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Abstract
We experimentally investigate near-infrared optical field generation through simultaneous three-wave mixing (TWM) and six-wave mixing (SWM) processes in room-temperature 85Rb atoms. The nonlinear processes are induced using three hyperfine levels in the D1 manifold, which cyclically interact with pump optical fields and an idler microwave field. The simultaneous appearance of TWM and SWM signals in discrete frequency channels is made possible by breaking the three-photon resonance condition. This gives rise to coherent population oscillations (CPO), which are observed experimentally. We explain through our theoretical model the role of CPO in the generation of the SWM signal and its enhancement due to parametric coupling with the input seed field in contrast to the TWM signal. Our experiment proves that a single tone microwave can be converted to multiple optical frequency channels. The simultaneous existence of TWM and SWM processes can potentially enable various types of amplification to be achieved with a single neutral atom transducer platform.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Nonlinear processes in a two-level atomic system, excited by a weak probe and a strong pump, have been exhaustively studied for inducing parametric amplification and oscillation in the probe waves [1–6]. These processes capitalize on the existence of four-wave parametric interactions, three-photon gain, and coherent population oscillations (CPO) in the presence of a transparency window in the probe’s absorption profile. A large class of these experiments were performed at high-light intensities and were limited to the observation of third-order nonlinear phenomena.
The emergence of electromagnetically induced transparency (EIT) has enabled the experimental realization of higher-order nonlinear processes with low-light intensities in atomic systems. By reducing linear absorption and simultaneously enhancing nonlinear susceptibilities at atomic resonances, EIT has been responsible for numerous studies of optical nonlinear processes in multi-level atomic systems [7–11]. The studies primarily constitute observation of four-wave mixing (FWM) and six-wave mixing (SWM) processes. Simultaneous observation of the FWM and SWM processes have also been reported, via dual EIT windows, by spatially aligning the input beams in a square-box pattern in a Y-type atomic system [12].
Nonlinear processes in atomic systems have also been utilized to interface microwave and optical frequencies, a requirement for realizing quantum hybrid systems to transmit quantum signals. Assisted by EIT, coherent microwave-to-optical conversion has been realized via an SWM process in cold Rydberg atoms [13] and via a three-wave mixing (TWM) process in room-temperature three-level atoms [14]. Recently, a coherent microwave-to-optical conversion of efficiency 82$\pm$2% has been achieved in Rydberg atoms [15] via off-resonant scattering. Opening up nonlinear channels in EIT systems has also aided in achieving phase-insensitive amplification (PIA) [16] and phase-sensitive amplification (PSA) [17,18] of optical signals. Particularly, the realization of a microwave-controlled phase-dependent amplifier [19], driven by a TWM process, has been utilized to perform digital communication from the microwave to optical spectrum [20].
In this letter, we experimentally investigate and theoretically analyze co-existing TWM and SWM processes in room-temperature $^{85}$Rb atoms. The scheme utilizes three hyperfine levels of the D1 manifold, cyclically connected by two co-propagating optical fields, the probe and coupling field, and one microwave field as shown in Fig. 1. The pump coupling field dresses the excited state ($|{3}\rangle$) and the meta-stable ($|{2}\rangle$) state causing EIT for the weak probe. The microwave field connects the ground ($|{1}\rangle$) and metastable levels. By suitable detuning of the microwave or probe field from its resonance, the three-photon resonance condition is broken and CPO is induced in the system. This causes the TWM and SWM processes to appear in distinct frequency channels simultaneously. The SWM process grows exponentially with propagation distance, due to its parametric coupling to the input probe while the TWM process grows linearly due to the absence of parametric coupling to the input probe field. Nevertheless, the simultaneous presence of EIT and CPO greatly enhances the TWM and SWM processes by reducing the linear absorption and enhancing the non-linear susceptibilities.
Fig. 1. Hyperfine energy levels of $^{85}$Rb in the D1 manifold used in the experiment. The detunings of probe field from $\vert 1 \rangle \rightarrow \vert 3 \rangle$ transition is denoted by $\delta _{\text {p}}$. The detunings $\delta _{\text {c}}$ and $\delta _{\mu }$ represent coupling and microwave field detunings from $\vert 2 \rangle \rightarrow \vert 3 \rangle$ and $\vert 1 \rangle \rightarrow \vert 2 \rangle$ respectively. The Rabi frequencies of the probe, coupling, and microwave fields are denoted as $\Omega _{\text {p}}$, $\Omega _{\text {c}}$ and $\Omega _{\mu }$ respectively. $\gamma _{ij}$ denote the natural decay widths connecting $\vert i \rangle \rightarrow \vert j \rangle$ transition. The dotted green arrow denotes the generated optical signal field.
To the best of our knowledge, this is the first observation of the simultaneous generation of a near-infrared optical field through a TWM and SWM process. The TWM process combines one electric dipole and one magnetic dipole transition while the SWM process combines three electric dipole and two magnetic dipole transitions. The novel magneto-optical interaction between the electric and magnetic dipoles, enhanced through a microwave cavity, breaks the centro-symmetry in our thermally broadened atomic system and opens up the TWM and SWM channels. We have theoretically analyzed the TWM and SWM phenomena and identified the requirement of a broken three-photon resonance and the resulting CPO as being responsible for the SWM process. The CPO oscillations are observed in real-time to validate our theory.
Moreover, the nonlinear channels also offer the possibility to realize an amplifier operating simultaneously in different regimes of linear amplification in distinct frequency channels. The phase-dependent amplification (PDA) resulting from the TWM process has been thoroughly analyzed in our previous manuscript [19]. Opening up the SWM channel offers the possibility to achieve PIA and PSA too simultaneously. Thus, the results reported in the manuscript substantially strengthen the versatility of the neutral atom-based transducer in not only converting signals from the microwave to optical domain but also in aiding various types of amplification to be enabled on a single physical platform.
2. Experiment
The experiment utilizes $\Delta$ configured three hyperfine energy levels in the D1 manifold of the $^{85}$Rb atoms. The coupling laser connecting the metastable state ($|{2}\rangle$) and the excited state ($|{3}\rangle$), an electric-dipole transition, is derived from an External Cavity Diode Laser (ECDL), tuned at 795 nm as shown in Fig. 2. The ground state ($|{1}\rangle$) is connected to the excited state, an electric-dipole transition, by a seed probe beam derived from the coupling laser after passing through a phase-modulated Electro-Optic Modulator (EOM). The EOM is driven by the microwave source (MS1). The metastable state and the ground state, a magnetic-dipole transition, are coupled by a 3.035 GHz microwave field, derived from another microwave source (MS2). The presence of two optical fields and one microwave field makes the system closed and cyclic ($\Delta$). The Rubidium vapor cell is kept inside a microwave cavity to enhance the magnetic dipole interaction between the atoms and the microwave field. The frequency tunable microwave cavity is designed to support the TE$_{011}$ mode with a quality factor of 9000 and a linewidth of 300 kHz. The microwave is fed to the cavity, after passing through an amplifier providing 30 dB gain, with a small loop antenna placed inside the cavity. By using two different microwave sources to drive the EOM and cavity, we control the three-photon detuning by suitably detuning the microwave or the probe field off-resonant.
Fig. 2. Experimental Setup for simultaneous observation of TWM and SWM signals. Here ECDL —External Cavity Diode Laser, MC —Microwave Cavity, MS —Microwave Source, SA —Spectrum Analyzer, PD —Photo-Detector, EOM —Electro-Optic Modulator, AOM —Acusto Optic Modulator, QWP —Quarter Wave Plate, M —Mirror, PBS —Polarising Beam Splitter, BS —Beam Splitter, LO —Local Oscillator.
The generated TWM and SWM beams, co-propagating along with seed probe and coupling beams, simultaneously exit the microwave cavity and fall on a high bandwidth photo-detector. In order to resolve the generated signals from other co-propagating fields, an optical heterodyning technique is used. The signals exiting the cavity mixes with a local oscillator (LO) beam, which is 120 MHz frequency detuned from the coupling beam. The photo-detector output is now recorded in a spectrum analyzer, where the TWM and SWM signals can be frequency resolved. The frequency-resolved TWM and SWM signals appear as shown in Fig. 3 in the spectrum analyzer in distinct frequency channels.
Fig. 3. Plot showing the simultaneous presence of both SWM(left) and TWM(right) peaks in different frequency bins.
3. Theoretical model
The experimental observation of the simultaneous TWM and SWM process in distinct frequency channels, as shown in Fig. 3, is achieved by breaking the three-photon resonance. The three-photon detuning $\delta _{3}$ is defined as,
The frequencies at which the TWM ($\omega _{\text {3}}$) and SWM ($\omega _{\text {6}}$) signals are generated, can be written as,
respectively. The TWM process absorbs one microwave photon and one coupling photon to generate a signal photon. The SWM process absorbs two microwave photons and two coupling photons and emits one probe photon to generate a signal photon. Furthermore, as inferred from Eqs. (2) and (3), it is necessary to break the three-photon resonance to distinguish the TWM and SWM process in the frequency spectrum. In our experiment, by changing $\delta _{\text {p}}$ using MS1 in Fig. 2, we introduce a non-zero $\delta _{3}$.The TWM and SWM process can be traditionally described as a non-linear interaction between the coupling, microwave, and probe fields, enabled by the hybrid $\chi ^{(2)}_{\text {p}}$ and $\chi ^{(5)}_{\text {p}}$ non-linearity respectively. In contrast to the TWM process, the SWM process is always accompanied by a probe photon emission, a consequence of the energy and momentum conservation, represented by Eq. (3). As will be shown subsequently, the interplay between the SWM and the probe field will have a significant effect on its propagation.
The SWM process and the probe photon emission can alternatively be viewed as a result of CPO in the system. By breaking three-photon resonance, the medium’s response at $\omega _{\text {p}}$ (due to the probe field) and at $\omega _{\text {c}}+\omega _{\mu }$ (due to the coupling and microwave field) is detuned causing CPO at the beat frequency $\delta _3$. The populations can follow the beat frequency if it is smaller than the decay rates of the transitions. The resulting CPO at $\delta _3$ acts as modulators to the medium’s response at $\omega _{\text {c}}+\omega _{\mu }$ to create the sidebands $\omega _{\text {c}}+\omega _{\mu }\pm \delta _3$. The sidebands are respectively the SWM signal and probe photon generation.
The growth of the SWM ($A_{6}$) and probe ($A_{\text {p}}$) amplitudes, as it propagates through the vapor cell, can be analyzed for the $\Delta _{\text {c}}=\Delta _{\mu }=0$ and $\Delta _{\text {p}}=\delta$ case. In the slowly varying amplitude approximation, the pertinent equations in the phase-matched condition are given by,
In the undepleted approximation of the coupling ($A_{\text {c}}$) and microwave ($A_{\mu }$) fields, the solution to Eq. (4) is given by,
In contrast to the exponential growth of the SWM with distance (Eq. (7)), the TWM grows linearly with the above approximations as [19],
Equation (8) predicts PIA for the probe field while Eq. (7) predicts PSA for an external probe at the frequency of the SWM signal [21]. The exponential growth of $A_6$ with sample length justifies its efficient generation, despite involving two magnetic-dipole transitions. Physically, the exponential growth can be illustrated by two factors. First, the generated $A_6$ can reinforce $A_{\text {p}}$ by acting as a seed probe in yet another SWM process. The reinforced $A_{\text {p}}$, in turn, builds up $A_6$ at an increased rate than previously. This results in parametric coupling between the input probe field ($A_{\text {p}}$) and the SWM signal ($A_{6}$), represented by Eq. (4). Second, the generation of the SWM signal when it acts as a seed to generate the probe field also contributes to the exponential growth. The absence of these parametric interactions in the TWM process leads to its linear growth. Nevertheless, the simultaneous presence of EIT and CPO greatly enhances the TWM and SWM process by reducing the linear absorption and enhancing the non-linear susceptibilities.
4. Results and discussions
Several features in this work are different from previous observations. Firstly, the TWM and SWM processes, co-propagating with the input fields, are easily distinguished in the frequency spectrum by breaking the three-photon resonance. This eliminates the need to align the input optical beams in a specific spatial geometry to separate out the nonlinearly generated beams [11,12]. Additionally, the co-propagation of the input optical beams, which are very close in wavelength, also ensures a simultaneous natural wave-vector phase matching for both the nonlinear processes.
Secondly, the simultaneous generation of the TWM and SWM signals opens up the possibility to convert quantum information from microwave to optical in distinct frequency channels simultaneously. Together, the TWM and SWM channels can simultaneously perform phase-dependent [19], phase-sensitive and phase-insensitive amplification of the probe signal. Thus, the simultaneous presence of these two channels can be used to realize a microwave-controlled optical amplifier operating in different regimes of linear amplification [21]. In the following discussions, we examine the TWM and SWM process’s dependence on the pump coupling field and the idler microwave field.
In Fig. 4(a), we observe SWM signal power’s linear dependence on the input probe power, a consequence of Eq. (7). The probe power in the experiment never reaches the power required to saturate the transition resulting in the absence of saturation region in Fig. 4. In Fig. 4(b) we observe a slight decrease in the amplitude of TWM as a function of probe power. Strictly speaking, from Eq. (9), TWM should be independent of the probe power as it requires only the coupling and microwave fields for its generation. But, since, the probe field is derived from the coupling field through an EOM, an increase in probe power would result in a decrease in coupling power which in turn decreases the TWM signal as seen in Fig. 4(b).
Fig. 4. Normalized signal of (a)SWM and (b)TWM as a function of EOM source power. With increasing EOM source power the seed probe power increases. The experiment is done for a microwave power of 1 Watt and total optical pump power of 100$\mu W$. Here $\delta _3 = 200\text {kHz}$. All data are normalized with respect to the first data point of the generated three wave signal (TWM).
In Figs. (5,6), we have plotted the dependence of the TWM and SWM processes on the powers of the pump coupling field and idler microwave field respectively. Both these pairs of plots have an initial linear growth of the TWM and SWM signal powers, with an increase in pump coupling power and idler microwave power. We also see in both these plots that the signal optical probe generation saturates beyond the threshold power of the pump coupling field and the microwave idler field.
Fig. 5. Normalized signal (a)SWM and (b)TWM as a function of total optical power entering in to the microwave cavity. Here $\delta _3=200 \text {kHz}$ and microwave power is 1 watt. All data are normalized with respect to the first data point of the generated three wave signal (TWM).
Fig. 6. Normalized signal of (a)SWM and (b)TWM as a function of input microwave power. Here $\delta _3=200\text {kHz}$ and optical power is 100 $\mu$W All data are normalized with respect to the first data point of the generated three-wave signal (TWM).
In Fig. 7, we observe the SWM signal’s perfect linear dependence on probe detuning and microwave detuning with slopes −1 and 2, respectively. This is a consequence of Eq. (3) and it is a signature of coherent generation due to the underlying nonlinear conversion process.
Fig. 7. Spectral detuning of SWM signal as a function of (a) seed probe field detuning($\delta _p$) (b) input microwave field detuning($\delta _{\mu }$). Here total optical power in to the cavity is 200$\mu$W and microwave power is 1 Watt. The values and signs of the linear dependence in both these plots are observed to be the same as predicted by Eq. (3)
As discussed earlier, the SWM signal generation can be understood as resulting from modulation of the TWM process whenever the three-photon condition is broken. In the absence of three-photon resonance, these modulations result in population modulations in all three levels in the $\Delta$ system resulting in CPO. In our experiment, we observe these CPO oscillations, in the time domain of our spectrum analyzer, in populations of the level connected by the probe laser. As predicted from our theoretical model, the frequency of the CPO is equal to the three-photon detuning, as shown in Figs. 8(a,b,c). In Fig. 8(d), we observe CPO’s dependence on three-photon detuning. The amplitude of the CPO decreases with increasing three-photon detuning, an indication that the populations are unable to follow the increasing beat frequency.
Fig. 8. CPO oscillations observed in the probe field as a function of time for $\delta _3$ values of (a)7 kHz (b)14 kHz and (c)28 kHz. The sub-plot(d) shows the CPO oscillation contrast as a function of $\delta _3$. The solid line is a fit to the experimental data points. The contrast is defined as the difference between the maximum and minimum value in the CPO time domain plot. Here total optical power going in to the cavity is 200$\mu$W and microwave power to the cavity is 1 Watt.
5. Conclusions
In this paper we experimentally observe for the first time, the simultaneous production of a three-wave (TWM) and a six-wave (SWM) mixing signal, from a room temperature collection of $^{85}$Rb atoms. Both the TWM and SWM signals generate an optical signal field from a microwave idler field and thus act as a dual-frequency transducer connecting microwave and optical frequencies. The very few number of interacting beams and the simplicity of their geometrical arrangement required to generate these signals make our system a robust platform for microwave-to-optical conversion. Thus our system, which is a traditional quantum memory system [22], can also act as a wavelength division multiplexer (WDM) distributing quantum information in the microwave domain to several frequencies in the optical domain. In addition, we have given a theoretical model for the SWM generation in terms of coherent population oscillations (CPO). These oscillations are experimentally observed in real-time validating the theoretical model. As a future application, we show through our theoretical model that the generation of TWM, SWM and probe signal fields make our system an amplifier that can potentially show PDA, PSA and PIA types of amplification in a single physical system.
Acknowledgements
The authors thank Meena M.S. from RRI for her support in instrumentation and Barry C. Sanders from IQST, University of Calgary for stimulating discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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