_1,2 as Eq. (24), and define the Bogoliubov modes for the spins

\begin{array}{l} π_{1} = σ_{1} cosh r - e^{i ϕ} σ_{2}^{+} sinh r, \\ π_{2} = σ_{2} cosh r - e^{i ϕ} σ_{1}^{+} sinh r, \end{array}

and rewrite the interaction Hamiltonian (21) in the form

H_{I} = \sum_{l = 1, 2} ℏ \bar{g} (a_{l} π_{l}^{†} + π_{l} a_{l}^{†}) .

Similar to the discussion above, the squeezing and entanglement is possible only when the absorption (dissipation) of Bogoliubov spin modes by the cavity reservoir dominates over that by the environmental reservoir. Thus, the squeezing and entanglement may occur in the regions of

\frac{Δ}{Ω} = (- 1, 0)

, (0,

\sqrt{\frac{2}{3}}

), (1, +∞). Utilizing the definition in Eq. (24), we find that the four-body interaction loop shown in Fig. 4(a) reveals the origin of the squeezing and entanglement when κ ≫ γ. The adiabatically evolving cavity fields a₁ (a₂) entangles itself with σ₂ (σ₁), and transfers immediately its state to σ₁ (σ₂). As a consequence, the dressed atom spin modes σ_1,2 are prepared in the two-mode squeezed and entangled state.

Fig. 4 Dissipation and entanglement of the dressed atom spins σ_1,2 by the cavity fields a_1,2. (a) The two-mode squeezing and entanglement of two cavity fields σ_1,2 are based on the synchronized transferring and squeezing interactions that are introduced by the engineered reservoirs a_1,2. (b) The Bogoliubov spin modes π_1,2 are dissipated by the cavity fields a_1,2.

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Parallel to the situation of γ ≫ κ, we can transform the four-body interaction loop into two much simpler separated two-body interactions as in Eq. (26). The two Bogoliubov spin modes π_1,2 are dissipated by the two cavity fields a_1,2 respectively, as shown in Fig. 4(b). What is more, the squeezing and entanglement of the dressed atom spins emerges when the dissipation of the Bogoliubov spin modes by the cavity reservoir dominates over that by the vacuum environmental reservoir.

4. Dependence of quantum correlations on system parameters

So far we have presented a physical analysis of the Bogoliubov mode dissipations for the cavity fields a_1,2 and for the dressed spins σ_1,2 by temporarily neglecting the vacuum damping. In this section we present a numerical verification by including both the spontaneous emission of the atoms and the cavity losses of the fields.

4.1. Two-mode field squeezing and two-mode spin squeezing

We exemplify the region of $\frac{Δ}{Ω} = (0, \sqrt{\frac{2}{3}})$ , and the other regions $\frac{Δ}{Ω} = (- \infty, 0)$ , ( $\sqrt{\frac{2}{3}}$ , +∞) are treated in similar way but not presented here. Following the standard technique, working in the dressed-state representation, and defining the c-number and operator correspondences α_1,2 ↔ a_1,2 and $v_{1, 2} \leftrightarrow \frac{i σ_{\pm 0}}{\sqrt{N_{\pm} - N_{0}}}$ , we derive the Langevin equations as follows [39]

\begin{array}{l} \frac{d α_{1}}{d t} = - κ α_{1} - \bar{g} v_{2} cosh r + \bar{g} v_{1}^{†} sinh r + F_{α_{1}}, \\ \frac{d α_{2}}{d t} = - κ α_{2} - \bar{g} v_{1} cosh r + \bar{g} v_{2}^{†} sinh r + F_{α_{2}}, \\ \frac{d v_{1}}{d t} = - Γ v_{1} - γ_{c} v_{2}^{†} + \bar{g} α_{2} cosh r + \bar{g} α_{1}^{†} sinh r + F_{v_{1}}, \\ \frac{d v_{2}}{d t} = - Γ v_{2} - γ_{c} v_{1}^{†} + \bar{g} α_{1} cosh r + \bar{g} α_{2}^{†} sinh r + F_{v_{2}}, \end{array}

where we have defined

\bar{g} = \tilde{g} \sqrt{N_{+} - N_{0}}

,

Γ = \frac{γ}{4} (2 + 3 {cos}^{2} θ - 2 {cos}^{4} θ)

, and

γ_{c} = \frac{γ}{8} {sin}^{2} (2 θ)

. The F’s are noise terms with zero means and correlations 〈F_O(t) F_O′ (t′)〉 = 2D_OO′δ (t − t′), where the nonzero diffusion coefficients are

2 D_{v_{l}^{†} v_{l}} = 2 Γ Π (Π = \frac{N_{0}}{N_{+} - N_{0}})

and

2 D_{α_{l} v_{l}} = 2 D_{α_{l}^{†} v_{l}^{†}} = \bar{g} sinh r

(l = 1, 2). It is seen clearly from Eq. (27) that the sinh r terms correspond to the parametric amplifier interactions while the cosh r terms describe the beam splitter interactions. The solutions at steady state are 〈α_1,2〉 = 〈v_1,2〉 = 0, and the stability analysis shows that the system is always stable in the involved region.

In order to describe the quantum correlations of the cavity fields and the dressed atom spins, we write the field and atomic fluctuation variables in terms of the position and moment quadratures as $δ x_{α_{l}} = \frac{1}{\sqrt{2}} (α_{l} + α_{l}^{†})$ , $δ p_{α_{l}} = \frac{- i}{\sqrt{2}} (α_{l} - α_{l}^{†})$ , $δ x_{v_{l}} = \frac{1}{\sqrt{2}} (v_{l} + v_{l}^{†})$ and $δ p_{v_{l}} = \frac{- i}{\sqrt{2}} (v_{l} - v_{l}^{†})$ , meanwhile the collective quadratures X_{α_±} = x_α₁ ± x_α₂, P_{α_±} = p_α₁ ± p_α₂, X_{v_±} = x_v₁ ± x_v₂ and P_{v_±} = p_v₁ ± p_v₂. Squeezing occurs when any of the variances of quadratures is less than unity [7]

{δ X_{α_{\pm}}^{2}, δ X_{v_{\pm}}^{2}, δ P_{α_{\pm}}^{2}, δ P_{v_{\pm}}^{2}} < 1,

where we have defined the variance δO² = 〈(δO)²〉 (O = X_{α_±}, P_{α_±}, X_{v_±}, P_{v_±}). Furthermore, the criteria for two-mode squeezing and entanglement for cavity fields and dressed atom spins are as follows [9]

\begin{array}{l} δ X_{α_{\pm}}^{2} + δ P_{α_{\pm}}^{2} < 2, \\ δ X_{v_{\pm}}^{2} + δ P_{v_{\mp}}^{2} < 2 . \end{array}

After obtaining the Langevin equations for the collective quadratures (X_{α_±}, P_{α_±}, X_{v_±}, P_{v_±}) from Eq. (27), we derive the steady-state variances

δ X_{α_{+}}^{2} = 1 - \frac{Γ_{1} (1 - e^{- 2 r}) - 2 Γ Π e^{- 2 r}}{(κ + Γ_{1}) (1 + C_{1}^{- 1})},

δ X_{v_{-}}^{2} = 1 - \frac{κ (1 - e^{- 2 r}) - 2 Γ Π (1 + \frac{κ + Γ_{2}}{C_{2} Γ_{2}})}{(κ + Γ_{2}) (1 + C_{2}^{- 1})},

where Γ_1,2 = Γ ± γ_c and C_1,2 = ḡ²/κΓ_1,2. Due to the symmetry of the two dressed interaction pathways

| + 〉 \overset{b_{1}}{\leftrightarrow} | 0 〉

and

| - 〉 \overset{b_{2}}{\leftrightarrow} | 0 〉

, the two dressed spins and the two cavity fields have the same variances

δ X_{α_{\pm}}^{2} = δ P_{α_{\mp}}^{2}, δ X_{v_{\pm}}^{2} = δ P_{v_{\mp}}^{2} .

Thus the conditions for the squeezing are the same as the conditions for the entanglement. The presence of the squeezing means the existence of the entanglement. The variances

δ X_{α_{\pm}}^{2}

and

δ X_{v_{\pm}}^{2}

are plotted, respectively in Figs. 5 and 6, for different decay rates. The main features are presented as follows.

Almost ideal two-mode spin squeezing is obtainable for particular parameters. Figs. 5 and 6 show respectively the optimal variances at $\frac{Δ}{Ω} \approx \sqrt{\frac{2}{3}}$ for the cavity fields $δ X_{α_{+}}^{2} = δ P_{α_{-}}^{2} \approx 0.06,$ and for the dressed spins $δ X_{v_{+}}^{2} = δ P_{v_{-}}^{2} \approx 0.07 .$
In order to make the squeezed and entangled states reach almost ideal squeezed states and Einstein-Podolsky-Rosen (EPR) entangled states, we need three conditions: a reasonable saturation Π ≲ 1, a large squeezing parameter r ≳ 1 and remarkably large cooperativity parameters C_1,2 ≫ 1. We stress that the saturation Π and the squeezing parameter r largely depend on the normalized detuning $\frac{Δ}{Ω}$ . The cooperative parameters C_1,2 depend on the decay rates of the atoms and the cavities. These conditions are well satisfied in the region of $\frac{Δ}{Ω} = (0, \sqrt{\frac{2}{3}})$ under respective adiabatic conditions. Thus, strong dissipation occurs for the Bogoliubov field or spin modes and leads to desirable squeezing and entanglement.
The squeezing and entanglement is confined in the regions of $\frac{Δ}{Ω} = (- 1, 0)$ , (0, $\sqrt{\frac{2}{3}}$ ), (1, +∞). The numerical results in Figs. 5 and 6 have verified the physical analysis in Sec. 3. The squeezing and entanglement occurs only when the dissipation of the Bogoliubov modes dominates. On the other hand, we notice that the squeezing and entanglement does not always exist in those regions. Further numerical analysis shows that the cooperative parameters are too small (C_1,2 ∼ 1) in such areas. Thus the dissipation of the Bogoliubov modes of dressed atom spins (cavity fields) by the cavity fields (by dressed atom spins) is not strong enough to dominate over the individual mode dissipation due to the vacuum environment.
The squeezing and entanglement is much better under adiabatic conditions (γ ≫ κ or κ ≫ γ) than that under non-adiabatic conditions (κ ∼ γ). In fact, when the atoms (cavity fields) decay rapidly, the dressed atom spins (cavity fields) reach their steady states much sooner than the cavity fields (dressed atom spins). Therefore, we treat the dressed atom spins (cavity fields) as engineered reservoirs. Due to the dissipation effects of Bogoliubov modes, each of the engineered reservoirs entangles with one system mode and meanwhile transfers its state to the other system mode. As a result, the faster the dissipation induced by engineered reservoir, the better the squeezing and entanglement we obtain.

Fig. 5 Two-mode field variances $δ X_{α_{\pm}}^{2}$ versus the normalized detuning $\frac{Δ}{Ω}$ for $g \sqrt{N} = 20 γ$ and κ = γ (dotted line), 0.1γ (dashed line), 0.01γ (solid line).

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Fig. 6 Two-mode field variances $δ X_{v_{\pm}}^{2}$ versus the normalized detuning $\frac{Δ}{Ω}$ for $g \sqrt{N} = 20 γ$ and γ = κ (dotted line), 0.1κ (dashed line), 0.01κ (solid line).

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As analyzed in the above section, two-mode squeezing appears in the regions $\frac{Δ}{Ω} = (- 1, 0)$ , (0, $\sqrt{\frac{2}{3}}$ ), (1, +∞) when we tune the cavity fields Δ_c₁ = −Δ_c₂ = Ω̄. This is because dissipation occurs in these regions and does no long happen in the other regions. The regions are changed to $\frac{Δ}{Ω} = (1, 0)$ , (0, $\sqrt{- \frac{2}{3}}$ ), (−1, −∞) when we tune the cavity fields Δ_c₁ = −Δ_c₂ = −Ω̄. Alike, we obtain two-mode spin squeezing and also two-mode light entanglement in our quantum beat system.

4.2. Excited-state spin squeezing

It is interesting to consider the correspondence of the above two-mode spin squeezing to the excited state spin correlation. In order to do this, we define the spin quadratures that involve excited states [41]

\begin{array}{l} J_{x} = σ_{12} + σ_{21}, \\ J_{y} = - i (σ_{12} - σ_{21}), \\ J_{z} = σ_{11} - σ_{22}, \end{array}

which follow the commutation relation [J_y, J_z] = 2iJ_x. At steady state, we have the mean values of the spin components 〈J_x〉 = (N₊ − N₀) cos² θ and 〈J_y〉 = 〈J_z〉 = 0. The spin squeezing occurs when either of the two inequalities 〈(δJ_y,z)²〉 < |〈J_x〉| is satisfied. We write the excited-state spins in terms of the dressed states as

\begin{array}{l} J_{y} = i sin θ (σ_{- +} - σ_{+ -}) - \frac{i cos θ}{\sqrt{2}} (σ_{+ 0} - σ_{0 +} + σ_{- 0} - σ_{0 -}), \\ J_{z} = sin θ (σ_{+ +} - σ_{- -}) - \frac{cos θ}{\sqrt{2}} (σ_{+ 0} + σ_{0 +} + σ_{- 0} + σ_{0 -}) . \end{array}

The variances of excited-state spins are obtainable for

\frac{Δ}{Ω} = (- \sqrt{\frac{2}{3}}, 0)

, (0,

\sqrt{\frac{2}{3}}

) as

\begin{array}{l} \frac{δ J_{y}^{2}}{| 〈 J_{x} 〉 |} = δ X_{v_{-}}^{2} + \frac{2 N_{+} {sin}^{2} θ}{| 〈 J_{x} 〉 |}, \\ \frac{δ J_{z}^{2}}{| 〈 J_{x} 〉 |} = δ X_{v_{+}}^{2} + \frac{2 N_{+} {sin}^{2} θ}{| 〈 J_{x} 〉 |}, \end{array}

where we have defined

δ J_{y, z}^{2} = 〈 {(δ J_{y, z})}^{2} 〉

. The first terms of Eq. (37) are the contributions of the two-mode variances of the spins, and the second terms are related to the atomic spontaneous emission. It should be aware that the excited-state spin squeezing in our quantum beat system emerges when either of the variances

\frac{δ J_{y, z}^{2}}{| 〈 J_{x} 〉 |}

is below the vacuum variance.

Numerical calculations of excited-state spin squeezing of different decay rates are shown in Fig. 7. The main features are presented as follows.

Excited-state spin squeezing emerges in the range ( $- \sqrt{\frac{2}{3}}$ , 0), (0, $\sqrt{\frac{2}{3}}$ ). As we can see from Eq. (37), the second terms on the right hand side are always positive. That means the existence of two-mode spin squeezing is a necessary condition for the excited-state spin squeezing. Thus, the excited-state spin squeezing should be confined in a narrower region than $\frac{Δ}{Ω} = (- 1, 0)$ , (0, $\sqrt{\frac{2}{3}}$ ), (1, +∞). For this reason, no excited spin squeezing occurs in the region of $\frac{Δ}{Ω} = (1, + \infty)$ while the two-mode spin squeezing still exists in that area. To explain this, we rewrite the first equation of Eq. (37) in the region of $\frac{Δ}{Ω} = (1, + \infty)$ as $\frac{δ J_{y}^{2}}{| 〈 J_{x} 〉 |} = δ X_{v -}^{2} + 2 Π {(\frac{Δ}{Ω})}^{2} .$ We notice that the atomic saturation is very large Π ≫ 1 when $\frac{Δ}{Ω} ~ 1$ . That makes the excited-state spin variances much greater than 1. On the other hand, in the far-off resonant system ( $\frac{Δ}{Ω} ≫ 1$ ), although the atomic saturation decreases Π ∼ 1, ${(\frac{Δ}{Ω})}^{2}$ increases rapidly and the two-mode spin squeezing is weak as shown in Fig. 6. Hence, there is no excited-state spin squeezing in the region of $\frac{Δ}{Ω} = (1, + \infty)$ .
The best squeezing of excited-state spins can approach 40% squeezing under adiabatic condition (κ ≫ γ). We notice that the second terms on the right hand side of Eq. (37) come from the first terms on the right hand side of Eq. (36). These terms correspond to the phase damping induced by the spontaneous emissions from the two excited states. Apparently, as shown in Eq. (37), the phase damping increases the fluctuation of the atomic spins. Thus, the excited-state spin squeezing is not as good as the two-mode spin squeezing and is limited to nearly 40%. Also, we know from Eq. (37) that the smaller the two-mode spin variances $δ X_{v_{\pm}}^{2}$ we achieve, the better the excited-state spin squeezing we obtain. Thus, as expected, the best excited-state spin squeezing appears under adiabatic condition.

Fig. 7 Excited-state spin variances $δ J_{y, z}^{2} / | 〈 J_{x} 〉 |$ versus the normalized detuning $\frac{Δ}{Ω}$ for $g \sqrt{N} = 20 κ$ and γ = κ (dotted line), 0.1κ (dashed line), 0.01κ (solid line).

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The present system is within the reach of the current experimental technology [24–27,33,34]. A large number of atoms can be used as candidates for the present system. For example, we can choose the alkali metal atom ⁸⁷Rb. The D₁ line (780 nm, |5S_1/2, F = 2〉 − |5P_3/2, F = 3〉) and the D₂ line (794 nm, |5S_1/2, F = 2〉 − |5P_1/2, F = 1〉) serve as the |3〉 − |1〉 and |3〉 − |2〉 transitions, respectively. The |5S_1/2, F = 1〉 − |5P_3/2, F = 0〉 transition is used as the repumping. By using the cold atoms we can remove the Doppler broadening.

5. Discussion and conclusion

So far, we have shown a surprising similarity of our results to those for the CPT system [15]. It is necessary to compare the two different systems and to find why it is the case. Here we look back on the CPT case. Two fields are applied to the three-level atoms in the Λ configuration as shown in Fig. 8(a). The Hamiltonian is written as follows

H = \sum_{l = 1, 2} ℏ (Ω_{l} σ_{3 l} + σ_{l 3} Ω_{l}^{*}),

where the operators and parameters have been defined as before. We introduce the superposition states |b〉 and |d〉 as in Eq. (6) and rewrite the Hamiltonian in the superposition state representation as

H = ℏ Ω (σ_{3 b} + σ_{b 3}) .

\frac{〈 σ_{d d} 〉}{N} = 1,

which corresponds to a maximum coherence

\frac{〈 σ_{12} 〉}{N} = - \frac{1}{2} .

Since the ground states are no-decaying states, the coherence 〈σ₁₂〉 is a long-lived coherence. This is so-called dark resonance. However, the two-mode squeezing and entanglement does not show up until we deviate appropriately from the exact dark resonance.

Fig. 8 Diagrammatic sketch of CPT system. (a) An ensemble of Λ-type atoms interacts with two cavity fields with Rabi frequencies Ω_1,2. CPT occurs when the system is resonant. (b) We use the superposition states |b, d〉 instead of the bare states |1, 2〉. All the population is trapped in the dark state |d〉 due to the transition |b〉 → |3〉 and the successive spontaneous emission |3〉 → |d〉 (dash line).

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Once the coherent excitation is introduced by tuning the applied fields [15], the coherence and nonlinearities cause the dissipation of Bogoliubov modes. Although the spontaneous emission is introduced, its noise effects on one of the two orthonormal quadratures are remarkably reduced due to the Bogoliubov dissipation. The parameter regions and the degree of the squeezing and entanglement are surprisingly similar to the present quantum-beat case although the maximal coherences are so different. Two-mode squeezing appears in the regions $\frac{Δ}{Ω} = (- 1, 0)$ , (0, $\sqrt{\frac{2}{3}}$ ), (1, +∞) when we tune the cavity fields Δ_c₁ = −Δ_c₂ = Ω̄ or in the regions $\frac{Δ}{Ω} = (1, 0)$ , (0, $- \sqrt{\frac{2}{3}}$ ), (−1, −∞) when we tune the cavity fields Δ_c₁ = −Δ_c₂ = −Ω̄. Almost ideal EPR entangled states are obtainable at $\frac{Δ}{Ω} \approx \sqrt{\frac{2}{3}}$ , as depicted in Figs. 5 and 6.

We stress that the maximal coherence for each atom is no more than $\frac{1}{4}$ in the quantum-beat case. That is significantly different from the coherence in CPT case. How does the similarities of two-mode squeezing and entanglement happens? The essence lies as follows. For the CPT system [15], the dissipation of the Bogoliubov modes occurs from the common state to two different states |0〉 → |±〉, as shown in Fig. 9(a). The Bogoliubov dissipation has good performance when $\frac{〈 σ_{\pm \pm} 〉}{N} ≲ \frac{〈 σ_{00} 〉}{N} ≲ 1$ (i.e., $0 ≲ \frac{| Δ |}{Ω} ≲ \sqrt{\frac{2}{3}}$ ). These conditions correspond to the case in which the system deviates from the exact dark resonance (the maximal coherence $- \frac{1}{2}$ ) but keeps coherence large enough. In sharp contrast, the dissipative transitions for the Bogoliubov modes, which are reversed in quantum-beat system, occur from the different states to the common state |±〉 → |0〉, as shown in Fig. 9(b). The Bogoliubov dissipation behaves well when $\frac{〈 σ_{00} 〉}{N} ≲ \frac{〈 σ_{\pm \pm} 〉}{N} ≲ \frac{1}{2}$ (i.e., $0 ≲ \frac{| Δ |}{Ω} ≲ \sqrt{\frac{2}{3}}$ ). These conditions are reached when the atoms deviate from the exact quantum-beat resonance (the maximal coherence $\frac{1}{4}$ ) but maintain coherence large enough.

Fig. 9 A comparison of dissipative transitions between the CPT case (a) and the quantum-beat case (b). The thickness of the levels represents the relative weights of the popuations of different dressed states. In the CPT case (a), the dissipative transitions occur from one common state to the other two different states |0〉 → |±〉, and in the quantum-beat case (b), the dissipative transitions happen from the two different states to the other common state |±〉 → |0〉.

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It is clear that the common essence of the two above systems lies in the quantum interference [20–23]. For the CPT based system, the quantum interference takes its effect when the atoms (which otherwise stay in the dark state) are properly excited by using the two-photon detuning [15]. For the quantum-beat based system, however, the quantum interference gives almost the same effect when the atoms (which otherwise excited maximally) are excited relatively weakly by using the two-photon detuning. It is for the above reasons that CPT and quantum-beat constitute two complementary models for the dissipation induced entanglement.

In conclusion, we have analyzed the dissipation effects of the atomic coherence and the non-linearities on the quantum correlations in a detuned quantum-beat system. The physical mechanism is given and the numerical calculation is performed. It has been shown that the two mode squeezing and entanglement is obtainable for the dressed atomic spins and for the cavity fields. The nearly ideal two-mode squeezing and the EPR entanglement appear at special value of $\frac{| Δ |}{Ω} \approx \sqrt{\frac{2}{3}}$ . The two-mode spin squeezing corresponds also to the excited-state spin squeezing and the best squeezing approaches 40% squeezing. Finally, a comparison with the CPT case shows that the quantum-beat based system represents the complementary model for the dissipation induced entanglement.

Funding

National Natural Science Foundation of China (NSFC) (Grants No. 11474118 and No. 61178021); National Basic Research Program (973 Program) of China (Grant No. 2012CB921604).

References and links

1. C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmermann, “On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. issues of principle,” Rev. Mod. Phys. 52, 341–392 (1980). [CrossRef]

2. J. Sherson, B. Julsgaard, and E. S. Polzik, “Deterministic atom-light quantum interface,” Adv. At., Mol., Opt. Phys. 54, 81–130 (2007). [CrossRef]

3. D. P. DiVincenzo, “Quantum computation,” Science 270, 255–261 (1995). [CrossRef]

4. M. A. Nielson and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

5. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wooters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1992). [CrossRef]

6. C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992). [CrossRef] [PubMed]

7. D. F. Walls, “Squeezed states of light,” Nature 306, 141–146 (1983). [CrossRef]

8. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935). [CrossRef]

9. L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000). [CrossRef] [PubMed]

10. S. Pielawa, G. Morigi, D. Vitali, and L. Davidovich, “Generation of Einstein-Podolsky-Rosen-entangled radiation through an atomic reservoir,” Phys. Rev. Lett. 98, 240401 (2007). [CrossRef] [PubMed]

11. A. S. Parkins, E. Solano, and J. I. Cirac, “Unconditional two-mode squeezing of separated atomic ensembles,” Phys. Rev. Lett. 96, 053602 (2006). [CrossRef] [PubMed]

12. H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, “Entanglement generated by dissipation and steady state entanglement of two macroscopic objects,” Phys. Rev. Lett. 107, 080503 (2011). [CrossRef] [PubMed]

13. E. G. Dalla Torre, J. Otterbach, E. Demler, V. Vuletic, and M. D. Lukin, “Dissipative preparation of spin squeezed atomic ensembles in a steady state,” Phys. Rev. Lett. 110, 120402 (2013). [CrossRef] [PubMed]

14. A. Orieux, M. A. Ciampini, P. Mataloni, D. Bruß, M. Rossi, and C. Macchiavello, “Experimental generation of robust entanglement from classical correlations via local dissipation,” Phys. Rev. Lett. 115, 160503 (2015). [CrossRef] [PubMed]

15. X. M. Hu, “Entanglement generation by dissipation in or beyond dark resonances,” Phys. Rev. A 92, 022329 (2015). [CrossRef]

16. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–44 (1997). [CrossRef]

17. J. P. Marangos, “Electromagnetically induced transparency,” J. Mod. Opt. 45, 471–503 (1998). [CrossRef]

18. M. D. Lukin, “Colloquium: trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457–472 (2003). [CrossRef]

19. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]

20. E. Arimondo, “Coherent population trapping in laser spectroscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, 1996), pp. 257–354. [CrossRef]

21. A. Dantan, J. Cviklinski, E. Giacobino, and M. Pinard, “Spin squeezing and light entanglement in coherent population trapping,” Phys. Rev. Lett. 97, 023605 (2006). [CrossRef] [PubMed]

22. A. Sinatra, “Quantum correlations of two optical fields close to electromagnetically induced transparency,” Phys. Rev. Lett. 97, 253601 (2006). [CrossRef]

23. P. B. Blostein and M. Bienert, “Opacity of electromagnetically induced transparency for quantum fluctuations,” Phys. Rev. Lett. 98, 033602 (2007). [CrossRef]

24. C. L. G. Alzar, L. S. Cruz, J. G. A. Gómez, M. F. Santos, and P. Nussenzveig, “Super-Poissonian intensity fluctuations and correlations between pump and probe fields in electromagetically induced transparency,” Europhys. Lett. 61, 485–491 (2003). [CrossRef]

25. V. A. Sautenkov, Y. V. Rostovtsev, and M. O. Scully, “Switching between photon-photon correlations and Raman anticorrelations in a coherently prepared Rb vapor,” Phys. Rev. A 72, 065801 (2005). [CrossRef]

26. J. -F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J. -Ph. Poizat, and Ph. Grangier, “Quantum nondemolition measurements using cold trapped atoms,” Phys. Rev. Lett. 78, 634–637 (1997). [CrossRef]

27. A. Kuzmich, K. Mølmer, and E. S. Polzik, “Spin squeezing in an ensemble of atoms illuminated with squeezed light,” Phys. Rev. Lett. 79, 4782–4785 (1997). [CrossRef]

28. M. O. Scully, “Correlated spontaneous-emission lasers: quenching of quantum fluctuations in the relative phase angle,” Phys. Rev. Lett. 55, 2802–2805 (1985). [CrossRef] [PubMed]

29. M. O. Scully and M. S. Zubairy, “Theory of the quantum-beat laser,” Phys. Rev. A 35, 752–758 (1987). [CrossRef]

30. J. Bergou, M. Orszag, and M. O. Scully, “Correlated-emission laser: phase noise quenching via coherent pumping and the effect of atomic motion,” Phys. Rev. A 38, 768–772 (1988). [CrossRef]

31. S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 78, 388–391 (1996). [CrossRef]

32. N. Lu, “Quantum theory of correlated-emission lasers: vacuum state for the mode of the relative phase and the relative amplitude,” Phys. Rev. A 45, 8154–8164 (1992). [CrossRef] [PubMed]

33. M. P. Winters and J. L. Hall, “Correlated spontaneous emission in a Zeeman laser,” Phys. Rev. Lett. 65, 3116–3119 (1990). [CrossRef] [PubMed]

34. I. Steiner and P. E. Toschek, “Quenching quantum phase noise: correlated spontaneous emission versus phase locking,” Phys. Rev. Lett. 74, 4639–4642 (1995). [CrossRef] [PubMed]

35. X. Zhang and X. M. Hu, “Entanglement between collective fields via atomic coherence effects,” Phys. Rev. A 81, 013811 (2010). [CrossRef]

36. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).

37. G. S. Agarwal, W. Lange, and H. Walther, “Intense-field renormalization of cavity-induced spontaneous emission,” Phys. Rev. A 48, 4555–4568 (1993). [CrossRef] [PubMed]

38. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-photon Interactions (Wiley, 1992).

39. M. Sargent III, M. O. Scully, and W. E. Lamb Jr., Laser Physics (Addison-Wesley, 1974).

40. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994). [CrossRef]

41. J. Ma, X. G. Wang, C. P. Sun, and F. Nori, “Quantum spin squeezing,” Phys. Rep. 509, 89–165 (2011). [CrossRef]

Quantum-beat based dissipation for spin squeezing and light entanglement

Abstract

1. Introduction

2. Model and master equation

2.1. Quantum beat

2.2. Detuned quantum beat system

3. Physical analysis of atom-photon interaction effects

3.1. Bogoliubov field dissipation by the atoms

3.2. Bogoliubov spin dissipation by the fields

4. Dependence of quantum correlations on system parameters

4.1. Two-mode field squeezing and two-mode spin squeezing

4.2. Excited-state spin squeezing

5. Discussion and conclusion

Funding

References and links

Cited By

Figures (9)

Equations (42)

Optics Express