Performance manipulation of the squeezed coherent light source based on four-wave mixing

Li Jin; Li Jin

doi:10.1364/OE.435735

1. Introduction

Optical frequency control is the foundation of precision measurement science. The laser with precise frequency control has the characteristics of narrow linewidth, high frequency stability and low frequency noise. Several applications require lasers with high phase coherence, ranging from improved metrology standards [1–4], to proposed gravitational wave detection [5–7], searches for variations of fundamental constants [8–9] and increased capabilities for positioning, navigation, and timing applications [10]. Its linewidth and frequency stability will directly affect the frequency resolution, accuracy and noise level of the above applications. For more accurate optical clocks and ultra-long laser interferometers used for gravitational wave detection, a laser with frequency instability of 10⁻¹⁷ or even higher is required to improve the detection sensitivity. Therefore, lasers with high spectral purity and frequency stability are urgently needed to further improve the accuracy of precision measurement.

Great efforts have been put into the development of extremely coherent sources based on ultra-stable optical Fabry-Pérot resonators [11–23]. The exceptionally low fractional frequency stability at short times was either obtained at room temperature with long cavities, which rely on a very carefully designed and optimized vibration-insensitive cavity support [11–15], or at cryogenic temperature with technically more complex systems involving bulky cryocoolers [20–23]. However, it ultimately limited by the thermal noise of the optical resonators, and the prestabilized laser also serves as oscillator to interrogate ultra-narrow optical transitions with linewidths of a few mHz. Several other novelty schemes have been investigated in parallel to circumvent the cavity thermal noise [24–32], such as electromagnetically induced transparency (EIT) [24], electromagnetically induced transparency and absorption (EITA) [25], phase-matching effect [26], nonlinear dispersion effect [27–28] and active optical clock schemes [29–32]. In some schemes, an available coherent laser with ultra-narrow linewidth is a prerequisite. As the most competitive proposal, active optical clock method can be highly immune to the fluctuations in the reference cavity length, and this superradiant laser was developed to operate in a continuous manner to improve the frequency stability in short timescales in the near future.

More recently it has been proposed that one could operate such a laser based on four-wave mixing in alkaline-earth atoms to achieve a linewidth of mHz level [33]. Within this work, we limit our attention to the underlying physics of bandpass filter behavior of the system, relevant manipulation technique and quantum noise properties. We model the dynamics of the system by optical-Bloch equations describing the coherent time evolution of atom ensembles and investigate the mechanism for lasing, which provides an overview of the underlying physics of wave-mixing. We have evaluated and analyzed the width broadening and light-shift effects on the wave-mixing lasing, and the results presented here represent the linewidth and frequency stability can be manipulated by choosing optimal operating parameters. This is a key step towards a useful frequency reference based on even narrower transitions. Within this approach, we also derive the quantum noise frequency spectra and quantify the quantum entanglement of the wave-mixing lasing. It not only provides a coherent laser source for precision measurements, but lays a theoretical and technical foundation for the establishment of a new generation of time-frequency metrology system.

2. Mechanism of wave-mixing lasing with confined ensembles

The scheme under consideration is a lifetime broadened four-state system. Here, we use lattice-confined ⁸⁷Sr atoms to make the presentation clearer and more concise. The energy-level diagram with relevant laser excitations is depicted in Fig. 1. The atoms interact with three optical fields simultaneously, pump laser fields ℰ₁ and ℰ₂ couple the $|{1\rangle \leftrightarrow |2\rangle } $ (¹S₀↔³P₁) and $|{2\rangle \leftrightarrow |3\rangle } $ (³P₁↔³S₁) transitions with one-photon detuning ${\Delta _2}$ and two-photon detuning ${\Delta _3}$, respectively, and a weak probe field drives $|{3\rangle \leftrightarrow |4\rangle } $ (³S₁↔³P₀) transition with detuning ${\Delta _4}$. The effective four-wave mixing process occurs when the phase-matching is fulfilled [34]. The FWM field ℰ_m generated with high spectral purity due to the long natural lifetime of the clock excited state in cold alkaline-earth atoms. For the sake of simplicity, we limit our study to a one-dimensional model with propagation along the z axis. We assume that the laser beams propagate undepleted through the atomic medium. Two pump and probe laser beams are treated as classical quantities and their interaction with the medium is described simiclassically.

Fig. 1. Energy diagram of a four-level ⁸⁷Sr atomic system and laser configuration for realizing the four-wave mixing. Two pump fields with Rabi frequencies ${\Omega _1}$ and ${\Omega _2}$ drive the $|1 \rangle \leftrightarrow |2 \rangle$ and $|2 \rangle \leftrightarrow |3 \rangle$ transitions, respectively. A weak probe field with Rabi frequency ${\Omega _3}$ couples the transition between states $|3\rangle $ and $|4\rangle $. ${\varDelta _2}$, ${\varDelta _3}$ and ${\varDelta _4}$ are the detunings for laser fields.

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Under the rotating-wave approximation, the four-state atomic system interacting with three laser fields can be described by the following density-matrix equations:

(1a)$$i{\partial _t}{\rho _{21}} + {d_{21}}{\rho _{21}} - {\Omega _m}{\rho _{24}} - {\Omega _1}({\rho _{22}} - {\rho _{11}}) + \Omega _2^\ast {\rho _{31}} = 0$$

(1b)$$i{\partial _t}{\rho _{\textrm{32}}} + {d_{\textrm{32}}}{\rho _{\textrm{32}}}\textrm{ + }{\Omega _\textrm{3}}{\rho _{\textrm{42}}} - {\Omega _\textrm{2}}({\rho _{\textrm{33}}} - {\rho _{\textrm{22}}}) - \Omega _\textrm{1}^\ast {\rho _{31}} = 0$$

(1c)$$i{\partial _t}{\rho _{\textrm{31}}} + {d_{\textrm{31}}}{\rho _{\textrm{31}}} - {\Omega _m}{\rho _{34}} - {\Omega _1}{\rho _{\textrm{32}}}\textrm{ + }{\Omega _\textrm{3}}{\rho _{\textrm{41}}}\textrm{ + }{\Omega _2}{\rho _{21}} = 0$$

(1d)$$i{\partial _t}{\rho _{41}} + {d_{41}}{\rho _{41}} - {\Omega _m}({\rho _{44}} - {\rho _{11}}) - {\Omega _1}{\rho _{\textrm{42}}}\textrm{ + }\Omega _3^\ast {\rho _{31}} = 0$$

(1e)$$i{\partial _t}{\rho _{42}} + {d_{42}}{\rho _{42}} + {\Omega _m}{\rho _{12}} + \Omega _3^\ast {\rho _{32}} - \Omega _1^\ast {\rho _{41}} - {\Omega _2}{\rho _{\textrm{43}}} = 0$$

(1f)$$i{\partial _t}{\rho _{43}} + {d_{43}}{\rho _{43}} - \Omega _3^\ast ({\rho _{44}} - {\rho _{33}}) - \Omega _2^\ast {\rho _{42}} + {\Omega _m}{\rho _{\textrm{13}}} = 0$$

where ${\rho _{kl}}$ is the atomic density-matrix element, ${d_{kl}} = {\Delta _k} - {\Delta _l} + i{\gamma _{kl}}\,({k,l = 1 - 4} )$. ${\Delta _{k(l )}}$ is the detuning of state $|k\rangle ({|l\rangle } )$ and ${\gamma _{kl}} = ({{\varGamma _k} + {\varGamma _l}} )/2 + \gamma _{kl}^{dph}$. ${\Gamma _{k(l )}}$ is the total decay rate of state $|k\rangle ({|l\rangle } )$ and $\gamma _{kl}^{dph}$ is the dephasing rate between states $|k\rangle $ and $|l\rangle $. ${\varOmega _k}({k = 1,2,3,m} )= {D_{kl}}{{{\cal E}}_k}/2\hbar $ is one-half of the Rabi frequency of the relevant atomic transition in a given laser field ℰ_k.

To predict the generation and propagation behavior of the FWM field, Eqs. (1a)-(1f) need to be solved simultaneously with the Maxwell equations. Within the plane wave, slowly varying amplitude, and phase approximation, the Maxwell’s equations for the probe field and FWM field can be expressed as:

(2a)$$i(\frac{\partial }{{\partial z}} + \frac{1}{c}\frac{\partial }{{\partial t}}){\Omega _3} + {\kappa _{34}}{\rho _{34}} = 0$$

(2b)$$i(\frac{\partial }{{\partial z}} + \frac{1}{c}\frac{\partial }{{\partial t}}){\Omega _m} + {\kappa _{41}}{\rho _{41}} = 0$$

where z is the propagation distance, ${\kappa _{kl}} = N{\omega _{kl}}{|{{D_{kl}}} |^2}/({2\hbar {\varepsilon_0}c} )$ with N, ${\omega _{kl}}$, and ${D_{kl}}$ being the concentration, transition frequency and the corresponding dipole moment of the relevant transition, respectively. $\hbar $ is the reduced Planck constant, and ${\varepsilon _0}$ is the vacuum permittivity.

However, due to the so small decoherent rate ${\gamma _{41}}$ that one could not treat the dynamic equation of motion for state $|4\rangle $ in adiabatic way. Therefore, Eq. (1d) should be treated by non-adiabatic way. Taking a non-depleted ground state approximation (${\rho _{11}} = 1$) and the weak-field approximation, and solving Eq. (1) using the time-Fourier-transform method, the set of equations of motion to be solved for the density matrix elements are given by:

(3a)$${\sigma _{41}} ={-} [\frac{{{\Lambda _m}}}{{\omega + d}} + \frac{{{\Omega _1}{\Omega _2}\Lambda _3^\ast }}{{(\omega + d){D_0}}}]$$

(3b)$${\sigma _{43}} ={-} [\frac{{\Omega _1^\ast \Omega _2^\ast {\Lambda _m}}}{{(\omega + d)D_0^\ast }} + \frac{{\Lambda _3^\ast }}{{\omega + d}}{\left|{\frac{{{\Omega _1}{\Omega _2}}}{{{D_0}}}} \right|^2}]$$

where ${\sigma _{kl}}$, ${\varLambda _3}$, and ${\varLambda _m}$ are the Fourier transforms of ${\rho _{kl}}$, ${\varOmega _3}$ and ${\varOmega _m}$, respectively. $\omega $ is the Fourier variable, and

(4a)$${D_0} = {d_{21}}{d_{31}} - \Omega _2^2$$

(4b)$$d = {d_{41}} - {d_{21}}\Lambda _3^2/{D_0} - {d_{43}}\Omega _1^2/D_0^\ast $$

Inserting the above equations into Eqs. (2a) and (2b), with given constant value of $\varLambda _3^\ast ({z = 0,\omega } )$ and ${\varLambda _m}({z = 0,\omega } )= 0$, we obtain analytical solution writes as:

(5)$${\Lambda _m} = \frac{{({\Lambda _ + } + {K_p})}}{{{D_p}}}\frac{{({\Lambda _ - } + {K_p})\Lambda _3^\ast (0,\omega )}}{{({\Lambda _ + } - {\Lambda _ - })}}[\exp ( - i{\Lambda _ + }z) - \exp ( - i{\Lambda _ - }z)]$$

where we have defined the new parameters:

(6a)$${\Lambda _ + } ={-} \frac{\omega }{c}$$

(6b)$${\Lambda _ - } ={-} \frac{\omega }{c} - \frac{{{\kappa _{43}}}}{{(\omega + d)}}{\left|{\frac{{{\Omega _1}{\Omega _2}}}{{{D_0}}}} \right|^2} + \frac{{{\kappa _{41}}}}{{(\omega + d)}}$$

(6c)$${K_p} = \frac{\omega }{c} + \frac{{{\kappa _{43}}}}{{(\omega + d)}}{\left|{\frac{{{\Omega _1}{\Omega _2}}}{{{D_0}}}} \right|^2}$$

(6d)$${D_p} = \frac{{{\kappa _{43}}}}{{(\omega + d)}}\frac{{\Omega _1^\ast \Omega _2^\ast }}{{D_0^\ast }}$$

To present the essential underlying physics of the generation of coherent radiations based on four-wave mixing, it is necessary to evaluate and analyze spectral characteristics and expected dependence of the emitted light. If the initial spectral density of the incident laser fields is Lorentzian in shape with linewidth $\varGamma $ and center frequency $\,{\omega _0}$, it writes as ${S_{in}} = {|{{\varOmega _1}{\varOmega _2}\Lambda _3^\ast } |^2} \cdot {\varGamma ^2}/[{4{{({\omega - {\omega_0}} )}^2} + {\varGamma ^2}} ]$, with the help of Eqs. (5) and (6), one could get the spectral distribution of the FWM field, yielding

(7)$${S_{out}} = {S_0}\frac{{{\Gamma ^2}}}{{4{{(\omega - {\omega _0})}^2} + {\Gamma ^2}}}{|{k(\omega )} |^2}$$

where ${S_0} = {|{{\varOmega _1}{\varOmega _2}\varLambda _3^\ast } |^2}{|{{\kappa_{41}}z/D_0^\ast } |^2}$, $k(\omega )= 1/({\omega + d} )$ denotes the dynamic response of atomic system. As seen from Eq. (7), the spectral bandwidth of the FWM laser is the combinations of the Lorentzian profile of the incident lasers and the atomic response. Note that the linewidth of the incident lasers is much larger than that of the atomic response, the full width at half maximum (FWHM) linewidth of the FWM laser is ultimately determined by the bandwidth of atomic response.

Figure 2 shows spectral distribution and exhibits special characteristics for the wave-mixing lasing. As seen from the spectral analysis of the Fig. 2, we can identify three spectral features: (1) The initial linewidth Γ and the frequency fluctuations of the incident lasers have a negligible effect on the properties of the FWM laser, except for the amplitude fluctuation, ensuring the robustness of both linewidth and stability of the FWM laser. (2) The emission bandwidth mainly depends on the response of the atomic medium. (3) It is clear that the spectral width of the FWM laser is always on the order of or even slightly less than the bandwidth of atomic medium. Those features indicate that atomic medium could be seen as a band-pass frequency filter, letting the mixing laser with band-pass spectrum pass through and eliminating the amplitude and phase fluctuations large than the bandwidth of atomic medium. The extraordinary coherence of the FWM laser will benefit from the ultra-narrow linewidth and stability of the clock transitions.

Fig. 2. The spectral distribution of the FWM field ${S_{out}}({{\omega_j},{\gamma_k}} )$. (a) The initial linewidth of incident lasers decreases from Γ₁=0.2 Hz × 2π to Γ₂=0.1 Hz × 2π, with ω₀=–0.3 Hz; (b) The frequency of incident lasers fluctuates from ω₁=0.2 Hz to ω₂=–0.5 Hz, with Γ=0.1 Hz × 2π. The term of ${|{k(\omega )/{k_0}} |^2}$ (blue solid line) represents the normalized response of the medium to all of the electromagnetic fields. Here, we assume the bandwidth of atomic response is 10 mHz × 2π.

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3. Characteristics and performance manipulation of wave-mixing lasing

3.1 Linewidth and frequency stability

In order to generate a coherent laser source with high accuracy and stability, we need to consider the width broadening and light-shift effects. Due to the band-pass filter behavior of the medium, the spectral features of the FWM laser mainly depends on the response of the atomic medium, the linewidth and ac Stark shift of FWM laser can be obtained as follows:

(8a)$$\gamma = {\mathop{\rm Im}\nolimits} [d]/\pi = [{\Gamma _{41}} + \frac{{({\Gamma _{21}}\Delta _3^2 + {\Gamma _3}\Omega _2^2)\Omega _1^2 + ({\Gamma _3}\Delta _2^2 + {\Gamma _{21}}\Omega _2^2)\Lambda _3^2}}{{{{({\Delta _2}{\Delta _3} - \Omega _2^2)}^2}}}]/\pi$$

(8b)$${\omega _j} = {\textrm{Re}} [d] = {\Delta _4} + \frac{{{\Delta _3}\Omega _1^2 - {\Delta _2}\Lambda _3^2}}{{{\Delta _2}{\Delta _3} - \Omega _2^2}}$$

Note that in the limit of large detunings the second term in the bracket of Eq. (8a), which corresponding to the width broadening induced by laser fields, become negligible and the width closed to the natural linewidth of the clock state. Choosing the optimum value for the detuning (i.e. ${\Delta _3}\varOmega _1^2 = {\Delta _2}\varLambda _3^2$) we find that the ac Stark shift is close to zero and the frequency stability ${\sigma _f}$ could be much improved, where ${\sigma _f} = Re[d ]\times {\xi _I}/{\omega _m}$ is linearly with power stabilized level ${\xi _I}$ and $Re[d ]$.

In Figs. 3(a) and (b), the magnitude of linewidth and ac Stark shift of the FWM laser are plotted in colored contour plots as functions of the probe field detuning ${\Delta _3}$ and Rabi frequency ${\Omega _3}$, respectively. System parameters are chosen as N = 10¹¹ /cm³, ${\varOmega _1}/({2\pi } )$=20 kHz, ${\varOmega _2}({2\pi } )$=31.8 MHz, and ${\Delta _2}/({2\pi } )$=–64.6 MHz. The equal-altitude contour curves visually indicate how parameters can be selected so that the tunable linewidth laser can be realized with the fixed frequency shift, thus achieving the required stability. In Fig. 3(c), we show the frequency stability of FWM laser as a function of power stabilized level ${\xi _I}$ and Stark shift $Re[d ]$. Note that, the power stabilized level being kept constant, the frequency stability depends on the value of the frequency shift. As $Re[d ]$ decreases, power stabilized level has little impact on the variation of the ac Stark shift of the clock transition, thus improving the frequency stability of the FWM laser. With this method, the width and the frequency stability of the FWM laser can, in principle, be continuously adjusted at will without loss of signal amplitude by varying the intensity and detuning of the probe laser, achieving high-precision performance manipulation of the wave-mixing lasing.

Fig. 3. Contour plots of (a) the linewidth and (b) ac Stark shift for the FWM laser as functions of the Rabi frequency ${\Omega _3}$ and detuning ${\Delta _3}$ of the probe field, respectively. (c) The frequency stability of the FWM laser as functions of power stabilized level of laser fields and Stark shift of the mixing field.

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3.2 Conversion efficiency

To provide a quantitative understanding of the role of population distribution during the mixing process, we also investigate the dynamic evolution of the wave-mixing lasing as the function of initial atomic coherent population prepared. According to the explicit Eqs. (1)–(2), under the assumption of short propagation regime, the dependence of the Rabi frequency of FWM laser is calculated as:

(9)$${\Lambda _m} \propto \left\{ {\begin{array}{c} {{\Omega _1}{\Omega _2}\Lambda _3^\ast (0,\omega )\frac{{\Delta \rho {\kappa_{41}}z}}{{\omega + d}}\;(if{\rho_{11}} > {\rho_{44}})}\\ {{\Omega _1}{\Omega _2}\Lambda _3^\ast (0,\omega )\frac{{\Delta \rho ({\kappa_{41}} - 2{\kappa_{43}}{{|{{\Omega _1}{\Omega _2}/{D_0}} |}^2})z}}{{\omega + d}}\;(if{\rho_{44}} > {\rho_{11}})} \end{array}} \right.$$

where $\Delta \rho = |{{\rho_{11}} - {\rho_{44}}} |$ is the population difference. The parameters are chosen as ${\varLambda _3}({2\pi } )$=10 kHz, ${\Delta _3}({2\pi } )$=–1.39 GHz, and other operating parameters are same as in Fig. 3. As shown in Fig. 4, it can be seen that the larger value of $|{{\rho_{11}} - {\rho_{44}}} |$, the more effective generation efficiency will be [panels (a) and (b)], which agrees well with the result of Eq. (9). In addition, the temporal evolution of the FWM laser are the same under the condition of the same value of $|{{\rho_{11}} - {\rho_{44}}} |$ [panels (a) and (c)], except for the decreased amplitude by a coupling factor of $({{\kappa_{41}} - 2{\kappa_{43}}{{|{{\varOmega _1}{\varOmega _2}/{D_0}} |}^2}} )/{\kappa _{41}}$. However, as the term of ${\Omega _1}{\Omega _2}/{D_0}$ decreases, the result shows a weak dependence on this factor.

Fig. 4. Impact of population distribution on conversion efficiency. Panels (a)-(d) show the dynamic evolution of the wave-mixing lasing versus propagation distance z for different values of steady-state populations: (a) ${\rho _{11}} = 1,\,{\rho _{44}} = 0$, (b) ${\rho _{11}} = 0.8,\,{\rho _{44}} = 0.2$, (c) ${\rho _{11}} = 0,\,{\rho _{44}} = 1$, and (d) ${\rho _{11}} = 0.2,\,{\rho _{44}} = 0.8$.

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The probe and wave-mixing fields the twin beams are cross coupled and are jointly amplified in the short propagation regime due to the coherent Raman gain. However, in the larger propagation regime, the destructive interference suppresses further production of the wave-mixing lasing even if a large fraction of the coherent population distribution still remain and a significant atomic coherence are still present in the medium [35], resulting in the saturation broadening at the exit of the medium, as shown in Figs. 4(a) and 4(c).

With experimentally achievable parameters, a controllable bandwidth of the wave-mixing lasing on the clock transition can be very efficiently generated. We further show that under suitable conditions, the peak intensity of the FWM laser can reach to nW/mm² level, as shown in Fig. 5. Here, we take N = 10¹¹ /cm³ and ${\mathrm{\xi }_\textrm{I}}$ = 10⁻⁴, and also note that the increased linewidth to some extent could lead to observable intensity growth. From the perspective of potential applications the most striking feature of wave-mixing lasing is its tunable linewidth and stability, with the linewidth from Hz to mHz level and frequency instability from 10⁻¹⁶ to 10⁻¹⁸ level. In this scheme, one also can achieve highly efficient generation of the wave-mixing lasing by choosing other alkaline-earth atoms, for instance, ¹⁷¹Yb and ¹⁹⁹Hg, as shown in Table 1. With the same as ⁸⁷Sr, the basic behavior is similar, but some analytical details are markedly different because Γ₄₁${\mathrm{\Gamma }_{41}}$ is different.

Fig. 5. The peak intensity of the wave-mixing lasing (in the units of nW/mm²) as a function of linewidth and frequency instability at 1 s, with the power stabilized level of order 10⁻⁴.

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Table 1. Atom, the corresponding clock wavelength, the full-width-half-maximum (FWHM), frequency instability and the peak intensity of the wave-mixing lasing

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3.3 Quantum noise

In the above analysis, we have employed a semi-classical description, i.e. the motion of atoms is governed by the density-matrix equation and the propagation of laser fields is described by the classical Maxwell equation. However, it is necessary to consider the role played by the quantum noise on the system, which requires an entire quantization treatment for both the probe and mixing laser fields, with Langevin noise $\hat{F}$ taken into account [36–39]. Using the Heisenberg-Langevin formalism, and based on the microscopic properties of the medium, one can obtain the annihilation operator of mixing laser $\hat{a}({z,t} )$. The general method to perform these calculations is detailed in Ref. [36].

Introducing the amplitude operator $\hat{X}({z,t} )= \hat{a}({z,t} )+ {\hat{a}^\dagger }({z,t} )$, the correlation of the amplitude operator is:

(10)$$\left\langle {\hat{X}(z,\omega )\hat{X}(z,{\omega^{\prime}})} \right\rangle = \frac{{2\pi L}}{c}\delta (\omega + {\omega ^{\prime}}){S_X}(z,\omega )$$

The correlation function of $\hat{F}({s,\omega } )$ can be calculated via the quantum regression theorem [40], the normalized output amplitude spectrum in the Fourier domain is:

(11)$${S_X}(z,\omega )\textrm{ = }{S_1}(z,\omega )\textrm{ + }{S_2}(z,\omega )\textrm{ + }{S_F}(z,\omega )$$

In Fig. 6, we show the amplitude noise of the FWM laser at the end of the medium as a function of the detuning ${\Delta _4}$. The amplitude noise consists of three contributions. The first part, S₁ (z,ω=0)${\textrm{S}_1}({\textrm{z},\mathrm{\omega } = 0} )$, is related to the amplitude noise spectrum of the input seed pulse S_{X_in}(0,ω). The second part, S₂ (z,ω=0), is related to the phase noise spectrum of the input seed pulse ${{S}_{{Y}\_{in}}}({0,\mathrm{\omega }} )$. The third part, S_F (z,ω=0) $ {{S}_{F}}({{z},\mathrm{\omega } = 0} )$, arises from the Langevin atomic noise. One can see that the Langevin atomic noise plays a negligible role in the contribution of the amplitude noise, verifying the Langevin forces can be neglected and can't reduce the performances of the system. In addition, the total amplitude noise of the FWM laser S_X(z,ω) reaches the minimum at ${\Delta _4}$=0. The result shows that the generation of wave-mixing lasing with a relative intensity noise spectrum well below the standard quantum limit, down to -4.7 dB. Entanglement and squeezing of the probe laser and wave-mixing lasing may occur in our system via the FWM, which is very promising to the generation of squeezed light in this system.

Fig. 6. The amplitude noise S_X (z = L,ω=0) of the wave-mixing lasing at the end of the medium as a function of the detuning ${\Delta _4}$. The black solid line shows the total output amplitude noise of the FWM field, the dotted, dashed, dash-dotted lines are three contributions S₁(L,0) , S₂(L,0) and S_F(L,0), respectively related to the amplitude noise spectrum of the input seed pulse S_{X_in}(0,ω), the phase noise spectrum of the input seed pulse ${{S}_{{Y}\_{in}}}({0,\mathrm{\omega }} )$, and the Langevin atomic noise. The normalized spectrum of the two quadrature components are taken as S_{X_in}(0,ω)= S_{Y_in}(0,ω) = 1.

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4. Conclusion

In conclusion, we have investigated the four-wave mixing scheme for realizing performance manipulation of the squeezed coherent light source in the alkaline-earth atoms. Based on the band-pass filter behavior of the atomic medium, the spectral characteristics of the wave-mixing lasing can be continuously manipulated by varying the intensities and detunings of the optical beams. Using the Heisenberg-Langevin formalism we evaluate and analyze the influence of the Langevin noise fluctuations on the system and predict that the generation of wave-mixing lasing with relative-intensity noise spectrum well below the standard quantum limit (down to -4.7 dB). Those results show that the FWM-based scheme represents a practical and robust approach for squeezed coherent laser with superior short-term stability and competitive accuracy, which promises great potential in quantum measurement, precision metrology, and quantum information.

Funding

National Natural Science Foundation of China (62005253); Shanxi Province Science Foundation for Youths (201901D211277); Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2020L0267); Science Foundation of North University of China (XJJ201901).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Atom	λ (nm)	Γ₄₁/(2π)	FWHM	σ_f	I_m (nW/cm²)
¹⁷¹Yb	555.8	7.5 mHz	15 mHz	5×10⁻¹⁸	5.5
⁸⁷Sr	698	1 mHz	2 mHz	5×10⁻¹⁸	10.3
¹⁹⁹Hg	265.5	0.1 Hz	0.11 Hz	5×10⁻¹⁸	1.5

Performance manipulation of the squeezed coherent light source based on four-wave mixing

Abstract

1. Introduction

2. Mechanism of wave-mixing lasing with confined ensembles

3. Characteristics and performance manipulation of wave-mixing lasing

3.1 Linewidth and frequency stability

3.2 Conversion efficiency

3.3 Quantum noise

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (1)

Equations (23)

Optics Express