# Intense, wideband optical waveform generation by self-balanced amplification
of fiber electro-optical sideband modulation

Yuzhuo Wang1<EMAIL_ADDRESS>Yizun He1 Lingjing Ji1 Jiangyong Hu1 Xing
Huang1 Yudi Ma1 Liyang Qiu1 Kaifeng Zhao2,3 Saijun Wu1<EMAIL_ADDRESS>1Department of Physics, State Key Laboratory of Surface Physics and Key
Laboratory of Micro and Nano Photonic Structures (Ministry of Education),Fudan
University, Shanghai 200433, China.
2Key Laboratory of Nuclear Physics and Ion-Beam Application (MOE), Fudan
University, Shanghai 200433, China.
3Institute of Modern Physics, Department of Nuclear Science and Technology,
Fudan University, China.

###### Abstract

We demonstrate a simple method to obtain accurate optical waveforms with a
GHz-level programmable modulation bandwidth and Watt-level output power for
wideband optical control of free atoms and molecules. Arbitrary amplitude and
phase modulations are transferred from microwave to light with a low-power
fiber electro-optical modulator. The sub-milliWatt optical sideband is co-
amplified with the optical carrier in a power-balanced fashion through a
tapered semiconductor amplifier (TSA). By automatically keeping TSA near
saturation in a quasi-continuous manner, typical noise channels associated
with pulsed high-gain amplifications are efficiently suppressed. As an example
application, we demonstrate interleaved cooling and trapping of two rubidium
isotopes with coherent nanosecond pulses.

††preprint: APS/123-QED

##  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 [1, 2, 3, 4, 5,
6, 7]. The long coherence time associated with the weakly dressed ground
states makes it possible to precisely control the dynamics using modulated CW
lasers, typically through acousto-optical modulation (AOM) with a MHz-level
bandwidth [8]. On the other hand, full control of the strong-transition
dynamics between ground and excited states becomes an emergent scenario, with
many applications in atomic and quantum optics, such as for ultrafast optical
acceleration of spinnor mattterwave [9, 10, 11, 12, 13], precise control of
light-assisted interactions [14, 15], and to access subradiant physics [16,
17, 18]. Since strong transitions have coherence time radiatively limited to
tens of nanoseconds, their full and coherent control requires optical
waveforms with modulation bandwidth at the GHz level beyond standard CW
modulation technology. Although ultrafast pulses can have bandwidth beyond
THz, the pulse spectral brightness is usually too weak to efficiently drive
the narrow transitions [19, 20, 21].

Efforts have been made to generate intense, coherent optical waveforms with
GHz modulation bandwidth for atomic physics applications [22]. For example,
coherent pulse trains with short inter-pulse delays are generated in the time
domain to excite atoms efficiently [11, 21]. Fiber electro-optical modulators
(fEOM) with $\sim$10 GHz bandwidths are exploited to transfer modulation from
microwaves to light [22, 23, 24, 18, 25, 26]. Compared with shaping ultrafast
pulses, modulation of CW lasers is more convenient for achieving long
coherence time for, e.g., complex optical control with composite pulses [18,
13, 27]. However, the integrated lithium-niobate-based fEOM suffers from
severe photo-refractive damage at short wavelengths, limiting the throughput
to a few tens of milliwatts or less in atomic physics applications [22, 23,
24, 18, 25]. The weak signal could be amplified into a watt-level output using
tapered semiconductor amplifiers (TSA) under a double-pass configuration [28,
22, 29]. However, unlike continuous seeding [28, 25], when the seeding
waveform is modulated in amplitude, the amplified spontaneous emission (ASE)
can be severe. The ASE problem is partially addressed in ref. [22] by
carefully managing the optical gain in both time and frequency domain,
although the remaining ASE that shares the time-frequency window with the
amplitude-modulated waveform output can still be detrimental. In refs. [24,
18] ASE is reduced by amplifying a CW laser first and then pulsing the high
power output into the fEOM with a low enough duty cycle for the wideband
modulation without damaging the fEOM. Although pulsed waveforms with hundreds
of milliwatts of power can be achieved this way, the procedure precludes the
possibility of generating continuous waveforms with high average power. Apart
from the ASE problem, it is known that when amplifying amplitude-modulated
light, the transient change of the gain saturation level in TSA associated
with the electron-hole density leads to self-phase modulation (SPM) [30, 31,
32, 33, 34, 35] and distortion of individual output waveforms.

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 $f_{\rm c}$. 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 $\sim 1$ W output power and $\omega_{\rm M}\sim 2\pi\times 4$ GHz
programmable waveform modulation bandwidth for fast cooling and control of
rubidium isotopes.

In the following, we first outline the operation principle of the amplified
optical waveform generation system. We then detail the performance of our
rubidium laser system and present an example application of the system for
cooling and trapping with interleaved nanosecond pulses.

##  II Methods

###  II.1 Sideband modulation

Figure 1:  The 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. The TSA1 is seeded from the side-port of
an optical isolator for double-pass amplification. Part of the output power
seeds TSA2 for an optional additional amplification. An optional Filter2
serves to remove the optical carrier from the final output. ECDL: External
cavity diode laser. O.I.: Optical isolator. PBS: Polarization beam splitter.
BS: Beam splitter.

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
$V(t)=V_{0}A(t)\cos(\omega_{\rm c}t+\varphi(t))$. Here $\omega_{\rm c}=2\pi
f_{\rm c}$ at the 10 GHz level is the microwave carrier frequency.
$A(t)\in[0,1]$ and phase $\varphi(t)\in[0,2\pi)$ are the amplitude and phase
modulation functions to be transferred to light, respectively. We use complex
$E_{\rm in}$ to represent the optical field. The output from fEOM can be
expressed as:

$\begin{array}[]{l}E_{\rm fEOM,out}(t)=e^{i\theta_{0}A(t)\cos(\omega_{\rm
c}t+\varphi(t))}E_{\rm in}(t)\\\
~{}~{}~{}~{}~{}~{}=\sum_{n}i^{n}J_{n}(\theta_{0}A(t))e^{in(\omega_{\rm
c}t+\varphi(t))}E_{\rm in}(t).\end{array}$ (1)

As described by the 2nd line of Eq. (1), the phase-modulated output can be
decomposed into an array of optical sidebands, with $n^{\rm th}$-order
$E_{n,{\rm out}}(t)=C_{n}(t)E_{\rm in}(t)$ subjected to complex modulation
function $C_{n}(t)=J_{n}(\theta_{0}A(t))e^{in\varphi(t)}$. Here $J_{n}$ is the
$n^{\rm th}$-order Bessel function. The maximum electro-optical phase shift
$\theta_{0}=\beta V_{0}$ is determined by the peak microwave voltage $V_{0}$
and the fEOM phase response coefficient $\beta$. By adjusting $V_{0}$ to let
$\theta_{0}=\theta_{n_{0}}\sim n_{0}+1$, the magnitude of
$|J_{n_{0}}(\theta_{0})|^{2}$ can be maximized to the $n_{0}^{\rm th}$-order
maximum as $J_{n_{0},{\rm max}}^{2}$ for efficient generation of an
$n_{0}^{\rm th}$ order sideband. To transfer a specific complex modulation to
the $n_{0}^{\rm th}$ order sideband, i.e., for
$C_{n_{0}}(t)=a(t)e^{i\phi(t)}$, one simply programs the rf waveform as

$\begin{array}[]{l}A(t)=J_{n_{0}}^{-1}(a(t)J_{n_{0},{\rm
max}})/\theta_{n_{0}}\\\ \varphi(t)=\phi(t)/n_{0}.\end{array}$ (2)

###  II.2 The first optical filter

We send the fEOM output $E_{\rm fEOM,out}$ through an optical filter (Filter1
in Fig. 1) to select the desired $n_{0}^{\rm th}$-order sideband and to
attenuate the optical carrier ($n=0$) to a suitable level. Here we consider
the simplest example of optical filtering by grating diffraction. A grating
constant $d<2\lambda$ is preferred to achieve good diffraction efficiency near
the Lithrow angle $\alpha\approx{\rm arcsin}(\lambda/2d)$ [36]. With the
filter input expanded into a Gaussian beam profile with Gaussian waist $w$,
and the diffraction order mode-matched by the single-mode fiber, the fiber
output power is approximately filtered in frequency by

$H(\omega)=\eta e^{-(\omega-\omega_{\rm A})^{2}/\Delta\omega_{1}^{2}}.$ (3)

Here $\eta$ is the maximum filter efficiency. The central frequency
$\omega_{\rm A}=\omega_{\rm L}+n_{0}\omega_{\rm c}$ resonant to an atomic
transition frequency is achieved by offset-locking the ECDL frequency
$\omega_{\rm L}$ and tuning the grating angle $\alpha$ to maximize
$H(\omega_{\rm A})$. By adjusting the laser beam polarization, a $\sim 70\%$
grating diffraction efficiency can be achieved in the near-infrared region,
leading to a typical $\eta\approx 40\%$ overall efficiency after fiber
coupling losses. The filter bandwidth $\Delta\omega_{1}=\frac{2\pi c}{w{\rm
tan}\alpha}=\frac{2\pi c}{w}\sqrt{4d^{2}/\lambda^{2}-1}$ can be adjusted with
$w$ to match

$H(\omega_{\rm L})/H(\omega_{\rm A})\approx J_{n_{0},{\rm max}}^{2}.$ (4)

As such, the resulting output $E_{\rm f1,out}$ is quasi-continuous during the
full-contrast $A(t)$ modulation with approximately constant average power.

###  II.3 Self-balanced amplification

At near-infrared, to avoid photo-refractive damage, the fEOM throughput is
limited to less than tens of milliwatts. With a sideband modulation efficiency
limited to $J_{n_{0},{\rm max}}^{2}$ (Eq. (1)) and after the Filter1 loss
$\eta$ (Fig. 1a, Eq. (3)), the filtered $E_{\rm f1,out}$ is typically less
than a milliwatt. The weak signal can be amplified into a watt-level output
using tapered semiconductor amplifiers under a double-pass configuration [28,
22, 29]. The ASE associated with the high gain can be efficiently suppressed
by saturating the TSA gain with a strong enough constant seeding [28]. Here,
for the amplification of seeding $E_{\rm f1,out}$ with a time-dependent
amplitude, the optical gain $g(t)$ and the effective index $n(t)$ of TSA is
expected to display time-dependent dynamics associated with semiconductor
carrier density [30, 31, 32, 33, 34, 35], leading to time-dependent ASE and
SPM to severely degrade the quality to the amplified $E_{\rm f1,out}$.

In this work, we realize that for a high enough microwave modulation carrier
frequency $\omega_{\rm c}\gg 1/\tau_{\rm c}$, where $\tau_{\rm c}$ is the
effective relaxation time of carrier density [30, 31, 33], the TSA carrier
density can hardly follow the microwave modulation, but stay at a saturation
level determined by the average seeding power. Therefore, when the quasi-
continuous seeding maintains a nearly constant average power during the full
$A(t)$ modulation in a self-balanced fashion, ASE can be suppressed in a way
similar to the case of CW seeding [28], while SPM can be suppressed in a way
similar to those achieved in multi-sideband seeding with large enough
frequency intervals [34, 35].

###  II.4 The second optical filter

With the microwave carrier frequency $\omega_{\rm c}$ at the 10 GHz level, the
amplified quasi-continuous $E_{\rm out}$ from TSAs can already be directly
applied to resonantly drive atomic systems with its $n_{0}^{\rm th}$ order
sideband. However, the off-resonant optical carrier and additional sidebands
could be detrimental to certain applications, such as for resonant imaging
applications. To further select the $n_{0}^{\rm th}$-order sideband from the
quasi-continuous output, an optional Filter2 with bandwidth
$\Delta\omega_{2}\ll|\omega_{\rm c}|$ can be constructed with a grating filter
(Fig. 1b) [36, 18], or with other types of narrow-line filters [37].

##  III The rubidium laser system

So far we have outlined the general operation principle and the 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 85Rb,
87Rb isotopes. Here, to cover the needed frequency range spanned by the 87Rb
ground-state hyperfine splitting, $f_{87,\rm hfs,g}=6.83$ GHz, we lock the
ECDL to the $F=1-F^{\prime}=0,1$ cross-over peak of the saturation
spectroscopy. Part of the output can be shift directly with AOM to address the
$F=1-F^{\prime}=2$ repumping transition. The majority of the ECDL output is
then modulated by fEOM with a carrier frequency of $\omega_{\rm
c}=2\pi\times(f_{87,\rm hfs,g}+\Delta f)=2\pi\times 6.36$ GHz, with the
$n_{0}=-1$ sideband to resonantly address the $F=2-F^{\prime}=3$ transition of
87Rb. The $\Delta f=-0.47$ GHz is determined by the ECDL locking point as well
as additional AOMs for the cooling laser control. We correspondingly set
Filter1 with a bandwidth $\Delta\omega_{1}\approx 6~{}$GHz. The filter
function $H(\omega)$ is estimated in Fig. 2a by measuring the $n_{0}=-1$ order
output with a Fabry-Perot (F-P) spectrometer.

With 30 mW output from the ECDL, we obtain $P_{\rm f1}\approx 1~{}$mW of the
filtered output $E_{\rm f1,out}$ to seed TSA1. To fully utilize the $E_{\rm
f1,out}$ for the double-pass seeding without optical damage, we limit the TSA1
driving current to $I_{1}=1.0A$ to obtain $P_{1}=140$ mW only. A 27 mW output
is fiber-coupled to seed TSA2 in the standard single-pass configuration. To
allow the TSA2 output to repump the 87Rb $F=1$ atoms, an AOM controlled
$\sim$0.5 mW of ECDL output is also combined to seed TSA2. By operating TSA2
at $I_{2}=1.7A$, a $P_{2}\approx 720$ mW output is obtained, which is directly
applied to the laser cooling and coherent control experiment without being
filtered by the optional Filter2.

###  III.1 ASE suppression

We use a home-made wavelength-meter [36] to characterize the ASE background
for both the TSA1 and TSA2 outputs. For this purpose, the laser power spectrum
is measured under three operation modes: by $E_{\rm f1,out}$ seeding under
full modulation, by $E_{\rm f1,out}$ seeding without modulation (with full
power in 0th order), and with TSA free-running (no seeding). We expose the
camera twice in each measurement to separately retrieve the spectrum density
of the coherent output and the ASE background. The background fluorescence is
measured with a 100 ms exposure time, while the central coherent spectrum
following the injected laser is measured with a 7 $\mu$s exposure time. The
two spectrum data are then combined in Fig. 5b after being relatively
normalized by the ratio of the exposure time. Here the $\sim 3~{}$GHz
frequency resolution of the spectrometer is beyond the $~{}4$ MHz laser
linewidth in this work, as verified by independent measurements. We normalize
the spectrum density according to the laser linewidth to obtain the peak
spectrum density relative to the ASE background. The spectrum density of the
coherent output is $\sim 75$ dB above the ASE background at the seeding
frequency which composes $\xi_{1}=90\%$ of the $P_{1}=140$ mW total output
power. Compared with TSA1 in the free-running mode, the $E_{\rm f1,out}$
seeding leads to $\sim 10$ dB suppression of ASE background, similar to Ref.
[28]. As discussed in Sec. IIB, with the Filter1 bandwidth
$\Delta\omega_{1}\approx\omega_{\rm c}$, the $E_{\rm f1,out}$ seeding
maintains the average seeding power $P_{\rm f1}\approx 0.6~{}$mW during fEOM
modulation, leading to a nearly identical level of ASE suppression regardless
of the $E_{\rm f1,out}$ modulation strength. Similar ASE suppression is also
obtained for TSA2, with the coherent output composing $\xi_{2}=84\%$ of the
720 mW total power and is $\sim 70$ dB beyond the ASE background in spectral
density.

Figure 2:  (a) Bandwidth measurement for the $n_{0}=-1$ order output when the
microwave carrier frequency $\omega_{\rm c}$ is sampled between $4-14$ GHz.
The data points are for $n_{0}=-1$ order outputs before and after TSA1
amplification, identified by a F-P spectrometer and normalized by the peak
values. The EOM phase modulation depth $\theta_{0}\approx 1.8$ is optimized
for the $n_{0}=-1$ order. The black and blue lines are $E_{\rm fEOM,out}$ and
$E_{\rm f1,out}$ according to a numerical model of the grating-based optical
filtering. The red curve is the Filter1 function $H(\omega)$. (b) The spectrum
of TSA1 output measured with a grating spectrometer. The ‘X’ markers estimate
the peak spectrum density according to the linewidth of the coherently
amplified laser output.

###  III.2 Accurate waveform generation

Figure 3:  Characterization of chirped pulses from TSA2 output without
Filter2. The frequency sweep range $\Delta f$ is 1 GHz and 4 GHz for the Figs.
(a)(b) data, respectively. The heterodyning beat notes are given in (i), from
which we derive the spectrogram (ii) and in-phase quadrature ${\rm
Re}[E_{-1}]$ (iv). Due to the self-balanced amplification, the total output
power stays approximately unchanged during the pulse modulation as suggested
in (iii).

The microwave amplitude $A(t)$ and $\varphi(t)$ are programmed according to
Eq. (2) to transfer specific waveforms to light. We then perform beat note
measurements to characterize the output waveforms from TSA1,2. In particular,
we mix $E_{\rm out}$ with a strong local field $E_{\rm r}$ through a
beamsplitter to measure the interference $s={\rm Re}\left(E_{\rm r}^{*}E_{\rm
out}e^{-i\omega_{\rm r}t}\right)$, where $\omega_{\rm r}=2\pi\times 1.2~{}$GHz
is the relative frequency shift between the local field and the unmodulated
ECDL output. All the measurements are performed with a fast photodetector
(Thorlabs PDA8GS) with a 9.5 GHz detection bandwidth.

Typical beat note measurements for the TSA2 output are given in Fig. 3 with
the waveforms programmed as chirp pulse modulation according to Eq. (2) with
$C_{-1}(t)=\sin(\pi t/\tau)e^{i\phi_{0}\sin(\pi t/\tau)}$ during $0<t<\tau$.
Here $\tau=10$ ns pulses are chirped with $\phi_{0}=10,40$ rads in Figs. 3(a,
i),(b,ii) respectively. The corresponding range of frequency sweep, $\Delta
f=\phi_{0}/\tau$, is 1 GHz and 4 GHz, respectively. By performing Fourier
transform of the beat note data within a shifting Blackman window with a
$T_{\mathrm{w}}=3$ ns width, the beat notes are demodulated into the $f-t$
spectrographs in Figs. 3(a,ii)(b,ii) in log scale. We further plot the target
$\dot{\phi}(t)-t$ curves in red lines onto the spectrographs to demonstrate
the accuracy of the frequency-phase control. It is important to note that
$E_{\rm out}$ has a constant total power approximately, as in Figs.
3(a,iii)(b,iii), during the full amplitude and phase modulation. The self-
balanced output power is a result of balanced $E_{\rm f1,out}$ by the Filter1
to seed TSA1,2.

To further confirm the accuracy of the modulated sideband, we use the known
target waveform phase $\phi(t)$ to demodulate $E_{-1}(t)$ from Fig. 3
(a,i)(b,i) data with a 250 MHz bandwidth. The real parts of $E_{-1}(t)$ are
plotted in Figs. 3(a,iv)(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. 4 where $\tau=20$ ns pulses are programmed with
interleaved phases $\\{\phi_{j=2n}=0,\phi_{j=2n+1}=n*\pi/6\\}$ for integer $j$
with a constant amplitude.

Figure 4:  Accurate phase modulation of the TSA2 output. The heterodyning beat
notes in Fig. (a) is digitally demodulated as described in the text to obtain
the time-dependent phase $\phi(t)$ in Fig. (b). The complex data is presented
in the Fig. (c) phasor digram.

###  III.3 SPM suppression

We verify the picture of SPM suppression at large carrier frequency
$\omega_{\rm c}$ with an independent measurement. In particular, we replace
the fEOM output in Fig. 1 with a seeding laser that contains two frequency
components with nearly equal amplitudes and a tunable difference $\delta f$.
The total 1 mW power is similar to the fEOM output. As schematically
illustrated in Fig. 5a, additional sidebands are generated during the optical
amplification through the SPM mechanism. An F-P spectrometer analyzes the
power distribution of the sidebands. The SPM efficiency is characterized by
the power of the additional sidebands normalized by the total power as
$\mathcal{E}=P_{\mathrm{nonli}}/P_{\mathrm{tot}}$. We measure $\mathcal{E}$ as
a function of $\delta f$. The results in Fig 5 demonstrate a $1/\delta
f^{2}$-scaling of suppressed SPM at large $\delta f$ , agreeing with similar
previous measurements [34]. Therefore, for the quasi-continuous seeding by
$E_{\rm f1,out}$, the frequency separation of $\delta f=\omega_{\rm
c}/2\pi=6.4~{}$GHz is large enough that the optical power distributed into
additional sidebands are limited to $\mathcal{E}\approx 2\sim 4\%$ during the
optical amplification.

Figure 5: Investigation of SPM as a function of modulation frequency $\delta
f$. (a): TSA is seeded with dual-frequency input with frequency difference
$\delta f$. The output is analyzed by a Fabry-Perot spectrometer. The
nonlinear side band power ratio $\mathcal{E}$ is recorded. (b): Sideband ratio
$\mathcal{E}$ vs frequency difference $\delta f$. The error bars reflect the
uncertainty of $\delta f$ and $\mathcal{E}$ measurements. The dashed line
gives the $1/\delta f^{2}$ scaling.

###  III.4 Interleaved cooling and trapping with (in)coherent nanosecond
pulses

Beyond 87Rb cooling, the high-power optical waveform generation system is
equipped in our lab more generally for cooling, trapping, coherent control and
laser spectroscopy of rubidium isotopes (Fig. 6) [27, 38]. To address a
specific atomic hyperfine transition, we set the microwave modulation
frequency $\omega_{c}$ to the corresponding value and program the slowly
varying amplitude and phase of optical pulses according to Eq (2). All the D2
transitions of both Rubidium isotopes, as shown in Fig. 6a, are able to be
generated by the waveform modulation, except for the repumping transition of
87Rb which is provided by the unmodulated sideband as mentioned before. To
simultaneously address multiple transitions , we simply program each component
independently according to Eq. (2). The final voltage signal $V(t)$ to drive
the fEOM sums over all the components weighted by the required amplitude
ratio, followed by proper normalization to ensure the maximum voltage still
optimally drives the -1th sideband in this work.

Here, we demonstrate the wideband performance of the system by magneto-optical
trapping (MOT) with interleaved nanosecond pulses. In particular, microwave
pulses with duration $\tau$, carrier frequency $\omega_{c,j}$, amplitude
$A_{j}$, and phase $\phi_{j}$ are applied with full duty cycle to fEOM in an
interleaved fashion to alternatively address 85Rb and 87Rb. To address 87Rb,
the carrier frequency $\omega_{c,j}=2\pi\times 6.36$ GHz is chosen as before
at even pulse number $j=2n$. To address 85Rb,
$\omega_{c,j}=2\pi\times\\{2.32,5.23\\}$ GHz are chosen at odd pulse number
$j=2n+1$ with two sidebands, with $A_{j}=\\{1,5\\}$ for repumping and cooling
respectively. A common “MOT detuning” from the cooling sidebands to the
hyperfine transitions is set as $\Delta_{e}=-2\pi\times 11$ MHz for both
isotopes. The microwave signal $V(t)$ to drive the fEOM is then synthesized as
described above. As such, both isotopes are subject to a repetitive train of
square pulses with $50\%$ duty cycle at a repetition rate of $f_{\rm
rep}=1/2\tau$. Importantly, for $\tau\Gamma_{e}<1$, the coherent dynamics
should be driven by the pulse train. Here the lifetime
$\tau_{e}=1/\Gamma_{e}\approx 26$ ns sets the “coherent memory time” for the
pulse-to-pulse excitation. To demonstrate the associated dynamics, we sample
the pulse duration $\tau$ from 0.15 ns up to 1 $\mu$s, and program the
interleaved pulses in a “coherent mode” with constant $\phi_{j}$. For
comparison, $\phi_{j}$ are randomized in a “random mode” to suppress the
inter-pulse-driven coherent dynamics.

The amplified TSA2 output waveforms are analyzed with the heterodyning method
described in Fig. 3. Examples for $\tau=5$ ns and $\tau=50$ ns are given in
Fig. 6(b). The unshown heterodyne beat notes 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 the
waveform programming. The 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 $j=2n+1$ 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-plus source MOT feeds a 2nd MOT in the standard
3D configuration. After loading the 2nd MOT for 1 second, we successively take
two fluorescence images for 85Rb and 87Rb , 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.6(c)(d) as a
function of pulse duration $\tau$ for both isotopes. Beyond $\tau=50~{}$ns,
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 $\tau<20$ ns where cooling
and trapping occur only for coherent pulses. Here, for the MOT operation, the
laser excitations are weak enough in the linear regime, the atomic dynamics is
largely decided by the spectrum of the pulses, which forms a frequency comb
with $f_{\rm rep}$ periodicity. In particular, as in Fig. 6(c), the locations
of fluorescence dips for both isotopes are coincide mostly with the spectral
analysis, which predicts efficient excitation of “open”, “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 [39, 40, 41].

Figure 6:  (a) The Level diagram and cooling related transitions on the 87Rb
(left) and ${}^{8}5$Rb (right) D2 line. (b) Spectragraphs derived from
heterodyning beat notes of interleaved nanosecond pulses with $\tau$=5 ns
(left) and $\tau=50$ ns (right) on log-scale. Fluorescence counts vs $\tau$
for 85Rb and 87Rb are shown in Fig. (c) and Fig. (d), respectively. Red arrows
mark the expected location of $\tau$, where multiple square pulses with
$T_{\rm rep}=2\tau$ period and coherent phases resonantly drive hyperfine
depumping transitions to degrade the MOT performance.

##  IV Summary and outlook

Novel research scenarios in atomic physics and quantum optics [9, 10, 11, 12,
18, 14, 15, 16, 17] require generation of powerful, arbitrarily shapeable
optical waveforms with the GHz-level modulation bandwith. A useful strategy
has been to optically amplify weak signals from high-speed integrated
modulators [22, 23, 25]. However, a common problem in this approach is
associated with amplification noise and signal distortions, particularly if
the signal level is not a constant so that the level of the amplifier’s gain-
saturation is modulated in time [30, 31, 32, 33, 34, 35].

In this work, we have explored a self-balancing technique in amplifying
sideband modulation to suppress signal distortion. Sub-milliWatt signal from a
fiber-EOM is amplified into Watt-level output with substantially suppressed
ASE and SPM noise. The technique exploits relatively slow carrier-relaxation
dynamics in TSAs, as has been observed previously [34]. We note similar
strategy is followed in refs. [11, 25] in a pre-designed manner instead of
being automatic. For the laser system demonstrated in this work, we remark
that the accuracy of the self-balanced seeding as well as the level of TSA
saturation drifts on a time-scale of hours to slightly affect the
repeatability of the waveforms. The discrepancies are about a fraction of the
distortion observed in Fig. 3. The drifts are likely due to the change of
optical injection and the TSA chip working condition. Improved output waveform
stability may be achievable in future work by combining inline optical
monitoring with active waveform-corrections.

## Acknowledgement

We acknowledge support from National Key Research Program of China under Grant
No. 2016YFA0302000 and No. 2017YFA0304204, from NSFC under Grant No. 12074083,
and the National Key Scientific Instrument and Equipment Development Project
(12027806).

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