Parametric amplification of Rydberg six- andeight-wave mixing processesZHAOYANG ZHANG,† JI GUO,† BINGLING GU, LING HAO, GAOGUO YANG, KUN WANG,AND YANPENG ZHANG*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*Corresponding author: ypzhang@mail.xjtu.edu.cn

Received 26 February 2018; revised 26 April 2018; accepted 26 April 2018; posted 27 April 2018 (Doc. ID 324899); published 18 June 2018

We study the parametric amplification of electromagnetically induced transparency-assisted Rydberg six- andeight-wave mixing signals through a cascaded nonlinear optical process in a hot rubidium atomic ensemble boththeoretically and experimentally. The shift of the resonant frequency (induced by the Rydberg–Rydberg inter-action) of parametrically amplified six-wave mixing signal is observed. Moreover, the interplays between thedressing effects and Rydberg–Rydberg interactions in parametrically amplified multiwave mixing signals are in-vestigated. The linear amplification of Rydberg multiwave mixing processes with multichannel nature acts againstthe suppression caused by Rydberg–Rydberg interaction and dressing effect. © 2018 Chinese Laser Press

OCIS codes: (020.5780) Rydberg states; (190.4223) Nonlinear wave mixing; (190.4380) Nonlinear optics, four-wave mixing;

(190.4410) Nonlinear optics, parametric processes.

https://doi.org/10.1364/PRJ.6.000713

1. INTRODUCTION

Atoms excited to Rydberg states have attracted considerable in-terest owing to their excellent properties, such as long lifetimes,large collision cross sections, large dipole moments, and gigan-tic spatial extension [1]. Besides, Rydberg–Rydberg interaction(RRI) between Rydberg atoms, such as dipole–dipole interac-tion and van der Waals interaction, can induce the dipoleblockade of nearby atoms [2,3]. These properties lead to a largeamount of prospective applications, such as quantum informa-tion processing [4,5], high-fidelity optical state control [6], andsingle-photon sources [7]. For these applications, detectingRydberg dressing-state effect and the interactions amongthe Rydberg atoms nondestructively is a basic requirement.Recently, electromagnetically induced transparency (EIT)[8–12] and multiwave mixing (MWM) [13–15] as nondestruc-tive optical detection methods [16,17] have been proposed andutilized for detecting Rydberg atoms [18,19] where theRydberg atoms are not ionized. The multichannel nature ofthe MWM process allows for multichannel informationprocessing related to Rydberg states. Furthermore, the MWMprocess can act as a multimode correlated light source [20].However, MWM signals generated from high-order nonlinearoptical 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 thesignal intensity and decrease the signal-to-noise ratio [22].

The optical parametric amplification (OPA) process, whichcan act as an optical amplifier and implement the linearamplification of the input signal [23], is proposed and exper-imentally achieved in the media of both gaseous and solid states[24,25]. Generally, the OPA process in an atomic medium ischaracterized by the so-called conical emission [26], where twocorrelated photons named as Stokes and anti-Stokes photonsare effectively generated [27]. The OPA process can be achievedby the cascaded nonlinear process, in which the generatedMWM signals coexist with the parametrical FWM process[20,28]. Intensity noise correlation [29] and intensity-differ-ence squeezing [30] of such a process have given rise to appli-cations in quantum metrology [31–33]. Inspired by such anOPA process, the high-order Rydberg MWM signal can alsobe injected into the Stokes or anti-Stokes port and then be lin-early and nondestructively amplified to enhance the signal-to-noise ratio of the Rydberg MWM signal. As a result, theperformance of applications related to Rydberg excitations suchas sensors [31–35] can be promisingly improved via OPA.

In this paper, we observed the Rydberg parametrically am-plified MWM (PA-MWM) signals assisted by the cascadednonlinear process in a K -type five-level system of 85Rb.With two EIT windows generated effectively, the parametri-cally amplified SWM (PA-SWM) and EWM (PA-EWM) sig-nals can be simultaneously detected. Meanwhile, the intensitiesof PA-MWM signals can be controlled by the detuning and the

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power of the coupling fields as well as the density of the atomicensemble. Moreover, the saturation of the signal intensity andthe shift of the resonant position caused by the RRI whenchanging the beam power and temperature can be detected.Finally, the suppression (enhancement) of coexisting PA-SWMand PA-EWM signals near resonance (far from resonance) isobserved. These phenomena in the Rydberg PA-MWM processcan improve the applications related to Rydberg atoms in thefields of quantum metrology and sensors.

2. BASIC THEORY AND EXPERIMENTAL SETUP

In the K -type five-level 85Rb atomic system shown in Fig. 1(a),two hyperfine energy levels, F � 3 (j0i) and F � 2 (j3i), ofthe ground state 5S1∕2, a Rydberg excited state nD5∕2 (j2i),and two lower excited states, 5P3∕2 (j1i) and 5D5∕2 (j4i),are connected by corresponding beams. The experimentalconfiguration is shown in Fig. 1(b). Five beams derived fromfour external cavity diode lasers (ECDLs) with linewidths<1 MHz are used to couple the following transitions. Thetransition j0i ↔ j1i is probed by beam E1 [wavelength of780.2 nm, frequency ω1, wave vector k1, and Rabi frequencyG1, defined as Gi � μijE i∕ℏ, where μij is the dipole momentbetween jii ↔ j ji (i, j � 1, 2, 3, 4)]. The Rydberg transitionj1i ↔ j2i is connected by a strong beam E2 (∼480 nm, ω2,k2, G2), which propagates opposite to beam E1. The transi-tion j1i ↔ j3i is connected by two beams E3 (780.2 nm, ω3,k3, G3) and E 0

3 (780.2 nm, ω 03, k

03, G

03), which are derived

from the same ECDL. In the atomic ensemble, E3 propagatesin the same direction with E2, while E 0

3 has a small angleof 0.3° with E3. The transition j1i ↔ j4i is driven by thebeam E4 (775.9 nm, ω4, k4, G4), whose propagation is sym-metrical to E 0

3 about E2.

A. Rydberg MWM ProcessBy turning these lasers on and off selectively, we can get differ-ent MWM signals with different orders. When the beams E2

and E4 are blocked, an FWM process satisfying the phase-matching condition kFWM � k1 � k3 − k 0

3 will occur in the

three-level subsystem j0i ↔ j1i ↔ j3i. When only the beamE2 with Rydberg dressing-state effect is blocked, an SWMprocess (denoted as SWM1) satisfying the phase-matching con-dition kSWM1 � k1 � k3 − k 0

3 � k4 − k4 will occur in a four-level subsystem j0i ↔ j1i ↔ j3i ↔ j4i, in which one photoneach from E1, E3, E 0

3, and two photons from E4 are involved.Similarly, by blocking E4, another SWM process (denoted asSWM2) with kSWM2 � k1 � k3 − k 0

3 � k2 − k2 can be ob-served in the subsystem j0i ↔ j1i ↔ j2i ↔ j3i. When allthe beams shown in Fig. 1(b) are on, a new EWM signalcan be generated with phase-matching condition kEWM �k1 � k2 − k2 � k3 − k 0

3 � k4 − k4 in the five-level atomic sys-tem j0i ↔ j1i ↔ j2i ↔ j3i ↔ j4i. It can be revealed from thephase-matching conditions that these MWM signals emit inthe direction opposite to E 0

3. These signals are detected by theavalanche photodiode detectors (APDs). To be specific, theEIT signal is received by D1, and the MWM signals arereceived by D2, as shown in Fig. 1(b).

Generally, the response of the atoms to the light isdescribed by the susceptibility. The generated MWM signalsare characterized by their nonlinear susceptibilities χ�2n�1� �ρ0μ10ρ

�2n�1�10 ∕ε0E1 (e.g., n � 1 for FWM, n � 2 for SWM,

and n � 3 for EWM). According to the perturbation chain

ρ�0�00 !E1ρ�1�10 !E3

ρ�2�30 !�E 0

3��ρ�3�10 via the Liouville pathway [36], the

density-matrix element for the EFWM is given by

ρ�3�10 � iG1jG3j2eikFWM·r

�d 01 � jG1j2∕Γ00�2d 0

3

, (1)

where d 01 � Γ10 � i�Δ1 � k1v� and d 0

3 � Γ30 � i�Δ1 − Δ3��i�k1v � k3v�, Γij is the decay rate between states jii andj ji, and Δi � Ωij − ωi is the detuning between the frequencyωi of beam E i and the resonant transition frequency Ωij

between jii ↔ j ji; kiv is the term of the Doppler effect.Considering the upper transition j1i ↔ j4i, the SWM1

signal is generated with the help of the EIT windows(Δ1 � Δ4 � 0). The perturbation chain for this process

can be written as ρ�0�00 !E1ρ�1�10 !E3

ρ�2�30 !�E 0

3��ρ�3�10 !E4

ρ�4�40 !E�4ρ�5�10 .

Fig. 1. (a) Five-level K -type energy level diagram depicting the generation of the MWM process in the 85Rb atomic system. (b) Experimentalsetup. 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 verticalpolarization 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 j0i ↔ j1i is coupled by the beam E 1 (780.2 nm). Rydberg transition j1i ↔ j2i is coupled by beam E 2 (480 nm), whichcounterpropagates with beam E1. j1i ↔ j3i is connected by beams E3 and E 0

3 (780.2 nm), which are derived from the same ECDL, and j1i ↔ j4iis coupled by beam E 4 (775.9 nm). The EIT signal and MWM spectrum signals are received by D1 and D2, respectively. (c1) Energy schematicdiagram for SP-FWM process; (c2) phase-matching condition of SP-FWM process.

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Considering the strong field dressing effect of E4 in the dressedperturbation chain [27,37], one can get

ρ�5�10 � iG1jG3j2jG4j2eikSWM1·r

�d 1 � jG1j2∕Γ00 � jG4j2∕d 4�3d 3d 4

, (2)

where d 1 � Γ10 � iΔ1, d 3 � Γ30 � i�Δ1 − Δ3� and d 4 �Γ40 � i�Δ1 � Δ4�.

The SWM2 signal is obtained in the EIT windows(Δ1 � Δ2 � 0), and the EWM is obtained in the twooverlapped EIT windows (Δ1 � Δ2 � 0 and Δ1 � Δ4 � 0).The SWM2 and EWM processes are described by the

perturbation chains ρ�0�00 !E1ρ�1�10 !E3

ρ�2�30 !�E 0

3��ρ�3�10 !E2

ρ�4�20 !E�2ρ�5�10

and ρ�0�00 !E1ρ�1�10 !E3

ρ�2�30 !�E 0

3��ρ�3�10 !E2

ρ�4�20 !�E02��ρ�5�10 !E4

ρ�6�40 !E�4ρ�7�10 ,

respectively.Similarly, we can get the density-matrix elements for E SWM2

and EEWM related to the Rydberg level j2i, while besides thestrong field dressing effect of E2 in the dressed perturbationchain, the RRI induced by E2 should be considered. Atomsexcited to the Rydberg energy level (j2i) can shift the energylevels of the nearby atoms, and thus significantly suppress therate of Rydberg transitions from j1i to j2i. To make our modeladaptive in the case of Rydberg excitation, we can substituteterms ρ0, G1, G2, and G3 in the classical model with ρ0.20 ,G0.2

1 , �G2∕n11�0.2, and G0.23 (see Appendix A). The respec-

tive density-matrix elements for E SWM2 and EEWM can begiven by

ρ�5�10 � iG0.21 �jG2j∕n11�0.4jG3j0.4eikSWM2·r

�d 1 � jG1j0.4∕Γ00 � �jG2j∕n11�0.4∕d 2��3d 2d 3

, (3)

and

ρ�7�10 � iG0.21 �jG2j∕n11�0.4jG3j0.4jG4j2eikEWM·r�

d 1 � jG1j0.4Γ00

� �jG2j∕n11�0.4d 2

� jG4j2d 4

�4d 2d 3d 4

, (4)

where d 2 � Γ20 � i�Δ1 � Δ2�.

B. OPA ProcessConsidering the degenerate two-level atomic configuration inFig. 1(c1) driven by E1, the spontaneous parametric four-wavemixing (SP-FWM) process, which generates two output weaksignals (Stokes signal E St and anti-Stokes signal EASt), will oc-cur in the subsystem j0i ↔ j1i [20,28], known as the conicalemission [26]. The signals in the Stokes port and the anti-Stokes port satisfy the phase-matching condition kSt � 2k1 −kAst and kASt � 2k1 − kSt, respectively, which are shown inFig. 1(c2).

According to the perturbation chains ρ�0�00 !E1ρ�1�10 !

EAStρ�2�00 !E1ρ�3�10�St� and ρ�0�00 !E1

ρ�1�10 !EStρ�2�00 !E1

ρ�3�10�ASt� of theStokes and anti-Stokes channels, their respective density-matrixelements can be given as

ρ�3�20�St� �−ijG1j2G�

ASt

d 1d 000d

010

, (5)

and

ρ�3�20�ASt� �−ijG1j2G�

St

d 1d 0 000d

0 010

, (6)

where d 000 � Γ00 � i�Δ1 − ΔASt�, d 0

10 � Γ10 � i�2Δ1 − ΔSt�,d 0 000 � Γ00 � i�Δ1 − ΔSt�, and d 0 0

10 � Γ10 � i�2Δ1 − ΔASt�.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 theStokes 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-Stokesfields of the OPA process are [38]

ha�outaouti � gha�in aini � �g − 1�, (7)

and

hb�outbouti � �g − 1�ha�in aini � �g − 1�, (8)

where a�a�� and b (b�) are the annihilation (creation) operatorof E St and EASt, and g � fcos�2t ffiffiffiffiffiffiffi

ABp

sin�φ1 � φ2�∕2� �cosh�2t

ffiffiffiffiffiffiffiAB

pcos�φ1 � φ2�∕2�g∕2 is the gain of the process

with the modules A and B (phases φ1 and φ2) defined inρ�3�10�St� � Aeiφ1 and ρ�3�10�ASt� � Beiφ2 for E St and EASt, respec-tively. From Eqs. (7) and (8), the output signal is amplified bythe 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 E4, as is shown inFig. 2(a1), the SWM2 signal (generated in the inverted-Y-typefour-level subsystem j0i ↔ j1i ↔ j2i ↔ j3i) with the effectof RRI as well as an FWM signal (serving as the back-ground) is injected into the Stokes port of the SP-FWM [inFigs. 2(a2) and 2(a3)] process. Then, the generated PA-SWM2 signal is detected by the APD. Such signal is observedby scanning Δ1 at different values of Δ2. The dashed lines inFigs. 2(b1) and 2(b2) represent the background PA-FWM sig-nal, upon which the prominent peaks indicate PA-SWM2 sig-nals. In Figs. 2(c1) and 2(c2), the SWM signals related to thefine structure of energy level (37D3∕2 and 54D3∕2) of the Rbatoms are revealed. In this case, the intensities of these PA-SWM2 signals are weaker, because the dipole moment μij be-tween 5P3∕2 and nD5∕2 is larger than that between 5P3∕2 andnD3∕2. For PA-SWM2 signals whose generation is related tonD3∕2, the fluctuation of the background PA-FWM signalplays a significant role with respect to the pure PA-SWM2signal, as shown in Figs. 2(c1) and 2(c2). The intensity ofthe PA-SWM2 signal transited from nD5∕2 is much strongerthan that transited from nD3∕2. Hence, the fluctuations ofthe same background PA-FWM signals are much weaker incontrast to pure PA-SWM2 signals, which can be seen inFigs. 2(b1) and 2(b2). Therefore, the PA-SWM2 signals trans-ited from nD5∕2 have a higher signal-to-noise ratio than thattransited from nD3∕2.

In Figs. 2(d1) and 2(d2), we investigate the effect of RRI bycomparing the detuning of the dressing field in the PA-SWM2process with that in the non-Rydberg PA-SWM1 process. Withthe non-Rydberg dressing field E4 turned on and Rydbergdressing field E2 turned off, the PA-SWM1 signal is obtained

Research Article Vol. 6, No. 7 / July 2018 / Photonics Research 715

by scanning Δ1 at the discrete value ofΔ4. To illustrate that thePA-SWM2 signals are affected by the RRI, here we mainlyfocus on the value of Δ1, where the strongest PA-SWM1 orPA-SWM2 signal is detected for each detuning of the dressingfield. From Eq. (2), when Δ4 is tuned to different values, thePA-SWM1 signals can reach the maximum values at the res-onant condition of Δ4 � Δ1 � 0. And when Δ4 � Δ1 � 0 isfulfilled, we can also find that the strongest signal appears atΔ1 � 0, which indicates Δ4 � 0. However, the resonant con-dition of PA-SWM2 should be rewritten as Δ2 � Δ1 � ε � 0by taking the energy shift ε caused by the RRI into con-sideration. In this case, the strongest signal is still observedat the condition of Δ1 � 0, while at this time Δ2 is given asΔ2 � −ε. The deviation between Δ2 and Δ4 can be foundin Figs. 2(d1) and 2(d2), which are obtained by considering theconditions of Δ1 � Δ2 � ε � 0 (for PA-SWM2 signals) andΔ1 � Δ4 � 0 (for PA-SWM1 signals). Here, Δ2 � 0 is deter-mined at the condition of low-beam power and temperature,where the RRI can be ignored. Compared with the detuning

Δ4, which is exactly at the zero point (marked as the blue ver-tical dashed line), detuning Δ2 of two PA-SWM2 processes(n � 37 and n � 54) when the maximum signal intensitiesare obtained (marked by the black and red vertical dashed lines,respectively) are far from the zero point. We can find that thedeviation of Δ2 between PA-SWM1 and PA-SWM2 signals atthe maximum value point also varies with the principal quan-tum number n of the Rydberg energy level. Comparing thedeviation gap of Δ2 at the condition of n � 37 with thegap at n � 54, it can be found that the deviation changesfrom approximately 75 to 100 MHz. This phenomenonindicates that the energy shift ε becomes larger for a higherRydberg energy level. According to the Appendix A, the energyshift is estimated to be 50 and 115 MHz for n � 37 andn � 54, respectively, which matches with our experimentalresult.

In Fig. 3, we show the intensity of Rydberg PA-SWM2signals subjected to changes in field power and temperature.We obtain the intensity of the PA-SWM2 signals at different

Fig. 2. (a1) Phase-matching diagram of the OPA process with E SWM1 injected into the Stokes port. (a2) Measured Stokes field E St 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 versusΔ1 at differentΔ2 for n � 37 andn � 54, respectively; (d1) PA-SWM1 signals (denoted as blue triangles) versus Δ1 at different Δ4 (Δ1 � Δ4 � 0); (d2) PA-SWM2 signals transitedfrom 37D5∕2 (denoted as black squares) and 54D3∕2 (denoted as red circles) versus Δ1 at different Δ2 (Δ1 � Δ2 � ε � 0).

Fig. 3. (a1) Measured PA-SWM2 signals versus Δ2 by increasing P2 for n � 37; (a2) intensity dependence of the PA-SWM2 signals corre-sponding to (a1) on P2; (b1) measured PA-SWM2 signals versus Δ2 by changing the temperature for n � 37; (b2) intensity dependence ofthe 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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power of E2 by scanningΔ2 as shown in Fig. 3(a1). Apparently,the intensity of SWM2 signal increases with the power P2 ofE2. Besides the demonstrated effects of principal quantumnumber n [16] and atomic density ρ0 [39], one can also findthat energy shift ε varies slightly with the power of the Rydbergdressing field E2 (which is related to the Rydberg atomicdensity given in the Appendix A). To illustrate the effect ofchanging the field intensity, we replace the term d 2 in theRydberg-modified fifth-order density-matrix element withd 02 � Γ20 � i�Δ1 � Δ2 � ε�. Then one can predict that laser

power P2 can exert impact on the intensity of the PA-SWM2signal from two terms, namely, Rabi frequency G2 and energyshift ε. In particular, G2 mainly determines the signal intensity,while ε affects not only the signal intensity but also the resonantposition. According to Eq. (3) and d 0

2, the maximum intensityof the SWM2 signal is found when the resonant condition(Δ2 � −Δ1 − ε) is satisfied. As shown in Fig. 3(a1), by scan-ning the detuning of Rydberg dressing field E2, we can findthat the value of Δ2 at the resonant position moves graduallyaway from zero with the increase of P2. This shift of detuningΔ2 at the resonant point caused by changing the power can beexplained quantitatively. The shifting rate of Δ2 on the powerof E2 can be represented by the first-order derivative of ε withrespect to P2. Such a derivative can be given as

dε

dP2

�2Cμ212ρ0.20 G−1.6

2

5ε0cAℏ2n0.44

�G2

1

Γ10�G22∕Γ20

�G23

Γ30

�−0.8

�G2

3

Γ30

−G2

1

2Γ10�G22∕Γ20�Γ2

10Γ20∕G22

� G21

Γ10�G22∕Γ20

�,

(9)

where A is the beam area, and the beam power P2 issubstituted by the corresponding Rabi frequency G2 withrelationship Gi � �2μ2Pi∕ℏ2ε0cA�1∕2 taken into considera-tion. Note that �G2

1∕�Γ10�G22∕Γ20� −G2

1∕�2Γ10�G22∕Γ20 �

Γ210Γ20∕G2

2��> 0, the result of this derivative is consistentlya positive number. Thus, the detuning Δ2 at resonant condi-tion (Δ2 � Δ1 � ε � 0) will decrease when P2 increases. Thismatches well with the experimental result in Fig. 3(a1).Therefore, the intensity of the Rydberg dressing field in thepower-controlled Rydberg signal contributes not only to thesignal intensity but also to the resonant position.

Subsequently, the influence of the temperature of the Rbatoms on the PA-SWM2 signals is investigated. The depend-ence of signal intensity on the temperature, which is propor-tional to the atomic density ρ, is clearly revealed by scanning Δ2

in Fig. 3(b1). The saturation of the PA-SWM2 signal intensitycan be observed in Fig. 3(b2). Particularly, along with the rise ofthe temperature, the growth rate of the PA-SWM2 signal in-tensity suffers a sharp decline. In order to interpret such anintensity saturation phenomenon, the atomic density should beconsidered together with the density-matrix element in Eq. (3)to describe the PA-SWM2 signal. Considering the Rydbergexcitation, the PA-SWM2 signal intensity is proportional toρ0.20 ρ�5�10 [38], in which ρ0 is the density of the atoms.Therefore, the saturation caused by the rise of temperaturecan be attributed to the term ρ0.20 . Similarly, in the case of in-creasing the field power, the saturation of the signal intensity

can be attributed to term �jG2j∕n11�0.2. However, owing to therelatively low power of E2 in Fig. 3(a2), which is far from thesaturation, the signal intensity keeps increasing with the in-crease of P2. One can find in Figs. 3(a2) and 3(b2) that whenthe parameters related to Rydberg excitation (such as the powerand the atomic density) are of relatively low value, the suppres-sion caused by RRI can be effectively eliminated. In the con-dition of low power and low temperature, the blockade domainin which only one excited Rydberg atom can exist is sufficientlylarge, considering Eq. (A4) in the Appendix A and ρ2V d � 1.Accordingly, Rydberg atoms can be so distant from each otherthat the RRI can be omitted. On the other hand, if the atomicpopulation at the ground state increases, the probability for theRydberg energy level to be occupied will be increased sub-sequently. However, the increase of the signal intensity will sat-urate because of the blockaded effect [40]. In addition, byenhancing the beam power, the number of Rydberg atoms willincrease according to Eq. (A4), and the blockade radius will bereduced; hence, the distance between Rydberg atoms will benarrowed. Therefore, the RRI will be enhanced, and the aver-age energy shift of the atoms will be larger. Consequently, thesaturation of the PA-SWM2 signal intensity will occur if thebeam power and the temperature increase continuously.

It can also be found that with the change of the temperature,the detuning Δ2 for resonance also changes. Likewise, we candifferentiate ε on the atomic density as

dε

dρ� 0.2Cρ−0.80

�jG2jn11

�0.4� jG1j2Γ10 � G2

2∕Γ20

� jG3j2Γ30

�0.2:

(10)

Obviously, the derivative in Eq. (10) is always positive,which illustrates that when the atomic density increases, ε willincrease so that Δ2 will decrease at the resonant condition(Δ2 � Δ1 � ε � 0). This result corresponds well with theexperimental result shown in Fig. 3(b1). The correspondingtheoretical simulation is shown in Fig. 3(b3). Therefore, thesaturation of the signal intensity and the shift of resonant po-sition of Rydberg signals can be achieved in the power-controlled and density-controlled PA-MWM process. It is alsoworth mentioning that the effect of parametric amplificationcan be affected by the temperature, since the Stokes fieldE St and anti-Stokes field EASt of the OPA process dependon the atomic population [23]. Hence, the PA-SWM signalintensity increases more rapidly than the SWM signal doeswithout parametric amplification, in comparison with our pre-vious work [17].

In Fig. 4, we discuss the generation of EWM and the depend-ence of coexisting PA-SWM and PA-EWM on detuning andfield intensity. When all the laser beams in the configurationin Fig. 1(b) are on, besides the aforementioned SWM1 andSWM2 signals, the EWM signal generated in the subsystemj0i ↔ j1i ↔ j2i ↔ j3i ↔ j4i 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 schematicdiagram in which the coexisting SWM and EWM signals areinjected into the Stokes port to realize the OPA process. Alongwith PA-SWM2 signals, the Rydberg PA-EWM signals are alsothe constituent part of the detected PA-MWM signals, whose

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intensities are shown by scanning Δ2 at several discrete valuesof Δ4 in Fig. 4(b) and P4 in Fig. 4(c). In Fig. 4(b), the Rydbergsignals are detected with the background of a broadenedFWM signal (generated in the Λ-type three-level subsystemj0i ↔ j1i ↔ j3i) and a non-Rydberg SWM1 signal. The co-existing PA-EWM and PA-SWM signals obtained by scanningΔ2 can be strengthened or suppressed at different values of Δ4

[42]. Considering the dressing effects of both E2 and E4, whenthe value of Δ4 is far-detuned from the resonance of j1i → j4i,coexisting SWM and EWM signals can only be enhanced, sincethe enhancement condition Δ2 � Δ− � Δ−� � 0 is the onlyextreme value condition that can be fulfilled. The additionaldetuning of primary and secondary dressing energy levels splitby E2 and E4 is given by Δ− � f�Δ2 � ε� − ��Δ2 � ε�2 �4jG2j2�1∕2g∕2 and Δ−� � �Δ 0

4 � �Δ 024 �4jG4j2��∕2, in which

Δ 04 � Δ4 − 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 j−G2i andj−G2 � G4i in Fig. 4(a1). However, when the value of Δ4

gradually approaches the resonance of j1i ↔ j4i, the influ-ence of dressing field is gradually transformed from enhance-ment into suppression. In addition, when Δ4 is near theresonant point, only the suppression of the output PA-MWMsignal can be observed, because only the suppression condi-tion Δ2 � Δ1 � 0 can be fulfilled. Meanwhile, it is possibleforΔ4 to be set to the value that both suppression and enhance-ment conditions can be satisfied; thus the half-enhancementand half-suppression of the output signal can be observed [42].In this way, the signal can be enhanced or suppressed when thecorresponding extreme value conditions (Δ2 � Δ− � Δ−� � 0and Δ2 � Δ1 � 0) are satisfied. Similarly, as shown in theright part (Δ4 > 0) of Fig. 4(b), when Δ4 continues to movefar away from the resonance, the coexisting PA-SWM2 andPA-EWM signals will be enhanced by the dressing field again.The OPA process acts against the dressing suppression andenables the enhancement of the coexisting PA-SWM andPA-EWM signals to be more obvious. Furthermore, one canalso notice that when we change the power of E4 in Fig. 4(c),the effect of E4 can also be switched between suppression andenhancement. When the power of the dressing field E4 is rel-atively low, the signal is enhanced in Fig. 4(c). In this case,the suppression caused by the dressing effect of E4 is notobvious, and both coexisting PA-SWM2 and PA-EWM signals

contribute to the peak in Fig. 4(c). Nevertheless, when P4 in-creases gradually, the PA-MWM signals near the resonantposition move gradually towards suppression with the transi-tion of half-enhancement and half-suppression, as shown inFig. 4(c). When the power P4 is high enough, the suppressionof the PA-MWM signals caused by the dressing effect of E4

gets dominant, and dip is observed, as shown in the bottomof Fig. 4(c).

4. CONCLUSIONS

We have studied the PA-SWM and PA-EWM signals in aRydberg EIT Rb atomic medium both experimentally andtheoretically. The intensity dependences of the MWM signalson the principal quantum number, the detuning of the probeand coupling fields, the power of coupling fields, and theatomic density are investigated. One can find that the MWMsignals can be effectively controlled by these parameters withthe combined effect of RRI and the parametric amplification.Moreover, the linear amplification of the Rydberg MWM sig-nals resulting from the OPA process acts against the suppres-sion of the Rydberg excitation, which can potentially improvethe logic gate devices and sensors related to Rydberg atoms.

APPENDIX A

SWM2 and EWM processes are related to the Rydberg tran-sition. The mean-field model is applied to modify the density-matrix elements of E SWM2 and EEWM. It is considered that theregion with only one excited Rydberg atom is a sphere with aradius of Rd , and the density of excited Rydberg atoms ρ2 issupposed to be locally uniform inside and around the sphere;thus, we get ρ2V d � 1, in which V d ∝ R3

d is the volume of thesphere. The level shift ε will be obtained as a function of itslocation r and the principal quantum number n as

ε � ρ2

ZV 0

U �r − r 0�d3r 0, (A1)

where U �r − r 0� is the van der Waals interaction amongRydberg atoms, and it can be given as

RV 0 U �r − r 0�d3r 0 ∝

1∕R6d for nD interaction. The optical Bloch equations for the

Rydberg excitation (5P3∕2 ↔ nD5∕2) are

id

dtcg �

G2

2eiβt2 ce , (A2)

Fig. 4. (a1) AT splitting in the five-level atomic system induced by E2 and E 4; (a2) phase-matching diagram of OPA injected with ESWM1,E SWM2, 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.

718 Vol. 6, No. 7 / July 2018 / Photonics Research Research Article

id

dtce � ε�r, t�ce �

G2

2e−iβt2cg , (A3)

where cg and ce are the probability amplitude for the groundstate and Rydberg state, respectively.

Solving Eqs. (A1)–(A3) under steady-state approximation,then considering ρ2V d � 1 and V d ∝ R3

d , the density ofthe atoms excited to Rydberg state j2i can be given by [2,14]

ρ2 � Cρ0.21

�jG2jn11

�0.4, (A4)

where C is a constant related to numerical integration outsidethe given sphere and the atom excitation efficiency between j0iand j1i, ρ1 is the density of the atoms at j1i, which is given by

ρ1 �ρ02

� jG1j2Re�d 1 � jG2j2∕d 2 � jG4j2∕d 4�

� jG3j2Re�d 3�

�,

(A5)

where ρ0 is the atomic density at the ground state j0i,d 1 � Γ10 � iΔ1, d 2 � Γ20 � i�Δ1 � Δ2�, d 3 � Γ30 �i�Δ1 − Δ3�, and d 4 � Γ30 � i�Δ1 � Δ4�. Term ρ0.21 indicatesthat the contribution of the particle number at j1i to the num-ber of atoms excited to j2i is suppressed. Term �jG2j∕n11�0.4shows that the excitation caused by E2 is also suppressed andlimited by principal quantum number n. The dependence of ρ2on n in Eq. (A4) shows that the probability of atoms beingexcited to j2i will decrease with a larger n. Now that the energyshift ε in Eq. (A1) is directly related to the atomic density ρ2, εis supposed to grow considerably larger with the increase of n orr within the given sphere, and accordingly, the Rydberg block-ade effect is enhanced. Finally, we can substitute ρ0, G1, G2,and G3 in the classical model with ρ0.20 , G0.2

1 , �G2∕n11�0.2, andG0.2

3 , respectively, as a modification in the situation of Rydbergexcitation.

Funding. National Key R&D Program of China(2017YFA0303700); National Natural Science Foundationof China (NSFC) (11474228, 11604256, 61605154); KeyScientific and Technological Innovation Team of ShaanxiProvince (2014KCT-10); Natural Science Foundation ofShaanxi Province (2017JQ6039); China PostdoctoralScience Foundation (2016M600776).

†These authors contributed equally to this work.

REFERENCES1. T. F. Gallagher, Rydberg Atoms (Cambridge University, 1994).2. D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R.

Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg exci-tation in an ultracold gas,” Phys. Rev. Lett. 93, 063001 (2004).

3. M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I.Cirac, and P. Zoller, “Dipole blockade and quantum informationprocessing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87,037901 (2001).

4. M. Saffman, T. G. Walker, and K. Molmer, “Quantum information withRydberg atoms,” Rev. Mod. Phys. 82, 2313–2363 (2010).

5. Y. O. Dudin and A. Kuzmich, “Strongly interacting Rydberg excitationsof a cold atomic gas,” Science 336, 887–889 (2012).

6. G. Günter, M. Robert-De-Saint-Vincent, H. Schempp, C. S. Hofmann,S. Whitlock, and M. Weidemüller, “Interaction enhanced imaging of

individual Rydberg atoms in dense gases,” Phys. Rev. Lett. 108,013002 (2012).

7. N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G.Hornecker, S. L. Portalupi, T. Grange, C. Anton, J. Demory, C.Gomez, I. Sagnes, N. D. L. Kimura, A. Lemaitre, A. Auffeves, A. G.White, L. Lanco, and P. Senellart, “Near-optimal single-photonsources in the solid state,” Nat. Photonics 10, 340–345 (2016).

8. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today50, 36–42 (1997).

9. M. Xiao, Y. Li, S. Jin, and J. Geabanacloche, “Measurement ofdispersive properties of electromagnetically induced transparencyin rubidium atoms,” Phys. Rev. Lett. 74, 666–669 (1995).

10. A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, andS. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Λ system,” Phys. Rev. Lett. 84, 5308–5311 (2000).

11. Z. Zhang, H. Tang, I. Ahmed, N. Ahmed, G. Khan, A. Mahesar, andY. Zhang, “Controlling Rydberg-dressed four-wave mixing via dualelectromagnetically induced transparency windows,” J. Opt. Soc.Am. B 33, 1661–1667 (2016).

12. C. Carr, M. Tanasittikosol, A. Sargsyan, D. Sarkisyan, C. S. Adams,and K. J. Weatherill, “Three-photon electromagnetically inducedtransparency using Rydberg states,”Opt. Lett. 37, 3858–3860 (2012).

13. E. Brekke, J. O. Day, and T. G. Walker, “Four-wave mixing in ultracoldatoms using intermediate Rydberg states,” Phys. Rev. A 78, 063830(2008).

14. Z. Zhang, J. Che, D. Zhang, Z. Liu, X. Wang, and Y. Zhang, “Eight-wave mixing process in a Rydberg-dressing atomic ensemble,” Opt.Express 23, 13814–13822 (2015).

15. Y. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spa-tial interference between four-wave mixing and six-wave mixing chan-nels,” Phys. Rev. Lett. 102, 013601 (2009).

16. A. Kölle, G. Epple, H. Kübler, R. Löw, and T. Pfau, “Four-wave mixinginvolving Rydberg states in thermal vapor,” Phys. Rev. A 85, 063821(2012).

17. H. B. Zheng, X. Yao, Z. Y. Zhang, J. L. Che, Y. Q. Zhang, Y. P. Zhang,and M. Xiao, “Blockaded six- and eight-wave mixing processes tail-ored by electromagnetically induced transparency scissors,” LaserPhys. 24, 045404 (2014).

18. M. D. Lukin, A. B. Matsko, M. Fleischhauer, and M. O. Scully,“Quantum noise and correlations in resonantly enhanced wave mixingbased on atomic coherence,” Phys. Rev. Lett. 82, 1847–1850 (1999).

19. T. Peyronel, O. Firstenberg, Q. Y. Liang, S. Hofferberth, A. V.Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinearoptics with single photons enabled by strongly interacting atoms,”Nature 488, 57–60 (2012).

20. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled im-ages from four-wave mixing,” Science 321, 544–547 (2008).

21. Z. Bai and G. Huang, “Enhanced third-order and fifth-order Kerr non-linearities in a cold atomic system via Rydberg-Rydberg interaction,”Opt. Express 24, 4442–4461 (2016).

22. Z. Zhang, H. Zheng, X. Yao, Y. Tian, J. Che, X. Wang, D. Zhu, Y.Zhang, and M. Xiao, “Phase modulation in Rydberg dressed multi-wave mixing processes,” Sci. Rep. 5, 10462 (2015).

23. Z. Zhang, F. Wen, J. Che, D. Zhang, C. Li, Y. Zhang, and M. Xiao,“Dressed gain from the parametrically amplified four-wave mixing pro-cess in an atomic vapor,” Sci. Rep. 5, 15058 (2015).

24. Y. Zhang, M. Belić, Z. Wu, H. Zheng, K. Lu, Y. Li, and Y. Zhang,“Soliton pair generation in the interactions of Airy and nonlinear accel-erating beams,” Opt. Lett. 38, 4585–4588 (2013).

25. M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson,and A. L. Gaeta, “Broad-band optical parametric gain on a silicon pho-tonic chip,” Nature 441, 960–963 (2006).

26. J. K. Thompson, J. Simon, H. Loh, and V. Vuletic, “A high-brightnesssource of narrowband, identical-photon pairs,” Science 313, 74–77(2006).

27. Y. P. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixingand six-wave mixing channels via dual electromagnetically inducedtransparency windows,” Phys. Rev. Lett. 99, 123603 (2007).

28. M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theoryof two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).

Research Article Vol. 6, No. 7 / July 2018 / Photonics Research 719

29. J. M. Wen and M. H. Rubin, “Transverse effects in paired-photongeneration via an electromagnetically induced transparency medium. II.Beyond perturbation theory,” Phys. Rev. A 74, 023809 (2006).

30. C. F. McCormick, A. M. Marino, V. Boyer, and P. D. Lett, “Strong low-frequency quantum correlations from a four-wave-mixing amplifier,”Phys. Rev. A 78, 043816 (2008).

31. R. C. Pooser and B. Lawrie, “Plasmonic trace sensing below thephoton shot noise limit,” ACS Photon. 3, 8–13 (2015).

32. R. C. Pooser and B. Lawrie, “Ultrasensitive measurement of micro-cantilever displacement below the shot-noise limit,” Optica 2, 393–399 (2015).

33. F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, andW. Zhang, “Quantummetrology with parametric amplifier-based photon correlation interfer-ometers,” Nat. Commun. 5, 3049 (2014).

34. J. A. Sedlacek, A. Schwettmann, H. Kübler, R. Löw, T. Pfau, andJ. P. Shaffer, “Microwave electrometry with Rydberg atoms in a vapourcell using bright atomic resonances,” Nat. Phys. 8, 819–824 (2012).

35. P. P. Herrmann, J. Hoffnagle, N. Schlumpf, V. L. Telegdi, and A. Weis,“Stark spectroscopy of forbidden two-photon transitions: a sensitiveprobe for the quantitative measurement of small electric fields,”J. Phys. B 19, 1271–1280 (1986).

36. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (OxfordUniversity, 1995).

37. H. Zheng, X. Zhang, Z. Zhang, Y. Tian, H. Chen, C. Li, and Y. Zhang,“Parametric amplification and cascaded-nonlinearity processes incommon atomic system,” Sci. Rep. 3, 1885 (2013).

38. H. Chen, Y. Zhang, X. Yao, Z. Wu, X. Zhang, Y. Zhang, and M. Xiao,“Parametrically amplified bright-state polariton of four-and six-wavemixing in an optical ring cavity,” Sci. Rep. 4, 3619 (2014).

39. Y. Li, G. Huang, D. Zhang, Z. Wu, Y. Zhang, J. Che, and Y.Zhang, “Density control of dressed four-wave mixing and super-fluorescence,” IEEE J. Quantum Electron. 50, 25–34 (2014).

40. J. Che, J. Ma, H. Zheng, Z. Zhang, X. Yao, Y. Zhang, and Y. Zhang,“Rydberg six-wave mixing process,” Europhys. Lett. 109, 33001(2015).

41. Y. P. Zhang, P. Y. Li, H. B. Zheng, Z. G. Wang, H. X. Chen, C. B. Li, R.Zhang, and Y. Zhang, “Observation of Autler-Townes splitting insix-wave mixing,” Opt. Express 19, 7769–7777 (2011).

42. P. Li, H. Zheng, Y. Zhang, J. Sun, C. Li, G. Huang, Z. Zhang, Y.Li, and Y. Zhang, “Controlling the transition of bright and darkstates via scanning dressing field,” Opt. Mater. 35, 1062–1070(2013).

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