Introduction

Cavity quantum electrodynamics has been intensively studied in recent years1,2,3,4, owing to the important applications in quantum optics and nonlinear optics. When a two-level atom is placed inside the optical cavity under the strong coupling condition, the “normal-mode splitting” have been experimentally demonstrated in many atomic systems. If the intracavity medium is a coherently-prepared three-level atomic medium, the so called “dark-state polariton” peak5,6,7 appears in the cavity transmission spectrum (CTS) due to the intracavity electromagnetically induced transparency (EIT). Such “dark-state polariton” have potential application in the long-lived storage of quantum information and quantum computation. Moreover, when the atomic system is in the double-Λ configuration, correlated photon pairs with high generation rate and narrow bandwidth have been generated8 in the cold atomic ensemble by using an optical parametric amplification (OPA) process operated below its oscillation threshold. Meanwhile, bright correlated anti-Stokes and Stokes twin beams with hot atoms inside an optical cavity have also been obtained by an OPA process above the threshold9. When an ensemble of cold atoms strongly coupled to a high-finesse optical cavity and the control light can be generated from the vacuum, then the vacuum-induced transparency was observed10. In addition, four-wave mixing (FWM) and six-wave mixing (SWM) processes based on third-order and fifth-order nonlinear processes in EIT media11,12,13 have also attracted lots of attention in recent years, in which a strong coupling beam renders a resonant, opaque medium nearly transparency while enhancing the nonlinearity and the coexistence of these two nonlinear processes due to double EIT windows and atomic coherences has been reported14,15. The nonlinear process plays important role in entanglement generation16,17 and cascade-nonlinear optical process18. To the best of our knowledge, the OPA seeded with FWM or SWM signal in an atom-cavity composite system has not been reported and the investigation of it would be important for building multi-channel nonlinear optical devices and ultra-narrow linewidth photon sources for long-distance quantum communication19.

In this letter, we report our investigations of bright-state polaritons of FWM and SWM signals through cascade OPA process in an atom-cavity composite system. The bright-state polaritons of FWM signal with 5 MHz linewidth are obtained. We also report the coexisting cavity modes of parametrically amplified FWM and SWM signals in an inverted-Y-type atomic system for the first time. Moreover, the electromagnetically induced absorption (EIA) peaks induced by the multiple cavity modes of the FWM signal are observed in the probe beam. These results are well explained by the presented theoretical model. The investigation will help us to better understand the interactions between the strongly coupled multi-level atoms and the optical cavity, which can find application in quantum information processing.

Results

Fig. 1(a) shows the experimental setup, while the energy levels of the atomic system used in the experiment is shown in Fig. 1(b1), where the fields E1 (frequency ω1, wave vector k1, Rabi frequency G1 and wavelength 780.2 nm), E2 (ω2, k2, G2 and 776.16 nm) & (ω2, , ) and E3 (ω3, k3, G3 and 780.2 nm) & (ω3, , ) are used as probe field, pumping fields and coupling fields, respectively. The resonant transition frequencies of |0〉→|1〉, |1〉→|2〉 and |1〉→|3〉 are Ω1, Ω2 and Ω3, respectively. Then the frequency detuning for each field can be defined as Δi = Ωi − ωi (i = 1, 2, 3). With all beams present in Fig. 1(b1), three phase-conjugate signals (FWM signals EF1 & EF2 and SWM signal ES), satisfying the phase-matching conditions of , and , can be generated simultaneously at the center of the atomic cell and propagate along the optical axis of the cavity (dashed line in Fig. 1(a)). Here, the narrow signals EF1 and ES are generated within the EIA window of δ = 0, where δ = Δ1 + Δ2 is two-photon detuning, while the broad signal EF2 will be generated due to no assistance of Doppler-free atomic coherence and EIA window. However, if the E1 beam has a sufficiently high power, as well as is far detuned from |0〉→|1〉, a spontaneous parametric FWM (SP-FWM) process will occur in the degenerate two-level atomic configuration (Fig. 1(b2)), which generates two weak fields (Stokes field ESt and anti-Stokes field EASt), satisfing 2k1 = kSt + kASt (Fig. 1(b3)), on a forward cone. The generated EF1 (or ES) signal is naturally injected into the input Stokes port of the SP-FWM process and is parametrically amplified, where the process will serve as an OPA. The parametrically amplified signal denoted as (or ) is still generated at the center of the atomic cell and propagate along the optical axis of the cavity, so they are mode-matched to the cavity and form the cavity modes.

Figure 1
figure 1

(a) Experimental setup. PBS: polarizing beam splitters; F: optical lens; APD: avalanche photodiode detectors; PZT: piezoelectric transducer; M1–M3: cavity mirrors; M4, high reflectivity mirror; λ/2: half-wave plate. (b1) Energy levels for the 85Rb atomic system. (b2) Energy schematic of SP-FWM process. (b3) phase-matching geometrical diagram of SP-FWM process. Calculated CTS versus Δ1 and Δac by using the expression of aF1 (c) in a FSR and (d) in multiple FSRs. (e) Calculated CTS versus Δ2 and Δac by using the expression of aF1 in multiple FSRs.

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According to the expression of aF1 given in the Method part, Fig. 1(c) shows the calculated normalized CTS versus Δac (defined as Δac = Ω1 − ωc with ωc being the resonant frequency of the cavity) and Δ1, which exhibits a double-peak structure along the Δ1 direction. This structure indicates the cavity polaritons7 of the parametrically amplified field , which results from the coupling between the cavity mode of and atoms with strength , where g is the single-atom-cavity coupling strength and N is the number of atoms in the cavity. When Δac changes from negative to positive continuously, the polaritons move, that is dictated by the condition of Δ1 − Δac = 0 and finally leads to the avoided-crossing plot in Fig. 1(c). Fig. 1(d) is plotted with the same variables as Fig. 1(c) but in a wider range, where multiple polaritons at a fixed Δ1 can be seen and their positions change as Δ1 changes. It is found that ΔF1 − Δac = lωFSR (l is an integer and ωFSR is the free-spectral range (FSR) with medium) is always fulfilled at all the polaritons, i.e. Δ1 − Δac = lωFSR (named cavity transmission window) is satisfied for all the polaritons. So, when Δ1 changes, the corresponding Δac for each transmission polaritons of changes accordingly. We also show normalized CTS versus Δ2 and Δac in Fig. 1(e), in which the polaritons nearly do not move with Δ2. The reason is that the energy shifts induced by dressing field E2 at different Δ2 values make the window Δ1 − Δac = lωFSR not move.

It is worth mentioning that the cavity polaritons of the parametrically amplified field and will have ultra-narrow linewidths, which would be useful for long-distance quantum communications. On the one hand, the linewidths of and are narrowed by the EIA window, on the other hand, and form cavity modes and then the cavity polaritons will be further narrowed due to the large dispersion change and reduced absorption accompanying EIA20,21.

Parametrically amplified bright-state polariton of FWM signal

We first measure the CTS without E3 and . In this case, the SP-FWM process generates a Stokes field ESt (measured and shown in Fig. 2(a1)) and an anti-Stokes field EASt (measured and shown in Fig. 2(a2)) on a forward cone. Such a process can also act as an OPA for the EF1 (in Fig. 2(a3)) injected into the Stokes port (in Fig. 2(a4)). The parametrically amplified signal denoted as forms cavity mode, which couples with the atoms. By scanning Δ1 across the transition F = 3↔F′ in 85Rb at different Δac values and taking Δ2 ≈ 1.2 GHz, the measured CTS of versus δ are shown in Figs. 2(b1)–(b3), where the lower curves (ii) and top curves (i) are the CTS of and the corresponding EIA window, respectively, with the atomic cell temperature T ≈ 77°C and powers of E1, E2 and P1 = 8 mW, P2 = 22 mW, P′2 = 22 mW, respectively. The CTS of without the atom-cavity mode coupling is shown in Fig. 2(b2) by the curve (iii), where one dip and two peaks can be seen clearly. The two peaks represent the Autler-Townes splitting of the signal , which derive from the two dressed states, namely, bright states. The dip comes from the dark state induced by the destructive interference between two dressed states. Tuning Δac with the atom-cavity mode coupling indicates that, when the Δ1 − Δac = 0 window is overlapped with the EIA window, the two peaks reach their maxima simultaneously on the curve (ii) of Fig. 2(b2). Comparing the curve (iii) with the curve (ii), the dip (corresponding to dark state) and the two peaks is amplified, which stems from the dressing splitting of the atom-cavity mode coupling to the energy level |1〉. So the two peaks on the curve (ii) represent the cavity polaritons of as shown in Fig. 1(c). When the Δ1 − Δac = 0 window deviates from EIA window by decreasing Δac, the two peaks become asymmetric and the left peak is amplified by the atom-cavity mode coupling as shown in Fig. 2(b1). The left peak corresponds to one of the bright states, so one intracavity bright state (“bright-state polariton” with a linewidth about 12 MHz) is obtained. The other narrow “bright-state polariton” (~5 MHz) is obtained when increasing Δac in Fig. 2(b3). The calculated CTS curves of (∝|aF1|2), which agree well with the measured results, are shown in Figs. 2(b4)–2(b6).

Figure 2
figure 2

(a1) Measured Stokes field ESt and (a2) anti-Stokes field EASt versus Δ1 in the SP-FWM process. (a3) Ladder-type atomic levels to generate EF1. (a4) Phase-matching diagram of OPA seeded with the EF1 in the Stokes port. (b1)–(b3) Measured probe absorption (i) and CTS (ii) versus δ at different Δac. (b4)–(b6) Theoretical simulation of the curves (ii) in Figs. (b1)–(b3), respectively. (c1) and (c2) Measured polaritons of versus Δac with increasing Δ1 and Δ2 from bottom to top, respectively.

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By scanning the voltage imposed on PZT, Fig. 2(c1) displays the parametrically amplified bright-state polaritons of versus Δac at different δ values, which is set as −12 MHz, 0 and 12 MHz from bottom to top by taking different Δ1 with fixed Δ2. Each CTS curve exhibits two bright polaritons, satisfying Δ1 − Δac = lωFSR, having a separation of ωFSR and changing their positions with Δ1 as predicted by the theoretical result in Fig. 1(d). Also, the heights of the polaritons change with Δ1, since the polaritons are enhanced at two-photon resonant. Next, by fixing Δ1 and taking different Δ2, the bright-state polaritons of versus Δac with δ set as −15 MHz, 0 and 15 MHz from bottom to top are shown in Fig. 2(c2), where the heights of the polaritons change with Δ2 due to two-photon detuning, but the positions are fixed. That is similar to the case shown in Fig. 1(e).

The polaritons of can be affected by the temperature of atomic cell, since increasing T can increase the atomic number N and then enlarge the collective coupling factor 7, which will yield dressing to the polaritons. According to the theoretical model presented in the method part, the splitting eigenvalues induced by the dressing of atom-cavity mode coupling to the polaritons can be given by , which correspond to the positions of the two polariton peaks in the CTS curves. So the splitting of the two polariton peaks will increase with T. Also, increasing temperature will lead to increased atomic number participating in the phase-conjugate FWM process as well as OPA process, which will enhance the height of the two polariton peaks. The measured CTS curves versus δ with increasing T from bottom to top are shown in Fig. 3(a1), where not only the height but also the splitting of the two polariton peaks increase with T. Fig. 3(A1) shows the calculated results corresponding to Fig. 3(a1), according to the expression of aF1. The experimentally measured (squares) and theoretically calculated (solid line) splittings of polaritons versus T are shown in Fig. 3(a2), where the splitting indeed increases with T. The experimental results agree with the theory.

Figure 3
figure 3

(a1) Measured polaritons of the versus δ with Δ2 ≈ 1.2 GHz and increasing T from bottom to top (65°C, 75°C, 85°C, 95°C). (A1) Theoretical simulation of the curves in (a1). (a2) Measured (squares) and Calculated (solid line) dressing splittings versus T. (b1) Measured polaritons with increasing P2 from bottom to top (3.3 mW, 8.6 mW, 17 mW). (b2) P2-dependence of the splitting with (top curves) and without (bottom curves) atom-cavity mode coupling. (c1) and (c2) are plotted in the same way as (b1) and (b2) but with increasing P1 (2.5 mW, 6 mW, 12 mW).

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The polaritons of can also be influenced by the powers P1 and P2. The measured CTS curves versus δ with increasing P2 from bottom to top are presented in Fig. 3(b1). The height and the splitting of the two polariton peaks become more pronounced as P2 increase. Figure 3(b2) shows the splitting versus P2 with the atom-cavity mode coupling (upper plots) and without the atom-cavity mode coupling (lower plots), where squares and dots are experimental results, while the solid curves are the corresponding fittings. The splitting on upper plots in Fig. 3(b2) is mainly induced by the dressing of the atom-cavity mode coupling and field E2 to polaritons, while the splitting on lower plots mainly results from the dressing of the field E2. By comparing the two cases, the dressing splitting of the atom-cavity mode coupling can be obtained. While the dressing splitting induced by field E2 to the polariton can be seen by increasing P2. Similar investigation has been done for the influence of P1 in Figs. 3(c1) and 3(c2). However, the upper squares (or solid line) in Fig. 3(c2) is much easier to saturate than that in Fig. 3(b2). The dressing splitting of the polariton in Fig. 3(c2) is mainly determined by the atom-cavity coupling and field E2 when the power of P1 is weak, which is similar to the case in Fig. 3(b2). As P1 is increased, the dressing of field E1 gets larger and has to be considered. So the dressing splitting of field E1 to polaritons is included in Fig. 3(c2) compared with Fig. 3(b2). Therefore, the saturation behavior with increasing P1 in Fig. 3(c2) mainly results from the balanced interactions between the destructive and constructive interferences of different dressing pathways22 induced by the atom-cavity mode coupling, field E1 and field E2.

Coexisting cavity modes of parametrically amplified FWM and SWM signals

If all laser beams are opened except (Fig. 1(b1)), we can get an inverted Y-type system, in which there will be ES and EF2 but no EF1. As indicated by Fig. 4(f), ES signal is naturally used as a seed injected into the Stokes port of SP-FWM process and amplified as . Then, the signal, mode-matched to the cavity, forms a cavity mode. In free space, the measured coexisting spectrum of and EF2 versus δ with Δ2 = 1.2 GHz is displayed as the bottom curve in the inset of Fig. 4(a), in which a narrow peak labeled as “S” (corresponding to EIA dip labeled as “E” on the top curve) and a broad signal can be seen. The frequency of peak “S” changes with Δ2, so peak “S” is the signal , while the broad signal is EF2 without the EIA window. With both and EF2 resonant in the cavity, the measured CTS versus δ is given by the bottom solid curve in Fig. 4(a), where we can see five peaks labeled as “P1”, “P2”, “ST” “P3” and “P4”, respectively and the peak “ST” corresponds to the EIA dip “E” on the top dash-dotted curve. Without E2, both the “ST” peak and “E” dip disappear in Fig. 4(b). So we can identify the “ST” peak is the transmission peak of and the other four peaks are the cavity modes of EF2 picked out by the cavity windows from the broad gain. Tuning the cavity length to make a cavity transmission peak of EF2 overlap with the “ST” peak (i.e. the cavity window overlaps with the EIA window), the resulted CTS (the solid curve) and the corresponding EIA (the dash-dotted curve) versus δ are shown in Fig. 4(c), where the coexisting cavity modes of the SWM () and FWM (EF2) signals are obtained. By blocking E2 at this time, the cavity mode of the SWM signal disappears and only the cavity modes of EF2 survive (dashed line of Fig. 4(c)). Comparing the results with and without E2 in Fig. 4(c), the enhancement (bright-state polariton of ) due to the coupling of cavity mode of with atoms can be seen. Also, the pulling of the cavity resonant frequency due to the dispersion change induced by E220,21 can be observed. By fixing Δ1 and Δ2 at the point of coexisting cavity modes and scanning Δac in Fig. 4(d), the measured coexisting cavity modes of and EF2 versus Δac are shown on the solid curve, while the dashed curve shows the cavity modes of EF2 without E2. Also, the pulling of the cavity resonant frequency is observed in Fig. 4(d). According to the coupled atom-cavity model, the CTS amplitude of is given by , where ΔS = Ω1 − ωS, d3 = Γ30 + i(Δ1 − Δ3) and is the Rabi frequency of . The transmission coefficient of intensity of EF2 is , where ri (ti) is the reflection (transmission) coefficient of the mirror Mi of the cavity with and ϕ(ωF2) = 2π(Δac − Δ1)/ωFSRE + (n − 1)LaωF2/c is the round-trip phase shift experienced by EF2, with light speed in vacuum c, length of the atomic cell La and FSR of the empty optical cavity ωFSRE = c/Lc with a cavity length Lc. The terms α = 2(ωF2/c)Im[(1 + χ)1/2] and n = Re[(1 + χ)1/2] are the intensity absorption coefficient and refractive index of the atomic medium, respectively, with χ = i2g2NLc/[LaωF2(d1 + |G2|2/d2 + |G3|2/d3)] when only the linear susceptibility is considered. Using the expressions of aS and TF2, the calculated normalized CTS of coexisting and EF2 versus δ are shown on the bottom curve in Fig. 4(e), which agrees with the experimental result.

Figure 4
figure 4

(a)–(c) Measured CTS of EF2 and signals versus δ at different Δac values. Coexisting and EF2 signals versus δ in free space shown in the inset of (a) and measured FWM-induced EIA dips shown in the insets of (b). (d) Measured CTS of EF2 and signals versus Δac with (solid curve) and without (dashed curve) E2. (e) Calculated FWM-induced EIA (top curve) and CTS (bottom curve) with coexisting EF2 and versus δ. (f) Phase-matching of OPA process seeded with ES.

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Besides, the EIA peaks induced by the multiple cavity modes of EF2 are observed in Fig. 4(b), where each inset is a measured EIA peak multiplied by 10. Because each cavity mode of EF2 couples to |0〉→|1〉 and propagates along the same direction as E1, the two-photon resonance (one photon from E1 and another from EF2) induces the Doppler-free atomic coherence23. Owing to the far detuning of the cavity modes of EF2, the EIA is induced instead of EIT. The probe absorption can be obtained by with dF1 = i(Δ1 − ΔF1) + Γ10, dac = i(Δ1 − Δac) + Γ10 and dS = i(Δ1 − ΔS) + Γ10, which indicates E2 field and the cavity modes of with far detuning can all induce EIA. However, the induced EIA by the cavity modes of are not easy to be observed separately, because they overlap with the EIA created by the E2 field, which is much stronger than the cavity modes. The induced EIA (insets of Fig. 4(b)) by the cavity modes of EF2 can be observed by tuning Δac or Δ2 to separate it from the EIA of E2. The calculated probe absorption by is shown on the top curve in Fig. 4(e), where the highest EIA peak is the sum from E2 and cavity modes of and the other peaks are induced by the cavity modes of EF2.

Discussion

We have experimentally studied the bright-state polaritons of multi-wave mixing signals through OPA processes in a multi-level atom-cavity composite system. It is demonstrated that the polaritons are much narrowed due to EIA and cavity window. It would be important for the narrow-band long-distance quantum communications. Besides, coexisting cavity modes of the generated SWM and FWM signals are also observed, which can help us to better understand the interactions between the strongly coupled multi-level atoms and the optical cavity. Moreover, the induced EIA by the multiple FWM cavity modes in the probe spectrum is observed. Such investigation in an atom-cavity coupling system may find potential applications in building multi-channel nonlinear optical devices for quantum information processing.

Methods

Experiment setup

The atom-cavity composite system is shown in Fig. 1(a), where a 7 cm long Rb atomic cell is placed inside a 38 cm long optical ring cavity. The concave mirrors M1 and M2 (with the same radius of curvature of 100 mm) have 99.9% and 97.5% reflectivities at 780 nm, respectively. The flat mirror M3 with a reflectivity of 97.5% at 780 nm is used as the output coupler. M1 is mounted on a piezoelectric transducer (PZT) to adjust and lock the cavity length. The atoms can be viewed as an inverted four-level Y-type system as shown in Fig. 1(b1), where the relevant energy levels are 5S1/2(F = 3)(|0〉), 5P3/2(|1〉), 5D3/2(|2〉) and 5S1/2(F = 2)(|3〉) in 85Rb. Three grating-stabilized diode lasers are used as the probe field E1 (frequency ω1, wave vector k1, Rabi frequency G1 and wavelength 780.2 nm), pumping fields E2 (ω2, k2, G2 and 776.16 nm) & (ω2, , ) and coupling fields E3 (ω3, k3, G3 and 780.2 nm) & (ω3, , ), where E1 drives the transition |0〉→|1〉, E2 & drive |1〉→|2〉 and E3 & drive |1〉→|3〉. The E2 (with horizontal polarization) and (with vertical polarization) beams come from the same laser. beam propagates along the optical axis of the cavity and E2 beam co-propagates with beam having a 2° angle between them. The E1 (vertically polarized) beam counter-propagates with E2 beam; that is the two-photon Doppler-free scheme for ladder-type system23. The CTS and absorption of E1 is detected by avalanche photodiode detector 1 (APD1) and APD2, respectively. The E3 and beams have the same propagation directions and polarizations with E2 and beams, respectively.

Theoretical model

According to the perturbation chains15, the density-matrix elements for EF1 and ES are and , where d1 = Γ10 + iΔ1, d2 = Γ20 + i(Δ1 + Δ2) and d3 = Γ30 + i(Δ1 − Δ3) with the decay rate Γij between states |i〉 and |j〉. The generated EF1 (or ES) signal is injected into the Stokes port of the SP-FWM process, then parametrically amplified and denoted as (or ). The photon numbers of the output Stokes ( or ) and anti-Stokes fields of the OPA are and , where is the annihilation operator of ESt (EASt) and is the gain of the process with the modules A and B (phases φ1 and φ2) defined in and for ESt and EASt, respectively18.

Then, we describe the theoretical model of such coupled atom-cavity system. Here, we just discuss the model with the cavity mode of coupling with atoms, since the case for can be discussed in the same way. In the limit of weak cavity field and with all the atoms initially in the ground state |0〉, the system can be described by the equations

where a is the amplitude of the cavity field , ρij the density-matrix element, γ the cavity loss, g the single-atom-cavity coupling strength, N the number of atoms in the cavity, the Rabi frequency of , ΔF1 = Ω1 − ωF1 and cavity frequency detuning Δac = Ω1 − ωc (with ωc being the resonant frequency of the cavity). So, the amplitude of CTS is given by , where Tc is the transmission function without dressing effect. The frequency ωF1 of the FWM signal is identical with ω1 to obey the energy conservation, giving ΔF1 = Δ1. According to the same coupled atom-cavity model, the CTS amplitude of is given by .