## Abstract

We investigate the nonreciprocity “∞”-shape optical bistability (OB) induced by the feedback dressing effect of six-wave mixing parametrically amplified process in a four-level atomic system. Compared to the traditional OB by scanning power, the “∞”-shape OB is scanning probe frequency and demonstrated by “∞”-shape non-overlapping region. More, this non-overlapping region in the *x* direction (frequency difference) and in the *y* direction (intensity difference) could demonstrate the degree of this OB phenomenon of dressed probe and conjugate signals, which can be changed by the intensity of feedback dressing. Further, we find the feedback intensity can be controlled by experimental parameters include powers of external-dressing, frequency detuning, incident phase and the nonlinear phase shift of internal-dressing beam. As a result, the nonreciprocity “∞”-shape OB is more sensitive and multiple than traditional OB. These outcomes have potential applications in logic-gate devices and quantum information processing.

© 2017 Optical Society of America

## 1. Introduction

Parametric multi-wave mixing (MWM) process based on atomic coherence [1] plays potential roles in low-noise imaging [2] and quantum communication [3], as well as the development of squeezing states [4]. A spontaneous parametric four-wave mixing (SP-FWM) process generates two weak fields (Stokes field and anti-Stokes field) on a forward cone, which can be injected by input signal and lead to optical parametrically amplification (OPA) [5,6]. Recently, narrow-band bright entangled light beams have been produced through PA four-wave mixing (PA-FWM) process [7]. Subsequently, several important applications of using narrow-band squeezing light in entangled images [8,9], quantum metrology includes low frequency and controllable bandwidth squeezing [10], phase-sensitive amplifier [11], ultrasensitive measurement and plasmonic sensors [12,13], have all been experimentally demonstrated. However, there exists strong nonlinearity and dispersion in OPA process. Coherent population trapping is one manifestation of electromagnetically induced transparency (EIT). For optically thick media, radiation trapping and optical pumping have been studied extensively in astrophysics, plasma physics, and atomic spectroscopy [14–16]. Due capacity of gain, oscillation and radiation trapping in FWM process, optical bistability (OB) behavior based on atomic coherence and quantum interference are practiced [17,18]. In recent decades, some schemes for realizing optical stability through OPA-MWM process in an optical cavity have been studied experimentally and theoretically [19–21]. More, OB has been demonstrated without a cavity using degenerated FWM in atomic vapor with two counter-propagating laser beams [22]. Subsequently, OB has been became the subject of many studies because of its broad application prospects in all-optical logic [23] and quantum information processing [20].

In this paper, we theoretically and experimentally investigate the nonreciprocity “∞”-shape OB phenomenon of dressed signals (probe and conjugate) from PA-SWM process in 85Rb atomic vapor cell. The nonreciprocity of the signals on the frequency-increasing and frequency-decreasing processes (corresponding to the rising and falling edges in one frequency scanning round trip, respectively) is caused by the feedback dressing effect induced by the parametrical amplification closed loop process. Therefore, the signals curves of two edges are not overlap by folding them from the maxima of the ramp curves point. And the non-overlapping region of OB can be approximately viewed with infinite sidebands, so we named this kind of OB as “∞”-shape OB [24]. Compared with traditional cavity-type OB, we implement the way that scanning frequency to improve the extent of OB from dressed PA-FWM closed loop configurations. In our experiments, the external-dressing fields, to significantly enhance the feedback dressing effect as a major benefit of combined action of PA-SWM and PA-FWM. When we change one of parameters, the frequency difference (along the horizontal x direction of non-overlapping region) is caused by the different feedback dressing while the intensity difference (along the vertical y direction of non-overlapping region) is caused by the difference on conditions for suppression and enhancement [21]. Therefore, we can obtain different output multi-states and realize the conversion between these states by controlling input parameters. These merits will greatly facilitate the potential applications of quantum logic-gate devices such as quantum flip-flop converter and quantum memory in quantum information processing.

## 2. Experimental setup

The experimental considers a four-level atomic system as shown in Fig. 1(b). The four relevant energy levels are 5*S*_{1/2}, *F* = 2 (|0>), 5*S*_{1/2}, *F* = 3 (|1>), 5*P*_{1/2}, *F* = 2 (|2>), and 5*P*_{3/2}, *F* = 4 (|3>) in ^{85}Rb. The three-level “double-Λ” type subsystem (|0>↔|1>↔|2>) is used to generate the SP-FWM process. With the laser frequency tuned to the *D*2 line transition of rubidium, the strong pump beam *E*_{1} (frequency *ω*_{1}, wave vector *k*_{1}, Rabi frequency *G*_{1}, wavelength 795 nm, power up to 200 mW) is coupled into the cell by polarization beam splitter. The weak probe beam *E** _{p}* (

*ω*,

_{p}

*k**,*

_{p}*G*

_{p}_{,}795 nm) with approximately 30 μW propagates in the same direction of

*E*_{1}with an angle of 0.26° and is detected by a branch of a balanced homodyne detector. The generated conjugate signal that can establish the coherence between the two ground states |0> and |1> co-propagates with

*E**symmetrically with respect to*

_{p}

*E*_{1}and is received by the other branch of the balanced detector. If external-dressing beam

*E*_{3}(

*ω*

_{3},

*k*_{3},

*G*

_{3}, 780 nm, 30 mW, from another ECDL) is injected in

*E*_{1}direction, the four-level “N-type” PA-SWM process is formed.

## 3. Basic theory

#### 3.1 Dressed SP-FWM and SP-SWM

Actually, dressed SP-FWM can be considered as a coherent superposition of one pure SP-FWM process and one pure SP-SWM process. First, with only beam *E*_{1} turned on, the three-level “double-Λ” type rubidium (Rb) subsystem (|0>↔|1>↔|2>) is formed. With the frequency detuning of *E*_{1}, the SP-FWM process will occur in the “double-Λ” configuration, which can generate the Stokes field *E** _{S}* and anti-Stokes field

*E**(satisfying the phase match conditions (PMCs)*

_{aS}

*k**= 2*

_{S}

*k*_{1}-

*k**and*

_{aS}

*k**= 2*

_{aS}

*k*_{1}-

*k**, respectively) on the forward cone shown in Fig. 1(a). Here the detuning ∆*

_{S}*=*

_{i}*Ω*-

_{i}*ω*is defined as the difference between the resonant transition frequency

_{i}*Ω*and the laser frequency

_{i}*ω*of

_{i}

*E**. The generated*

_{i}

*E**and*

_{S}

*E**field could be obtained by the via perturbation chains ${\rho}_{11}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{21}^{(1)}\stackrel{{\omega}_{aS}}{\to}{\rho}_{01}^{(2)}\stackrel{{\omega}_{1}^{\ast}}{\to}{\rho}_{21(S)}^{(3)}$ (*

_{aS}

*E**) and ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{20}^{(1)}\stackrel{{\omega}_{S}}{\to}{\rho}_{10}^{(2)}\stackrel{{\omega}_{1}^{\ast}}{\to}{\rho}_{20(aS)}^{(3)}$ (*

_{S}

*E**), respectively. Then, we add the strong external-dressing field*

_{aS}

*E*_{3}at

*E*_{1}direction which is coupling to the transition |1>↔|3> in Fig. 1(b). Taking into account the external-dressing effect of

*E*_{3}, the corresponding third-order density matrix elements

*ρ*

^{(3)}

_{21(}

_{S}_{)}and

*ρ*

^{(3)}

_{20(}

_{aS}_{)}of dressed SP-FWM can be rewritten as:

*G*=

_{i}*μ*(

_{ij}E_{i}/ħ*i, j*= 1, 2, 3,

*S*,

*aS*) is the Rabi frequency between levels |

*i*>↔|

*j*>, and

*μ*is the dipole momentum;

_{ij}*d*

_{20}= Γ

_{20}+

*i*∆

_{1},

*d*

_{10}

*=*

_{D}*d*

_{10}+

*G*

_{3}

^{2}/

*d*

_{30},

*d*

_{10}= Γ

_{10}+

*i*(∆

_{1}-∆

*),*

_{S}*d*

_{30}= Γ

_{13}+

*i*(∆

_{1}-∆

*+ ∆*

_{S}_{3}),

*d*'

_{20}= Γ

_{20}+

*i*(∆

_{1}-∆

*+ ∆'*

_{S}_{1}),

*d*

_{21}= Γ

_{21}+

*i*∆'

_{1},

*d*

_{01}

*=*

_{D}*d*

_{01}+

*G*

_{3}

^{2}/

*d*

_{31},

*d*

_{01}= Γ

_{01}+

*i*(∆'

_{1}-∆

*),*

_{aS}*d*

_{31}= Γ

_{31}+

*i*(∆'

_{1}-∆

*+ ∆*

_{aS}_{3}),

*d*'

_{21}= Γ

_{21}+

*i*(∆'

_{1}-∆

*+ ∆*

_{aS}_{1}); Γ

*= (Γ*

_{ij}*+ Γ*

_{i}*)/2 is the de-coherence rate between |*

_{j}*i*> and |

*j*>. More interestingly, under the weak field limit (|

*G*

_{3}|

^{2}<<Γ

_{21}Γ

_{31}or Γ

_{20}Γ

_{30}), Eqs. (1) and (2) can be expanded to be:

*ρ*

^{(5)}

_{21(}

_{S}_{)}= (-

*G*

_{3}

^{2}/

*d*

_{01}

*d*

_{31})

*ρ*

^{(3)}

_{21(}

_{S}_{)}and

*ρ*

^{(5)}

_{20(}

_{aS}_{)}= (-

*G*

_{3}

^{2}/

*d*

_{10}

*d*

_{30})

*ρ*

^{(3)}

_{20(}

_{aS}_{)}are the corresponding fifth-order density matrix elements of formed SP-SWM process which can be deduced by the via perturbations${\rho}_{11}^{(0)}\stackrel{{\omega}_{3}}{\to}{\rho}_{31}^{(1)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{11}^{(2)}\stackrel{{\omega}_{1}}{\to}{\rho}_{21}^{(3)}\stackrel{-{\omega}_{aS}}{\to}{\rho}_{01}^{(4)}\stackrel{{\omega}_{1}^{\ast}}{\to}{\rho}_{21(S)}^{(5)}$ and ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{20}^{(1)}\stackrel{-{\omega}_{S}}{\to}{\rho}_{10}^{(2)}\stackrel{{\omega}_{3}}{\to}{\rho}_{30}^{(3)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{10}^{(4)}\stackrel{{\omega}_{1}^{\ast}}{\to}{\rho}_{20(aS)}^{(5)}$, respectively. This means that the density-matrix element of the dressed SP-FWM can be considered as a coherent superposition of one pure SP-FWM process and one pure SP-SWM process under the weak field limit.

#### 3.2 Dressed PA-FWM and PA-SWM

Finally, we add the weak probe beam *E** _{p}* propagates in the

*E*_{1}direction with an angle of 0.26°. The presence of

*E**can be viewed as being injected into the Stokes or anti-Stokes port of the dressed SP-FWM process, and the injection will serve as an optical parametric amplification (OPA) process (with PMCs*

_{p}

*k**= 2*

_{aS}

*k*_{1}-

*k**and*

_{p}

*k**= 2*

_{S}

*k*_{1}-

*k**) assisted by the cascaded nonlinear process. When*

_{p}

*E**is injected into the Stokes port of the dressed SP-FWM process, it can amplify the seeded signal in an appropriate condition. The photon numbers of the output Stokes and anti-Stokes fields in the amplification process with injection are described as:*

_{p}*A*and

*B*(phase angles

*φ*

_{1}and

*φ*

_{2}) defined in

*ρ*'

^{(3)}

_{21(}

_{S}_{)}=

*Ae*

^{iφ}^{1}and

*ρ*'

^{(3)}

_{20(}

_{aS}_{)}=

*Be*

^{iφ}^{2}for

*E**and*

_{S}

*E**, respectively. With*

_{aS}

*E**injected, dressed PA-FWM process is formed. This moment, the output $\u3008{\widehat{a}}_{D}^{+}{\widehat{a}}_{D}\u3009$ and $\u3008{\widehat{b}}_{D}^{+}{\widehat{b}}_{D}\u3009$ are named probe and conjugate fields of dressed PA-FWM, respectively. For another, if*

_{p}

*E**is injected into the Stokes port of the SP-SWM process, the gain*

_{p}*g*can be defined in

*ρ*

^{(5)}

_{21(}

_{S}_{)}=

*Ce*

^{iφ}^{3}and

*ρ*

^{(5)}

_{20(}

_{aS}_{)}=

*De*

^{iφ}^{4}, and the output $\u3008{\widehat{a}}_{6}^{+}{\widehat{a}}_{6}\u3009$ and $\u3008{\widehat{b}}_{6}^{+}{\widehat{b}}_{6}\u3009$ are part of PA-SWM process which can be deduced by the via perturbations ${\rho}_{11}^{(0)}\stackrel{{\omega}_{3}}{\to}{\rho}_{31}^{(1)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{11}^{(2)}\stackrel{{\omega}_{1}}{\to}{\rho}_{21}^{(3)}\stackrel{-{\omega}_{Conj}}{\to}{\rho}_{01}^{(4)}\stackrel{{\omega}_{1}^{\ast}}{\to}{\rho}_{21(S)}^{(5)}$ (

*E**) and ${\rho}_{00}^{(0)}\stackrel{{\omega}_{1}}{\to}{\rho}_{20}^{(1)}\stackrel{-{\omega}_{P}}{\to}{\rho}_{10}^{(2)}\stackrel{{\omega}_{3}}{\to}{\rho}_{30}^{(3)}\stackrel{-{\omega}_{3}}{\to}{\rho}_{10}^{(4)}\stackrel{{\omega}_{1}^{\ast}}{\to}{\rho}_{20(aS)}^{(5)}$ (*

_{S}

*E**), respectively.*

_{aS}#### 3.3 Feedback dressing of OPA process

Specially, considering OPA process with enhanced nonlinearity, there exists an un-neglected feedback effect (also a self-dressing effect) [21]. Clearly, the generated probe and conjugate fields have an self-dressing effect of *|G _{F}|*

^{2}(

*|**G*

_{P}

*|*^{2}and

*|**G*

_{C}

*|*^{2}), which are derived from the relatively strong feedback effect. These self-dressings effect have a similar influence with the internal-dressing effect of

*E*_{1}and the external-dressing effect of

*E*_{3}, which can together result in OPA process. Besides, we have demonstrated the dressed SP-FWM can be considered as a coherent superposition of two pure processes (SP-FWM and SP-SWM). Therefore, with the injection of

*E**, the output $\u3008{\widehat{a}}_{D}^{+}{\widehat{a}}_{D}\u3009$and $\u3008{\widehat{b}}_{D}^{+}{\widehat{b}}_{D}\u3009$ of dressed PA-FWM can be expanded by corresponding third-order density matrix elements:*

_{p}*ρ*'

^{(5)}

_{21(}

_{S}_{)}and

*ρ*'

^{(5)}

_{20(}

_{aS}_{)}of PA-SWM process can be written as:

*e*

^{i}^{∆}

*is related to the orientations of induced dipole moments*

^{α}

*μ*_{1}and

*μ**, which can be manipulated by means of altering the incident angle*

_{p}*α*between the pump field

*E*_{1}and the probe field

*E**. The*

_{p}*e*

^{i}^{∆}

*is nonlinear phase shift introduced as*

^{Φ}*Φ*= 2

*k*

_{P}_{/}

_{C}n_{2}

*I*

_{1}

*e*/

^{-ξ· ξ}z*n*

_{0}, where

*k**=*

_{p/C}

*k**=*

_{S}

*k**,*

_{aS}*n*

_{2}is cross-Kerr of internal-dressing field

*E*_{1}. In order to research the propagate effect in this system, we consider the internal-dressing effect of the pump field

*E*_{1}and introduce an additional phase factor

*e*

^{i}^{(∆}

^{α+}^{∆}

^{Φ}^{)}into the dressing term. Also, with

*E**and*

_{p}

*E*_{1}viewed as probe and coupling fields, respectively, the first-order density matrix

*ρ*

^{(1)}

_{20}of the probe transmission signal with dressing effect is:

*I*∝ (

_{P}*I*

_{0}

*-*Im

*ρ*

^{(1)}

_{20}+ |

*ρ”*

^{(3)}

_{21(}

_{S}_{)}|

^{2}) [

*I*∝ (

_{C}*I*

_{0}+ |

*ρ”*

^{(3)}

_{20(}

_{aS}_{)}|

^{2})], where

*I*

_{0}is the intensity of the probe field without Doppler absorption.

#### 3.4 Non-overlapping region and suppression (enhancement) conditions of nonreciprocity “∞”-shape OB

What’s more, because OPA process in the above mainly produce an un-neglected feedback dressing *|G _{F}|*

^{2}(

*|G*

_{P}|^{2}and

*|G*

_{C}|^{2}), the folded signals of probe and conjugate could exist “∞”-shape non-overlapping region includes the frequency difference in

*x*direction and the intensity difference in

*y*direction. Firstly, there exists frequency difference (

*δ*) between the two peaks or dips (Figs. 2-4) in the same baseline. Whereas the nonreciprocity reflected from the change of nonreciprocity phase$\Delta \phi $is as follow:

*δ*is the frequency difference that can reflect the OB phenomenon directly,

*n*

_{1}is the linear refractive index of the Rb cell. The feedback intensity

*I*(

_{up}*I*) is generated at the same frequency scan. n

_{down}_{2}

*(n*

_{up}_{2}

*) is the nonlinear refractive index coefficient that can be generally expressed as*

_{down}*n*

_{2}

*≈*

_{up}*n*

_{2}

*≈*

_{down}*n*

_{2}= Re[

*χ*

^{(3)}/(

*ε*)], which is the mainly dominated by the Kerr coefficient of

_{0}cn_{0}

*E*_{1}. Besides, the intensity difference at the frequency-rising and frequency-falling edges can also advocate the OB phenomenon in the OPA process. Physically, the intensity difference of the probe (conjugate) signals can be understood through requirement for the dressing suppression and enhancement. When considering the different feedback dressing on the rising and falling edges, the signal will meet different enhancement (or suppression) conditions, which results in the nonreciprocity of intensity difference ∆

*I*.

Additionally, the suppression and enhancement of these signals play a very important role in the OPA process. For instance, the primary Autler-Townes (AT) splitting is caused by external-dressing field *E*_{3} and the corresponding eigenvalues are *λ* _{±} = [∆_{3} ± (∆_{3}^{2} + 4|*G*_{3}|^{2})^{1/2}]/2. The secondary AT splitting is caused by self-dressing effect *E** _{F}* whose corresponding eigenvalues are

*λ*

_{+}_{±}= [∆'

*± (∆'*

_{F}

_{F}^{2}+ 4|

*G*|

_{F}^{2})

^{1/2}]/2 (∆'

*= ∆*

_{F}*-*

_{F}*λ*). With the feedback dressing and external-dressing effects considered, the corresponding suppression and enhancement conditions of

_{+}

*E**and*

_{S}

*E**are ∆*

_{aS}*∆'*

_{S}-*= 0, ∆*

_{F}*∆ʺ*

_{aS}-*= 0 and ∆*

_{F}*-*

_{S}*λ*-

_{+}*λ*= 0, ∆

_{++}*-*

_{aS}*λ*-

_{+}*λ*= 0, respectively.

_{++}## 4. Results and discussions

In our experiment, the signals of dressed probe and conjugate fields are shown on Figs. 1(d) and 1(e), respectively. Where *E*_{1}, *E*_{3} and *E** _{p}* are turned on together, each combination of electromagnetically induced absorption dip (

*ρ*

_{20}

^{(1)}) and gain peak (

*ρ”*

^{(3)}

_{P}_{(}

_{S}_{/}

_{aS}_{)}) can be seen in the dressed probe signal (

*I*∝ (

_{P}*I*

_{0}

*-*Im

*ρ*

^{(1)}

_{20}+ |

*ρ”*

^{(3)}

_{21(}

_{S}_{)}|

^{2})), but the dressed conjugate signal only shows the gain peaks (

*ρ”*

^{(3)}

_{C}_{(}

_{S}_{/}

_{aS}_{)}). Besides, there exists six peaks and dips at the rising edge both in dressed probe and conjugate signals. If probe beam

*E**connect upper transition |2> to |1> in Fig. 1(b), with*

_{p}*ω*

_{p}-ω_{1}<0, the output probe signal is injected into Stokes in Fig. 1(a), and there will generate dip (d6)

^{87}Rb,

*F*= 2, peak (d5)

^{85}Rb,

*F*= 3 and peak (d4) near-degenerate PA-FWM Stokes signal in Fig. 1(d). At the same time, the output conjugate signal is poured into anti-Stokes, then peak (e6)

^{87}Rb,

*F*= 2, peak (e5)

^{85}Rb,

*F*= 3 and peak (e4) near-degenerate Stokes PA-FWM signal will produce in Fig. 1(e). Analogously, when

*E**connect upper transition |2> and |0>, with*

_{p}*ω*

_{p}-ω_{1}>0, the output probe signal is injected into anti-Stokes, and there will generate peak (d3) near-degenerate PA-FWM anti-Stokes signal, peak (d2)

^{85}Rb,

*F*= 2, peak (d1)

^{87}Rb,

*F*= 1 in Fig. 1(d). Meanwhile, the output conjugate signal is poured into Stokes, then peak (e3) near-degenerate PA-FWM anti-Stokes signal, peak (e2)

^{85}Rb,

*F*= 2 and peak (e1)

^{87}Rb,

*F*= 1 will produce in Fig. 1(e). At

*ω*

_{p}-ω_{1}= 0, probe and pump fields are resonant. Besides, the dressed probe and conjugate signals at the falling edge is symmetrical with the rising edge signals in Figs. 1(d) and 1(e).

In Fig. 1(c), there are the two peaks (^{85}Rb, *F* = 2) of probe signal that *E*_{3} is dressed in Fig. 1(c1) while *E*_{3} is blocked in Fig. 1(c2). If taking into account the dressing effect of *E*_{3}, Stokes or anti-Stokes of dressed PA-FWM can be expanded as: ${{\rho}^{\u2033}}_{20(as)}^{(3)}={{\rho}^{\u2034}}_{20(as)}^{(3)}\text{+}{{\rho}^{\prime}}_{20(as)}^{(5)}$ at Eq. (8), we set ${{\rho}^{\u2034}}_{}^{(3)}=B{e}^{i{\phi}_{2}}$ and ${{\rho}^{\prime}}_{}^{(5)}=D{e}^{i{\phi}_{4}}$. We make *I _{F}* =

*B*

^{2}and

*I*=

_{S}*D*

^{2}represent the intensity of PA-FWM and PA-SWM, respectively. Besides,

*I*= 2

_{x}*BD*cos∆

*φ*(∆

*φ*=

*φ*

_{2}-

*φ*

_{4}) represent interference term. When we intentionally set experiment satisfies the suppression condition ∆'

_{1}-∆

_{3}= 0, the relative nonlinear phase between PA-FWM and PA-SWM changed to ∆

*φ*=

*π*. Thus the cross term

*I*= 2

_{x}*BD*cos∆

*φ*is negative, leading to ${|{\rho}^{\u2034}}_{}^{(3)}+{{\rho}^{\prime}}_{}^{(5)}\text{|}=\sqrt{{I}_{F}+{I}_{S}-2BD}=\text{|}{{\rho}^{\u2034}}_{}^{(3)}|-|{{\rho}^{\prime}}_{}^{(5)}|$. It can be concluded that intensity of dressed PA-FWM is greatly suppressed when suppression condition is satisfied. This is the reason that the signal of Fig. 1(c2) is obvious suppression where

*E*_{3}is dressed. Further, the experiment results (Figs. 1(d), 1(e) and 2-4) all have dressing field

*E*_{3}, which means the dressed signals of OPA process can be considered as suppressed PA-FWM (PA-FWM - PA-SWM).

Besides, the signals on the rising and falling edges in one complete cycle by folding from the turning point of the round trip do not overlap. In the following experimental results (Figs. 2-4), we investigate the nonreciprocity “∞”-shape OB phenomena by comparing the folded dressed signals of ^{85}Rb, *F* = 3 and ^{85}Rb, *F* = 2 (probe and conjugate) and analyzing the size of non-overlapping region.

Figure. 2 represents “∞”-shape OB phenomena by scanning detuning Δ* _{p}* at different power of external-dressing field

*E*_{3}. Besides, the incident angle ∆α between pump field

*E*_{1}and probe field

*E**is large and the diameter of*

_{p}

*E*_{1}is small. In Fig. 2(a), there is dressed probe signal, which can be expressed as ${\widehat{a}}_{D}={\widehat{a}}_{4}+{\widehat{a}}_{6}$ (dressed PA-FWM = PA-FWM + PA-SWM) in Eq. (7). Similarly, dressed conjugate signal in Fig. 2(b) also can be expressed as ${\widehat{b}}_{D}={\widehat{b}}_{4}+{\widehat{b}}_{6}$ (dressed PA-FWM = PA-FWM + PA-SWM) in Eq. (8). Then, we will analyze the size of folded signals about “∞”-shape non-overlapping region that we have illustrated in section 3.4 of theory.

In *x* direction, frequency difference (*δ*) is shown in Fig. 2(a1). The intensity of Stokes signal ^{85}Rb, *F* = 3 is *I _{P}* ∝ (

*I*

_{0}

*-*Im

*ρ*

^{(1)}

_{20}+ |

*ρ”*

^{(3)}

_{21(}

_{S}_{)}|

^{2}). Since the two peaks in Fig. 2(c1) have the same baseline, the feedback dressing (also a self-dressing) term

*|G*

_{P}|^{2}on the rising and falling edges are not equal in Eq. (12). Therefore, the intensity

*I*

_{p(}_{up)}is not equal to

*I*

_{p(}_{down)}while

*n*

_{2}

*≈*

_{up}*n*

_{2}

*,so Eq. (12) can be changed to ∆*

_{down}*φ = Nn*(

_{2}*I*-

_{up}*I*)

_{down}*ω*, it is obvious that

_{p}l/c = n_{1}δl/c*δ*is mainly induced by the difference between

*I*

_{up}and

*I*

_{down}. Further, it can be seen that

*δ*wax from bottom to top in Fig. 2(c). Where (

*I*-

_{up}*I*) is fixed in Eq. (12), hence

_{down}*δ*is proportional to

*n*

_{2}. Meanwhile,

*n*

_{2}= Re[

*χ*

^{(3)}/(

*ε*)]∝

_{0}cn_{0}*ρ*

^{(3)}is related to

*E*_{3}, it is clear that

*δ*vary due to the change in power of

*E*_{3}in Fig. 2(c). For the same reason attributes the anti-stokes signal

^{85}Rb,

*F*= 2 (

*I*∝ (

_{P}*I*

_{0}

*-*Im

*ρ*

^{(1)}

_{20}+ |

*ρ”*

^{(3)}

_{20(}

_{aS}_{)}|

^{2})) in Fig. 2(d), the

*δ*is induced by the feedback dressing term

*|G*

_{P}|^{2}, where

*δ*wax from bottom to top is also caused by different power of

*E*_{3}. In Fig. 2(b), the intensity of dressed conjugate signal is

*I*∝ (

_{C}*I*

_{0}+ |

*ρ”*

^{(3)}

_{(}

_{S}_{/}

_{aS}_{)}|

^{2}), where

*δ*is induced by different self-dressing

*|G*on the rising (falling) edge in Eqs. (9) and (10). Especially, the change of

_{C}|^{2}*δ*from bottom to top is very tiny in Figs. 2(e) and 2(f), where

*δ*of dressed probe signal is more sensitive than dressed conjugate signal for different power of

*E*_{3}. This is caused by the different feedback intensity that

*|G*

_{P}|^{2}is stronger than

*|G*

_{C}|^{2}, where the input probe beam with a intensity of

*I*is amplified to produce an output probe with intensity

_{p}*gI*and an output conjugate with intensity (

_{p}*g*-1)

*I*.

_{p}What’s more, it is obvious that the left peaks is higher than right peaks in the same baseline in Fig. 2(c), which is caused by the different enhancement conditions of dressed probe signals. Here we consider second-order splitting caused by dressing field of *E*_{3} and feedback dressing effect of *|G _{P}|*

^{2}that we have been described in section 3.4 of theory. Because of different feedback dressing term

*|G*on the rising (falling) edges, the enhancement conditions ∆

_{P}|^{2}*S-λ*

_{+(up)}

*-**λ*0 and ∆

_{+ + (up)}=*S*

*-**λ*

_{+(down)}

*-**λ*0 of left and right peaks, respectively, are not same while there are different corresponding eigenvalues

_{+ + (down)}=*λ*. As a result, one can find that the enhancement of left peaks are bigger than right peaks as shown in Fig. 2(c) and vice verse in Fig. 2(d), this contrary phenomenon relates to the energy-level of hyperfine states of

^{85}Rb atomic system. It depends on the different dipole moment between

^{85}Rb 5S

_{1/2}(

*F =*2) and

^{85}Rb 5S

_{1/2}(

*F =*3). Moreover, in Fig. 2(b), the enhancement condition is ∆

*aS/S-λ*0, and the enhancement of left peaks is bigger than right peaks in Fig. 2(e) while the enhancement of left peaks is similar with right peaks in Fig. 2(f).

_{+}-λ_{++}=In Fig. 3, we demonstrate “∞”-shape OB phenomena by changing the pump detuning ∆1. In Figs. 3(a) and 3(b), there are dressed probe signals (${\widehat{a}}_{D}={\widehat{a}}_{4}+{\widehat{a}}_{6}$) and dressed conjugate signals (${\widehat{b}}_{D}={\widehat{b}}_{4}+{\widehat{b}}_{6}$), respectively. Besides, the power of external-dressing E3 in Fig. 3 is bigger than the power of E3 in Fig. 2. In order to better analyze “∞”-shape non-overlapping region, we subtract a common part from Figs. 3(c)-3(f). Besides, it is particular that generating mechanism of multi-peaks can find in PA-SWM process in Fig. 3.

The multi-peaks of probe and conjugate signals in Figs. 1(d) and 1(e) are dominantly controlled by the amount of the incident angle *∆α* between probe field and pump field. The multi-peaks accompanied with near-degenerate PA-FWM process that is hard to be induced by large *∆α* in Fig. 2. But with small *∆α* in Figs. 1(d) and 1(e), the process of near-degenerate PA-FWM is triggered easily due to pumping effect. Under the signals of multi-peaks in Figs. 1(d2)-1(d5), there are correspondent energy-level diagrams for the multi-peaks in Figs. 1(f1)-1(f4). As a result, when we set *∆α* same as Figs. 1(d) and 1(e) in Fig. 3, there appears “∞”-shape OB of multi-peaks and non-overlapping region induced by different feedback dressing *|G _{F}|*

^{2}on the rising (falling) edge and difference on conditions for suppression and enhancement. What’s more, some multi-peaks are so small that we cannot find them easily, which is caused by strong suppression effect.

Secondly, frequency difference in *x* direction of “∞”-shape non-overlapping region can also be observed in Fig. 3. Same with Fig. 2, *δ* is induced by different feedback dressing term *|G _{F}|*

^{2}on the rising (falling) edge in Fig. 3. From bottom to top in Figs. 3(c1)-3(c5), the

*δ*is increased gradually, where (

*I*

_{up}-

*I*

_{down}) is fixed in Eq. (12), so the change of

*δ*is caused by the increases of

*n*

_{2}. In Fig. 3, with decreased pump detuning Δ

_{1},

*n*

_{2}is increasing from bottom to top gradually, which will enhance the feedback intensity. For this reason, the

*δ*is increased from bottom to top gradually in Fig. 3(e). But the change of

*δ*is very small in Figs. 3(d) and 3(f), it demonstrates that the

*δ*of

^{85}Rb,

*F*= 2 is more sensitive than the

*δ*of

^{85}Rb,

*F*= 3 for the changed ∆

_{1}. Besides, the small

*∆α*in Fig. 3 also enhance the feedback intensity of probe and conjugate signals, so comparing with Fig. 2, the change of

*δ*is saturated in Fig. 3(d) and is larger in Fig. 3(e).

Finally, intensity difference in *y* direction can also be found in Fig. 3. For one thing, the right peaks are higher than left peaks in Figs. 3(c)-3(f), which is caused by the different enhancement conditions of right peaks and left peaks. For another, from bottom to top in Figs. 3(c)-3(f), the change of intensity at one side go thought a gradually process from low to high and then to low again, where the enhancement of peaks is from weak to strong and then to weak again.

In Fig. 4, we analyze the “∞”-shape OB phenomena of dressed probe (${\widehat{a}}_{D}={\widehat{a}}_{4}+{\widehat{a}}_{6}$) and conjugate (${\widehat{b}}_{D}={\widehat{b}}_{4}+{\widehat{b}}_{6}$) signals at different diameter of pump beam *E*_{1}. Similar to Fig. 3, the power of *E*_{3} in Fig. 4 is bigger than Fig. 2, so we subtract a common part in Figs. 4(c)-4(f) to analyze “∞”-shape OB phenomena preferably. And the incident angle *∆α* is small in Fig. 4, so there also exists multi-peaks. Same with Figs. 2 and 3, Figure. 4 investigate “∞”-shape OB phenomena also by comparing the non-overlapping region of signals. Specially, what does attract here is the difference of full width at half maxium (FWHM) between the rising edge signals and falling edge signals in Fig. 4.

Specifically, by adjusting the pump beam diameter in our experiment, the relative position of pump and probe beams will be changed, so an additional nonlinear phase shift factor *e ^{iΔΦ}* is introduced via cross-Kerr effect in Eqs. (9) and (10). Meanwhile, when taking thestrong internal dressing effect of

*E*_{1}, the excited state energy level 5

*P*

_{1/2}(|2>) would be split into

*G*

_{2 ±}and

*G*'

_{2 ±}. The corresponding eigenvalues are

*λ*

_{S}_{±}= [∆'

_{1}± (∆'

_{1}

^{2}+ 4|

*G*

_{1}|

^{2}

*cos*(

*Φ*))

^{1/2}]/2, (|

*G*

_{2}± ›), and

*λ*

_{aS}_{±}= [∆

_{1}± (∆

_{1}

^{2}+ 4|

*G*

_{1}|

^{2}

*cos*(

*Φ*))

^{1/2}]/2, (|

*G*'

_{2}± ›). The secondary AT splitting is caused by the self-dressing effect of

*E**whose corresponding eigenvalues are*

_{F}*λ*

_{S}_{+ ±}= [∆'

*± (∆'*

_{F}

_{F}^{2}+ 4|

*G*|

_{F}^{2})

^{1/2}]/2, [∆'

*∆*

_{F}=*-*

_{F}*λ*

_{S}_{+}], (|

*G*

_{2}+ ± ›) and

*λ*

_{aS}_{+ ±}= [∆ʺ

*± (∆ʺ*

_{F}

_{F}^{2}+ 4|

*G*|

_{F}^{2})

^{1/2}]/2, [∆ʺ

*∆*

_{F}=*-*

_{F}*λ*

_{aS}_{+}], (|

*G*'

_{2}+ ± ›). So the suppression and enhancement conditions of

*E**and*

_{S}

*E**are ∆*

_{aS}*∆'*

_{S}-*= 0, ∆*

_{F}*∆ʺ*

_{aS}-*= 0 and ∆*

_{F}*-*

_{S}*λ*

_{+}-

*λ*

_{++}= 0, ∆

*-*

_{aS}*λ*

_{+}-

*λ*

_{++}= 0, respectively. Theoretically, the change of

*Ф*leads to the move of

*E*_{1}-split energy levels, which means the switch of enhancement and suppression. As a result, in same baseline at Fig. 4(c5), the FWHM of right peak is bigger than left peak’s attributed to different feedback dressing term

*|G*

_{P}|^{2}at rising and falling edges. The FWHM of right peaks is bigger than left peaks’ in Fig. 4, which means the feedback intensity at rising edge is stronger than falling edge. From top to bottom, the FWHM of peaks on one side are obvious increasing in Figs. 4(c)-4(f), where the enhancement of peaks is stronger with increased diameter of

*E*_{1}.

The frequency difference in *x* direction of “∞”-shape non-overlapping region is also obvious in Fig. 4. From bottom to top in Figs. 4(c)-4(f), the change of *δ* is minute, where the change of FWHM is offsetting the influence of different diameter of *E*_{1}. For the same reason, intensity difference in *y* direction also is little-changed.

## 5. Conclusion

In summary, generalized nonreciprocity “∞”-shape OB of the dressed probe (conjugate) signals caused by self-dressing effect in PA-SWM process (${\widehat{a}}_{6}$and ${\widehat{b}}_{6}$) was experimentally and theoretically observed. We found that there exists “∞”-shape non-overlapping region includes frequency difference and intensity difference, whereas the nonreciprocity of frequency difference is much more obviously. Besides, we also found that the feedback intensity can be controlled by experimental parameters of dressing fields. As a result, increasing the power of external-dressing can make frequency difference more obvious, and changing the phase includes incident angle α and the diameter of internal-dressing fields can induce multi-peaks and FWHM difference, respectively. In brief, our experiment reveals that nonreciprocity “∞”-shape OB can be controlled by feedback intensity attributed to feedback dressing. This research can provide and novel methodology for the applications of logic-gate devices and quantum information processing.

## Funding

National Key R&D Program of China (2017YFA0303700); National Natural Science Foundation of China (NSFC) (11474228, 61605154, 61308015); Key Scientific and Technological Innovation Team of Shaanxi Province (2014KCT-10).

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