Parametric amplification of Rydberg six- and eight-wave mixing processes

Zhaoyang Zhang; Ji Guo; Bingling Gu; Ling Hao; Gaoguo Yang; Kun Wang; Yanpeng Zhang

doi:doi:10.1364/PRJ.6.000713

Photonics Research, 2018, 6 (7): 07000713, Published Online: Jul. 4, 2018

Parametric amplification of Rydberg six- and eight-wave mixing processes

论文大纲

Zhaoyang Zhang ^1†Ji Guo ^1†Bingling Gu Ling Hao Gaoguo Yang Kun Wang Yanpeng Zhang ^*

Author Affiliations

Key Laboratory for Physical Electronics and Devices of the Ministry of Education & Shaanxi Key Laboratory of Information Photonic Technique, Xi’an Jiaotong University, Xi’an 710049, China

Rydberg states Nonlinear wave mixing Nonlinear optics, four-wave mixing Nonlinear optics, parametric processes

Abstract

We study the parametric amplification of electromagnetically induced transparency-assisted Rydberg six- and eight-wave mixing signals through a cascaded nonlinear optical process in a hot rubidium atomic ensemble both theoretically and experimentally. The shift of the resonant frequency (induced by the Rydberg–Rydberg interaction) of parametrically amplified six-wave mixing signal is observed. Moreover, the interplays between the dressing effects and Rydberg–Rydberg interactions in parametrically amplified multiwave mixing signals are investigated. The linear amplification of Rydberg multiwave mixing processes with multichannel nature acts against the suppression caused by Rydberg–Rydberg interaction and dressing effect.

1. INTRODUCTION

Atoms excited to Rydberg states have attracted considerable interest owing to their excellent properties, such as long lifetimes, large collision cross sections, large dipole moments, and gigantic spatial extension ^[1]. Besides, Rydberg–Rydberg interaction (RRI) between Rydberg atoms, such as dipole–dipole interaction and van der Waals interaction, can induce the dipole blockade of nearby atoms ^[2,3]. These properties lead to a large amount of prospective applications, such as quantum information processing ^[4,5], high-fidelity optical state control ^[6], and single-photon sources ^[7]. For these applications, detecting Rydberg dressing-state effect and the interactions among the Rydberg atoms nondestructively is a basic requirement. Recently, electromagnetically induced transparency (EIT) ^{[8–12" target="_self" style="display: inline;">–12]} and multiwave mixing (MWM) ^{[1315" target="_self" style="display: inline;">–15]} as nondestructive optical detection methods ^[16,17] have been proposed and utilized for detecting Rydberg atoms ^[18,19] where the Rydberg atoms are not ionized. The multichannel nature of the MWM process allows for multichannel information processing related to Rydberg states. Furthermore, the MWM process can act as a multimode correlated light source ^[20]. However, MWM signals generated from high-order nonlinear optical processes such as six-wave mixing (SWM) and eight-wave mixing (EWM) are much weaker than EIT signals ^[21]. In addition, Rydberg excitation can further suppress the signal intensity and decrease the signal-to-noise ratio ^[22].

The optical parametric amplification (OPA) process, which can act as an optical amplifier and implement the linear amplification of the input signal ^[23], is proposed and experimentally achieved in the media of both gaseous and solid states ^[24,25]. Generally, the OPA process in an atomic medium is characterized by the so-called conical emission ^[26], where two correlated photons named as Stokes and anti-Stokes photons are effectively generated ^[27]. The OPA process can be achieved by the cascaded nonlinear process, in which the generated MWM signals coexist with the parametrical FWM process ^[20,28]. Intensity noise correlation ^[29] and intensity-difference squeezing ^[30] of such a process have given rise to applications in quantum metrology ^{[3133" target="_self" style="display: inline;">–33]}. Inspired by such an OPA process, the high-order Rydberg MWM signal can also be injected into the Stokes or anti-Stokes port and then be linearly and nondestructively amplified to enhance the signal-to-noise ratio of the Rydberg MWM signal. As a result, the performance of applications related to Rydberg excitations such as sensors ^{[31–35" target="_self" style="display: inline;">–35]} can be promisingly improved via OPA.

In this paper, we observed the Rydberg parametrically amplified MWM (PA-MWM) signals assisted by the cascaded nonlinear process in a $K$ -type five-level system of ${}^{85}{Rb}$ . With two EIT windows generated effectively, the parametrically amplified SWM (PA-SWM) and EWM (PA-EWM) signals can be simultaneously detected. Meanwhile, the intensities of PA-MWM signals can be controlled by the detuning and the power of the coupling fields as well as the density of the atomic ensemble. Moreover, the saturation of the signal intensity and the shift of the resonant position caused by the RRI when changing the beam power and temperature can be detected. Finally, the suppression (enhancement) of coexisting PA-SWM and PA-EWM signals near resonance (far from resonance) is observed. These phenomena in the Rydberg PA-MWM process can improve the applications related to Rydberg atoms in the fields of quantum metrology and sensors.

2. BASIC THEORY AND EXPERIMENTAL SETUP

In the $K$ -type five-level ${}^{85}{Rb}$ atomic system shown in Fig. 1(a), two hyperfine energy levels, $F = 3$ ( $| 0 ⟩$ ) and $F = 2$ ( $| 3 ⟩$ ), of the ground state $5 S_{1 / 2}$ , a Rydberg excited state $n D_{5 / 2}$ ( $| 2 ⟩$ ), and two lower excited states, $5 P_{3 / 2}$ ( $| 1 ⟩$ ) and $5 D_{5 / 2}$ ( $| 4 ⟩$ ), are connected by corresponding beams. The experimental configuration is shown in Fig. 1(b). Five beams derived from four external cavity diode lasers (ECDLs) with linewidths $< 1 MHz$ are used to couple the following transitions. The transition $| 0 ⟩ \leftrightarrow | 1 ⟩$ is probed by beam $E_{1}$ [wavelength of 780.2 nm, frequency $ω_{1}$ , wave vector $k_{1}$ , and Rabi frequency $G_{1}$ , defined as $G_{i} = μ_{i j} E_{i} / ℏ$ , where $μ_{i j}$ is the dipole moment between $| i ⟩ \leftrightarrow | j ⟩$ ( $i$ , $j = 1$ , 2, 3, 4)]. The Rydberg transition $| 1 ⟩ \leftrightarrow | 2 ⟩$ is connected by a strong beam $E_{2}$ ( $\sim 480 nm$ , $ω_{2}$ , $k_{2}$ , $G_{2}$ ), which propagates opposite to beam $E_{1}$ . The transition $| 1 ⟩ \leftrightarrow | 3 ⟩$ is connected by two beams $E_{3}$ (780.2 nm, $ω_{3}$ , $k_{3}$ , $G_{3}$ ) and $E_{3}^{'}$ (780.2 nm, $ω_{3}^{'}$ , $k_{3}^{'}$ , $G_{3}^{'}$ ), which are derived from the same ECDL. In the atomic ensemble, $E_{3}$ propagates in the same direction with $E_{2}$ , while $E_{3}^{'}$ has a small angle of 0.3° with $E_{3}$ . The transition $| 1 ⟩ \leftrightarrow | 4 ⟩$ is driven by the beam $E_{4}$ (775.9 nm, $ω_{4}$ , $k_{4}$ , $G_{4}$ ), whose propagation is symmetrical to $E_{3}^{'}$ about $E_{2}$ .

(a) Five-level K-type energy level diagram depicting the generation of the MWM process in the Rb85 atomic system. (b) Experimental setup. D, photodetector; L, lens; PBS, polarized beam splitter at corresponding wavelength; FD, frequency doubler; HR, high-reflectivity mirror; HW, half-wave plate at corresponding wavelength. Transverse double-headed arrows and filled dots indicate the horizontal polarization and vertical polarization of incident beams, respectively. Five beams derived from the four laser systems are coupled into the 10 mm long Rb cell wrapped with μ-metal sheets. The transition |0⟩↔|1⟩ is coupled by the beam E1 (780.2 nm). Rydberg transition |1⟩↔|2⟩ is coupled by beam E2 (480 nm), which counterpropagates with beam E1. |1⟩↔|3⟩ is connected by beams E3 and E3′ (780.2 nm), which are derived from the same ECDL, and |1⟩↔|4⟩ is coupled by beam E4 (775.9 nm). The EIT signal and MWM spectrum signals are received by D1 and D2, respectively. (c1) Energy schematic diagram for SP-FWM process; (c2) phase-matching condition of SP-FWM process.

Fig. 1. (a) Five-level $K$ -type energy level diagram depicting the generation of the MWM process in the ${}^{85}{Rb}$ atomic system. (b) Experimental setup. D, photodetector; L, lens; PBS, polarized beam splitter at corresponding wavelength; FD, frequency doubler; HR, high-reflectivity mirror; HW, half-wave plate at corresponding wavelength. Transverse double-headed arrows and filled dots indicate the horizontal polarization and vertical polarization of incident beams, respectively. Five beams derived from the four laser systems are coupled into the 10 mm long Rb cell wrapped with μ-metal sheets. The transition $| 0 ⟩ \leftrightarrow | 1 ⟩$ is coupled by the beam $E_{1}$ (780.2 nm). Rydberg transition $| 1 ⟩ \leftrightarrow | 2 ⟩$ is coupled by beam $E_{2}$ (480 nm), which counterpropagates with beam $E_{1}$ . $| 1 ⟩ \leftrightarrow | 3 ⟩$ is connected by beams $E_{3}$ and $E_{3}^{'}$ (780.2 nm), which are derived from the same ECDL, and $| 1 ⟩ \leftrightarrow | 4 ⟩$ is coupled by beam $E_{4}$ (775.9 nm). The EIT signal and MWM spectrum signals are received by D1 and D2, respectively. (c1) Energy schematic diagram for SP-FWM process; (c2) phase-matching condition of SP-FWM process.

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2.3 A. Rydberg MWM Process

By turning these lasers on and off selectively, we can get different MWM signals with different orders. When the beams $E_{2}$ and $E_{4}$ are blocked, an FWM process satisfying the phase-matching condition $k_{FWM} = k_{1} + k_{3} - k_{3}^{'}$ will occur in the three-level subsystem $| 0 ⟩ \leftrightarrow | 1 ⟩ \leftrightarrow | 3 ⟩$ . When only the beam $E_{2}$ with Rydberg dressing-state effect is blocked, an SWM process (denoted as SWM1) satisfying the phase-matching condition $k_{SWM 1} = k_{1} + k_{3} - k_{3}^{'} + k_{4} - k_{4}$ will occur in a four-level subsystem $| 0 ⟩ \leftrightarrow | 1 ⟩ \leftrightarrow | 3 ⟩ \leftrightarrow | 4 ⟩$ , in which one photon each from $E_{1}$ , $E_{3}$ , $E_{3}^{'}$ , and two photons from $E_{4}$ are involved. Similarly, by blocking $E_{4}$ , another SWM process (denoted as SWM2) with $k_{SWM 2} = k_{1} + k_{3} - k_{3}^{'} + k_{2} - k_{2}$ can be observed in the subsystem $| 0 ⟩ \leftrightarrow | 1 ⟩ \leftrightarrow | 2 ⟩ \leftrightarrow | 3 ⟩$ . When all the beams shown in Fig. 1(b) are on, a new EWM signal can be generated with phase-matching condition $k_{EWM} = k_{1} + k_{2} - k_{2} + k_{3} - k_{3}^{'} + k_{4} - k_{4}$ in the five-level atomic system $| 0 ⟩ \leftrightarrow | 1 ⟩ \leftrightarrow | 2 ⟩ \leftrightarrow | 3 ⟩ \leftrightarrow | 4 ⟩$ . It can be revealed from the phase-matching conditions that these MWM signals emit in the direction opposite to $E_{3}^{'}$ . These signals are detected by the avalanche photodiode detectors (APDs). To be specific, the EIT signal is received by D1, and the MWM signals are received by D2, as shown in Fig. 1(b).

Generally, the response of the atoms to the light is described by the susceptibility. The generated MWM signals are characterized by their nonlinear susceptibilities $χ^{(2 n + 1)} = ρ_{0} μ_{10} ρ_{10}^{(2 n + 1)} / ϵ_{0} E_{1}$ (e.g., $n = 1$ for FWM, $n = 2$ for SWM, and $n = 3$ for EWM). According to the perturbation chain $ρ_{00}^{(0)} \overset{E_{1}}{\to} ρ_{10}^{(1)} \overset{E_{3}}{\to} ρ_{30}^{(2)} \overset{{(E_{3}^{'})}^{*}}{\to} ρ_{10}^{(3)}$ via the Liouville pathway ^[36], the density-matrix element for the $E_{FWM}$ is given by $ρ_{10}^{(3)} = \frac{i G_{1} {| G_{3} |}^{2} e^{i k_{FWM} \cdot r}}{{(d_{1}^{'} + {| G_{1} |}^{2} / Γ_{00})}^{2} d_{3}^{'}},$ (1)where $d_{1}^{'} = Γ_{10} + i (Δ_{1} + k_{1} v)$ and $d_{3}^{'} = Γ_{30} + i (Δ_{1} - Δ_{3})$ $+ i (k_{1} v + k_{3} v)$ , $Γ_{i j}$ is the decay rate between states $| i ⟩$ and $| j ⟩$ , and $Δ_{i} = Ω_{i j} - ω_{i}$ is the detuning between the frequency $ω_{i}$ of beam $E_{i}$ and the resonant transition frequency $Ω_{i j}$ between $| i ⟩ \leftrightarrow | j ⟩$ ; $k_{i} v$ is the term of the Doppler effect.

Considering the upper transition $| 1 ⟩ \leftrightarrow | 4 ⟩$ , the SWM1 signal is generated with the help of the EIT windows ( $Δ_{1} + Δ_{4} = 0$ ). The perturbation chain for this process can be written as $ρ_{00}^{(0)} \overset{E_{1}}{\to} ρ_{10}^{(1)} \overset{E_{3}}{\to} ρ_{30}^{(2)} \overset{{(E_{3}^{'})}^{*}}{\to} ρ_{10}^{(3)} \overset{E_{4}}{\to} ρ_{40}^{(4)} \overset{E_{4}^{*}}{\to} ρ_{10}^{(5)}$ . Considering the strong field dressing effect of $E_{4}$ in the dressed perturbation chain ^[27,37], one can get $ρ_{10}^{(5)} = \frac{i G_{1} {| G_{3} |}^{2} {| G_{4} |}^{2} e^{i k_{SWM 1} \cdot r}}{{(d_{1} + {| G_{1} |}^{2} / Γ_{00} + {| G_{4} |}^{2} / d_{4})}^{3} d_{3} d_{4}},$ (2)where $d_{1} = Γ_{10} + i Δ_{1}$ , $d_{3} = Γ_{30} + i (Δ_{1} - Δ_{3})$ and $d_{4} = Γ_{40} + i (Δ_{1} + Δ_{4})$ .

The SWM2 signal is obtained in the EIT windows ( $Δ_{1} + Δ_{2} = 0$ ), and the EWM is obtained in the two overlapped EIT windows ( $Δ_{1} + Δ_{2} = 0$ and $Δ_{1} + Δ_{4} = 0$ ). The SWM2 and EWM processes are described by the perturbation chains $ρ_{00}^{(0)} \overset{E_{1}}{\to} ρ_{10}^{(1)} \overset{E_{3}}{\to} ρ_{30}^{(2)} \overset{{(E_{3}^{'})}^{*}}{\to} ρ_{10}^{(3)} \overset{E_{2}}{\to} ρ_{20}^{(4)} \overset{E_{2}^{*}}{\to} ρ_{10}^{(5)}$ and $ρ_{00}^{(0)} \overset{E_{1}}{\to} ρ_{10}^{(1)} \overset{E_{3}}{\to} ρ_{30}^{(2)} \overset{{(E_{3}^{'})}^{*}}{\to} ρ_{10}^{(3)} \overset{E_{2}}{\to} ρ_{20}^{(4)} \overset{{(E_{2}^{'})}^{*}}{\to} ρ_{10}^{(5)} \overset{E_{4}}{\to} ρ_{40}^{(6)} \overset{E_{4}^{*}}{\to} ρ_{10}^{(7)}$ , respectively.

Similarly, we can get the density-matrix elements for $E_{SWM 2}$ and $E_{EWM}$ related to the Rydberg level $| 2 ⟩$ , while besides the strong field dressing effect of $E_{2}$ in the dressed perturbation chain, the RRI induced by $E_{2}$ should be considered. Atoms excited to the Rydberg energy level ( $| 2 ⟩$ ) can shift the energy levels of the nearby atoms, and thus significantly suppress the rate of Rydberg transitions from $| 1 ⟩$ to $| 2 ⟩$ . To make our model adaptive in the case of Rydberg excitation, we can substitute terms $ρ_{0}$ , $G_{1}$ , $G_{2}$ , and $G_{3}$ in the classical model with $ρ_{0}^{0.2}$ , $G_{1}^{0.2}$ , ${(G_{2} / n^{11})}^{0.2}$ , and $G_{3}^{0.2}$ (see Appendix A). The respective density-matrix elements for $E_{SWM 2}$ and $E_{EWM}$ can be given by $ρ_{10}^{(5)} = \frac{i G_{1}^{0.2} {(| G_{2} | / n^{11})}^{0.4} {| G_{3} |}^{0.4} e^{i k_{SWM 2} \cdot r}}{{[d_{1} + {| G_{1} |}^{0.4} / Γ_{00} + {(| G_{2} | / n^{11})}^{0.4} / d_{2})]}^{3} d_{2} d_{3}},$ (3)and $ρ_{10}^{(7)} = \frac{i G_{1}^{0.2} {(| G_{2} | / n^{11})}^{0.4} {| G_{3} |}^{0.4} {| G_{4} |}^{2} e^{i k_{EWM} \cdot r}}{{(d_{1} + \frac{{| G_{1} |}^{0.4}}{Γ_{00}} + \frac{(| G_{2} | / n^{11})^{0.4}}{d_{2}} + \frac{{| G_{4} |}^{2}}{d_{4}})}^{4} d_{2} d_{3} d_{4}},$ (4)where $d_{2} = Γ_{20} + i (Δ_{1} + Δ_{2})$ .

2.4 B. OPA Process

Considering the degenerate two-level atomic configuration in Fig. 1(c1) driven by $E_{1}$ , the spontaneous parametric four-wave mixing (SP-FWM) process, which generates two output weak signals (Stokes signal $E_{St}$ and anti-Stokes signal $E_{ASt}$ ), will occur in the subsystem $| 0 ⟩ \leftrightarrow | 1 ⟩$ ^[20,28], known as the conical emission ^[26]. The signals in the Stokes port and the anti-Stokes port satisfy the phase-matching condition $k_{St} = 2 k_{1} - k_{Ast}$ and $k_{ASt} = 2 k_{1} - k_{St}$ , respectively, which are shown in Fig. 1(c2).

According to the perturbation chains $ρ_{00}^{(0)} \overset{E_{1}}{\to} ρ_{10}^{(1)} \overset{E_{ASt}}{\to} ρ_{00}^{(2)} \overset{E_{1}}{\to} ρ_{10 (St)}^{(3)}$ and $ρ_{00}^{(0)} \overset{E_{1}}{\to} ρ_{10}^{(1)} \overset{E_{St}}{\to} ρ_{00}^{(2)} \overset{E_{1}}{\to} ρ_{10 (ASt)}^{(3)}$ of the Stokes and anti-Stokes channels, their respective density-matrix elements can be given as $ρ_{20 (St)}^{(3)} = \frac{- i {| G_{1} |}^{2} G_{ASt}^{*}}{d_{1} d_{00}^{'} d_{10}^{'}},$ (5)and $ρ_{20 (ASt)}^{(3)} = \frac{- i {| G_{1} |}^{2} G_{St}^{*}}{d_{1} d_{00}^{''} d_{10}^{''}},$ (6)where $d_{00}^{'} = Γ_{00} + i (Δ_{1} - Δ_{ASt})$ , $d_{10}^{'} = Γ_{10} + i (2 Δ_{1} - Δ_{St})$ , $d_{00}^{''} = Γ_{00} + i (Δ_{1} - Δ_{St})$ , and $d_{10}^{''} = Γ_{10} + i (2 Δ_{1} - Δ_{ASt})$ .

When the generated MWM waves propagate along with the Stokes beam and have the same frequency as the Stokes signal, it is considered that the MWM signals are injected into the Stokes port and then parametric amplification is achieved ^[23]. Such amplified signals are termed as PA-MWM signals. The photon numbers of the output Stokes and anti-Stokes fields of the OPA process are ^[38] $⟨ {\hat{a}}_{out}^{+} {\hat{a}}_{out} ⟩ = g ⟨ {\hat{a}}_{in}^{+} {\hat{a}}_{in} ⟩ + (g - 1),$ (7)and $⟨ {\hat{b}}_{out}^{+} {\hat{b}}_{out} ⟩ = (g - 1) ⟨ {\hat{a}}_{in}^{+} {\hat{a}}_{in} ⟩ + (g - 1),$ (8)where $\hat{a} ({\hat{a}}^{+})$ and $\hat{b}$ ( ${\hat{b}}^{+}$ ) are the annihilation (creation) operator of $E_{St}$ and $E_{ASt}$ , and $g = {\cos [2 t \sqrt{A B} \sin (φ_{1} + φ_{2}) / 2] + \cosh [2 t \sqrt{A B} \cos (φ_{1} + φ_{2}) / 2]} / 2$ is the gain of the process with the modules $A$ and $B$ (phases $φ_{1}$ and $φ_{2}$ ) defined in $ρ_{10 (St)}^{(3)} = A e^{i φ_{1}}$ and $ρ_{10 (ASt)}^{(3)} = B e^{i φ_{2}}$ for $E_{St}$ and $E_{ASt}$ , respectively. From Eqs. (7) and (8), the output signal is amplified by the factor $g$ in the Stokes port and $g - 1$ in the anti-Stokes port.

3. EXPERIMENTAL RESULTS

At first, with all of the beams on except $E_{4}$ , as is shown in Fig. 2(a1), the SWM2 signal (generated in the inverted-Y-type four-level subsystem $| 0 ⟩ \leftrightarrow | 1 ⟩ \leftrightarrow | 2 ⟩ \leftrightarrow | 3 ⟩$ ) with the effect of RRI as well as an FWM signal (serving as the background) is injected into the Stokes port of the SP-FWM [in Figs. 2(a2) and 2(a3)] process. Then, the generated PA-SWM2 signal is detected by the APD. Such signal is observed by scanning $Δ_{1}$ at different values of $Δ_{2}$ . The dashed lines in Figs. 2(b1) and 2(b2) represent the background PA-FWM signal, upon which the prominent peaks indicate PA-SWM2 signals. In Figs. 2(c1) and 2(c2), the SWM signals related to the fine structure of energy level ( $37 D_{3 / 2}$ and $54 D_{3 / 2}$ ) of the Rb atoms are revealed. In this case, the intensities of these PA-SWM2 signals are weaker, because the dipole moment $μ_{i j}$ between $5 P_{3 / 2}$ and $n D_{5 / 2}$ is larger than that between $5 P_{3 / 2}$ and $n D_{3 / 2}$ . For PA-SWM2 signals whose generation is related to $n D_{3 / 2}$ , the fluctuation of the background PA-FWM signal plays a significant role with respect to the pure PA-SWM2 signal, as shown in Figs. 2(c1) and 2(c2). The intensity of the PA-SWM2 signal transited from $n D_{5 / 2}$ is much stronger than that transited from $n D_{3 / 2}$ . Hence, the fluctuations of the same background PA-FWM signals are much weaker in contrast to pure PA-SWM2 signals, which can be seen in Figs. 2(b1) and 2(b2). Therefore, the PA-SWM2 signals transited from $n D_{5 / 2}$ have a higher signal-to-noise ratio than that transited from $n D_{3 / 2}$ .

(a1) Phase-matching diagram of the OPA process with ESWM1 injected into the Stokes port. (a2) Measured Stokes field ESt and (a3) anti-Stokes field EASt versus Δ1; (b1) and (b2) intensity of PA-SWM2 signals transited from nD5/2 versus Δ1 at different Δ2 for n=37 and n=54, respectively; (c1) and (c2) intensity of PA-SWM2 signals transited from fine structure of energy level nD3/2 versus Δ1 at different Δ2 for n=37 and n=54, respectively; (d1) PA-SWM1 signals (denoted as blue triangles) versus Δ1 at different Δ4 (Δ1+Δ4=0); (d2) PA-SWM2 signals transited from 37D5/2 (denoted as black squares) and 54D3/2 (denoted as red circles) versus Δ1 at different Δ2 (Δ1+Δ2+ϵ=0).

Fig. 2. (a1) Phase-matching diagram of the OPA process with $E_{SWM 1}$ injected into the Stokes port. (a2) Measured Stokes field $E_{St}$ and (a3) anti-Stokes field $E_{ASt}$ versus $Δ_{1}$ ; (b1) and (b2) intensity of PA-SWM2 signals transited from $n D_{5 / 2}$ versus $Δ_{1}$ at different $Δ_{2}$ for $n = 37$ and $n = 54$ , respectively; (c1) and (c2) intensity of PA-SWM2 signals transited from fine structure of energy level $n D_{3 / 2}$ versus $Δ_{1}$ at different $Δ_{2}$ for $n = 37$ and $n = 54$ , respectively; (d1) PA-SWM1 signals (denoted as blue triangles) versus $Δ_{1}$ at different $Δ_{4}$ ( $Δ_{1} + Δ_{4} = 0$ ); (d2) PA-SWM2 signals transited from $37 D_{5 / 2}$ (denoted as black squares) and $54 D_{3 / 2}$ (denoted as red circles) versus $Δ_{1}$ at different $Δ_{2}$ ( $Δ_{1} + Δ_{2} + ϵ = 0$ ).

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In Figs. 2(d1) and 2(d2), we investigate the effect of RRI by comparing the detuning of the dressing field in the PA-SWM2 process with that in the non-Rydberg PA-SWM1 process. With the non-Rydberg dressing field $E_{4}$ turned on and Rydberg dressing field $E_{2}$ turned off, the PA-SWM1 signal is obtained by scanning $Δ_{1}$ at the discrete value of $Δ_{4}$ . To illustrate that the PA-SWM2 signals are affected by the RRI, here we mainly focus on the value of $Δ_{1}$ , where the strongest PA-SWM1 or PA-SWM2 signal is detected for each detuning of the dressing field. From Eq. (2), when $Δ_{4}$ is tuned to different values, the PA-SWM1 signals can reach the maximum values at the resonant condition of $Δ_{4} + Δ_{1} = 0$ . And when $Δ_{4} + Δ_{1} = 0$ is fulfilled, we can also find that the strongest signal appears at $Δ_{1} = 0$ , which indicates $Δ_{4} = 0$ . However, the resonant condition of PA-SWM2 should be rewritten as $Δ_{2} + Δ_{1} + ϵ = 0$ by taking the energy shift $ϵ$ caused by the RRI into consideration. In this case, the strongest signal is still observed at the condition of $Δ_{1} = 0$ , while at this time $Δ_{2}$ is given as $Δ_{2} = - ϵ$ . The deviation between $Δ_{2}$ and $Δ_{4}$ can be found in Figs. 2(d1) and 2(d2), which are obtained by considering the conditions of $Δ_{1} + Δ_{2} + ϵ = 0$ (for PA-SWM2 signals) and $Δ_{1} + Δ_{4} = 0$ (for PA-SWM1 signals). Here, $Δ_{2} = 0$ is determined at the condition of low-beam power and temperature, where the RRI can be ignored. Compared with the detuning $Δ_{4}$ , which is exactly at the zero point (marked as the blue vertical dashed line), detuning $Δ_{2}$ of two PA-SWM2 processes ( $n = 37$ and $n = 54$ ) when the maximum signal intensities are obtained (marked by the black and red vertical dashed lines, respectively) are far from the zero point. We can find that the deviation of $Δ_{2}$ between PA-SWM1 and PA-SWM2 signals at the maximum value point also varies with the principal quantum number $n$ of the Rydberg energy level. Comparing the deviation gap of $Δ_{2}$ at the condition of $n = 37$ with the gap at $n = 54$ , it can be found that the deviation changes from approximately 75 to 100 MHz. This phenomenon indicates that the energy shift $ϵ$ becomes larger for a higher Rydberg energy level. According to the Appendix A, the energy shift is estimated to be 50 and 115 MHz for $n = 37$ and $n = 54$ , respectively, which matches with our experimental result.

In Fig. 3, we show the intensity of Rydberg PA-SWM2 signals subjected to changes in field power and temperature. We obtain the intensity of the PA-SWM2 signals at different power of $E_{2}$ by scanning $Δ_{2}$ as shown in Fig. 3(a1). Apparently, the intensity of SWM2 signal increases with the power $P_{2}$ of $E_{2}$ . Besides the demonstrated effects of principal quantum number $n$ ^[16] and atomic density $ρ_{0}$ ^[39], one can also find that energy shift $ϵ$ varies slightly with the power of the Rydberg dressing field $E_{2}$ (which is related to the Rydberg atomic density given in the Appendix A). To illustrate the effect of changing the field intensity, we replace the term $d_{2}$ in the Rydberg-modified fifth-order density-matrix element with $d_{2}^{'} = Γ_{20} + i (Δ_{1} + Δ_{2} + ϵ)$ . Then one can predict that laser power $P_{2}$ can exert impact on the intensity of the PA-SWM2 signal from two terms, namely, Rabi frequency $G_{2}$ and energy shift $ϵ$ . In particular, $G_{2}$ mainly determines the signal intensity, while $ϵ$ affects not only the signal intensity but also the resonant position. According to Eq. (3) and $d_{2}^{'}$ , the maximum intensity of the SWM2 signal is found when the resonant condition ( $Δ_{2} = - Δ_{1} - ϵ$ ) is satisfied. As shown in Fig. 3(a1), by scanning the detuning of Rydberg dressing field $E_{2}$ , we can find that the value of $Δ_{2}$ at the resonant position moves gradually away from zero with the increase of $P_{2}$ . This shift of detuning $Δ_{2}$ at the resonant point caused by changing the power can be explained quantitatively. The shifting rate of $Δ_{2}$ on the power of $E_{2}$ can be represented by the first-order derivative of $ϵ$ with respect to $P_{2}$ . Such a derivative can be given as where $A$ is the beam area, and the beam power $P_{2}$ is substituted by the corresponding Rabi frequency $G_{2}$ with relationship $G_{i} = {(2 μ^{2} P_{i} / ℏ^{2} ϵ_{0} c A)}^{1 / 2}$ taken into consideration. Note that $[G_{1}^{2} / (Γ_{10} + G_{2}^{2} / Γ_{20}) - G_{1}^{2} / (2 Γ_{10} + G_{2}^{2} / Γ_{20} + Γ_{10}^{2} Γ_{20} / G_{2}^{2})] > 0$ , the result of this derivative is consistently a positive number. Thus, the detuning $Δ_{2}$ at resonant condition ( $Δ_{2} + Δ_{1} + ϵ = 0$ ) will decrease when $P_{2}$ increases. This matches well with the experimental result in Fig. 3(a1). Therefore, the intensity of the Rydberg dressing field in the power-controlled Rydberg signal contributes not only to the signal intensity but also to the resonant position.

(a1) Measured PA-SWM2 signals versus Δ2 by increasing P2 for n=37; (a2) intensity dependence of the PA-SWM2 signals corresponding to (a1) on P2; (b1) measured PA-SWM2 signals versus Δ2 by changing the temperature for n=37; (b2) intensity dependence of the PA-SWM2 signals corresponding to (b1) on resonant condition on temperature; (b3) theoretically simulated PA-SWM2 signals to (b1). The dots indicate the experimental data, and the solid curve represents the theoretical simulation. The dashed lines are a guide for the eyes.

Fig. 3. (a1) Measured PA-SWM2 signals versus $Δ_{2}$ by increasing $P_{2}$ for $n = 37$ ; (a2) intensity dependence of the PA-SWM2 signals corresponding to (a1) on $P_{2}$ ; (b1) measured PA-SWM2 signals versus $Δ_{2}$ by changing the temperature for $n = 37$ ; (b2) intensity dependence of the PA-SWM2 signals corresponding to (b1) on resonant condition on temperature; (b3) theoretically simulated PA-SWM2 signals to (b1). The dots indicate the experimental data, and the solid curve represents the theoretical simulation. The dashed lines are a guide for the eyes.

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Subsequently, the influence of the temperature of the Rb atoms on the PA-SWM2 signals is investigated. The dependence of signal intensity on the temperature, which is proportional to the atomic density $ρ$ , is clearly revealed by scanning $Δ_{2}$ in Fig. 3(b1). The saturation of the PA-SWM2 signal intensity can be observed in Fig. 3(b2). Particularly, along with the rise of the temperature, the growth rate of the PA-SWM2 signal intensity suffers a sharp decline. In order to interpret such an intensity saturation phenomenon, the atomic density should be considered together with the density-matrix element in Eq. (3) to describe the PA-SWM2 signal. Considering the Rydberg excitation, the PA-SWM2 signal intensity is proportional to $ρ_{0}^{0.2} ρ_{10}^{(5)}$ ^[38], in which $ρ_{0}$ is the density of the atoms. Therefore, the saturation caused by the rise of temperature can be attributed to the term $ρ_{0}^{0.2}$ . Similarly, in the case of increasing the field power, the saturation of the signal intensity can be attributed to term ${(| G_{2} | / n^{11})}^{0.2}$ . However, owing to the relatively low power of $E_{2}$ in Fig. 3(a2), which is far from the saturation, the signal intensity keeps increasing with the increase of $P_{2}$ . One can find in Figs. 3(a2) and 3(b2) that when the parameters related to Rydberg excitation (such as the power and the atomic density) are of relatively low value, the suppression caused by RRI can be effectively eliminated. In the condition of low power and low temperature, the blockade domain in which only one excited Rydberg atom can exist is sufficiently large, considering Eq. (A4) in the Appendix A and $ρ_{2} V_{d} = 1$ . Accordingly, Rydberg atoms can be so distant from each other that the RRI can be omitted. On the other hand, if the atomic population at the ground state increases, the probability for the Rydberg energy level to be occupied will be increased subsequently. However, the increase of the signal intensity will saturate because of the blockaded effect ^[40]. In addition, by enhancing the beam power, the number of Rydberg atoms will increase according to Eq. (A4), and the blockade radius will be reduced; hence, the distance between Rydberg atoms will be narrowed. Therefore, the RRI will be enhanced, and the average energy shift of the atoms will be larger. Consequently, the saturation of the PA-SWM2 signal intensity will occur if the beam power and the temperature increase continuously.

It can also be found that with the change of the temperature, the detuning $Δ_{2}$ for resonance also changes. Likewise, we can differentiate $ϵ$ on the atomic density as

Obviously, the derivative in Eq. (10) is always positive, which illustrates that when the atomic density increases, $ϵ$ will increase so that $Δ_{2}$ will decrease at the resonant condition ( $Δ_{2} + Δ_{1} + ϵ = 0$ ). This result corresponds well with the experimental result shown in Fig. 3(b1). The corresponding theoretical simulation is shown in Fig. 3(b3). Therefore, the saturation of the signal intensity and the shift of resonant position of Rydberg signals can be achieved in the power-controlled and density-controlled PA-MWM process. It is also worth mentioning that the effect of parametric amplification can be affected by the temperature, since the Stokes field $E_{St}$ and anti-Stokes field $E_{ASt}$ of the OPA process depend on the atomic population ^[23]. Hence, the PA-SWM signal intensity increases more rapidly than the SWM signal does without parametric amplification, in comparison with our previous work ^[17].

In Fig. 4, we discuss the generation of EWM and the dependence of coexisting PA-SWM and PA-EWM on detuning and field intensity. When all the laser beams in the configuration in Fig. 1(b) are on, besides the aforementioned SWM1 and SWM2 signals, the EWM signal generated in the subsystem $| 0 ⟩ \leftrightarrow | 1 ⟩ \leftrightarrow | 2 ⟩ \leftrightarrow | 3 ⟩ \leftrightarrow | 4 ⟩$ is introduced into this system. The energy-level diagram with Autler–Townes (AT) splitting ^[41] is shown in Fig. 4(a1). Figure 4(a2) shows the schematic diagram in which the coexisting SWM and EWM signals are injected into the Stokes port to realize the OPA process. Along with PA-SWM2 signals, the Rydberg PA-EWM signals are also the constituent part of the detected PA-MWM signals, whose intensities are shown by scanning $Δ_{2}$ at several discrete values of $Δ_{4}$ in Fig. 4(b) and $P_{4}$ in Fig. 4(c). In Fig. 4(b), the Rydberg signals are detected with the background of a broadened FWM signal (generated in the $Λ$ -type three-level subsystem $| 0 ⟩ \leftrightarrow | 1 ⟩ \leftrightarrow | 3 ⟩$ ) and a non-Rydberg SWM1 signal. The coexisting PA-EWM and PA-SWM signals obtained by scanning $Δ_{2}$ can be strengthened or suppressed at different values of $Δ_{4}$ ^[42]. Considering the dressing effects of both $E_{2}$ and $E_{4}$ , when the value of $Δ_{4}$ is far-detuned from the resonance of $| 1 ⟩ \to | 4 ⟩$ , coexisting SWM and EWM signals can only be enhanced, since the enhancement condition $Δ_{2} + Δ_{-} + Δ_{- +} = 0$ is the only extreme value condition that can be fulfilled. The additional detuning of primary and secondary dressing energy levels split by $E_{2}$ and $E_{4}$ is given by $Δ_{-} = {(Δ_{2} + ϵ) - [{(Δ_{2} + ϵ)}^{2} + 4 {| G_{2} |}^{2}]^{1 / 2}} / 2$ and $Δ_{- +} = [Δ_{4}^{'} + (Δ_{4}^{' 2} + 4 {| G_{4} |}^{2})] / 2$ , in which $Δ_{4}^{'} = Δ_{4} - Δ_{-}$ . It should be noted that the position of split energy level is also related to the term $ϵ$ induced by RRI. These two split energy levels are denoted as $| - G_{2} ⟩$ and $| - G_{2} + G_{4} ⟩$ in Fig. 4(a1). However, when the value of $Δ_{4}$ gradually approaches the resonance of $| 1 ⟩ \leftrightarrow | 4 ⟩$ , the influence of dressing field is gradually transformed from enhancement into suppression. In addition, when $Δ_{4}$ is near the resonant point, only the suppression of the output PA-MWM signal can be observed, because only the suppression condition $Δ_{2} + Δ_{1} = 0$ can be fulfilled. Meanwhile, it is possible for $Δ_{4}$ to be set to the value that both suppression and enhancement conditions can be satisfied; thus the half-enhancement and half-suppression of the output signal can be observed ^[42]. In this way, the signal can be enhanced or suppressed when the corresponding extreme value conditions ( $Δ_{2} + Δ_{-} + Δ_{- +} = 0$ and $Δ_{2} + Δ_{1} = 0$ ) are satisfied. Similarly, as shown in the right part ( $Δ_{4} > 0$ ) of Fig. 4(b), when $Δ_{4}$ continues to move far away from the resonance, the coexisting PA-SWM2 and PA-EWM signals will be enhanced by the dressing field again. The OPA process acts against the dressing suppression and enables the enhancement of the coexisting PA-SWM and PA-EWM signals to be more obvious. Furthermore, one can also notice that when we change the power of $E_{4}$ in Fig. 4(c), the effect of $E_{4}$ can also be switched between suppression and enhancement. When the power of the dressing field $E_{4}$ is relatively low, the signal is enhanced in Fig. 4(c). In this case, the suppression caused by the dressing effect of $E_{4}$ is not obvious, and both coexisting PA-SWM2 and PA-EWM signals contribute to the peak in Fig. 4(c). Nevertheless, when $P_{4}$ increases gradually, the PA-MWM signals near the resonant position move gradually towards suppression with the transition of half-enhancement and half-suppression, as shown in Fig. 4(c). When the power $P_{4}$ is high enough, the suppression of the PA-MWM signals caused by the dressing effect of $E_{4}$ gets dominant, and dip is observed, as shown in the bottom of Fig. 4(c).

(a1) AT splitting in the five-level atomic system induced by E2 and E4; (a2) phase-matching diagram of OPA injected with ESWM1, ESWM2, and EEWM into the Stokes port. (b) Measured MWM versus Δ2 with discrete Δ4 for n=37; the range of Δ4 is from −150 to 150 MHz. (c) Measured MWM signals versus Δ2 with increasing P4 for n=37; the range of P4 is from 10 to 18 mW.

Fig. 4. (a1) AT splitting in the five-level atomic system induced by $E_{2}$ and $E_{4}$ ; (a2) phase-matching diagram of OPA injected with $E_{SWM 1}$ , $E_{SWM 2}$ , and $E_{EWM}$ into the Stokes port. (b) Measured MWM versus $Δ_{2}$ with discrete $Δ_{4}$ for $n = 37$ ; the range of $Δ_{4}$ is from $- 150$ to 150 MHz. (c) Measured MWM signals versus $Δ_{2}$ with increasing $P_{4}$ for $n = 37$ ; the range of $P_{4}$ is from 10 to 18 mW.

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4. CONCLUSIONS

We have studied the PA-SWM and PA-EWM signals in a Rydberg EIT Rb atomic medium both experimentally and theoretically. The intensity dependences of the MWM signals on the principal quantum number, the detuning of the probe and coupling fields, the power of coupling fields, and the atomic density are investigated. One can find that the MWM signals can be effectively controlled by these parameters with the combined effect of RRI and the parametric amplification. Moreover, the linear amplification of the Rydberg MWM signals resulting from the OPA process acts against the suppression of the Rydberg excitation, which can potentially improve the logic gate devices and sensors related to Rydberg atoms.

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1. INTRODUCTION

2. BASIC THEORY AND EXPERIMENTAL SETUP

2.3 A. Rydberg MWM Process

2.4 B. OPA Process

3. EXPERIMENTAL RESULTS

4. CONCLUSIONS

Zhaoyang Zhang, Ji Guo, Bingling Gu, Ling Hao, Gaoguo Yang, Kun Wang, Yanpeng Zhang. Parametric amplification of Rydberg six- and eight-wave mixing processes[J]. Photonics Research, 2018, 6(7): 07000713.

Parametric amplification of Rydberg six- and eight-wave mixing processes

1. INTRODUCTION

2. BASIC THEORY AND EXPERIMENTAL SETUP

2.3 A. Rydberg MWM Process

2.4 B. OPA Process

3. EXPERIMENTAL RESULTS

4. CONCLUSIONS

Article Outline

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Parametric amplification of Rydberg six- and eight-wave mixing processes

1. INTRODUCTION

2. BASIC THEORY AND EXPERIMENTAL SETUP

2.3 A. Rydberg MWM Process

2.4 B. OPA Process

3. EXPERIMENTAL RESULTS

4. CONCLUSIONS

Article Outline

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