Abstract
Keywords
I. Introduction
Optical control of atomic motion is traditionally accomplished by weakly dressing atoms in their ground-state manifolds, such as laser cooling, atom interferometry, and ion-based quantum information processing[
Efforts have been made to generate intense, coherent optical waveforms with GHz modulation bandwidth for atomic physics applications[
In this work, we introduce a simple method to achieve intense, wideband programmable optical waveforms with substantially suppressed noise associated with ASE and SPM effects. The method starts with phase-modulating a CW laser with fEOM at a microwave carrier frequency . Complex waveforms are transferred from the microwave to the optical sidebands of the fEOM output, which is then filtered and amplified with TSA. To suppress the SPM waveform distortion and ASE noise, we tune the optical filter to balance the power between the desired sideband (when the fEOM microwave modulation is on) and the optical carrier (when the modulation is off) so as to maintain a nearly constant seeding power and a consistent level of TSA saturation. The optical carrier off-resonant to the atomic transition can be subsequently filtered away. We demonstrate the method with a wideband optical waveform generation system with output power and programmable waveform modulation bandwidth for fast cooling and control of rubidium (Rb) isotopes. As to be clarified shortly, the modulation bandwidth is limited by the carrier frequency in the sideband modulation scheme. By increasing the microwave carrier frequency, waveform modulation bandwidth beyond 10 GHz can be achieved[
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In the following, we first outline the operation principle of the amplified optical waveform generation system. We then detail the performance of our Rb laser system and present an example application of the system for cooling and trapping with interleaved nanosecond pulses.
2. Methods
2.1. Sideband modulation
The setup of our laser system is schematically illustrated in Fig. 1. A frequency-stabilized CW laser (an external-cavity-diode laser, ECDL) is phase-modulated through fEOM by an amplitude and phase-modulated microwave signal . Here, at the 10 GHz level is the microwave carrier frequency. and phase are the amplitude and phase modulation functions to be transferred to light, respectively. We use complex to represent the optical field. The output from fEOM can be expressed as
Figure 1.Schematic setup of the waveform generation system. The spectrum of the optical waveform is illustrated at each stage of the amplified modulation. (a) Schematic diagram of fEOM modulation and first optical filtering. CW laser from ECDL is modulated by fEOM with a programmable microwave signal. The fEOM output is collimated into a suitable size and filtered by grating diffraction before being coupled into a single-mode fiber. (b) High-gain optical amplification. TSA1 is seeded from the side port of an optical isolator for double-pass amplification. Optional Filter2 serves to remove the optical carrier from the final output. ECDL, external cavity diode laser; OI, optical isolator; PBS, polarization beamsplitter; BS, beamsplitter.
As described by the second line of Eq. (1), the phase-modulated output can be decomposed into an array of optical sidebands, with the th-order subjected to complex modulation function . Here, is the th-order Bessel function. The maximum electro-optical phase shift is determined by the peak microwave voltage and the fEOM phase response coefficient . By adjusting to let , the magnitude of can be maximized to the (n0)th-order maximum as for efficient generation of an (n0)th-order sideband. To transfer a specific complex modulation to the (n0)th-order sideband, i.e., for , one simply programs the rf waveform as
The desired sideband is optically selected and amplified as the output, as to be discussed in the following. The frequency span of the signal [Eq. (2)] is limited by the requirement that the single sideband has to be isolated during the output, leading to the -limited modulation bandwidth in the sideband modulation scheme[
2.2. The first optical filter
We send the fEOM output through an optical filter ( in Fig. 1) to select the desired (n0)th-order sideband and to attenuate the optical carrier () to a suitable level. Here, we consider the simplest example of optical filtering by grating diffraction. A grating constant is preferred to achieve good diffraction efficiency near the Littrow angle [
Here, is the maximum filter efficiency. The central frequency resonant to an atomic transition frequency is achieved by offset-locking the ECDL frequency and tuning the grating angle to maximize . By adjusting the laser beam polarization, an grating diffraction efficiency can be achieved in the near-infrared region, leading to a typical overall efficiency after fiber-coupling losses. The filter bandwidth can be adjusted with to match
As such, the resulting output is quasi-continuous during the full-contrast modulation with approximately constant average power.
2.3. Self-balanced amplification
At near-infrared wavelengths, to avoid photo-refractive damage, the fEOM throughput is limited to less than tens of milliwatts. With a sideband modulation efficiency limited to [Eq. (1)] and after the loss [Fig. 1(a), Eq. (3)], the filtered is typically less than 1 mW. The weak signal can be amplified into a watt-level output using TSAs under a double-pass configuration[
In this work, we realize that for a high enough microwave modulation carrier frequency , where is the effective relaxation time of carrier density[
2.4. The second optical filter
With the microwave carrier frequency at the 10 GHz level or higher[
3. The Rubidium Laser System
So far, we have outlined the general operation principle and key elements of the amplified laser system. In the following, we provide additional details of the laser system designed for cooling and coherent manipulation of , isotopes. Here, to cover the needed frequency range spanned by the ground-state hyperfine splitting, , we lock the ECDL to the cross-over peak of the saturation spectroscopy. Part of the output can be shifted directly with AOM to address the repumping transition. The majority of the ECDL output is then modulated by fEOM with a carrier frequency of , with the sideband to resonantly address the transition of . The is determined by the ECDL locking point as well as additional AOMs for the cooling laser control. We correspondingly set with a bandwidth . The filter function is estimated by measuring the (−1st-order) output with a Fabry–Perot (F-P) spectrometer (Supplementary Material).
We heat the fEOM (EOSpace, Model PM-0S5-20-PFA-PFA-780-UL) to to increase the photo-refractive damage threshold[
As detailed in the Supplementary Material, the spectrum density of the coherent output is above the ASE background at the seeding frequency, which comprises of the total output power P1. Compared with free-running, the seeding leads to suppression of ASE background, similar to Ref. [28]. Similarly, ASE suppression is also obtained for , with the coherent output comprising of P2 and is beyond the ASE background in spectral density. As discussed in Sec. 2.2, with bandwidth , the seeding maintains the average seeding power during modulation of fEOM, leading to a nearly identical level of ASE suppression regardless of the modulation strength. In addition, for the quasi-continuous seeding by , the 6.4 GHz frequency separation is large enough that SPM is largely suppressed too, as being characterized by merely 2%–4% of optical power redistributed into additional sidebands during the optical amplification.
3.1. Accurate waveform generation
The microwave amplitudes and are programmed according to Eq. (2) to transfer specific waveforms to light. We then perform beat note measurements to characterize the output waveforms from . In particular, we mix with a strong local field through a BS to measure the interference , where is the relative frequency shift between the local field and the unmodulated ECDL output. All measurements are performed with a fast photodetector (Thorlabs PDA8GS) with a 9.5 GHz detection bandwidth.
Typical beat note measurements for the output are given in Fig. 2 with the waveforms programmed as chirp pulse modulation according to Eq. (2) with during . Here, pulses are chirped with rad in Figs. 2(a-i) and 2(b-ii), respectively. The corresponding range of frequency sweep, , is 1 GHz and 4 GHz, respectively. By performing Fourier transform of the beat note data within a shifting Blackman window with a width, the beat notes are demodulated into the spectrographs in Figs. 2(a-ii) and 2(b-ii) in log scale. We further plot the target curves in red lines onto the spectrographs to demonstrate the accuracy of the frequency-phase control. It is important to note that has an approximate constant total power, as in Figs. 2(a-iii) and 2(b-iii), during the full amplitude and phase modulation. The self-balanced output power is a result of balanced by to seed .
Figure 2.Characterization of chirped pulses from TSA2 output without Filter2. The frequency sweep range Δf is 1 GHz and 4 GHz for data in (a) and (b), respectively. The heterodyning beat notes are given in (i), from which we derive the (ii) spectrogram and (iv) in-phase quadrature Re(E−1). As in (iii), due to the self-balanced amplification, the total output power stays approximately unchanged, with a fractional deviation <15% during the full-pulse modulation.
To further confirm the accuracy of the modulated sideband, we use the known target waveform phase to demodulate from Figs. 2(a-i) and 2(b-i) data with a 250 MHz bandwidth. The real parts of are plotted in Figs. 2(a-iv) and 2(b-iv) to compare with the target waveforms. Here, we see small deviations of waveform amplitudes from their target values. The deviation is primarily due to the nonlinearity of rf and optical amplification, which can, in principle, be compensated for by correcting the Eq. (2) model. On the other hand, the optical frequency and phase are programmed with remarkable accuracy. The accurate phase programmability is further demonstrated in Fig. 3, where pulses are programmed with interleaved phases for integer with a constant amplitude.
Figure 3.Accurate phase modulation of TSA2 output. The heterodyning beat notes in (a) are digitally demodulated as described in the text to obtain the time-dependent phase ϕ(t) in (b). The complex data is presented in (c) the phasor diagram.
3.2. Interleaved cooling and trapping with (in)coherent nanosecond pulses
Beyond cooling, the high-power optical waveform generation system is equipped in our lab more generally for cooling, trapping, coherent control, and laser spectroscopy of Rb isotopes (Fig. 4)[
Figure 4.(a) Level diagram and cooling related transitions on the 87Rb (left) and 85Rb (right) D2 line. (b) Spectragraphs derived from heterodyning beat notes of interleaved nanosecond pulses with τ = 5 ns (left) and τ = 50 ns (right) on log-scale. Fluorescence counts versus τ for 85Rb and 87Rb are shown in (c) and (d), respectively. Red arrows mark the expected location of τ, where multiple square pulses with Trep = 2τ period and coherent phases resonantly drive hyperfine depumping transitions to degrade the MOT performance.
Here, we demonstrate the wideband performance of the system by magneto-optical trapping (MOT) with interleaved nanosecond pulses. In particular, microwave pulses with duration , carrier frequency , amplitude , and phase are applied with a full duty cycle to fEOM in an interleaved fashion to alternatively address and . To address , the carrier frequency is chosen as before at even pulse number . To address , are chosen at odd pulse number with two sidebands, with for repumping and cooling, respectively. A common “MOT detuning” from the cooling sidebands to the hyperfine transitions is set as for both isotopes. The microwave signal to drive the fEOM is then synthesized as described above. As such, both isotopes are subject to a repetitive train of square pulses with a 50% duty cycle at a repetition rate of . Importantly, for , the coherent dynamics should be driven by the pulse train. Here, the lifetime sets the “coherent memory time” for the pulse-to-pulse excitation. To demonstrate the associated dynamics, we sample the pulse duration from 0.15 ns up to 1 µs and program the interleaved pulses in a “coherent mode” with constant . For comparison, is randomized in a “random mode” to suppress the inter-pulse-driven coherent dynamics.
The amplified output waveforms are analyzed with the heterodyning method described in Fig. 2. Examples for and are given in Fig. 4(b). The heterodyne beat notes that are not shown are recorded in the “coherent mode,” although the phase coherence does not appear in the time-resolved spectrum. The gain saturation by TSAs has not been compensated for during waveform programming. SPM suppression is not perfect either. The nonlinearity during the TSA optical amplification leads to weak but visible additional sidebands on the log scale, particularly for the pulses when two seeding sidebands are injected to the TSAs. These additional sidebands hardly impact the operation of the mixed MOT.
The amplified nanosecond pulses with a total power of 700 mW are sent to a double-MOT system, where a 2D-pulse source MOT feeds a second MOT in the standard 3D configuration. After loading the second MOT for 1 s, we successively take two fluorescence images for and , each with 5 ms exposure time with a CCD camera, by setting the MOT beams resonant to the respective cooling transitions. Typical fluorescence counts are plotted in Figs. 4(c) and 4(d) as a function of pulse duration for both isotopes. Beyond , the MOT driven by interleaved pulses behaves similarly in the “coherent” and “random” modes in terms of atom number, as suggested by fluorescence imaging. The critical role of phase coherence emerges for , where cooling and trapping occur only for coherent pulses. Here, for the MOT operation, the laser excitations are weak enough in the linear regime, and the atomic dynamics is largely decided by the spectrum of the pulses, which forms a frequency comb with periodicity. In particular, as in Fig. 4(c), the locations of fluorescence dips for both isotopes coincide mostly with the spectral analysis, which predicts efficient excitation of “open” and “depumping” transitions by the frequency combs in both isotopes. Without further exploring the cooling scheme in this work, we note that cooling and trapping by the nanosecond coherent pulse train can be an interesting topic for future broadband cooling and trapping[
4. Summary and Outlook
Novel research scenarios in atomic physics and quantum optics[
In this work, we have explored a self-balancing technique in amplifying sideband modulation to suppress signals. Sub-milliwatt signals from an fEOM are amplified into watt-level output. The ASE noises are suppressed to a level similar to those achieved in constant seeding[
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