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Abstract
Squeezed light near an atomic resonance is beneficial for efficient atom-light quantum interfaces. It is desirable but challenging to directly generate in atoms due to excess noise from spontaneous emission and reabsorption. Here, we report on the use of energy-level modulation to actively control atomic coherence and interference in degenerate four-wave mixing (DFWM) and then to enhance the DFWM gain process for the generation of near-resonant squeezed twin beams. With this technique, we obtain a −2.6 dB intensity-difference squeezing detuned 100 MHz from the D1 F = 4 to F’ = 4 transition of 133Cs.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Squeezed states of light with wavelength matching on atomic transition have become valuable resources in atom-based quantum interfaces [1,2], quantum information manipulation [3], and quantum-enhanced sensors [4–9]. The common techniques for the generation of continuous variable squeezed light are nonlinear crystal-based optical parametric oscillators (OPO) [10] and atomic-based four-wave mixings (FWM) [11,12]. The OPO proved to be a mature technology to prepare flexible squeezed light that can be operated at the desired wavelength and bandwidth [13], e.g., the generation of vacuum squeezed light on resonance with the Rb D1 line [14,15] and Cs D1/ D2 line [16,17]. The FWM process in atoms provides attractive alternatives with unique features, such as the very narrowband, bright, and high-degree squeezing resources without cavities [18–22]. The demand for strong atom-squeezed light coupling in atom-based quantum network protocols is stimulating the efforts to prepare narrowband quantum squeezing matching exactly to the atomic transition [23–25]. One of the difficulties for the generation of squeezing with atom-based FWM is the detection of resonant squeezed light, which is restrained by the excess noises mainly from spontaneous emissions and losses due to the strong re-absorption at resonance [24,26]. Therefore, studies focusing on how to suppress the resonant excess noise and improve the nonlinear FWM efficiency become very important.
Shifting the generated squeezed light beyond the atomic Doppler background can be used to avoid these excess noises. Pumped by a 750-MHz blue detuned light, the first investigation of strong intensity-difference squeezing with the degree of −3.5 dB is obtained from non-degenerate FWM in hot rubidium vapor [18], where the probe field is naturally detuned 750 MHz from relevant resonant transition because of two-photon resonance excitation. In the subsequent experiments, the degree of squeezing is improved to −9.2 dB [19]. Similar results have also been obtained using cesium atoms [27–29], with the best result of −6.5 dB by pumping the FWM with 1.6 GHz blue detuned light. However, this method of detuning pump light does not work for sodium and potassium, whose hyperfine splitting between ground states is lower than the Doppler-broadened linewidth. Thus, all of the atomic transitions are completely overlapped even at room temperature. Because of competition between the gain and absorption, a pair of bright twin-beams with squeezing of only −1.1 dB was obtained using a 500 MHz-blue detuned pump light [30]. In the above-mentioned non-degenerate FWM system [18,28,30], although a double-Λ electromagnetically induced transparency (EIT) configuration was introduced to form the coherence between hyperfine electronic ground states, the non-negligible residual absorption makes it hard to observe quantum effects when the probe frequency is tuned closer to resonance.
In our previous works [27,28], we pointed out that the large hyperfine splitting of the ground states results in the weak Raman coupling strength for the generation of the anti-stokes field [31]. Therefore, it is difficult to generate strong quantum correlation in the system with large hyperfine splitting. The Zeeman-coherence-based degenerate FWM (DFWM) is a promising system for generating stronger correlated twin beams. In comparison with the light-induced coherence between hyperfine split ground states in double-Λ EIT configuration, an EIT-type degenerate two-level configuration allows for the generation of long-lived Zeeman coherences to enhance the FWM process [32,33]. So degenerate configuration is appealing as a potential near-resonant squeezed light source because stronger FWM gain is achievable by tuning the frequency of the optical fields close to the atomic resonance under the same power of optical pumping and the same atomic number density [26]. In this paper, we choose Zeeman-coherence-based degenerate FWM (DFWM) to study the generation of cesium near-resonant squeezed light. The generated squeezing, because of the light being near atomic resonance, is likely to be limited by excess noise from the atoms and cannot be measured. So, we are exploring a way to further enhance the FWM nonlinearity.
The key to improving the nonlinear efficiency of FWM is to manipulate the quantum coherence of coupled light-atom systems actively. It has been proposed and demonstrated that multiple EIT schemes within a multi-level atom exhibit strong quantum coherence [34–38]. In this multilevel involved scheme, multiple optical transition channels are opened via dressing the ground- or excited-energy level for EIT configuration [39,40]. Quantum interference from these transition channels makes the transparency of EIT windows, as well as atomic nonlinearity, able to be actively controlled because of destructive or constructive interference [41]. The improved atomic nonlinearity gives us the ability to significantly increase the FWM efficiency and then enhance the squeezing level [42,43], e.g., Da Zhang et al measured enhanced intensity-difference squeezing via energy-level modulation FWM in a dual EIT configuration, with the noise reduction from initial −3.6 dB gets up to −7.0 dB or −9.0 dB [42].
In this paper, an additional pump field is applied to dress the excited energy level in dual-EIT configuration and realize enhanced DFWM. Moreover, this dressing field also plays a role in optical repumping, which further increases the nonlinear gain and makes the squeezing larger. Consequently, the FWM gain dominates over the excess noise from spontaneous emission and atomic reabsorption at the near-resonant case, so intensity-difference squeezed twin beams with frequency detuning only several tens of MHz from resonant cesium hyperfine transition is obtained. Last but not least, this method of near-resonant squeezed light generation via dressing energy levels to enhance FWM gain is likely to be generalized to other atoms because it is not limited by the ground-state hyperfine splitting of the atoms.
2. Basic theory
To observe intensity-difference squeezed light in the Zeeman-coherence-based DFWM system, we study two configurations: the standard DFWM configuration and the enhanced DFWM configuration with the dressing field.
2.1 Standard DFWM configuration
We consider an open two-level atomic system as shown in Fig. 1(a), consisting of two ground states $|0 \rangle$, $|1 \rangle$ and one excited state $|2 \rangle$. Specifically, the three states are the hyperfine states of the cesium atom with $|{{6^2}{S_{1/2}},F = 3} \rangle$, $|{{6^2}{S_{1/2}},F = 4} \rangle$, and $|{{6^2}{P_{1/2}},F^{\prime} = 4} \rangle$, respectively (F is the total angular momentum of hyperfine states). The frequency separation between the two ground states is ${\omega _{10}} = 9.2{\kern 1pt} {\kern 1pt} GHz$. A strong x-polarized pump beam ${E_1}$ (frequency ${\omega _1}$, wave vector ${{\boldsymbol k}_1}$, Rabi frequency ${\Omega _1} = {{{\mu _{12}}{E_1}} / \mathrm{\hbar }}$) and a weak y-polarized probe beam ${E_p}$ (${\omega _p}$, ${{\boldsymbol k}_p}$, ${\Omega _p}$) couple the same transition of $|1 \rangle \leftrightarrow |2 \rangle$ with the detuning ${\Delta _1}$ and ${\Delta _p}$, respectively. The detuning is defined as the difference between the laser frequency ${\omega _i}$ of field ${E_i}$ and the corresponding resonant transition frequency, i.e., ${\Delta _1} = {\omega _1} - {\omega _{21}}$ and ${\Delta _p} = {\omega _p} - {\omega _{21}}$, where ${\omega _{21}}$ is the resonant frequency of transition $|1 \rangle \leftrightarrow |2 \rangle$. The pump and probe fields drive ${\mathrm{\sigma }^ \pm }$ Zeeman transitions $|{{6^2}{S_{1/2}},F = 4,{m_\textrm{F}}} \rangle \leftrightarrow |{{6^2}{P_{1/2}},F = 4,{m_\textrm{F}} \pm \textrm{1}} \rangle$ and $\mathrm{\pi }$ Zeeman transitions $|{{6^2}{S_{1/2}},F = 4,{m_\textrm{F}}} \rangle \leftrightarrow |{{6^2}{P_{1/2}},F = 4,{m_\textrm{F}}} \rangle$[see Fig. 1(b)], respectively, establishing the coherence between the Zeeman sublevels of the hyperfine state $|{{6^2}{S_{1/2}},F = 4} \rangle$. This configuration can be regarded as a complex multiple N-type coupled system. In a typical N-type subsystem as shown in the dashed block in Fig. 1(b), the probe field is amplified and the conjugate field is generated via the FWM process. The conjugate beam has the same polarization as the probe beam. In the standard DFWM configuration shown in Fig. 1(a), the DFWM process is a collective effect from multiple N-type transition circles (${\mathrm{\sigma }^ + } \to \mathrm{\pi } \to {\mathrm{\sigma }^ - } \to \mathrm{\pi }$) in Zeeman sublevels. It means that the atoms absorb two pump photons of ${E_1}$ to create one probe photon ${E_p}$ and one conjugate photon ${E_c}$. Governed by energy conversion $2{\omega _1} = {\omega _p} + {\omega _c}$, the three beams have identical frequencies, i.e., ${\Delta _1} = {\Delta _p}$. A portion of atoms in the excited state is possible to transfer spontaneously to the other ground state $|0 \rangle$ and leave the DFWM process, so this configuration is an open atomic system. As shown in Fig. 1(c), the pump beam ${E_1}$ propagates through the cell along the z-axis, and the probe beam ${E_p}$ intersects with ${E_1}$ in the y-z plane at a small angle. Finally, a conjugate beam ${E_c}$ is generated on the opposite side of ${E_1}$ due to the phase-matched DFWM vector configuration of ${{\boldsymbol k}_\textrm{p}} + {{\boldsymbol k}_\textrm{c}} = 2{{\boldsymbol k}_\textrm{1}}$. Such a DFWM process can be described using a perturbative chain: $\rho _{11}^{(0 )}\buildrel {{\omega _1}} \over \longrightarrow \rho _{21}^{(1 )}\buildrel {{\omega _\textrm{p}}} \over \longrightarrow \rho _{11}^{(2 )}\buildrel {{\omega _1}} \over \longrightarrow \rho _{21}^{(3 )}$ for the creation of a conjugate beam ${E_c}$, and its intensity is given by ${I_\textrm{c}} \propto {|{\rho_{21}^{(3 )}} |^2}$. The density matrix element $\rho _{21}^{(3 )}$ can be solved using perturbation theory with
Fig. 1. (a) The diagram of the standard DFWM energy levels. (b) The DFWM in Zeeman-sublevels. (c) The spatial alignments for incident beams.
2.2 Enhanced DFWM configuration
When a strong field ${E_\textrm{2}}$ (${\omega _\textrm{2}}$, ${{\boldsymbol k}_p}$, ${\Omega _p}$) couples to the transition of $|\textrm{0} \rangle \leftrightarrow |2 \rangle$ with detuning ${\Delta _2}$(${\Delta _2} = {\omega _2} - {\omega _{20}}$, where ${\omega _{21}}$ is the resonant frequency of transition $|\textrm{0} \rangle \leftrightarrow |2 \rangle$). Taking into account the dressing effect of E2, Eq. (1) can be rewritten as
Fig. 2. (a) The diagram of the enhanced DFWM energy levels. (b) The dressed-state picture of the enhancement of DFWM for the Λ-dressing system. (c) The spatial alignments for incident beams.
Fig. 3. The theoretical simulation of FWM intensity gain spectrum when ${\Delta _\textrm{1}} = {\Delta _\textrm{p}}$. The blue solid curve refers to the standard DFWM case with ${\Omega _\textrm{2}} = 0$, and the red dashed curve refers to the enhanced DFWM case with ${{{\Omega _\textrm{2}}} / {2\pi }} = 100{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$. Other Parameters: ${{{\Omega _\textrm{1}}} / {2\pi }} = 80{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$, ${{{\Omega _\textrm{p}}} / {2\pi }} = 5{\kern 1pt} {\kern 1pt} \textrm{MHz}$, ${{{\Delta _\textrm{2}}} / {2\pi }} = 150{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$, ${{{\Gamma _{21}}} / {2\pi }} = 4.6{\kern 1pt} {\kern 1pt} \textrm{MHz}$, ${{{\Gamma _{01}}} / {2\pi }} = 600{\kern 1pt} {\kern 1pt} \textrm{kHz}$, and ${{{\Gamma _{11}}} / {2\pi }} = 30{\kern 1pt} {\kern 1pt} \textrm{kHz}$.
3. Experimental setup
Our experiment is performed in cesium vapor. The pump bean E1 and seeded probe beam Ep are from the same laser (Ti: sapphire laser), whose frequency is locked to near the resonant transition $|1 \rangle \leftrightarrow |2 \rangle$. The dressing beam ${E_\textrm{2}}$ is from another laser (diode laser, Toptica DL pro), whose frequency is locked to near the resonant transition $|0 \rangle \leftrightarrow |2 \rangle$. The pump beam ${E_\textrm{1}}$ and the dressing beam ${E_\textrm{2}}$, both polarized along the x direction, are coupled by BS1 and then incident on the reflection port of GT1 together. The probe beam ${E_\textrm{p}}$, polarized along the y direction, enters through the transmission port of GT1. These three beams are focused with 1/ e2 diameters of ${w_1} = 730\textrm{ }\mathrm{\mu m}$, ${w_2} = 740\textrm{ }\mathrm{\mu m}$, and ${w_\textrm{p}} = 280\textrm{ }\mathrm{\mu m}$ at the center of the Cs cell, respectively, and overlap over almost the full length-25 mm of the cell. The Cs cell is placed in a heating oven to control the atom temperature and coated with anti-reflection end windows to maximize transmission. Considering the FWM phase matching as well as spatial separation for beams, the geometry is arranged as follows: ${E_\textrm{1}}$ propagates through the vapor cell along the z-axis, ${E_\textrm{p}}$ intersects with ${E_\textrm{1}}$ in the y-z plane at an angle of 3.4 mrad, while ${E_\textrm{2}}$ intersects with ${E_\textrm{1}}$ in the x-z plane with an angle of 4.6 mrad. As expected, the generated conjugate beam ${E_\textrm{c}}$ carries the identical frequency and polarization with the seeded probe beam, while propagating on the other side of ${E_\textrm{1}}$ with a cross angle of 3.4 mrad in the y-z plane. At the exit of the vapor cell, the two x-polarized beams ${E_\textrm{1}}$ and ${E_\textrm{2}}$ are reflected out of the experimental system by GT2, and a movable light screen S and CCD are used to capture the pattern of FWM output beams. For the measurement of gain spectra of ${E_\textrm{p}}$ and ${E_\textrm{c}}$, direct detection from PD1 and PD2 are used, where the seeded probe beam is from a third laser (tunable diode laser, Toptica DL pro) with frequency scanned ${\pm} 500\textrm{ }\textrm{MHz}$ across the $|1 \rangle \leftrightarrow |2 \rangle$ resonant transition. For the measurement of intensity-difference squeezing between ${E_\textrm{p}}$ and ${E_\textrm{c}}$ as well as the corresponding Shot Noise Level (SNL), two sets of self-homodyne detection (SHD) systems SHD1 and SHD2 are employed to record the photocurrent fluctuation from the probe beam and the conjugate beam, respectively. Here, each SHD consists of a 50/50 beam splitter and two balanced low-noise-amplified photodetectors with a trans-impedance gain of 104 V/A and 85% quantum efficiency. SHD is a popular way to measure the quadrature amplitude fluctuation of bright fields [44]. In Fig. 4, the sum of photocurrent from SHD is proportional to the light field’s intensity noise, and the difference of that shows the corresponding SNL, see the appendix for details. To obtain the maximum squeezing, a classical electric attenuator g is put on the probe arm by optimizing electric gain. It is theoretically predicted that the detection system shown in Fig. 4 is capable of maximizing the squeezing measurement if the electric attenuator is set to be $g = \sqrt {{G_\textrm{p}}/({G_\textrm{p}} - 1)}$, as shown in Fig. 8 in the Appendix. In our experiment, the value of the optimal electric attenuator g is adjusted precisely by balancing the fluctuations of the sum-photocurrent of the two SHD systems.
Fig. 4. Schematic of the experimental setup. The bidirectional arrows in propagation points for the polarization vector of light fields; GT, Glan–Taylor prism; BS, 50/50 Beam Splitter; HR, High-Reflectivity mirror; CCD, Camera; S, Screen; PD, Photodiode Detector; g, Electric Attenuator; SA, Spectrum Analyzer.
4. Experimental results and discussion
For comparison, both the standard DFWM phenomena and enhanced DFWM phenomena with the dressing field ${E_\textrm{2}}$ absence and presence have been shown in Fig. 5. The nonlinear optical gain spectrum at settled atom temperature of $57\textrm{ }^\circ \textrm{C}$ are present in Fig. 5(a) and Fig. 5(b), respectively. To record the normalized transmission spectra of the twin beams, a tunable diode laser (a third laser, which have not shown in Fig. 4) with frequency scanned ${\pm} 500\textrm{ }\textrm{MHz}$ across the $|1 \rangle \leftrightarrow |2 \rangle$ resonant transition is used as the seeded probe field (with power of ${P_{\textrm{p - in}}} = 70{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathrm{\mu W}$). The gray curves in Fig. 5(a) and Fig. 5(b) are the saturated absorption spectrum of cesium $|{{6^2}{S_{1/2}},F = 4} \rangle \leftrightarrow |{{6^2}{P_{1/2}},F^{\prime} = 4} \rangle$ transition (used for frequency reference in the experiment), where the peak at ${\Delta _\textrm{p}} = 0$ shows atomic resonance. When the dressing field ${E_\textrm{2}}$ is absent, it is an opened two-level standard DFWM configuration, an EIT window with weak probe gain of ${G_\textrm{p}} \approx 0.84$(with the power of ${P_{\textrm{p - out}}} = 58.8{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathrm{\mu W}$), accompanying a conjugate gain peak with the amount of ${G_\textrm{c}} \approx 0.20$(${P_{\textrm{c - out}}} = 14{\kern 1pt} {\kern 1pt} \mathrm{\mu W}$) occurs at two-photon resonance ${\Delta _\textrm{p}} = {\Delta _\textrm{1}}$ for fixed pump detuning ${{{\Delta _\textrm{1}}} / {2\pi }} ={-} 100{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$ and fixed pump power ${P_\textrm{1}} = 115{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{mW}$, see the red and blue curves in Fig. 5(a). When the dressing field ${E_\textrm{2}}$ (with power of ${P_\textrm{2}} = 115{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{mW}$, Rabi frequency ${{{\Omega _\textrm{2}}} / {2\pi }} = 334{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$) is turned on, a dual EIT system is formed, one is for Zeeman coherence-based two-level EIT driven by ${E_\textrm{p}}$ and ${E_\textrm{1}}$, the other one is for hyperfine coherence-based Λ-type EIT driven by ${E_\textrm{p}}$ and ${E_\textrm{2}}$. Fixing the frequency of ${E_\textrm{2}}$ on blue detuned $150{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}{E_\textrm{2}}$ to the atomic resonant transition of $|0 \rangle \leftrightarrow |2 \rangle$ (i.e., ${{{\Delta _\textrm{2}}} / {2\pi }} = 150{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$), two gain peaks are shown for the probe field in dual EIT system [see the red curve in Fig. 5(b)]. The left one with an improved gain of ${G_\textrm{p}} \approx 2.59$(${P_{\textrm{p - out}}} = 181{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathrm{\mu W}$) and ${G_\textrm{c}} \approx 1.65$(${P_{\textrm{c - out}}} = 116{\kern 1pt} {\kern 1pt} \mathrm{\mu W}$) at the ${\Delta _\textrm{p}} = {\Delta _\textrm{1}}$ EIT subsystem, shows the enhanced DFWM phenomena in contrast with the gain peaks in Fig. 5(a). As mentioned in the theory part, the strong field ${E_\textrm{2}}$ dresses the excited state $|2 \rangle$ into $|+ \rangle$ and $|- \rangle$ with splitting about ${{{{\tilde{\Omega }}_\textrm{2}}} / {2\pi }} = 366{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$, and then opens two dressed DFWM channels $\rho _{11}^{(0 )}\buildrel {{\omega _1}} \over \longrightarrow \rho _{ {\pm} 1}^{(1 )}\buildrel {{\omega _\textrm{p}}} \over \longrightarrow \rho _{11}^{(2 )}\buildrel {{\omega _1}} \over \longrightarrow \rho _{ {\pm} 1}^{(3 )}$. Constructive interference occurs between two dressed DFWM channels when the frequency is close to ${\Delta _\textrm{p}} = {\Delta _1} = ({{\Delta _\textrm{2}} - {{\tilde{\Omega }}_2}} )/2 \approx{-} 108{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$. This comparison with standard DFWM and enhanced DFWM phenomenon can also be displayed by the output beams’ pattern in the illustrations of Fig. 5(a) and Fig. 5(b). The residual y-polarized beam ${E_\textrm{1}}$ in the illustration of Fig. 5(a) is from imperfect extinction of the GT2. The spot of ${E_\textrm{1}}$ in the illustration of Fig. 5(b) becomes brighter than that in Fig. 5(a) due to the larger rotation of polarization [45]. The right EIT window with nearly 100% transmissivity for the probe field occurs at the ${\Delta _\textrm{p}} = {\Delta _\textrm{2}}$ EIT subsystem, as shown in Fig. 5(b), where multi-wave mixing or other nonlinear phenomenon occurs. There is a big background envelope around the probe gain peak (red curve) in Fig. 5(a), but isn’t in Fig. 5(b). It can be explained by the population depletion effect of optical pumping [46] in the open two-level system for the standard DFWM configuration. Next, we focus on the intensity quantum correlation between the probe-conjugate twin beams in the left gain peak.
Fig. 5. The Normalized transmission spectrum for (a) the standard DFWM with ${P_\textrm{2}} = 0{\kern 1pt} {\kern 1pt} \textrm{mW}$ and (b) the enhanced DFWM with ${P_\textrm{2}} = 115{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{mW}$. The red and blue curves are for the probe and conjugate, respectively. The gray curve is for the saturated absorption spectrum of cesium. Experimental parameters: ${P_\textrm{1}} = 115{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{mW}$, ${P_{\textrm{p - in}}} = 70{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathrm{\mu W}$, ${{{\Delta _\textrm{1}}} / {2\pi }} ={-} 100{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$, ${{{\Delta _\textrm{2}}} / {2\pi }} = 150{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$, and $T\textrm{ = 57 }^\circ \textrm{C}$.
The optical gain of the standard DFWM case is so small that the noise cannot be detected accurately by the photodetector in our experiment, i.e., the intensity noise is close to the electronic noise of the detection system. So, we improve the optical gain in the standard DFWM scheme to the same magnitude as that in the enhanced DFWM scheme by increasing atom temperature. Figure 6 gives a comparison of the intensity-difference noise for the two cases of the standard DFWM and the enhanced DFWM. The green trace and green squares show the SNL, the blue trace and blue triangles show the intensity-difference noise of the standard DFWM case, and the orange trace and orange dots show the intensity-difference noise of the enhanced DFWM case. To compare the squeezing values between the two DFWM configurations directly, the noise traces have been normalized by the SNL. Moreover, the true squeezing value is shown by subtracting the electronic noise of the detection system. Although the gain peaks in Fig. 5(a) and Fig. 5(b) are close to atomic resonance and within the Doppler broadened absorption background. Intensity-difference noise between the twin beams is changed from un-correlated ($\textrm{6}\textrm{.4}{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{d}\textrm{B}$ above the SNL) for the standard DFWM case to quantum correlated with about $- \textrm{2}\textrm{.6}{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{d}\textrm{B}$ squeezing at $1.5{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$ analysis frequency for the enhanced DFWM case, shown in Fig. 6(a). Note that the results are obtained via the photodetectors with 85% quantum efficiency. The squeezing extends only to about $4.2{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$, it is limited mainly by the pump intensity [47] and the atomic absorption near resonance. To better quantify the degree of squeezing, we vary the seeded probe power and record the intensity-difference noise power versus the total optical power of the twin beams, see the blue triangles and the orange dots in Fig. 6(b). Similarly, we also record the noise power of difference between the subtracted photocurrents from SHD1 and SHD2 at 1.5 MHz as the SNL, see the green squares in Fig. 6(b). Fitting the standard DFWM, the enhanced DFWM and SNL noise power curves to straight lines, their slopes are equal to ${k_{\textrm{standard DFWM}}} = 0.894$, ${k_{\textrm{enhanced DFWM}}} = 0.114$, and ${k_{\textrm{SNL}}} = 0.204$, respectively. It shows that the degrees of intensity-difference squeezing of the twin beams generated through enhanced DFWM is $- 10\log ({{{{k_{\textrm{SNL}}}} / {{k_{\textrm{enhanced DFWM}}}}}} )={-} \textrm{2}\textrm{.6}{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{d}\textrm{B}$, the intensity-difference noise obtained from the standard DFWM is higher than the SNL about $\textrm{6}\textrm{.4}{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{d}\textrm{B}$. In addition, the electronic noise from the detection system have been subtracted from all the data points in Fig. 6, and the same processing has been done on all the subsequent experimental data of squeezing in this paper.
Fig. 6. The comparison of intensity-difference noise for the standard DFWM and the enhanced DFWM cases. (a) The normalized noise traces. (b) The noise power at $1.5{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$ versus the total optical power of the twin beams. The green trace and green squares show the SNL, the blue trace and blue triangles show the intensity-difference noise of the standard DFWM case, and the orange trace and orange dots show the intensity-difference noise of the enhanced DFWM case. In the measurement, the spectrum analyzer was set to a resolution bandwidth of $300{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{kHz}$ and a video bandwidth of $300{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{Hz}$. The value of electric attenuator g is $1{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{dB}$. The atom temperature is $T\textrm{ = 105 }^\circ \textrm{C}$ for the standard DFWM case and $T\textrm{ = 57 }^\circ \textrm{C}$ for the enhanced DFWM case. Other experimental parameters are the same as those in Fig. 5.
The experimental results in Fig. 6 show that, we have successfully obtained squeezed-twin beams near the atomic resonant transition frequency through the enhanced DFWM. This generated squeezing benefits from the enhanced optical nonlinearity because of energy-level modulation as well as the population transfer effect by the dressing field E2. In the following, we explore the best degree of squeezing we may obtain and the manipulation of squeezing by changing other parameters. To do this, we record the maximum difference between the intensity-difference noise power and the corresponding SNL at analysis frequency region 0∼5 MHz as the squeezing value. Firstly, when set the dressing field ${E_\textrm{2}}$ with optical power of ${P_\textrm{2}} = 115{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{mW}$ and blue detuned with ${{{\Delta _\textrm{2}}} / {2\pi }} = 150{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$, and changing the pump detuning from ${{{\Delta _\textrm{1}}} / {2\pi }} ={-} 150{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$ to $- 50{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$, the nonlinear optical gain shows increasing because of stronger atom-light nonlinear coupling with frequency closer to atomic resonance, see the black triangles in Fig. 7(a). However, the squeezing is not increased monotonically but with an optimal degree of squeezing about $- \textrm{2}\textrm{.6}{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{d}\textrm{B}$ occurs at ${{{\Delta _\textrm{1}}} / {2\pi }} ={-} 100{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$, see the red circles in Fig. 7(a). The main reason is that the enhanced nonlinear gain from constructive interference in the dressed state occurs at the nearby frequency ${\Delta _\textrm{p}} = {\Delta _1} = ({{\Delta _\textrm{2}} - {{\tilde{\Omega }}_2}} )/2 \approx{-} 108{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$. On the other hand, close to atomic resonance the re-absorption from the atom dominates over the DFWM process, the squeezing is significantly reduced but $- \textrm{1}\textrm{.4}{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{d}\textrm{B}$ squeezing can still be measured at ${{{\Delta _\textrm{1}}} / {2\pi }} ={-} 50{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$[see the red circles in Fig. 7(a)].
Fig. 7. Dependence of the intensity-difference squeezing (red circles) and enhanced DFWM gain on (a) pump detuning ${\Delta _\textrm{1}}$, (b) atom temperature T, pump power (c)${P_\textrm{1}}$, and (d)${P_\textrm{2}}$. The blue dotted line at 0 dB shows the corresponding SNL. Electronic noise has been subtracted from all the data points. The dressing field detuning is set to be ${{{\Delta _\textrm{2}}} / {2\pi }} = 150{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$, and the pump detuning is set to be ${{{\Delta _\textrm{1}}} / {2\pi }} ={-} 100{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$ except for the data in (a).
We then turn our attention to the dependence of intensity-difference squeezing as well as enhanced DFWM gain on the temperature of the vapor cell [shown in Fig. 7(b)], with the pump power fixed at ${P_\textrm{1}} = {P_\textrm{2}} = 115{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{mW}$ and the pump detuning ${{{\Delta _\textrm{1}}} / {2\pi }} ={-} 100{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$, ${{{\Delta _\textrm{2}}} / {2\pi }} = 150{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$. Changing the atom temperature from $\textrm{45 }^\circ \textrm{C}$ to $\textrm{69 }^\circ \textrm{C}$ with the step of $\textrm{3 }^\circ \textrm{C}$, the nonlinear optical gain increases dramatically from 1.1 to 6.7, due to the rapidly improving atomic number densities. In contrast, the degree of intensity-difference squeezing is not varied monotonically but displays a maximum value of $- \textrm{2}\textrm{.8}{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{d}\textrm{B}$ dB at $\textrm{57 }^\circ \textrm{C}$. At lower temperatures, weaker nonlinearity induces lower enhanced DFWM gain, thus the quantum correlation between the twin beams is weak. As the temperature further increases from $\textrm{57 }^\circ \textrm{C}$, the quantum correlation degrades because atomic absorption loss from the optically thicker cesium vapor becomes stronger and stronger. As shown in the red circles in Fig. 7(b), the intensity-difference quantum correlation vanishes after $\textrm{66 }^\circ \textrm{C}$.
Lastly, the effect of pump power on the enhanced DFWM gain as well as squeezing is investigated under the optimum pump detuning ${{{\Delta _\textrm{1}}} / {2\pi }} ={-} 100{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$, ${{{\Delta _\textrm{2}}} / {2\pi }} = 150{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{MHz}$, and optimum atom temperature $T\textrm{ = 57 }^\circ \textrm{C}$, shown in Fig. 7(c) and Fig. 7(d). Raising the pump power ${P_\textrm{1}}$ from $60{\kern 1pt} {\kern 1pt} \textrm{mW}$ to $115{\kern 1pt} {\kern 1pt} \textrm{mW}$ with fixed ${P_\textrm{2}} = 115{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{mW}$, the nonlinear optical gain grows monotonically, and the degree of squeezing keeps up a steady increase until the maximum value $- \textrm{2}\textrm{.6}{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{d}\textrm{B}$ at ${P_\textrm{1}} = 115{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{mW}$, as expected. As for the dressing field ${E_\textrm{2}}$, the dressing effect as well as the population transfer effect can improve the DFWM nonlinear optical gain and intensity-difference squeezing. In the experimental data in Fig. 7(d), the DFWM efficiency grows monotonically and exhibits an improving degree of squeezing and enhanced DFWM gain with increasing pump power ${P_\textrm{2}}$ from 0 to $115{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{mW}$ when ${P_\textrm{1}} = 115{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{mW}$.
5. Conclusions
We explored the relative-intensity noise between near-resonant twin beams from dressed DFWM using a cesium vapor cell. Compared with the uncorrelated case from DFWM, intensity-difference squeezing with the maximum value of $- \textrm{2}\textrm{.6}{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{d}\textrm{B}$ has been obtained due to the active enhanced atomic coherence and interference via energy-level modulation by dressing field, which can control the DFWM gain dominates over excess noises from spontaneous emission and atomic reabsorption at near-resonant frequency. The generated bright squeezed light lies in an atomic strongly absorbed Doppler broaden background (detuned from atomic resonance with 100 MHz) and shows potential application for atomic-based sensors, atomic interferometer, quantum information as well as exploring the interaction between squeezed light and quantum storage medium.
Appendix: the detection of the maximum squeezing
The DFWM process can be viewed as a parametric amplification with the effective Hamiltonian
where $\hat{a}_\textrm{p}^\dagger$ and $\hat{a}_\textrm{c}^\dagger$ are the creation operators of the probe and conjugate fields, respectively; $\kappa = |{{\chi^{(3 )}}E_1^2} |= |{N\mu_{12}^2\rho_{21}^{(3 )}{\boldsymbol /}{\varepsilon_0}\hbar {\Omega _\textrm{p}}} |$ is the pumping parameter; N is the atomic number density; $\hbar$ is the reduced Planck constant; and ${\varepsilon _0}$ is the vacuum permittivity. It means that the atoms absorb two pump photons of E1 to create one probe photon Ep and one conjugate photon Ec. The input-output relation for quantized light fields Ep and Ec areSHD is a popular way to measure the quadrature amplitude fluctuation of bright fields [43]. In Fig. 3, the fluctuation of the sum- and difference- photocurrents for the PD3 and PD4 in SHD1, and that for the PD5 and PD6 in SHD2 are respectively written as
Finally, the noise figure for intensity-difference squeezing is defined as
If $g = \sqrt {G/(G - 1)}$, the squeezing holds maximum with
This is the result of direct detection [19,30]. The Eqs. (11) and (12) show that the larger the magnitude of nonlinear gain G, the higher level of squeezing will be obtained. In Fig. 3, we plot the noise figure for both cases of $g = \sqrt {G/(G - 1)}$ (blue solid curve) and $g = 1$ (red dashed curve), where the negative value represents the appearance of squeezing. It is theoretically predicted that the former one is capable of maximizing the squeezing measurement at the same FWM gain G, as shown in the blue solid curve in Fig. 8. In the experiment, the value of the optimal electric attenuator g is adjusted precisely by balancing the fluctuations of the sum-photocurrent of the two SHD systems.
Funding
Fund for Shanxi Key Subjects Construction; Applied Basic Research Project of Shanxi Province, China (201901D211166, 20210302123437); National Natural Science Foundation of China (11704235, 12004334, 92065108).
Acknowledgments
Thanks for the useful discussion with Xudong Yu and Dehuan Cai.
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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