# Composite picosecond control of atomic state through a nanofiber interface

Yudi Ma<EMAIL_ADDRESS>Ruijuan Liu  Lingjing Ji  Liyang Qiu  Saijun Wu
<EMAIL_ADDRESS>Department 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.  Dianqiang Su  Yanting
Zhao<EMAIL_ADDRESS>1State Key Laboratory of Quantum Optics and Quantum
Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan
030006, China.
2Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan
030006, China  Ni Yao  Wei Fang<EMAIL_ADDRESS>Interdisciplinary Center
for Quantum Information, State Key Laboratory of Modern Optical
Instrumentation, College of Optical Science and Engineering, Zhejiang
University, Hangzhou 310027, China

(today)

###### Abstract

Atoms are ideal quantum sensors and quantum light emitters. Interfacing atoms
with nanophotonic devices promises novel nanoscale sensing and quantum optical
functionalities. But precise optical control of atomic states in these devices
is challenged by the spatially varying light-atom coupling strength, generic
to nanophotonic. We demonstrate numerically that despite the inhomogenuity,
composite picosecond optical pulses with optimally tailored phases are able to
evanescently control the atomic electric dipole transitions nearly perfectly,
with $f>99\%$ fidelity across large enough volumes for e.g. controlling cold
atoms confined in near-field optical lattices. Our proposal is followed by a
proof-of-principle demonstration with a 85Rb vapor – optical nanofiber
interface, where the excitation by an $N=3$ sequence of guided picosecond D1
control reduces the absorption of a co-guided nanosecond D2 probe by up to
$\sim 70\%$. The close-to-ideal performance is corroborated by comparing the
absorption data across the parameter space with first-principle modeling of
the mesoscopic atomic vapor response. Extension of the composite technique to
$N\geq 5$ appears highly feasible to support arbitrary local control of atomic
dipoles with exquisite precision. This unprecedented ability would allow
error-resilient atomic spectroscopy and open up novel nonlinear quantum
optical research with atom-nanophotonic interfaces.

††preprint: APS/123-QED

##  I Introduction

Atoms are ideal quantum sensors and quantum light emitters. With atomic
spectroscopy, the centers and linewidths of optical transitions are precisely
measured for probing external potentials and unknown interactions [1, 2, 3, 4,
5, 6, 7]. In quantum optics, collective absorption and emission by ensemble of
atoms are controlled for applications ranging from nonlinear frequency
generation [8, 9] to quantum information processing [10, 11]. Interfacing
atoms with nanophotonic devices promises exciting perspectives for atomic
spectroscopy at nanoscales [3, 4, 5, 6, 7] and for implementing efficient
nonlinear optics [12, 13, 14, 15]. Furthermore, with the development of laser
cooling technology [16], atoms are cooled and optically trapped in the near
field of nanophotonic structures [17, 18, 19, 20] to support many-body
interaction mediated by exchange of confined photons [21, 22, 23, 20, 24, 25].
Quite obviously, in all these scenarios that exploit the strong, electric
dipole light-atom interaction, it would be highly useful if one can precisely
control the 2-level “optical spins” defined on the strong transitions, similar
to those achieved in nuclear magnetic resonance (NMR) [26, 27] or with
2-photon narrow transitions [28, 29]. For example, such ability would support
efficient excitation in nanoscale atomic spectroscopy [3, 4, 5, 6, 7] while
suppressing the light shift and broadening [30, 31]. The precise optical spin
control would enable arbitrary, high contrast optical modulation in nonlinear
spectroscopy [12, 13, 14, 15] to improve the sensitivity to time-dependent
perturbations [32, 33]. For atomic ensembles, precise control of the atomic
dipoles can be useful for steering collective couplings between atoms and the
guided light [22, 23, 24, 25], or even to coherently suppress the couplings
[34, 35] for accessing the subradiant physics in the confined geometry [36,
37, 38, 39, 40, 41, 42]. However, unlike NMR or narrow line controls, high
precision optical control of electric dipole strong transitions is itself an
outstanding challenge [43]. The technical difficulty is amplified at
nanophotonic interfaces where a uniform optical control seems prohibited by a
none-uniform light intensity and polarization distribution.

Figure 1:  (a): Schematic setup for composite picosecond control of atomic
states through an atom-optical nanofiber (ONF) interface, with exemplary $N=1$
and $N=5$ picosecond ${\bf E}_{c}$ sequence sent through ONF to resonantly
drive the D1 transition of free 85Rb in the near field. The atomic population
in the $F=3$ ground-state hyperfine level ($|g\rangle$) can be monitored by a
co-guided D2 probe (See Fig. 2a). The red (blue) lines stand for the intensity
$|{\bf E_{c}}|^{2}$ (real envelope ${\rm Re}(E_{c})$) of the composite pulse.
(b): The 85Rb level diagram and optical coupling scheme in this work. (c):
$i_{c}(x,y)$ and $\varepsilon(x,y)$ distributions near the $d=500$ nm ONF for
the HE${}_{11}^{y}$-coupled control pulses ($\varepsilon_{\rm in}=0$, also see
Appendix B). The normalized intensity $i_{c}(x,y)=|{\bf E}_{c}(x,y)|^{2}/|{\bf
E}_{c}({\bf 0})|^{2}$ decays radially with a characteristic field decay length
$\xi\approx 1/\sqrt{\beta_{\rm c}^{2}-k_{c}^{2}}$ of 200 nm ($\beta_{\rm
c}\approx 1.2k_{c}$ is the propagation constant. $k_{c}=2\pi/795$ nm). The
local ellipticity is defined as $\varepsilon(x,y)=|{\bf E}_{c}^{*}\times{\bf
E}_{c}|/|{\bf E}_{c}|^{2}$. (d): Population inversion by the highly
inhomogenuous HE${}_{11}^{y}$ coupled $N=1$ and $N=5$ control pulses. For
stationary atoms [17, 18, 19], $f=\rho_{aa}(\tau_{c})>99\%$ efficiency can be
achieved locally by optimizing the relative phases $\\{\varphi_{j}\\}$ of the
$N=5$ composite pulse. See the $f=95\%$ (red curves), $f=99\%$ (purple curves,
with $w\approx$110 nm width) contours in the Fig. (d,ii) simulation, with
$\\{\varphi_{j}\\}=\\{0,5\pi/6,2\pi/6,5\pi/6,0\\}$ according to ref. [44]. For
comparison, for the single pulse “$\pi$” excitation, even the $f=95\%$ contour
in Fig. (d,i) is limited to a $w\approx$40 nm width. The $f=99\%$ contour is
hardly visible, which completely vanishes with $\varepsilon_{\rm in}=0.1$ (not
shown). (e): Bloch sphere representation of the simplified 2-level
$|g\rangle-|a\rangle$ dynamics subjected to $N=1$ single pulse (represented by
Fig. (e,i) alone), and $N=5$ composite pulses [44] (Trajectories driven by
each of the $N=5$ pulses are highlighted by Fig. (e,i-v) respectively.), with
equal pulse area $\mathcal{A}_{j}=\mathcal{A}\in(0.8\pi,1.2\pi)$.

We propose to achieve precise nanophotonic control of atomic states by
implementing a class of composite control schemes [44, 45] with picosecond
optical excitations [43]. We demonstrate numerically that despite the near-
field inhomogenuity, electric dipole transitions of alkaline atoms can be
controlled nearly perfectly, with $f>99\%$ fidelity across large enough
volumes for near-field samples such as cold atoms confined in near-field
optical lattices [17, 18, 19]. Our proposal is followed by a proof-of-
principle demonstration of the composite picosecond control where an $N=3$
sequence of guided picosecond pulses robustly invert the D1 population of
free-flying 85Rb atoms trespassing an ONF interface. The effect of population
inversion is monitored by the transiently enhanced transmission of a co-guided
nanosecond probe pulse. By optimizing the relative phases among the sequence
of transform-limited picosecond pulses [43], a $\sim 70\%$ enhancement of the
probe transmission is observed (even though microscopically this enhancement
is after all the near-field thermal averaging). The measurements are compared
with first-principle modeling of light-atom interaction. The agreements across
the parameter space of the composite control strongly suggest that our $N=3$
sequence reaches a close-to-ideal performance [44].

Our work paves a practical pathway toward large $N$, composite optical control
of strong transitions for atoms confined in the near field [17, 18, 19], with
an exquisite precision similar to those achieved in magnetic resonances [44,
46]. The nearly perfect dipole control could open up a variety of research
opportunities with nanophotonic interfaces, such as for error-resilient near-
field spectroscopic sensing [3, 4, 5, 6, 7] with composite pulses [30, 31, 32,
33], for studying nanophotonic nonlinear optics [12, 13, 14, 15] with
unprecedented control precision and flexibility, and for developing novel
quantum optical devices by harnessing the collective couplings between the
atomic ensemble and the nanophotonically confined light [34, 35, 21].

In the following the paper is structured into three sections. First, in Sec.
II, we outline our theoretical proposal for the picosecond composite control
at nanophotonic interfaces. Using the D lines of alkaline atoms and a
composite population inversion scheme [44] as example, we numerically
demonstrate that composite techniques can be implemented to evanescently
control atomic states in a nearly perfect manner, over a large volume in the
near field. We highlight the control power efficiency enabled by the
nanophotonic optical confinement, to potentially support $N\sim 100$ sub-
pulses for realizing highly sophisticated maneuvers [45] using moderately
strong pulses from a mode-locked laser [43]. Next, in Sec. III, we detail our
experimental measurements with a thermal vapor-ONF interface to confirm the
practicality of this proposal. We present first-principle numerical
simulations to compare with the measurements, from which we infer the
performance for an $N=3$ robust population inversion at the interface. The
technical limitation to the $N\leq 3$ control in this demonstration is
discussed in Sec. IVA. We conclude this paper in Sec. IVB with an outlook into
future research opportunities opened up by the picosecond composite control
technique.

##  II Composite picosecond control at nanophotonic interface

Precise control of strong optical transitions can be achieved at a carefully
chosen time scale. As in Fig. 1b, we consider arbitrary control of a
$|g\rangle-|a\rangle$ “optical spin” defined on the D1 line of an alkaline
atom, using a control pulse with Rabi frequency $\Omega_{c}$ and duration
$\tau_{c}$. Clearly, $\tau_{c}\gg 1/\omega_{\rm fs}$ to resolve the fine-
structure-split optical lines, and $\tau_{c}\ll 1/\Gamma_{a}$ to avoid
spontaneous emission ($\Gamma_{a}$ the natural linewidth of the excited
$|a\rangle$ states), are required for achieving the coherent 2-level “spin”
control. Meanwhile, $\tau_{c}\ll 1/\omega_{{\rm hfs},g},1/\omega_{{\rm
hfs},a}$ is preferred so that the multi-level dynamics associated with
hyperfine Raman excitations [47, 48] are relatively easy to manage (Appendix
A). The effect of the D1 control can be monitored by a weak D2 probe, if
needed. The laser of choice for the D1 control is a picosecond laser [49, 50,
43]. The $\sim 100$ GHz bandwidth is narrow enough to resolve atomic lines
with multi-THz separations, wide enough to cover the typical hyperfine
splitting at the GHz level, and support quick enough pulsed control to avoid
radiation damping. Furthermore, a moderate Rabi frequency $\Omega_{\rm c}$ at
the 100 GHz level is sufficient for the picosecond control. The moderate
strength helps avoiding e.g., photo-ionization during multiple controls [51],
and practically enable efficient implementation with low-energy pulses.

The light confinement enables efficient optical control at nanophotonic
interfaces [52, 53, 12, 13, 14, 15]. We consider the specific example of the
ONF interface illustrated in Fig. 1, where the D1 transition of the proximate
85Rb atoms is controlled by transform-limited picosecond pulses of
$\lambda_{\rm c}=795~{}$nm light guided through a step-index silica nanofiber
[19] in the fundamental HE11 mode (Appendix B) [54]. At a $d=500$ nm fiber
diameter, about $20\%$ of the light power propagates evanescently in vacuum to
interact with the atoms. The electric dipole control is remarkably efficient.
A resonant picosecond pulse energy of merely $\mathcal{E}_{1}(\pi)\sim
1~{}{\rm pJ\cdot ps}/\tau_{c}$ is sufficient for the $|g\rangle-|a\rangle$
Rabi frequency $\Omega_{c}\propto E_{c}$ (Eq. (11)) to reach $\pi/\tau_{c}$
near the ONF surface, for locally driving a “$\pi$” inversion within
$\tau_{c}$. Here $E_{c}({\bf r},t)$ is the amplitude of the complex electric
field envelop function ${\bf E}_{c}({\bf r},t)$ in the near field.

We are interested in the actual quality of the D1 inversion. Unfortunately,
the strong optical confinement (Fig. 1c) is associated with light intensity
and polarization inhomogenuities [54, 55, 56] to prevent a simple pulse from
achieving a uniform inversion in the near field, particularly in presence of
the $|g\rangle-|d\rangle$ Raman couplings [47]. Nevertheless, it is important
to note that when the control pulse is short enough, $\tau_{c}\ll
1/\omega_{{\rm hfs},g},1/\omega_{{\rm hfs},a}$, the hyperfine
$|g\rangle-|a\rangle$ transition in the D1 example reduces to a
$J_{g}=1/2\leftrightarrow J_{a}=1/2$ transition (Appendix C.4). By choosing
the quantization axis along the local helicity axis ${\bf e}_{h}={\bf
E}_{c}^{*}\times{\bf E}_{c}/|{\bf E}_{c}|^{2}$, the D1 excitation is
decomposed into 2-level $\sigma^{\pm}$ couplings, with a relative coupling
strength determined by the local ellipticity of light $\varepsilon(x,y)=|{\bf
e}_{h}|$ (Eq. (9)). For the limiting case of a linearly polarized field (with
quantization axis perpendicular to ${\bf E}_{c}$), the coupling strengths
become degenerate. One thus expects nearly perfect inversion efficiency
insensitive to initial $|g\rangle$ Zeeman sub-levels (Fig. 7), at specific
intensities $|{\bf E}_{c}|^{2}$. This is confirmed with our full-level
simulations (Appendix C) [57, 58, 48], illustrated in Fig. 1(d,i) with an
example 2D distribution of $|g\rangle-|a\rangle$ inversion efficiency $f(x,y)$
for stationary, initially unpolarized atoms in $|g\rangle$ (see Eq. (3)).
Here, for a control pulse ${\bf E}_{c}$ in the HE${}_{11}^{y}$ mode with an
incident ellipticity $\varepsilon_{\rm in}=0$ (Fig. 1a), the near-field
ellipticity is minimized near $y=0$ (Fig. 1c), $\varepsilon(x,y)\ll 1$ to
support the nearly perfect inversion within $\tau_{c}=6$ ps time, with
$\mathcal{E}_{1}=0.4~{}$pJ here. However, the inversion is not robust. A
nanoscale shift of atomic position can degrade the performance substantially.
Furthermore, the $f=99\%$ contour (which is barely visible in Fig. 1(d,i))
vanishes at a moderate $\varepsilon_{\rm in}=0.1$, as suggested by additional
numerical simulations. We note such a slight change of the HE11 polarization
state [54] could easily be induced by the birefringence of a stressed fiber.

To improve the control robustness, composite pulse sequence with $N$ sub-
pulses can be programmed to achieve highly accurate control with uniform
efficiency by exploiting the geometric phase of 2-level transitions [59]. Such
composite techniques are well developed in the research field of nuclear
magnetic resonance [27, 44, 45]. In particular, a sequence of $N$-pulses with
equal amplitude $\\{a_{j}\\}$ and optimal phase $\\{\varphi_{j}\\}$,
$j=1,...,N$ can be applied to invert the population of a 2-level spin in a
manner which is highly resilient to the field strength and frequency detuning
errors [44].

In the optical domain, the NMR-inspired composite technique can be exploited
for driving ensemble of atoms illuminated by an inhomogeneous laser field, as
well as for driving ensemble of transitions with different transition
strengths (and slight different transition frequencies) for a same atom. The
ONF-based atomic state control in this work exploits both aspects of the
control technique (Appendix C.4). For the aforementioned reasons associated
with quantum control timescales, the composite pulses need to be synthesized
within picoseconds with full waveform programmability, a requirement within a
technical gap between the laser modulation technologies for continuous wave
and ultrafast lasers [60]. In a recent work [43], we developed a “direct ${\bf
k}$-space-to-time pulse shaping” method [61, 62, 63] to precisely generate
$\tau_{c}\sim 100$ ps waveforms with tens to hundreds of GHz bandwidth, only
limited by that of the seeding mode-locked laser. With the method, as detailed
in Appendix E.1, an incident single pulse with energy $\mathcal{E}_{0}$ and a
full pulse width at half maximum $\tau_{0}$ can be shaped into $j=1,...N$,
$\tau_{j}=(j-1)\tau_{\rm d}$ even-delayed sub-pulses with energy
$\\{\mathcal{E}_{j}\\}$ and phases $\\{\varphi_{j}\\}$, with fully
programmable amplitudes and phases of the sub-pulses. Taking into account the
sech2-shaped soliton pulses from the mode-locked laser [64], we associate a
total control duration $\tau_{c}=(N-1)\tau_{\rm d}+\tau$ to the composite
pulse with $\tau=2\tau_{0}$ to characterize the control in the time domain.

A particular example of robust atomic state control is illustrated in Fig.
1(d,ii) according to numerical simulation of 85Rb atom driven by the near
field composite D1 excitation (Sec. III.3). The robust control is best
illustrated on a Bloch sphere first, as in Fig. 1e, which simplifies the
hyperfine $|g\rangle-|a\rangle$ transition (Appendix C.4) with 2-level
dynamics. The pulse areas are defined as $\mathcal{A}_{j}=|\int\Omega_{j}{\rm
d}t|$ near the ONF surface, with
$\Omega_{j}\propto\sqrt{\mathcal{E}_{j}}e^{i\varphi_{j}}$ to be the Rabi
frequency of the sub-pulse $j$ (Eqs. (11)(32)). By optimizing the relative
phases $\\{\varphi_{j}\\}$ among the sub-pulses of an $N=5$
equal-$\mathcal{E}_{j}$ sequence, according to ref. [44], the seemingly
redundant rotations of the state vector along the $\vec{\Omega}_{j}\equiv({\rm
Re}(\Omega_{j}),{\rm Im}(\Omega_{j}),0)$ axes (Fig. 1(e,i-v)) are phased in a
way to cancel the inhomogenuous $\delta\mathcal{A}\in(-0.2\pi,0.2\pi)$
deviation from $\mathcal{A}_{j}=\pi$. The final state inversion fidelity in
Fig. 1(e,v) reaches $f>99.97\%$ [44]. This is in contrast to the simple
single-pulse control represented by Fig. 1(e,i) alone, which only reaches
$f\approx 90\%$ on average due to the same $\delta\mathcal{A}$ broadening,
hardly avoidable in the near field.

We now apply the $N=5$ optimal composite excitation [44] to the 85Rb D1
interaction in the near field. For the atom starting from an arbitrary Zeeman
sub-level of $|g\rangle$ (Fig. 7) at $t=0$, with $\mathcal{E}_{j}=0.4$ pJ
here, $f>99\%$ population inversion around $y=0$ is achieved across a
substantial area where the near field is approximately linear (the purple
contour in Fig. 1(d,ii)). The width $w\approx 110$ nm along the narrower
$x-$direction with $\delta\mathcal{A}\in(-0.3\pi,0.3\pi)$ is substantially
larger than the 1-pulse case, where even for the $95\%$ contour the width is
only $w\approx 40$ nm (Fig. 1(d,i)). The total control time is $\tau_{c}=66$
ps in the Fig. 1(d,ii) simulation, with a $\tau_{0}=3~{}$ps pulse width. We
note longer $\tau_{c}$ leads to reduced width for the $f=99\%$ contour along
$y$, with the $w$ along $x$ unaffected, as long as $\tau_{c}\ll 1/\omega_{{\rm
hfs},a}$ so that 2-photon Raman transitions by linear polarized excitations
are suppressed [65] (Appendix C.4). The population inversion is robust against
small incident polarization variations. In particular, the $f=99\%$ contour in
Fig. 1(d,ii) is hardly changed when the incident polarization is increased to
$\varepsilon_{\rm in}=0.1$ in the simulation. We thus expect robust
implementation of the $N=5$ composite sequence to e.g. arrays of trapped atoms
through the ONF interface [17, 18, 19] to achieve ultra-precise population
inversion.

Beyond simple population inversion, composite pulses can be shaped to
implement more sophisticated controls, such as to geometrically shift the
optical phases of the D2 dipoles with double D1 inversions [34, 35], or even
for driving arbitrary qubit gates [45] on the optical transitions. For
composite pulses with nearly equal amplitudes, the sub-pulse energy
$\mathcal{E}_{j}<\mathcal{E}_{0}/N^{2}$ is limited by conservation of optical
spectrum density. Nevertheless, with the
$\mathcal{E}_{0}=N^{2}\mathcal{E}_{1}(\pi)$ scaling and at a moderate pulse
shaping efficiency [43], the required input pulse energy $\mathcal{E}_{0}$ for
sophisticated controls [45, 30, 31, 32, 33] with up to $N=10^{2}$ sub-pulses
is below 100 nJ. The required seeding pulse energy $\mathcal{E}_{0}$ is
reduced further for nano-structures with better atom-light cooperativity [66,
67, 68, 15].

##  III A Proof-of-principle demonstration

The picosecond composite control technique is applicable not only to
microscopically confined cold atoms, but also to thermal atoms that
transiently trespass the nanophotonic interface. In this section, we exploit
the thermal vapor-ONF interface to demonstrate the picosecond atomic state
control.

Figure 2: (a): Schematic of the control-probe scheme in this work to
investigate the picosecond atomic state controlability at the ONF interface.
The red and blue lines describe the composite picosecond pulse envelop,
similar to Fig. 1a but with $N=3$ here. HWP: half-wave plate. QWP: quarter-
wave plate. (b): A $\tau_{p}=2$ ns probe pulse resonant to the D2
$|g\rangle-|e\rangle$ transition is evanescently attenuated through the
thermal vapour-ONF interface, leading to reduced transmission by $A$ (an
integration of the blue shaded area). See Fig. 1b for the full level diagram
including $|d\rangle$. (c): By firing the D1 control pulses
($|g\rangle-|a\rangle$) to deplete the ground state population $\rho_{gg}$,
the $\Delta t$-delayed D2 probe attenuation is reduced by $\delta T$ (an
integration of the red shaded area). The sub-$\tau_{p}$ $A$ and $\delta T$
transients are explained in Appendix D.4. The $\overline{\delta T}\equiv\delta
T/A$ ratio reflects reduced $|g\rangle$ population in the near field. (d):
Schematic illustration of thermal atoms trespassing the ONF evanescent field.
Here $i_{\rm p}(x,y)=|{\bf E}_{\rm p}(x,y)|^{2}/|{\bf E}_{\rm p}({\bf
0})|^{2}$. (e): Experimental and simulated (Eq. (5)) transient transmission
$\overline{\delta T}$ vs $\Delta t$, for $N=1$ single D1 pulse control (See
Appendix E.4.). The error bars represent the statistical uncertainties from 10
repeated measurements.

###  III.1 Measurement principle

The experimental setup illustrated schematically in Fig. 2a follows a control-
probe strategy. A nanosecond probe pulse with duration $\tau_{\rm p}$,
resonant to the $|g\rangle-|e\rangle$ D2 transition of 85Rb, is combined with
the picosecond $|g\rangle-|a\rangle$ D1 control pulses with duration
$\tau_{\rm c}$ through a polarization-dependent beamsplitter and sent to the
ONF interface. The evanescent coupling between the ONF-guided probe pulse with
the thermal atoms surrounding ONF leads to the probe attenuation. As in Fig.
2b, we denote the fractional attenuation to the probe transmission $T$ by the
atomic absorption as $A=\Delta T/T$. In presence of the resonant D1 control
pulses that also interact with the atoms evanescently (Fig. 2c), the ground
state population $\rho_{gg}$ in the near field is depleted, leading to
transiently enhanced transmission $\delta T=-\delta A$.

Optically, our thermal vapor-ONF system is well within a mesoscopic regime
featuring transient spectral broadening and non-local responses [69]. In
particular, at a ${\rm T}=360$ K temperature to be discussed shortly, the 85Rb
atoms has a thermal velocity of $v_{\rm T}=\sqrt{k_{\rm B}{\rm T}/M}\approx
190$ m/sec ($M$ is the atomic mass). For the ONF-guided probe with a
propagation constant $\beta_{\rm p}\approx 1.2k_{\rm p}$ [54] ($k_{\rm
p}=2\pi/780$ nm), the Doppler linewidth [70] $\sqrt{2{\rm ln}2}\beta_{\rm
p}v_{\rm T}/\pi\sim 700$ MHz is broadened to $\sim 800$ MHz according to
absorption spectroscopy, which, as unveiled by simulation (Appendix D.4), is
due to near-field atomic motion associated with the nanosecond $\delta
t=\xi_{\rm p}/v_{\rm T}$ ($\xi_{\rm p}=1/\sqrt{\beta_{\rm p}^{2}-k_{\rm
p}^{2}}=200~{}$nm is the probe field decay length. The control field decay
length $\xi$ is similarly defined, see Fig. 1.). As schematically illustrated
in Fig. 2(b,c), for the pulsed control and probe, we expect both $A$ and
$\delta T$ to vary rapidly within a nanosecond due to the Doppler broadening
and the $x-y$ motion (Appendix D.4). Here, we record the $\tau_{\rm
p}$-integrated $\delta T$ and $A$ to obtain the normalized transient
transmission,

$\overline{\delta T}\equiv\delta T/A.$ (1)

Intuitively, $\overline{\delta T}$ is the fractional reduction of the light
scattering power during $\tau_{\rm p}$ by the atoms surrounding the ONF, due
to the excitation by the control pulse that transfers the atomic population
away from the state $|g\rangle$. As detailed in Sec. III.3, $\overline{\delta
T}$ is compared with a numerical model to infer the ground-state depletion
efficiency (also see Eqs. (18)(19))

$f_{g}({\bf r})\equiv\Delta\rho_{gg}({\bf r},\tau_{c})/\rho_{gg}^{(0)},$ (2)

as well as the $|g\rangle-|a\rangle$ population inversion efficiency

$f({\bf r})\equiv\rho_{aa}({\bf r},\tau_{c})/\rho_{gg}^{(0)},$ (3)

by the picosecond control applied during $0\leq t\leq\tau_{c}$. Here the
$\rho_{gg}^{(0)}$ at $t=0$ and $\rho_{aa}{(\tau_{c})}$ at $t=\tau_{c}$ are the
atomic population summed over the Zeeman and hyperfine sublevels. We note that
in presence of the hyperfine $|d\rangle$-level (Fig. 1b) and 2-photon
picosecond Raman couplings [47], we generally expect $f({\bf r})\neq
f_{g}({\bf r})$.

###  III.2 Experimental setup

Experimentally, the rubidium partial pressure at the nanofiber location is
maintained around $p=10^{-5}$ Torr by electronically heating a dispenser
$\sim$30 cm away in the vacuum. Depending on the actual vapor pressure during
different periods of this project, attenuation of $A\approx 0.15\sim 0.2$ to
the ONF transmission is obtained for the probe locked to the F=3-F′=4 D2
hyperfine transition of 85Rb. A heater attached to the ONF mount helps to
maintain a 90∘C or ${\rm T}=360$ K temperature to suppress condensation of Rb
atoms to the ONF surface. The probe beam is pulsed with a duration $\tau_{\rm
p}=2$ ns to match the transiently broadened optical response of atoms to be
detailed shortly, and is sent to ONF after a $\Delta t$ delay relative to the
picosecond control (Fig. 2c). With a pair of computer-motorized half and
quarter waveplates (Appendix E.3), the adjustments of the polarization states
of the orthogonal HE11 modes for the control and probe pulses are
computerized. To separate the probe pulse from the pump background in a
polarization-independent manner at the ONF output, we split the combined beams
with another PBS (not in Fig. 2a) to filter each path with a holographic
grating, both at the “p” polarization with $\sim 70\%$ diffraction efficiency.
The grating-filtered probe signals are then overlapped to an avalanche
photodiode (APD) module (Hamamatsu C5658, 1 GHz detection bandwidth). An
additional interference filter to bandpass the 780 nm probe is inserted to
fully remove the 795 nm control photon background.

As detailed in Appendix E.2, the nanosecond APD output is analyzed by a home-
made analogue signal integrator which compares $10^{7}$ interleaved
measurements with and without the control pulses in two seconds. The high
speed differential measurements lead to quality signals capable of resolving
$\delta T$ at the $0.1\%$ level, even though the probe power is kept at
$P_{\rm p}\approx 10$ nW to avoid saturating the D2 absorption. Normalized
transient transmission $\overline{\delta T}(\Delta
t,\\{\mathcal{E}_{j},\varphi_{j}\\})$ is recorded for various composite pulse
sequence at certain $\Delta t$ delay. The “steady-state” absorption $A$ for
the $\overline{\delta T}$ normalization is monitored instead with a slow
photo-multiplier tube (Hamamatsu CR131), by comparing the resonant
transmission with the off-resonant values during frequency scans of the
D2-laser (See Appendix E.4 discussions). Notice here and in the following $A$
and $\delta T$ are normalized by the generic ONF transmission $T$ estimated to
be better than $90\%$ [19].

We verify the transient broadening picture with a simple delayed-probe
experiment. As in Fig. 2e, while a $\tau_{c}=2\tau_{0}=24$ picosecond D1
control pulse (Appendix E.1) with strong enough $\mathcal{E}_{1}\approx 0.5$
pJ is able to reduce the atomic absorption by $\overline{\delta T}\approx
50\%$ at $\Delta t=0$ (Also see Sec. III.4, Appendix D.4,E.4), the
transmission recovers rapidly with a time constant as short as $2$ ns. After
the picosecond excitation, atoms at the vicinity of ONF transferred to either
the excited $|a\rangle$ or the other $F=2$ hyperfine ground states $|d\rangle$
(Fig. 1b) become invisible to the probe, leading to the reduced absorption.
These atoms gradually leave the near field and are replaced by incoming atoms
in $|g\rangle$ both from far away and from ONF surface desorption, within a
few nanoseconds, as being verified numerically (Fig. 2e, the circles are from
the numerical model detailed in Appendix D.4). This rapid recovery of the
optical responses ensures that the aforementioned $10^{7}$ measurements within
seconds are independent to each other.

###  III.3 Modeling the ONF interface

We numerically model the control and probe dynamics at the ONF interface (Fig.
1, Fig. 2) both to predict the atomic state control efficiency $f({\bf r})$
(Eq. (3)) as those in Fig. 1d, and experimentally to infer the efficiencies
$f({\bf r})$, $f_{g}({\bf r})$ (Eq. (2)) from the $\overline{\delta T}$
measurements (Eq. (1)). The model is based on optical Bloch equations (OBE)
[71] (Appendix C, D). Since the atomic state dynamics of interest is within
the short nanosecond window, in this work we ignore the transition frequency
shifts due to the surface interactions, which, averaged by the probe
evanescent coupling, are at the MHz-level [4, 72].

The numerical model starts with calculating the guided HE11 field profiles
${\bf E}_{c}({\bf r})$, ${\bf E}_{\rm p}({\bf r})$ for the control and probe
beams respectively [54] (Appendix B). Next, for atom at location ${\bf r}$ in
the near field, we integrate the Schrödinger equation for the resonant dipole
D1-line interaction (Appendix C) to numerically obtain the evolution operator
$U_{c}({\bf r},\\{\mathcal{E}_{j},\varphi_{j}\\})$ for the picosecond
composite pulses of interest. Since $v_{\rm T}\tau_{c}\ll\xi$,
$\Gamma_{a}\tau_{c}\ll 1$, atomic motion and spontaneous decay can be ignored
during the picosecond control. With in mind the guided HE11 profiles are
invariant along $z$, we sample the control field in the $x-y$ plane (Fig. 1c)
at a fixed $z$. We then evaluate the $|g\rangle-|a\rangle$ inversion
efficiency $f({\bf r})$ (Eq. (2), Eq. (18)) as those in Fig. 1d for atoms
populating $|g\rangle$ with initial $\rho_{gg}^{(0)}=1$. On the other hand,
for evaluating the ground-state depletion efficiency $f_{g}({\bf r})$ (Eq.
(2)), which is directly linked to the $\overline{\delta T}$ measurements, we
set $\rho_{gg}^{(0)}=7/12$ and $\rho_{dd}^{(0)}=5/12$ for the 85Rb atom evenly
populating the twelve ground-state Zeeman sublevels
$|g_{m}\rangle,|d_{m}\rangle$ according to the experimental expectations. Here
$m=m_{F}$ is the magnetic quantum number, see Fig. 7 in the Appendix.

In contrast to the picosecond “impulse” interaction, one cannot simply ignore
the atomic motion during the nanosecond D2 probe. In fact, with $v_{\rm
T}\tau_{\rm p}\approx 380~{}{\rm nm}>\xi,\xi_{\rm p}$, we expect the thermal
vapor-ONF system to be within a mesoscopic regime [69], invalidating
macroscopic effective media theory based on local optical responses. We set up
an “exact” model and a diffusive average model to describe the D2 optical
response at the ONF interface.

For the “exact” model, we sample the phase space $({\bf r}_{0},{\bf v}_{0})$
of the thermal vapor uniformly surrounding the ONF, and evaluate the dipole
interaction between the guided ${\bf E}_{\rm p}({\bf r}(t),t)$ and the atoms
following the ballistic trajectories ${\bf r}(t)={\bf r}_{0}+{\bf v}_{0}t$. As
mentioned, initially $\rho_{0}(0)=\rho^{(0)}$ uniformly populates all the 12
ground state Zeeman sublevels, which, after being subjected to the picosecond
$U_{c}$ control, is denoted as
$\rho(\tau_{c})=U_{c}\rho_{0}(0)U_{c}^{\dagger}$. Then, during
$\tau_{c}<t<\tau_{c}+\Delta t+\tau_{p}$, the $\rho(t)$ for the moving atoms,
as well as $\rho_{0}(t)$ for those without experiencing the control pulses,
all evolve according to the OBE on the D1-D2 manifold. To evaluate the probe
scattering rates, we ignore the moderate ONF-modification to the atomic
response [73, 72] to simply have

$\displaystyle\bar{\gamma}(t)$ $\displaystyle=\frac{1}{\hbar}\left\langle{\rm
Im}[\langle{\bf E}^{*}_{\rm p}({\bf r(t)},t)\cdot{\bf
d}\rangle_{\rho(t)}]\right\rangle_{{\bf r}_{0},{\bf v}_{0}},$ (4)
$\displaystyle\bar{\gamma}_{0}(t)$
$\displaystyle=\frac{1}{\hbar}\left\langle{\rm Im}[\langle{\bf E}^{*}_{\rm
p}({\bf r(t)},t)\cdot{\bf d}\rangle_{\rho_{0}(t)}]\right\rangle_{{\bf
r}_{0},{\bf v}_{0}},$

with and without the control pulses. Here the inner
$\langle...\rangle_{\rho(t)}$ represents ${\rm tr}(\rho(t)...)$ for the
quantum mechanical average of a single trajectory. The outer
$\langle...\rangle_{{\bf r}_{0},{\bf v}_{0}}$ sums the Monte Carlo
trajectories according to the thermal distribution. The probe absorption $A$
without the control pulses, as well as the transient transmission
$\overline{\delta T}$ induced by the control, are evaluated as

$\displaystyle A$ $\displaystyle=\frac{\hbar\omega_{\rm
p}\int\bar{\gamma}_{0}(t){\rm d}t}{\int P_{\rm p}(t){\rm d}t},$ (5)
$\displaystyle\overline{\delta T}$
$\displaystyle=\frac{\int(\bar{\gamma}_{0}(t)-\bar{\gamma}(t)){\rm
d}t}{\int\bar{\gamma}_{0}(t){\rm d}t},$

in the limit of weak absorption $A\ll 1$. Here $P_{\rm
p}(t)=\frac{1}{2}\int\varepsilon_{0}\mathcal{N}^{2}v_{\rm p}|{{\bf E}_{\rm
p}({\bf r},t)}|^{2}{\rm d}^{2}{\bf r}_{\perp}$ is the transient optical power
of the probe pulse, $v_{\rm p}=\omega_{\rm p}/\beta_{\rm p}$ is the phase
velocity of the ONF guided probe light ($\beta_{\rm p}$ is the associated
propagation constant), and $\mathcal{N}({\bf r}_{\perp})$ is the transverse
ONF refractive index profile.

We note that with the external trajectories ${\bf r}(t)={\bf r}_{0}+{\bf
v}_{0}t$ at the 100 m/s level speed, mechanical forces associated with the
surface interaction and the light pulses can be safely ignored. For
trajectories that hit the ONF surface during the simulation time, we assume
immediate desorption with a randomized emission direction, with the internal
state reset to a random ground state. The simulations typically require
$10^{6}$ sampling trajectories to converge for a specific control pulse
configuration. We refer readers to Appendix D.4 for details of the Monte Carlo
simulations.

The Monte Carlo simulation of the “exact model” becomes too resource-demanding
for us when trying to explore the composite control configurations across the
$\\{\mathcal{E}_{j},\varphi_{j}\\}$ parameter space. To enhance the speed, a
simpler diffusive average model is proposed for efficient evaluation of
$\overline{\delta T}$ as

$\displaystyle\overline{\delta T}$ $\displaystyle\approx\overline{f_{g}({\bf
r})}$ (6) $\displaystyle=\frac{\int R({\bf r}_{\perp}-{\bf r}_{\perp,0})i_{\rm
p}({\bf r}_{\perp,0})f_{g}({\bf r}_{\perp,0}){\rm d}^{2}{\bf r}_{\perp}}{\int
R({\bf r}_{\perp}-{\bf r}_{\perp,0})i_{\rm p}({\bf r}_{\perp,0}){\rm
d}^{2}{\bf r}_{\perp}},$

using the ground state depletion ratio $f_{g}({\bf r})$ defined by Eq. (2).
Here $i_{\rm p}({\bf r}_{\perp})\propto|{\bf E}_{\rm p}({\bf r}_{\perp})|^{2}$
is the transverse intensity profile of the guided D2 probe in the near field
(Fig. 2d). The thermal diffusion kernel is set as

$R({\bf r}_{\perp})=\frac{1}{\pi\xi_{\rm p}^{\prime 2}}e^{-|{\bf
r}_{\perp}|^{2}/\xi_{\rm p}^{\prime 2}}$ (7)

with a phenomenological $\xi^{\prime}_{\rm p}\approx\sqrt{2}v_{\rm T}\tau_{\rm
p}/2$ to account for atomic diffusion in the $x-y$ plane during the $\tau_{\rm
p}$ probe time (see Appendix D.5). To account for the presence of ONF wall, we
simply set $i_{\rm p}=0$ inside ONF.

Physically, Equation (6) assumes that for evaluating the $\overline{\delta T}$
ratio in Eq. (5), the coupling strengths between the guided probe and the
surrounding atoms is decided by the local probe intensity $i_{\rm p}({\bf r})$
(Fig. 2d) and the ground state population $\rho_{gg}$ only. The picosecond
controls hardly modify the velocity distribution of the mesoscopic vapor.
Therefore, while the aforementioned Doppler and transient broadening weaken
the atom-light coupling strengths on average, the reductions are largely
shared by the mesoscopic vapor with and without the control pulses. The
ignorance of coherence transients, as detailed in Appendix D.4, is justified
by the fact that the effects are either averaged out during the $\tau_{\rm p}$
integration or largely canceled in the $\overline{\delta T}$ ratio too. In
Appendix D.5 we show that the Eq. (6) model, with all the quite strong
approximations, generates results that agree fairly well with the “exact
model” by Eq. (5). The Eq. (6) approximation helps us to efficiently simulate
$\overline{\delta T}$ with the numerical model to compare with the
experimental measurements, with which the actual $U_{c}({\bf
r},\\{\mathcal{E}_{j},\varphi_{j}\\})$ parameters and the $f_{g}({\bf r})$,
$f({\bf r})$ efficiencies in the experiments are inferred.

###  III.4 Saturation of picosecond excitation

Figure 3: Polarization-dependent picosecond D1 single excitation at the ONF
interface. Figs. (a)(c) give the simulated transient population depletion
$f_{g}({\bf r})$ by a single pulse excitation with energy $\mathcal{E}_{1}$
(Eqs. (2) (19)). The incident HE11 mode is with linear ($\varepsilon_{\rm
in}=0$) and circular ($\varepsilon_{\rm in}=1$) incident ellipticity for the
Figs. (a)(c) simulations respectively. A $f_{g}=80\%$ contour is highlighted
in Fig. (a,ii). Simulated $\overline{\delta T}$ according to Eq. (6) at
various incident ellipticities are given by Fig. (b) with the rainbow plot,
color-coded by $\varepsilon_{\rm in}$. The $\Omega_{1}$-axis is specified
according to Eqs. (11)(33). The experimentally measured $\overline{\delta T}$
as a function of $\mathcal{E}_{1}$ and associated peak $\Omega_{1}$ are
scatter-plotted with blue square ($\varepsilon_{\rm in}=0$) and red disk
($\varepsilon_{\rm in}=1$) symbols, with error-bars estimated from three
repeated measurements. To obtain the theoretical-experimental match in Fig.
(b), the full experimental data set is uniformly rescaled (Appendix E.4)
within the measurement uncertainties suggested by the double-sided arrows on
the top right.

We first benchmark the ONF interface with an $N=1$, single-pulse control-probe
experiment. Enhanced atom-light interactions have been demonstrated previously
at the nanofiber interface [12, 13, 14, 15]. Here we demonstrate a full
saturation of the control-induced change of probe transmission to
$\overline{\delta T}\approx 50\%$ with merely sub-pico-Joule single control
pulses. We highlight the polarization $\varepsilon_{\rm in}$-dependence of the
transient optical response.

Intuitively, in presence of the inhomogeneous light intensity and ellipticity
distributions (Fig. 1c, Fig. 6), we expect the impulse D1 excitation with
large enough $\mathcal{E}_{1}$ to deplete the ground state population in a
spatial dependent matter, lead to $f_{g}({\bf r})$ oscillating between 0 and 1
in the near field. Consequently, the evanescent coupling induced probe
attenuation is expected to be transiently halved, $\overline{\delta T}\approx
50\%$ [74], as in Fig. 2e at $\Delta t=0$.

In this section, the simple picture of optical saturation is confirmed by
detailed measurements of $\mathcal{E}_{1}$-dependent $\overline{\delta T}$. To
optimally retrieve the nonlinear signal, the probe delay is reduced to $\Delta
t=0$ ps (Fig. 2c). Typical transient transmission $\overline{\delta T}$ data
are plotted in Fig. 3b as a function of pulse energy $\mathcal{E}_{1}$ and the
peak Rabi frequency $\Omega_{1}$ estimated at the ONF surface (Appendix E.4).
In Appendix E.3 we detail the HE11 polarization control with automated
polarization adjustments (Fig. 2a). For the $\overline{\delta T}$ data here,
the polarization states of control and probe beams are set to be linear
($\varepsilon_{\rm in}=0$, blue square symbols) and circular
($\varepsilon_{\rm in}=1$, red disk symbols) respectively. In both cases we
find $\overline{\delta T}$ increases linearly with small $\mathcal{E}_{1}$.
Furthermore, nearly complete saturation of $\overline{\delta T}$ to $50\%$
occurs at $\mathcal{E}_{1}$ as small as 1 pJ.

Following Sec. III.3, we numerically evaluate the D1 atomic state dynamics
subjected to the vectorial light-atom interaction in the near field. Typical
ground-state depletion efficiency $f_{g}({\bf r})$ are plotted in Fig. 3(a)(c)
for the case of linear ($\varepsilon_{\rm in}=0$) and circular
($\varepsilon_{\rm in}=1$) HE11 modes of control respectively. Notice for the
$N=1$ pulse here, $\tau_{c}\ll 1/\omega_{{\rm hfs},g},1/\omega_{{\rm hfs},g}$
is fairly well satisfied, so that the $J_{g}=1/2\leftrightarrow J_{a}=1/2$
intuition in Appendix A can be applied. In particular, as in Fig. 3a, the
ground state population can be nullified in the near field where the
polarization is purely linear. For comparison, the reduced peak $f_{g}$ for
the circular incident polarization in Fig. 3c is associated with local
ellipticity $\varepsilon\approx 0.95$ (Appendix B) so the $\sigma^{\pm}$
coupling strengths are different substantially (Eq. (9)), making perfect
$|g\rangle-|a\rangle$ inversion impossible in presence of population in
different Zeeman sublevels. Nevertheless, regardless of the $\varepsilon_{\rm
in}$ value, the near-field chirality [55] prevents completely circular
polarization (Fig. 5), i.e. $\varepsilon(x,y)=1$, from uniformly occurring in
the near field. In this case, as illustrated by Fig. 5, all ground state
sublevels have chance to be strongly excited by the control pulse, leading to
$\varepsilon_{\rm in}$-independent $\overline{\delta T}$ saturation to $\sim
50\%$ at large enough $\mathcal{E}_{1}$ in Fig. 3b.

We present simulated $\overline{\delta T}$ according to Eq. (6) in the same
Fig. 3b as color domain plot, with various ellipticity color-coded for the
incident control (and the orthogonal probe) polarizations. The experimentally
measured data are matched to the simulation, by linearly rescaling the
$\overline{\delta T}$, $\mathcal{E}_{1}$ and $\Omega_{1}$ axes within the
uncertainties by the “steady state” absorption $A$ and laser power
measurements (Appendix E.4). Fairly good agreements are found between
theoretical and experimental $\overline{\delta T}$. The notable discrepancies
near $\Omega_{1}=2\pi\times 100~{}$GHz could be due to imperfect control/probe
polarization control in the experiment (Appendix E.3).

###  III.5 Robust composite control at the ONF interface

Figure 4: Three-pulse atomic state control through the ONF interface. (a) and
(d): The top plots give the intensity $|{\bf E}_{c}|^{2}$ (red) and the real
envelope ${\rm Re}[E_{c}]$ (blue) of the $N=3$ pulse sequences shaped from a
transform-limited $\tau_{0}=12$ ps pulse, with $\tau_{\rm d}$=24 ps inter-
pulse delay, and $\\{\varphi_{j}\\}=0$ and
$\\{\varphi_{j}\\}=\\{0,\pm\pi/2,0\\}$ respectively. The bottom plots give the
associated Bloch sphere 2-level dynamics with a broadened sub-pulse area
$\mathcal{A}_{j}=\mathcal{A}$. (b)(c): Simulated $f_{g}({\bf r})$ and $f({\bf
r})$ for the Fig. (d) $\\{\varphi_{j}\\}$ combinations. The red capital
letters in Figs. (a-d) mark the parameter combinations in Figs. (e-i). (e-i):
Simulated transient transmission $\overline{\delta T}$ vs
$\Delta\varphi_{1,3}$ at various peak $\Omega_{j}$, according to Eq. (11) with
the associated $\mathcal{E}_{j}=\mathcal{E}=2,6,17,50,20\times 10^{-3}$ pJ
sub-pulse energy (Eq. (33)). (j-n): Corresponding experimental
$\overline{\delta T}$ data. The experimental $\Omega_{j}$, $\overline{\delta
T}$ values are globally re-scaled to match the simulations, detailed in
Appendix E.4.

We now demonstrate robust population inversion at the ONF interface. This is
achieved by implementing the composite technique prescribed by ref. [44] with
our picosecond pulse sequence generator (Fig. 10). Due to technical reasons to
be discussed in Sec. IV.1, we limit the sub-pulse number to $N=3$ in this
demonstration.

As outlined in Appendix E.1, the picosecond sequence is generated by shaping
$\tau_{0}=12$ ps pulses into three sub-pulses, with $\tau_{\rm d}=24$ ps
inter-pulse equal delay, equal pulse energy $\\{\mathcal{E}_{j}\\}$, and
independently programmable phases $\\{\varphi_{j}\\}$. At a fixed
$\mathcal{E}_{j}=\mathcal{E}$, we perform 2D scan of the relative phases
$\Delta\varphi_{1,3}=\varphi_{1}-\varphi_{3}$ and
$\Delta\varphi_{2,3}=\varphi_{2}-\varphi_{3}$ in small steps to record the
transient transmission $\overline{\delta T}$ (Fig. 4(j-n)). As in Figs.
4(a)(d), the moderate $\tau_{\rm d}=2\tau_{0}$ leads to partially overlapping
sub-pulses. With accurate modeling in Appendix C to account for the overlap,
the composite scheme is effectively captured by non-overlapping pulses, as
assumed in the following.

At low pulse energy, we expect the picosecond control to be most efficient
when $\Delta\varphi_{1,3}=\Delta\varphi_{2,3}=0$ so the resonant spectra
density to $|g\rangle-|a\rangle$ excitation is maximized. This is the case in
Fig. 4e according to the simulation where the equal-phase point with optimal
$\overline{\delta T}$ is marked with “O”. The corresponding optical waveform
is plotted in Fig. 4a on the top. Indeed, for the equal-phase case, the
3-pulse control is equivalent to a single-pulse control with an elongated
duration $\tau_{c}$. Therefore, the control dynamics is not robust against
variation of optical intensity (nor polarization), as suggested by the
simplified 2-level Bloch sphere dynamics in the bottom plot of Fig. 4a.

However, with increased single-pulse energy $\\{\mathcal{E}_{j}\\}$ and the
associated $\\{\Omega_{j}\\}$, a transition of the optimal $\\{\varphi_{j}\\}$
occurs around $\mathcal{A}_{j}\approx\pi/3$. Beyond the point, the optimal
$\varphi_{2}$ takes none-zero values relative to the equal $\varphi_{1,3}$.
This is suggested by the simulations in Figs. 4(e-h), which agree globally
with the Figs. 4(j-m) experimental measurements without freely adjustable
parameters (Sec. E.4). The subtle difference between Fig. 4(e-i) and Fig.
4(j-n) are likely associated with slightly varying $\mathcal{E}_{j}$ during
the experimental phase scan (Sec. IV.1). With $\Omega_{j}\approx 2\pi\times
75$ GHz and linear $\varepsilon_{\rm in}=0$ (Fig. 4(h,m)), transient
transmission $\overline{\delta T}\sim 70\%$ are found at two phase
combinations with $\varphi_{2}=\pm\pi/2,\varphi_{1,3}=0$. For the case of
$\varphi_{2}=\pi/2$, our full level simulations suggest $\overline{\delta T}$
integrated over $\tau_{p}=2$ ns is supported by transient depletion of the
$|g\rangle$ state population with $f_{g}({\bf r})>80\%$ (Fig. 4b), over a
connected area at the ONF proximity by the picosecond impulse control to be
substantially larger than the 1-pulse case (Fig. 3a). This population
depletion is largely due to the $|g\rangle-|a\rangle$ inversion with $f({\bf
r})>80\%$ albeit across a slightly smaller area.

The efficient population inversion across the highly inhomogenuous near field,
as suggested by the Fig. 4(h,m) data, is a result of geometric robustness
inherent to the 2-level composite control [59] (also see Appendix A). As
illustrated in Fig. 4d, the “redundant” SU(2) rotation enables robust
$|g\rangle-|a\rangle$ inversion for the equal-area 3-pulse control with
$\mathcal{A}_{j}=\mathcal{A}\in(0.5\pi,\pi)$ when $\varphi_{2}$ takes the
value of $\pm\pi/2$ relative to $\varphi_{1,3}$. In both cases, the rotation
by the 2nd sub-pulse automatically cancels out the extra rotations by the
first and the third sub-pulses. As a result, an $f>80\%$ inversion efficiency
is uniformly achieved in the near field (Fig. 4b) to support the
$\overline{\delta T}\approx 70\%$ observation (Fig. 4m) for the thermal atoms.
We also refer readers to Fig. 1(d,ii)(e) for the $N=5$ example [44], where the
full-level simulation suggests nearly perfect inversion over a substantial
volume in the near field.

We finally remark on Raman transitions driven by the composite pulse with the
fairly long $\tau_{c}\sim 1/\omega_{{\rm hfs},g}$ duration (Appendix C.4). As
in Fig. 4b, compared to the expected inversion efficiency $f({\bf r})$, the
$\rho_{gg}$-depletion efficiency $f_{g}({\bf r})$ is larger, which is a result
of directional $|g\rangle\rightarrow|d\rangle$ transfer at
$\varphi_{2}=\pi/2$. Similarly, not shown in Fig. 4b is the “A” point with
$\varphi_{2}=-\pi/2$, where an opposite $|d\rangle\rightarrow|g\rangle$
transfer leads to reduced $f_{g}({\bf r})$ relative to $f({\bf r})$. However,
in neither case these Raman contributions notably affect the apparent
$\pm\pi/2$ symmetry in the transmission $\overline{\delta T}$ in Figs. 4(e-h)
and Figs. 4(j-m), since when the local polarization is approximately linear
($\varepsilon({\bf r})\ll 1$, Fig. 1c), the Raman transitions are largely
suppressed as long as $\omega_{{\rm hfs},a}\tau_{c}\ll 1$ [65] (Appendix C.4).
On the other hand, in Fig. 4(i)(n) $\overline{\delta T}$ at
$\varphi_{2}=-\pi/2$ (the “C” point) is substantially smaller than that for
$\varphi_{2}=\pi/2$ (the “D” point). The broken sign symmetry is associated
with substantial $|g\rangle\leftrightarrow|d\rangle$ Raman transfer, as
unveiled by comparing $f_{g}({\bf r})$ with $f({\bf r})$ in Fig. 4c according
to the full-level simulations. Here, with the control light in the circularly
polarized HE${}_{11}^{\sigma}$ mode, the ellipticity in the near field is
substantially larger (Appendix B). With atoms randomly initialized in
$|g\rangle$ and $|d\rangle$, the apparently inefficient depletion $f_{g}({\bf
r})$ at $\varphi_{2}=-\pi/2$ in Fig. 4c is associated with substantial
$|d\rangle\rightarrow|g\rangle$ transfer which negatively offsets the
depletion by $|g\rangle-|a\rangle$ inversion. Similarly, not shown in Fig. 4
are $f_{g}({\bf r})>f({\bf r})$ at “D” point with $\varphi_{2}=\pi/2$, as
those in Fig. 4b, but by a wider margin, due to the more efficient
$|g\rangle\rightarrow|d\rangle$ transfer. We note that the picosecond Raman
excitation is also expected to induce $|g\rangle-|d\rangle$ 2-photon
coherence, resulting in $\omega_{{\rm hfs},g}$-frequency transients in the
nanosecond probe transmission (Fig. 8). These transients are efficiently
averaged out with the $\tau_{\rm p}=2$ nanosecond integration (Fig. 2(b,c)),
not resolved by the $\overline{\delta T}$ measurements in this work.

##  IV Discussions

###  IV.1 Toward large $N$

The $N=3$ composite picosecond control demonstrated in Sec. III.5 relies on an
ergodic search for optimal pulse parameters. As in Fig. 4, the method supports
detailed investigation of the control dynamics across the parameter space by
comparing the experimental measurements with theory. On the other hand, the
“brutal force” approach becomes impractical at larger $N$, particularly when
the search time is constrained by slow experimental cycles. Ideally, the
relative amplitude and phase $\\{a_{j},\varphi_{j}\\}$ should be directly
programmed into an $N$-pulse sequence generator according to the optimal
control theory for the accurately modeled experimental system. When the
physical model of either the interaction or the pulse shaper itself is not
accurate, then a close-loop approach should be followed for in situ optimizing
of pulse parameters, similar to the pioneer works in nonlinear optics [75, 76]
and quantum information processing [77].

Efforts toward composite control at larger $N$ in this work is frustrated by a
pulse shaper parameter cross-talk, as mentioned in ref. [43]. As being
discussed there, the cross-talk is associated with acousto-optical
transduction, in particular the nonlinearity of multi-frequency rf
amplification for driving the single AOM in this work (Appendix E.1). The
cross-talk leads to enough complexity to prevent us from precisely modeling
the shaper itself when operating at the required efficiency for $N\geq 5$. The
cross-talk was also large enough to prevent a successful “close-loop”
optimization with our ONF setup. Toward directly programming optimal
picosecond control, we are currently working on improving the pulse shaper for
efficient arbitrary sequence generation at large $N$ [78].

###  IV.2 Summary and outlook

A fundamental quest in nanophotonics is to enhance optical nonlinearities
through confinements. The enhancements are not only instrumental to realizing
efficient nonlinear optics and spectroscopy [52, 53], but also may support
controllable interaction mediated by single confined photons [21]. In this
work, we suggest that highly precise, arbitrary control of atomic electric
dipole transitions can be achieved within picoseconds at nanophotonic
interfaces, despite the coupling strength inhomogenuity, using the NMR-
inspired composite technique in a power-efficient manner. It is important to
note that this kind of atomic state control is rarely achieved before, even in
free space [43].

Experimentally, this work takes a first step toward precise nanophotonic
control with the composite picosecond scheme. An optimally phased $N=3$
sequence is demonstrated to robustly invert the population of free-flying
atoms across an optical nanofiber. In particular, the $\sim 70\%$ reduction of
the evanescently coupled probe absorption, integrated over two nanoseconds of
mesoscopic atomic motion [69], strongly suggests $f>80\%$ population inversion
uniformly achieved around the nanofiber in the near field. We confirm the
accurate implementation of the geometric scheme [44, 59] by matching the
measurements with first-principle modeling of the mesoscopic light-atom
interaction. Our experimental work paves a practical pathway toward $N\geq 5$
composite control [44, 45] of atomic state, for atoms confined in the near
field [17, 18, 19], with exquisite precision.

Composite pulses are widely applied across fields to achieve robust control of
meta-stable quantum states [79, 80, 81, 82, 48, 83]. We expect many novel
applications by extending the technique to control two-level atoms at
nanophotonic interfaces. For example, it is well known that the optical
transition properties are modified at nanophotonic interfaces. By measuring
the line centers and widths and to compare with the free-space values, the
nanoscopic electro-magnetic perturbations including the van der Waals
interaction can be inferred [3, 4, 5, 6, 7]. Here, instead of relying on
regular linear spectroscopy, precise $\pi/2$ and $\pi$ pulses can be combined
to optimize the measurement efficiency and to auto-balance the light shifts
[30, 31]. More generally, the highly precise optical dipole control should
facilitate novel developments of integrated nonlinear optics [12, 13, 14, 15]
by improving the contrast of nonlinear optical modulation to the limit set by
2-level atoms. Repetitive application of the fast, arbitrary controls might
even enable one to dynamical decouple or amplify time-dependent perturbations
on demand, similar to those envisioned in NMR spectroscopy [32, 33].

Finally, we note the composite picosecond control can be applied to trapped
array of atoms [17, 18, 19] for efficient steering of collective couplings of
the atomic ensemble to the waveguide [34, 35, 84]. To illustrate the effect,
we come back to the $N=5$ scheme in Fig. 1 and consider a 1D lattice gas
trapped near ONF [17, 18, 19]. Following the $|g\rangle-|a\rangle$ inversion
by the $N=5$ composite pulse [44], a second $N=5$ composite pulse can drive a
nearly perfect return of atomic population back to the ground states. If the
second composite pulse is sent from the opposite direction of the ONF guide,
then a sub-wavelength-scale $e^{2i\beta_{\rm c}z}$ optical phase is patterned
to the state $|g\rangle$. Here $\beta_{\rm c}$ is the propagation constant of
the guided control pulses. As such, the delocalized $|g\rangle-|e\rangle$
dipole spin wave excited beforehand by the guided ${\bf E}_{\rm p}$ pulse to
the lattice [17, 18, 19] can be reversibly shifted into the subradiant domain
on demand [34, 35, 84], for accessing the nearly dissipationless, long-range
optical dipolar interaction dynamics at the ONF interface [37, 36, 21, 38, 39,
40, 85, 86, 87].

## Acknowledgement

We thank Professor Darrick Chang for very helpful discussions. We acknowledge
support from National Key Research Program of China under Grant No.
2022YFA1404204 and No. 2017YFA0304204, from National Natural Science
Foundation of China under Grant No. 12074083, 61875110, 62105191, 62035013,
62075192.

## Data availability

Data and simulation codes underlying the results presented in this paper are
not publicly available at this time but may be obtained from the authors upon
reasonable request.

##  Appendix A Composite control of an $I=0$ alkaline atom

Figure 5: D1 population inversion by resonant composite excitations, for an
alkaline atom without hyperfine structure. The level diagram for the
degenerate $S_{1/2}-P_{1/2}$ transition is given in (b). The quantization axis
is set along the light helicity vector ${\bf e}_{h}={\bf E}_{c}^{*}\times{\bf
E}_{c}/|{\bf E}_{c}|^{2}$, which is parallel to ${\bf k}$ for the free field
example. The ellipticity is defined as $\varepsilon=|{\bf e}_{h}|$. Fig. (d)
gives numerical results of inversion efficiency $f$ at various $\varepsilon$
for $N=$1, 3, and 5 pulses, as a function of single pulse area
$\mathcal{A}=|\int\Omega_{c}{\rm d}t|$, according to Eq. (11). The Fig. (c)
plots on the left give the time-dependent intensity $i$ (red line) and real
envelope ${\rm Re}[E_{c}]$ (blue line) of the pulses, with relative phases
according to ref. [44] marked on the top.

In this Appendix, we show that for alkaline atoms and if the hyperfine
splitting can be ignored, then the composite picosecond control can operate
well within a fine-structure manifold. We consider the exemplary composite
technique to invert the state population of a fictitious alkaline atom without
hyperfine structure ($I=0$), by driving the $J_{g}=1/2\rightarrow J_{a}=1/2$
D1 line with free-space optical pulses. The conclusions are applicable to the
D2 line, and to more general SU(2) controls [45] in a straightforward manner.
The simple picture is also applicable locally in the near field (Appendix B),
and to $I\neq 0$ atom in the $\tau_{c}\ll 1/\omega_{{\rm
hfs},g},1/\omega_{{\rm hfs},a}$ limit (Appendix C.4).

As in Fig. 5, in free space the quantization axis for the light-atom
interaction is naturally chosen along the ${\bf k}$ direction. More generally,
for the incident control electric field characterized by a slowly varying
complex envelope ${\bf E}_{c}$ and a helicity vector,

${\bf e}_{h}={\bf E}_{c}^{*}\times{\bf E}_{c}/|{\bf E}_{c}|^{2},$ (8)

the D1 transition is decomposed into $\sigma^{+}$ and $\sigma^{-}$ transitions
along the ${\bf e}_{h}$ quantization axis, with the associated Rabi
frequencies

$\begin{array}[]{l}\tilde{\Omega}_{\rm c}^{+}({\bf r},t)=\sqrt{2}{\rm
cos}(\theta/2)\Omega_{\rm c}({\bf r},t)\\\ \tilde{\Omega}_{\rm c}^{-}({\bf
r},t)=\sqrt{2}{\rm sin}(\theta/2)\Omega_{\rm c}({\bf r},t).\end{array}$ (9)

Here $\theta\in[0,\pi/2]$ is associated with the field ellipticity

$\varepsilon=|{\bf e}_{h}|$ (10)

as $\varepsilon={\rm cos}\theta$. The Rabi frequency $\Omega_{c}$ is defined
as

$\Omega_{c}({\bf r},t)=\frac{E_{\rm c}({\bf
r},t)}{\hbar}\frac{1}{\sqrt{3}}|\langle J_{g}||{\bf d}||J_{a}\rangle|.$ (11)

It is important to note that to apply Eqs. (8)(9) for the linearly polarized
${\bf E}_{c}$, the quantization axis needs to be perpendicular to ${\bf
E}_{c}$, i.e., as a limiting case of small $\varepsilon$.

Here we consider $\Omega_{\rm c}({\bf r},t)=\Omega_{\rm c}(t)$ and ${\bf
e}_{h}({\bf r})={\bf e}_{h}$ to be spatially uniform. Our goal is to design
certain pulse sequences to invert the population of the reduced D1 system
initialized in the unpolarized ground state,
$\rho(0)=\frac{1}{2}(|g_{+}\rangle\langle g_{+}|+|g_{-}\rangle\langle
g_{-}|)$. This is investigated with the simulation method outlined in Appendix
C by solving the Schrödinger equation for a control time $\tau_{c}$ and then
evaluate $f=\rho_{aa}\equiv\rho_{a_{+}a_{+}}+\rho_{a_{-}a_{-}}$, for the
$J_{g}=1/2\leftrightarrow J_{a}=1/2$ transition.

Clearly, the Rabi frequencies for the $\sigma^{\pm}$ couplings by Eq. (9) are
equal only for linearly polarized light. For general polarization state, it is
not possible to simultaneously invert the two sub-spins with $N=1$ single
“$\pi$”-pulse (Fig. 5(c,i)). In fact, in the limiting case of circular
polarized $\pi$-pulse (Fig. 5(d,i), the yellow curve with $\varepsilon=1$),
only $50\%$ of ground-state population can be inverted, albeit with a
$\sqrt{2}$-times larger Rabi oscillation frequency relative to the linear
polarized case (Fig. 5(d,i), the blue curve with $\varepsilon=0$). For the
linear polarization, the inversion near $\mathcal{A}=\pi$ scales as
$\rho_{aa}(\tau_{c})={\rm sin}^{2}(\mathcal{A}/2)$ and is therefore quite
sensitive to the pulse area $\mathcal{A}=|\int\Omega_{c}{\rm d}t|$, requiring
perfect control of light intensity.

For comparison, in Fig. 5(d,ii) and Fig. 5(d,iii) the population inversion
efficiencies are shown for a composite 3-pulse (Fig. 5(c,ii)) and 5-pulse
(Fig. 5(c,iii)) sequence respectively. Both sequences follow prescription by
ref. [44] as $\mathcal{A}_{j}=\mathcal{A}$ close to $\pi$ and phase
$\\{\varphi_{j}\\}$=$\\{0,\pi/2,0\\}$, $\\{0,5\pi/6,\pi/3,5\pi/6,0\\}$ for the
3- and 5-pulses. The 3-pulse sequence is experimentally exploited in Sec.
III.5. The geometric origin of the intensity-error resilience for the $N=3,5$
composite pulses are illustrated with the Bloch sphere picture in Fig. 4d,
Fig. 1e respectively. Here, comparing with the single pulse inversion in Fig.
5(d,i) that is perfected only for linearly polarized light at
$\mathcal{A}=\pi$, the composite pulse schemes are much more tolerant to
deviation of $\mathcal{A}$ from $\pi$, and can achieve $f\rightarrow 1$ even
by elliptically polarized excitations.

##  Appendix B The HE11 mode

Figure 6: Near field distribution of the normalized light intensity $i(x,y)$
and ellipticity $\varepsilon(x,y)$ for the circular HE${}_{11}^{+}$ mode of
the ONF in this work at $\lambda_{\rm c}=795$ nm.

Aided by precise knowledge of the nanofiber geometry [19], we follow ref. [54]
to calculate the field distribution around the nanofiber for the guided HE11
mode at specific wavelengths. The intensity and ellipticity distribution for
the control laser at $\lambda_{\rm c}=795$ nm, for the case of linearly
polarized HE${}_{11}^{y}$ mode with $\varepsilon_{\rm in}=0$ is presented in
Fig. 1c. For the ${\bf e}_{y}$ polarized incidence, we see the near field
${\bf E}_{{\rm HE}_{11}^{y}}$ is linearly polarized around $y=0$ with
$\varepsilon({\bf r})\ll 1$. On the other hand, for control pulses with
circular polarized HE${}_{11}^{+}$ mode with incident ellipticity
$\varepsilon_{\rm in}=1$, the electric field ${\bf E}_{{\rm HE}_{11}^{+}}={\bf
E}_{{\rm HE}_{11}^{x}}+i{\bf E}_{{\rm HE}_{11}^{y}}$ has a uniform local
ellipticity $\varepsilon({\bf r})\approx 0.95$ which is substantially larger
as shown here in Fig. 6. Similar mode intensity and polarization distributions
are numerically evaluated for the probe pulse at $\lambda_{\rm p}=780$ nm.

More generally, for ${\rm HE}_{11}$ mode with an incident $\varepsilon_{\rm
in}={\rm cos}(\Theta)$ ellipticity, with an angle $\phi$ between the incident
elliptical axis and the $x$-axis, the field is a coherent superposition of
that for the HE${}_{11}^{+}$ and HE${}_{11}^{-}$ modes weighted by ${\rm
cos}(\Theta/2)$ and ${\rm sin}(\Theta/2)$, given by

${\bf E}({\bf r})={\rm cos}(\frac{\Theta}{2})e^{i\frac{\phi}{2}}{\bf E}_{{\rm
HE}_{11}^{+}}+{\rm sin}(\frac{\Theta}{2})e^{-i\frac{\phi}{2}}{\bf E}_{{\rm
HE}_{11}^{-}}.$ (12)

The normalized intensity distribution $i({\bf r})$ and ellipticity
distribution $\varepsilon({\bf r})$ can be evaluated accordingly. The spatial
profile of the evanescent field is equipped to evaluate the light-atom
interaction to be detailed in the following.

##  Appendix C Modeling the D1 control

###  C.1 Level diagram

The level diagram for the full D1/D2 electric dipole interaction is summarized
in Fig. 7a. For the convenience of numerical calculation, we choose the ONF
guiding direction ${\bf e}_{z}$, along which the field intensity and
polarization distributions are invariant, as the fixed atomic quantization
axis. The choice of local helicity axis as quantization axis will be discussed
in Appendix C.4. To conveniently formulate the multi-level vectorial
interactions, we introduce Dirac kets $|g_{m}\rangle$,$|d_{m}\rangle$ to label
the Zeeman sublevels of the hyperfine ground states. Similarly, the excited
state Zeeman sublevels are label by $|a_{m}\rangle$, $|e_{m}\rangle$. The
$a,e$ symbols also index the total angular momentum $F$ of the corresponding
hyperfine levels [48].

###  C.2 The D1 Hamiltonian

We consider spectrum transform-limited picosecond D1 pulses from a mode-locked
laser with a temporal amplitude profile $\mathcal{P}(t)={\rm
sech}(1.76t/\tau_{0})$ [64]. After time-domain pulse shaping [43], the
composite sequence is sent through ONF to interact with atom at location ${\bf
r}$. The pulsed optical field at the ONF interface is described by a slowly-
varying envelop function

${\bf E}_{\rm c}({\bf r},t)={\bf E}({\bf r})S_{N}(\\{a_{j},\varphi_{j}\\},t),$
(13)

with a spatial profile according to Eq. (12), and temporally following the
composite profile according to Eq. (32),

$S_{N}(\\{a_{j},\varphi_{j}\\},t)=\sum_{j=1}^{N}a_{j}e^{i\varphi_{j}}\mathcal{P}(t-(j-1)\tau_{\rm
d}),$ (14)

with $|a_{j}|\leq 1$ and $\tau_{d}\gg\tau_{0}$.

With the atomic states defined earlier, the Rabi frequencies to drive the
$|g\rangle-|a\rangle$ and $|d\rangle-|a\rangle$ transitions are written as

$\Omega_{a_{n}s_{m}}^{l}({\bf r},t)=\langle a_{n}|{\bf E}_{\rm c}({\bf
r},t)\cdot{\bf d}_{l}|s_{m}\rangle/\hbar$ (15)

for all the $s=g,d$ states. Here ${\bf d}_{l}$ with $l=-1,0,1$ are the
electric dipole operators of the atom along $\\{{\bf e}_{-},{\bf e}_{z},{\bf
e}_{+}\\}$ directions respectively. Therefore, $n=m+l$ is required by
conservation of the magnetic quantum number. More generally, with the Clebsch-
Gordan coefficients $\mathcal{C}_{a_{n}c_{m}}^{l}$ and the D1 Rabi frequency
defined in Eq. (11), the Rabi frequencies can also be written as
$\Omega_{a_{n}s_{m}}^{l}({\bf r},t)=\sqrt{3}\Omega_{\rm c}({\bf
r},t)\mathcal{C}_{a_{n}s_{m}}^{l}$.

The D1 resonant dipole interaction under the rotating wave approximation is
written as

$\begin{array}[]{l}H_{\rm D1}({\bf
r},t)=\hbar\sum_{a}(\omega_{a}-\omega_{a0})\sigma^{a_{n}a_{n}}+\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hbar\sum_{s=g,d}(\omega_{s}-\omega_{g0})\sigma^{s_{m}s_{m}}+\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{\hbar}{2}\sum_{s={g,d}}\sum_{l}\Omega^{l}_{a_{n}s_{m}}({\bf
r},t)\sigma^{a_{n}s_{m}}+{\rm h.c.}\end{array}$ (16)

Here an implicit summation of repeated $n,m$ indices are assumed. The
$\omega_{a0},\omega_{g0}$ are decided by the energy of reference level in the
5P1/2 and 5S1/2 electronic states respectively, chosen as the top hyperfine
levels in this work. The control Rabi frequencies are accordingly written in
the $\omega_{a0,g0}$ frame with the resonant frequency
$\omega_{c}=\omega_{a0,g0}$ canceled. The Pauli matrices are defined as
$\sigma^{a_{n}s_{m}}=|a_{n}\rangle\langle s_{m}|$, similarly for
$\sigma^{a_{n}a_{n}}$ and $\sigma^{s_{m}s_{m}}$.

###  C.3 Numerical integration

With the Eq. (16) Hamiltonian, we integrate the evolution operator for the
composite pulse,

$U_{\rm c}({\bf
r},\\{\mathcal{E}_{j},\varphi_{j}\\})=\hat{T}e^{-i\int_{0}^{\tau_{c}}H_{\rm
D1}({\bf r},t^{\prime})dt^{\prime}/\hbar}$ (17)

to propagate electronic state of stationary atom at location ${\bf r}$. We are
particularly interested in high quality $|g\rangle-|a\rangle$ inversions [35],
which is characterized by an average inversion efficiency

$f({\bf r})={\rm tr}(U_{c}({\bf r})\rho^{(0)}U_{c}^{\dagger}({\bf r}){\bf
1}_{a}).$ (18)

Here ${\bf 1}_{a}=\sum_{a,n}|a_{n}\rangle\langle a_{n}|$ is the projection
operator into the 5P1/2 “$|a\rangle$” manifold. We similarly define ${\bf
1}_{g}=\sum_{m}|g_{m}\rangle\langle g_{m}|$ and ${\bf
1}_{d}=\sum_{m}|d_{m}\rangle\langle d_{m}|$. Here the initial atomic state is
described by the density matrix $\rho^{(0)}=\frac{1}{7}{\bf 1}_{g}$ for 85Rb
so that $\rho^{(0)}_{gg}=1$.

In addition, related to the experimental observation in this work is a
normalized ground state depletion efficiency for an initially unpolarized
atom, defined by Eq. (2) in the main text. The ground-state population after
the $U_{c}$ control is evaluated as

$\rho_{gg}({\bf r},\tau_{c})={\rm tr}(U_{c}({\bf
r})\rho^{(0)}U_{c}^{\dagger}({\bf r}){\bf 1}_{g}).$ (19)

Here the initial atomic state is described by the density matrix
$\rho^{(0)}=\frac{1}{12}({\bf 1}_{g}+{\bf 1}_{d})$ for 85Rb so that
$\rho_{gg}^{(0)}=7/12$.

Numerical evaluation of $f({\bf r}_{\perp})$, $f_{g}({\bf r}_{\perp})$
according to Eqs. (12-19), as those in Fig. 1, Fig. 3, Fig. 4 in the main
text, are implemented in a straightforward manner with Matlab [48].

Figure 7: (a): The full level representation of the Fig. 1b scheme in the main
text. With quantization axis along $z$, the local ${\bf E}_{\rm p,c}({\bf r})$
is decomposed into ${\bf e}_{\pm,z}$ directions to drive the $\Omega_{\rm
p,c}^{\pm,\pi}$ hyperfine couplings respectively. The fine and hyperfine
mixing lead to 2-photon Raman couplings among the ground-state hyperfine
sublevels (not shown in the diagram). (b): With $\tau_{c}\ll 1/\omega_{{\rm
hfs},g},1/\omega_{{\rm hfs},a}$, the light-atom interaction is decomposed into
2-level $\sigma^{\pm}$ couplings in the
$|nl_{J},I,\tilde{m}_{J},\tilde{m}_{I}\rangle$ basis ($n=5$, $I=5/2$ are
omitted in the figure). The tilde signs emphasize the choice of quantization
axis along the local helicity axis of light instead. The relative strength is
determined by the ellipticity $\varepsilon({\bf r})$ (Eq. (9)).

###  C.4 Composite control of hyperfine atom

Although the picosecond D1 interaction can be evaluated numerically, the
underlying physics can be obscured by the apparently complicated multi-level,
multi-photon couplings. To understand the control robustness enabled by
composite pulse techniques [44, 45], a simpler picture of D1 transition
dynamics without the hyperfine structure has been outlined in Appendix A. The
picture remains valid for atoms with hyperfine splittings if the optical
excitation is short enough: $\tau_{c}\ll 1/\omega_{{\rm hfs},g},1/\omega_{{\rm
hfs},a}$. In this limit, the light-atom interaction can be written in the
$|nl_{J},I,\tilde{m}_{J},\tilde{m}_{I}\rangle$ basis with separately
“conserved” electron and nuclear angular momenta ${\bf J}$ and ${\bf I}$ (Fig.
7b), with a local quantization axis along the helicity axis of the elliptical
field (Eq. (8)) for the HE11 field ${\bf E}_{c}({\bf r})$. Since the hyperfine
mixing of the $\tilde{m}_{J}+\tilde{m}_{I}=\tilde{m}_{F}$ levels are
negligible during $\tau_{c}$, the Fig. 7b 2-level $\sigma^{\pm}$ transitions
are decoupled like Fig. 5b. As such, the control of “optical spin” defined on
any pair of $|g_{m}\rangle$ (or $|g_{m}\rangle$ superposition) and the
corresponding $|a_{m}\rangle$ levels is decomposed into simultaneous SU(2)
control of the $|nl_{J},I,\pm
1/2,\tilde{m}_{I}\rangle\leftrightarrow|nl_{J},I,\mp 1/2,\tilde{m}_{I}\rangle$
spins, albeit with unequal $\tilde{\Omega}_{c}^{\pm}$ in general.
Nevertheless, for small $\varepsilon$ so the
$|\tilde{\Omega}^{+}/\tilde{\Omega}^{-}|$ ratio is moderate (Eq. (9)), then
the relative control errors can be suppressed, quite naturally, by the
intensity-error-resilient composite pulse techniques [44, 45] as in Fig. 5.

On the other hand, if the field polarization is close to be linear with
$\varepsilon\ll 1$, then the short pulse requirement for achieving efficient
2-level composite control can be relaxed to $\tau_{c}\ll 1/\omega_{{\rm
hfs},a}$ alone, since the tensorial Raman couplings are suppressed in alkaline
atoms [65]. This situation is particularly relevant when longer control pulses
with $\tau_{c}\sim 1/\omega_{{\rm hfs},g}\ll 1/\omega_{{\rm hfs},a}$ are
applied, such as for the $N=3$ composite pulses in Sec. III.5.

Beyond the short pulse limits, the composite pulses may be tailored to manage
the phase shifts associated with hyperfine splitting [35] and to suppress
multi-level dynamics [48], so as to effectively achieve few-level precise
control on the hyperfine manifold.

##  Appendix D Modeling the D2 absorption

Our goal in this section is to set up an “exact” model for predicting the
attenuation of the nanosecond probe pulse in the ONF - atomic vapor setup
outlined by Fig. 1 and Fig. 2.

###  D.1 The D2 Hamiltonian

Similar to Eq. (13), we consider the probe pulse with a near-field profile

${\bf E}_{\rm p}({\bf r},t)={\bf E}({\bf r})S_{\rm p}(t).$ (20)

The spatial profile ${\bf E}({\bf r})$ is again evaluated according to Eq.
(12), but at an orthogonal incident polarization according to the Fig. 2a
setup ($\Theta\rightarrow\Theta+\pi$). The nanosecond temporal profile, as in
Fig. 2, is described by $S_{\rm p}(t)$ with $|S_{\rm p}(t)|\leq 1$. The Rabi
frequencies to drive the $|g\rangle-|e\rangle$ and $|d\rangle-|e\rangle$
transitions are written as

$\Omega_{e_{n}s_{m}}^{l}({\bf r},t)=\langle e_{n}|{\bf E}_{\rm p}({\bf
r},t)\cdot{\bf d}_{l}|s_{m}\rangle/\hbar$ (21)

for all the $s=g,d$ states and with $l=-1,0,1$ for the $\sigma^{+}$, $\pi$,
and $\sigma^{-}$ transitions respectively.

The Hamiltonian during the D2 interaction under the rotating wave
approximation is written as

$\displaystyle H_{\rm D2}({\bf r},t)=$
$\displaystyle\hbar\sum_{e}(\omega_{e}-\omega_{e0})\sigma^{e_{n}e_{n}}+$ (22)
$\displaystyle\hbar\sum_{s=g,d}(\omega_{s}-\omega_{g0})\sigma^{s_{m}s_{m}}+$
$\displaystyle\frac{\hbar}{2}\sum_{s={g,d}}\sum_{l}\Omega^{l}_{e_{n}s_{m}}({\bf
r},t)\sigma^{e_{n}s_{m}}+{\rm h.c.}$

The notation follows the same conventions as those for Eq. (16). The probe
Rabi frequencies are accordingly written in the $\omega_{e0,g0}$ frame with
the resonant frequency $\omega_{\rm p}=\omega_{e0,g0}$ canceled. With
$\omega_{gd}=\omega_{{\rm hfs},g}\gg 1/\tau_{\rm p}$ for the nanosecond pulse
duration $\tau_{\rm p}$, the $|d\rangle$ states are invisible to the probe
pulse. We therefore effectively set $\Omega_{e_{n}d_{m}}^{l}=0$ to reduce the
computational cost.

###  D.2 Spontaneous emission

We account for spontaneous emission for both the D1 and D2 lines with a
stochastic wavefunction method [88, 89, 90]. For the purpose, six quantum jump
operators associated with D1 and D2 emissions are introduced as

$\displaystyle C_{a}^{l}$ $\displaystyle=\sqrt{\Gamma_{\rm
D1}}\sum_{a,s=g,d}\mathcal{C}_{s_{m},a_{n}}^{l}\sigma^{s_{m}a_{n}},$ (23)
$\displaystyle C_{e}^{l}$ $\displaystyle=\sqrt{\Gamma_{\rm
D2}}\sum_{e,s=g,d}\mathcal{C}_{s_{m},e_{n}}^{l}\sigma^{s_{m}e_{n}}.$

The $\mathcal{C}^{l}_{s,a}$ $\mathcal{C}^{l}_{s,e}$ are decided by the
Clebsch-Gordan coefficients for the hyperfine transitions. As suggested in
Sec. III.3 in the main text, the impact of surface interactions to the probe
absorption are negligible within the nanosecond evolution. We therefore set
$\Gamma_{\rm D1}=(27.7~{}{\rm ns})^{-1}$, $\Gamma_{\rm D2}=(26.2~{}{\rm
ns})^{-1}$ as the natural linewidths of the free atom [91].

The atomic density matrix $\rho(t)$ evolves according to the optical Bloch
equation as

$i\hbar\dot{\rho}=[H_{\rm
D2},\rho]-\frac{i\hbar}{2}\\{\hat{\Gamma},\rho\\}+\hbar\sum_{l}C_{a}^{l}\rho
C_{a}^{l\dagger}+C_{e}^{l}\rho C_{e}^{l\dagger}$ (24)

with

$\displaystyle\hat{\Gamma}=\hat{\Gamma}_{\rm D1}+\hat{\Gamma}_{\rm D2},{\rm
with}$ (25) $\displaystyle\hat{\Gamma}_{\rm D1}=\Gamma_{\rm
D1}\sigma^{a_{m}a_{m}}$ $\displaystyle\hat{\Gamma}_{\rm D2}=\Gamma_{\rm
D2}\sigma^{e_{n}e_{n}}$

As to be detailed next, we evaluate Eq. (24) by averaging the stochastic
wavefunctions [90] with the near-field optical parameters decided by classical
trajectories.

###  D.3 Sampling the thermal atomic distribution

As outlined in Sec. III.3 in the main text, we treat the center-of-mass motion
of thermal atoms classically with prescribed ballistic trajectories ${\bf
r}(t)={\bf r}_{0}+{\bf v}_{0}t$, which enter Eq. (24) as time-dependent
parameters.

In the Monte Carlo simulation to be introduced next, the initial $({\bf
r}_{0},{\bf v}_{0})$ is randomly sampled according to the phase-space
distribution $g({\bf r}_{0},{\bf v}_{0})$ for the thermal atomic vapor
uniformly surrounding the ONF as

$g({\bf r}_{0},{\bf v}_{0})=\sqrt{\left(\frac{M}{2\pi k_{\rm B}{\rm
T}}\right)^{3}}\mu({\bf r}_{0})e^{-M|{\bf v}_{0}|^{2}/2k_{\rm B}{\rm T}}.$
(26)

We set ${\rm T}=360$ K according to the in-vacuum thermometer readout. $M$ is
atomic mass of 85Rb and $k_{\rm B}$ is the Boltzmann constant. The stationary
$g({\bf r},{\bf v})$ distribution is maintained by a detailed balance of
microscopic transportation across the ONF near field. We ignore the impact of
ONF surface and optical forces to $g({\bf r}_{0},{\bf v}_{0})$.

###  D.4 Monte Carlo simulation

As in Fig. 2d in the main text, we consider a cylindrical volume around ONF
with a radius of $R=1~{}\mu$m and a length of $L=3$ mm to fully cover the
near-field interaction. Within the volume, we assume a uniform atomic density
$\mu=P/k_{\rm B}{\rm T}$ maintained by mesoscopic thermal transportation with
$P\approx 10^{-3}$ pascal. Numerically, individual atoms enter the volume
through the $r_{\perp}=R$ surface at random time. The atomic velocity,
incident angle, and flux density obey the Maxwell’s distribution at ${\rm
T}=360$ K. We choose the simulation time interval of $-t_{\rm
w}<t<\tau_{c}+\Delta t+t_{\rm p}$, with the control pulse at $0<t<\tau_{c}$
and probe pulse at $\tau_{c}+\Delta t<t<\tau_{c}+\Delta t+\tau_{\rm p}$. A
$t_{\rm w}=30$ ns time window is chosen for the classical Monte Carlo
trajectories to reach thermal equilibrium within the cylinder, before the
control pulse is fired.

To efficiently simulate the mesoscopic optical response including both the
classical and quantum randomnesses, we sample many classical atomic
trajectories ${\bf r}(t)={\bf r}_{0}+{\bf v}_{0}t$ and evaluate a single
stochastic wavefunction [90] for each ${\bf r}(t)$. The optical response is
then evaluated by the ensemble average of expectation values. The numerical
method is detailed as following.

The atomic initial state $|\psi(t=0)\rangle$ is set as one of the
$\\{|g_{m}\rangle,|d_{m}\rangle\\}$ internal ground states. If the trajectory
hits the nanofiber surface later, we assume the atom is immediately scattered
back to the volume with a new random velocity according to the Maxwell
velocity distribution, and with the internal state reset to one of the
$\\{|g_{m}\rangle,|d_{m}\rangle\\}$ states.

During $0<t<\tau_{c}$, we ignore atomic motion and evolve $|\psi(t)\rangle$
unitarily with $H_{\rm D1}({\bf r}_{0},t)$, leading to

$|\psi(\tau_{c})\rangle=U_{c}({\bf
r}(0),\\{\mathcal{E}_{j},\varphi_{j}\\})|\psi(0)\rangle$ (27)

with $U_{c}$ from Eq. (17).

We now consider the evolution of the stochastic wavefunction,
$|\psi_{S}(t)\rangle$, during $\tau_{c}<t<\tau_{c}+\Delta t+\tau_{\rm p}$
subjected to the $H_{\rm D2}$ Hamiltonian and $C_{a,e}^{l}$ quantum jumps.
Here, taking advantage of the fact that the D1 excited states $|a\rangle$ are
not affected by the D2 probe couplings, we reduce the simulation complexity by
restricting $|\psi_{S}(t)\rangle$ within the D2 manifold, initiated at a
“quantum jump” time $t^{j}_{S}$ which is the solution to [90]

$\rho_{aa}(\tau_{c})e^{-\Gamma_{a}t^{j}_{S}}=r.$ (28)

Here $r\in[0,1]$ is a random number.
$\rho_{aa}(\tau_{c})=\langle\psi(\tau_{c})|{\bf 1}_{a}|\psi(\tau_{c})\rangle$
is the atomic population in states $|a\rangle$ immediately after the control
pulses. Given $\rho_{aa}(\tau_{c})>r$ to guarantee a $t^{j}_{S}$ solution, the
internal state for the D2 simulation is initialized as

$|\psi_{S}(t^{j}_{S})\rangle=C_{a}^{l}|\psi(\tau_{c})\rangle,$ (29)

heralded by a $l-$polarized emission of D1 photon. The branching ratio for the
$l=\sigma^{-},\pi,\sigma^{+}$ emission is decided by the relative
probabilities
$p_{l}=\langle\psi(\tau_{c})|C_{a}^{l\dagger}C_{a}^{l}|\psi(\tau_{c})\rangle$.

On the other hand, for $\rho_{aa}(\tau_{c})<r$ so that there is no solution to
$t^{j}_{S}$, then we set $t^{j}_{S}=\tau_{c}$ with atomic state simply being
projected as

$|\psi_{S}(t^{j}_{S})\rangle=({\bf 1}_{g}+{\bf 1}_{a})|\psi(\tau_{c})\rangle.$
(30)

During $t^{j}_{S}<t<\tau_{c}+\Delta t+\tau_{p}$, the stochastic wavefunction
$|\psi_{S}(t)\rangle$ evolves according to the effective Hamiltonian $H_{\rm
eff}=H_{\rm D2}-i\hat{\Gamma}_{\rm D2}/2$ and is probabilistically interrupted
by the D2 emission associated with $C_{e}^{l}$ [90].

Following Eq. (4) in the main text, the D2 probe scattering rate is
numerically evaluated as

$\bar{\gamma}(t)=\frac{1}{N_{S}}\sum_{S=1}^{N_{s}}\theta(t,t^{j}_{S}){\rm
Im}\left[\sum_{e,l}\langle\psi_{S}|e_{m}\rangle\langle
g_{n}|\psi_{S}\rangle\Omega_{e_{m}g_{n}}^{l}\right]$ (31)

with normalized state vectors $|\psi_{S}(t)\rangle$. The step function
$\theta(t,t_{S}^{j})$ is 1 for $t>t_{S}^{j}$, and 0 otherwise. By removing the
control evolution, straightforward simplification of Eq. (31) can evaluate the
scattering rate $\bar{\gamma}_{0}(t)$ for the atomic vapor in absence of the
control pulses. For a particular experimental configuration,
$\bar{\gamma}(t)$, $\bar{\gamma}_{0}(t)$ are typically evaluated with
$N_{S}=10^{6}$ trajectories, each takes about 1 h time on a PC cluster
(Intel-i7 34 cores). The probe absorption $A$ and the normalized difference
$\overline{\delta T}$ are then evaluated according to Eq. (5) in the main
text.

The Monte Carlo simulation enables us to look into nanosecond absorption
dynamics during the nanosecond probe itself (Fig. 2b,c) , which is not
resolved experimentally (Appendix E.2). As in Fig. 8 numerical example for the
linear $\varepsilon_{\rm in}=0$ and circular $\varepsilon_{\rm in}=1$ HE11
couplings, in absence of the picosecond control pulses (black lines), the
probe transmission $T(t)$ always relax from $T(\tau_{c})=1$ to the steady-
state $T_{\rm ss}\approx 85\%$ in this example within one nanosecond.
Physically, the initial $T=1$ is associated with the zero atomic dipole
$\langle{\bf d}\rangle$ initially, which is then driven by ${\bf E}_{\rm p}$
in a time-dependent, off-resonant fashion due to the 100 MHz-level Doppler
shift and nanosecond transient associated with the atomic motion along $z$ and
$(x,y)$ respectively. Thermal ensemble average of the oscillatory scatterings,
as by Eq. (4), leads to the rapid relaxation of $T$ to $T_{\rm ss}$ within a
nanosecond. The relaxation time is predominantly decided by the $2\sqrt{2{\rm
ln}2}\beta_{\rm p}\sqrt{k_{B}{\rm T}/M}\approx 2\pi\times 700$ MHz Doppler
width and is further shortened by the transient broadening of the mesoscopic
gas. Indeed, the choice of $\tau_{\rm p}=2~{}$ns in this work was made to
balance the noise level with the rapid decay of the transient absorption.

On the other hand, as the colored solid-curve examples in Fig. 8, after
subjecting an $N=1$ picosecond control ($\Delta t=0$), the probe transmission
$T(t)$ decreases much slower. Physically, the picosecond excitation breaks the
atomic distribution from the Eq. (26) thermal equilibrium, and it takes time
for “new” atoms in $|g\rangle$ to transport to the interface, albeit with a
random “time of arrival” microscopically and therefore with fast dipole
transients being self-suppressed on average. Similar relaxations are found in
the simulation for the $N=3$ composite excitations given by the dashed lines
in Fig. 8(a)(b), where the simulation parameters are according to the “A”-“D”
combinations in the Fig. 4 measurements.

Similar to the slowed relaxation in the Fig. 8 curves of $T(t)$ subjected to
the control pulses, for the $\tau_{\rm p}$-integrated probe absorption (the
normalized transient transmission $\overline{\delta T}$ by Eq. (1)) in the
$\Delta t$-delayed control-probe measurements (Fig. 2e), the recovery to the
steady-state value is also decided by the refilling of ground state atoms into
the ONF near-field region, primarily by the mesoscopic transportation from far
away. With in mind the exponential form of the evanescent tail (Fig. 1), it is
easy to show that the “refilling time constant” depends logarithmically on the
control strength $\Omega_{\rm c}$. We note the hyperfine Rabi frequency is
itself polarization dependent (Fig. 5b, Fig. 7). As a result, the recovery
dynamics is slightly more complex in the $\varepsilon_{\rm in}=0$ linear
incidence case (Fig. 8a) due to the more strongly varying near-field
$\varepsilon(x,y)$ (Fig. 1c). By setting $\mathcal{E}_{1}=0.5$ pJ in the
simulation according to the Fig. 2e measurements ($\varepsilon_{\rm in}=0$)
and by rescaling the experimental $\overline{\delta T}$ as explained in Sec.
E.4, fairly good agreement is obtained between the experimental and numerical
$\Delta t$-$\overline{\delta T}$ data Fig. 2e.

Notably, in Fig. 8(a)(b) we see weak modulation in all the $T(t)$ curves with
$\sim 0.3$ ns periodicity, a subtle interference between the transient ${\bf
d}_{eg}$ and ${\bf d}_{ed}$ dipoles. As discussed in Sec. III.5, the ${\bf
E}_{\rm c}$ picosecond control is long enough to induce $|g\rangle-|d\rangle$
hyperfine excitation, particularly if the optical polarization is circular. On
the other hand, the rapidly rising nanosecond probe pulse, even though with
$\omega_{\rm p}$ centered to the $\omega_{eg}$ hyperfine transition (Fig. 1b,
Fig. 7a), has enough spectrum component to off-resonantly excite the
$|d\rangle-|e\rangle$ transition. The “quantum beat” in all the Fig. 8 curves
is a result of the $|d\rangle-|g\rangle$ Raman coherence induced by the
picosecond control pulse, so that the tiny ${\bf d}_{ed}$ dipole is phased to
interfere with the ${\bf d}_{eg}$ dipole at $\omega_{{\rm hfs},g}=2\pi\times
3.04$ GHz. The beat note is more pronounced for circular HE11 excitation with
$\varepsilon_{\rm in}=1$, as expected according to Sec. III.5. Nevertheless,
these $\omega_{{\rm hfs},g}$-scale optical transients average out in the
$\delta T$ measurements with $\tau_{\rm p}$-integration. The rapid average of
coherent transients supports a much simpler diffusive average model, to be
discussed in the following.

Figure 8: Monte Carlo simulation of the transient transmission $T(t)$ during
the $\tau_{\rm p}=2$ ns probe (Also see Fig. 2(b)(c)). Figs. (a)(b) are with
linear ($\varepsilon_{\rm in}=0$) and circular ($\varepsilon_{\rm in}=1$) HE11
incidences respectively. For $N=1$ control, the pulse energy is
$\mathcal{E}_{1}=0,0.01,0.05,0.15,0.3,0.6$ pJ, associated with peak Rabi
frequencies $\Omega_{1}=2\pi\times 0,30,75,125,180,255$ GHz (associated with
Fig. 3b in the main text). For $N=3$ control, the phase combination is set
according to the marks “A”-“D” in Fig. 4(h)(i) in the main text. The pulse
energy are $\mathcal{E}_{j}=0.05$ pJ (“A”,“B”) and $\mathcal{E}_{j}=0.02$ pJ
(“C”,“D”) respectively with the associated $\Omega_{j}=2\pi\times 75,50$ GHz.

###  D.5 The diffusive average model

The diffusive average model by Eq. (6) in the main text starts with
calculating $f_{g}({\bf r})$ according to Eqs. (19)(2) and $i_{\rm p}=|{\bf
E}_{\rm p}|^{2}$ according to Eqs. (20)(12). The model is based on the
observation that for atom with velocity ${\bf v}$ and location ${\bf r}(t)$,
its contribution to the overall photon scattering rate (Eq. (4) in the main
text) is on-average proportional to the population $\rho_{gg}(t)$ and the
probe intensity $i_{\rm p}({\bf r}(t))$ in the linear probe regime, where we
further have $\rho_{gg}(t>\tau_{\rm c})\approx\rho_{gg}(\tau_{c})$ within the
nanosecond $\tau_{\rm p}$ time. For thermal atoms uniformly sampling the phase
space, we expect errors associated with the coherence transients to average
out quite efficiently. The residual transients that survive the average, as
those exemplified in Fig. 8, have additional chance to be suppressed by the
$\tau_{\rm p}$-average and then cancel each other in the $\overline{\delta T}$
ratio (Eq. (5)). Finally, since the “impulse” excitation hardly change the
atomic velocity distribution, the reductions of average atomic absorption by
the Doppler and transient broadening are largely shared by the unperturbed and
the excited vapors. The effects are thus expected to be largely cancelled in
the $\overline{\delta T}$ ratio (Eq. (5)) too.

To confirm the validity of the Eq. (6) approximation we repeat the Fig.
4(h)(i) calculation along the $\Delta\varphi_{1,3}=0$ line using the Monte
Carlo method (Eq. (5)). The results are presented in Fig. 9(c)(d) with
circular symbols. By adjusting the diffusive length to be $\xi_{\rm
p}^{\prime}=0.9(\sqrt{2}v_{\rm T}\tau_{\rm p}/2)$ (with $\sqrt{2}$ to account
for the 2D motion), the difference of $\overline{\delta T}$ between the
predictions by the “exact” and diffusive average models is typically less than
0.05.

Figure 9: Comparison of the $\overline{\delta T}$ simulations based on the
diffusive average (Eq. (5)) and the Monte Carlo method (Eq. (6)). Here (a,b)
are the same simulation data as those in Fig. 4(h,i). The solid lines in (c,d)
are extracted from (a,b) along the $\Delta\varphi_{1,3}=0$ dashed lines. The
circles in (c,d) are from the Monte Carlo simulations.

##  Appendix E Experimental Detail

###  E.1 Pulse sequence generation system

Figure 10: Schematic of the picosecond pulse sequence generation system [43]
for the ONF-interface atomic state control in this work. A transform-limited
picosecond pulse is diffracted by a multi-frequency-driven acousto-optical
modulator (AOM) into multiple paths, retro-diffracted by a grating to double-
pass the AOM with tunable delay $\\{\tau_{j}\\}$. The output pulses along the
time-reversed direction is separated from the input using a polarization beam
splitter (PBS), repetition rate prescaled and pulsed picked (not shown),
before being coupled to a single-mode fiber toward the nanofiber experiment.
Here $\tau_{j}=(j-1)\tau_{d}$ with $\tau_{d}=24$ ps. The amplitude and phase
of each pulse, $\\{a_{j},\varphi_{j}\\}$, is programmed by the amplitude and
phase of radio-frequency (rf) sideband signals driving the AOM.

Our nanofiber-atom interface technique relies on coherent generation of
composite sequence of picosecond pulses with precisely tunable amplitude and
phase, $\\{a_{j},\varphi_{j}\\}$, to optimize the atomic state contrl. In this
work, the composite picosecond pulse generator is based on a time-domain pulse
shaping method developed recently [43], as schematically illustrated in Fig.
10 and briefly summarized in the following.

We use a Ti-Sapphire mode-locked laser (Spectra-Physics Tsunami system) to
generate transform-limited picosecond pulses with $\tau_{0}=12~{}$ps at a
repetition rate of $f^{(0)}_{\rm rep}=80$ MHz. The pulsed output, referred to
as ${\bf E}_{\rm in}(t)$ in the following, is directed to a multi-frequency
$\\{\omega_{j}\\}$-driven double-pass acousto-optical modulator (AOM). Instead
of retro-reflecting the multi-diffracted beams, we use a high-density grating
(2400 line/mm) to retro-diffract each path backward for the second AOM
diffraction. The grating retro-diffraction introduces a path-dependent delay
$\tau_{j}=2L_{j}/c\propto\omega_{j}-\omega_{1}$ relative to $\tau_{1}$. Among
the AOM double-diffracted beams, the direction-reversed beam is picked by a
polarization beamsplitter, repetition rate pre-scaled [92] and pulse-picked
(not shown in Fig. 10) to $f_{\rm rep}=2$ MHz, single-mode selected, before
being combined with the probe and coupled into ONF (Fig. 1). In this work, the
tunable delay for $\tau_{j}=(j-1)\tau_{d}$ is set by $\tau_{d}=24$ ps. Taking
into account an overall loss coefficient $\kappa$, the shaped composite pulse
output, as a sum of individually delayed sub-pulse ${\bf E}_{j,\rm out}(t)$,
has a complex envelop function [43]

$\displaystyle{\bf E}_{\rm c}(t)$ $\displaystyle=\kappa\sum_{j}^{N}{\bf
E}_{j,\rm out}(t),$ (32) $\displaystyle=\sum_{j}^{N}a_{j}e^{i\varphi_{j}}{\bf
E}_{\rm in}(t-\tau_{j})$

at the ONF interface. The amplitudes and phases of the composite pulses
$\\{a_{j},\varphi_{j}\\}$ are controlled by that of the $f_{\rm
rep}$-synchronized rf waveforms $\\{a^{c}_{j},\varphi^{c}_{j}\\}$ as
$a_{j}\propto\kappa(a_{j}^{\rm c})^{2}$ and $\varphi_{j}=2\varphi_{j}^{\rm
c}+$constant for the weakly driven AOM, as described in ref. [43]. With the
multiple beams sharing a common optical path, the composite pulse shaped by
the rf-programmed multi-frequency AOM is amplitude and phase stable over
(many) hours. Inspired by related work for “direct space-to-time pulse
shaping” [61, 62, 63], we refer this method as “direct ${\bf k}$-space-to-time
pulse shaping”.

###  E.2 High speed signal acquisition and averaging

Figure 11: High speed signal acquisition and averaging. (a): The schematic
setup, with the processed signals at each stages represented by the blue
curves in (b), from the nanosecond voltage pulse pairs at the APD output, to
the slowly varying voltage levels sampled by the DAQ card. The synchronized
signals to control the two multipliers (Mlp1 and Mlp2) are generated by an
arbitrary sequence generator (CIQTEK ASG8000) and an arbitrary waveform
generator (CIQTEK AWG4100) respectively. The control pulse is combined with
every other probe pulse for $t_{\rm on}=1.25$ ms and then turned off for
$t_{\rm off}=1.25$ ms for removing the voltage offset.

In the control-probe experiment, we keep the peak power of the nanosecond D2
probe at $P_{\rm p}\approx 10$ nW level to avoid saturation. The signal is
close to the noise-equivalent power of the 1 GHz avalanche photodiode (APD)
module (Hamamatsu C5658). Fortunately, taking advantage of rapid probe
repetition at 4 MHz, enabled by the rapid recovery of the thermal vapor-ONF
system (Fig. 2e), it is still possible to retrieve transient transmission
$\delta T$ with a $0.1\%$ sensitivity within seconds by rapidly averaging the
difference of transmission induced by the control pulses.

We use off-the-shelf rf components to construct a signal averager
schematically illustrated in Fig. 11. The $\tau_{\rm p}=2~{}$ns probe pulse is
sent to probe the ONF system with a $t_{\rm rep}^{\prime}=250$ ns repetition
time. A synchronized control pulse is fired immediately before every other
probe pulse to form a $\delta T$ measurement cycle with the nanosecond pulse
pairs. As outlined in Fig. 11, after the APD module receive the pulse pairs,
they are amplified (Mini-Circuits ZFL-500+) and subjected to two multipliers
(AD834) for time-domain windowing and pulse sign reversals. The processed
signals are then averaged by a 10 kHz bandwidth low-pass filter. The
integrated signal level $\delta V$ reflects the difference of nanosecond probe
transmission induced by the picosecond control. The control pulse is combined
with every other probe pulse for $t_{\rm on}=1.25$ ms and then turned off for
$t_{\rm off}=1.25$ ms for removing the voltage offset. The alternating
measurement and calibration cycles ensure any slowly varying electronic offset
is removed. To compare $\delta V$ with the probe signal level itself, we
remove the second pulse of the probe pulse pair during a $t_{0}=20$ ms
interval of $t_{\rm int}=200$ ms integration cycle to record $V_{0}$. The low-
passed signal is send to computer through a data acquisition (DAQ) card (NI
USB-6363). We integrate $10^{7}$ differential measurements in two seconds to
obtain a $\delta T=\delta V/V_{0}$ readout. The measurement is remarkably
accurate with a $0.1\%$ sensitivity, which is inferred from the rms deviation
from repeated $\delta T$ readouts.

###  E.3 HE11 mode polarization control and measurement

Figure 12: Simulated (a) (Eq. (6)) and experimentally measured (b)
$\overline{\delta T}$ at $\mathcal{E}_{1}=0.2$ pJ (Appendix E.4), when the
polarizations of the orthogonally polarized control and probe pulses (Fig. 2a)
are scanned by the HWP and QWP plates before they are coupled to the HE11 mode
of ONF. Two angular combinations for realizing linear and circular polarized
HE11 modes are marked with “C” and “L” in the experimental data graph in (b).

We use a pair of automated half-wave plate (HWP) and quarter-wave plate (QWP)
in front of the nanofiber coupler (Fig. 2a) to control the polarization states
of the HE11 mode, for the orthogonally polarized control and probe pulses. An
example polarization-dependent $\overline{\delta T}$ measurement with a 2D
polarization scan is shown in Fig. 12. As suggested by Fig. 3 of the main text
and according to the numerical simulations, the polarization dependence of the
transient probe transmission with single control pulses is most pronounced
near $\mathcal{E}_{1}\sim$0.2 pJ, which is also the pulse energy value set in
this measurement. Representative waveplate angular combinations with
$\varepsilon_{\rm in}=0$ and $\varepsilon_{\rm in}=1$ are marked with “L” and
“C” respectively. The angular combinations are sampled in picosecond controls
with single and composite pulses as those in Fig. 3 and Fig. 4.

###  E.4 $A$ and $\mathcal{E}_{j}$ estimations in presence of slow drifts

To obtain normalized $\overline{\delta T}$ as those in Figs. 2 3 4 12 from the
transient transmission $\delta T$ measurements as outlined in Appendix E.2,
the atomic absorption $A$ across ONF (Fig. 2b) needs to be accurately measured
for the normalization (Eq. (1)). In addition, to compare the experimental
$\overline{\delta T}$ with simulations, we need to estimate individual pulse
energy $\mathcal{E}_{j}$ and the peak Rabi frequency $\Omega_{j}$ for the
composite pulses. In this work, both $A$ and $\mathcal{E}_{j}$ are estimated
with moderate accuracies. In the following we detail the procedure to measure
these parameters and to rescale their peak values against unknown offsets.

As outlined in the main text, we record the “steady state” absorption $A$,
i.e., the nanosecond probe absorption by the mesoscopic vapor in equilibrium
in absence of the control pulse excitation, with CW absorption spectroscopy
[93]. For all the experiments in this work, the $A$ recordings were made only
at the beginning of the control-probe measurements, due to technical reasons.
Fortunately, taking advantage of the simplicity in the single-pulse absorption
depletion measurement as those in Fig. 3, we are able to infer $A$ in situ,
using the approximate $A\approx 2\delta T$ relation from the $\delta T$
measurement itself at strong enough $\mathcal{E}_{1}$-excitations (Sec.
III.4). By analyzing multiple data sets taken during different periods of this
project, we conclude that during the initial hour of the rubidium dispenser
operation, the ONF local vapor pressure could vary substantially to affect
$A$. The slow pressure relaxation is likely associated with surface atomic
adsorption across the vacuum chamber. After the initial hour, the vapor
pressure tends to stabilize for hours, during which most of the data presented
in this work were taken. The stable measurement condition supports faithful
retrieval of $\overline{\delta T}$ features as a function of the control
parameters, as those in Figs. 2 3 4 12. However, we expect the absolute value
of $A$ to be scaled by the unknown pressure drift after hours since the
initial measurement. To counter the effect we hand-rescale $A$ as detailed
below to best match the numerical simulations.

To estimate the control pulse parameters $\Omega_{j}$, we measure the incident
power $P$ of the control pulses with a calibrated power meter (Thorlabs
PM160), before the pulses are coupled to ONF (Fig. 10). Taking into account
the pre-calibrated fiber coupling efficiency $\eta_{C}\approx 50\%$, the pulse
energy $\mathcal{E}_{j}$ is estimated as $\mathcal{E}_{j}=\eta_{C}t_{\rm
rep}P/N$ for the equal-amplitude $N-$pulse sequence. Here $t_{\rm rep}=500$ ns
is the period of the pulse picking. Automatic adjustments of the pulse energy
is achieved by scanning the rf signal amplitudes $a_{j}^{\rm c}$ in the pulse
generation system, which are pre-calibrated to the output pulse power.

We define $\Omega_{j}$ as the peak Rabi frequency of the picosecond pulse at
the nanofiber surface averaging over the angular direction $\phi$,
$\Omega_{j}\equiv a_{j}\langle(\Omega_{\rm c}({\bf
r}))_{r=d/2}\rangle_{\phi}$. The $\Omega_{j}$ is estimated according to Eqs.
(13)(11), with ${\bf E}_{c}({\bf r},t)$ inferred from

$\frac{1}{2}\int\varepsilon_{0}\mathcal{N}^{2}|{\bf E_{\rm c}({\bf
r},t)}|^{2}v_{c}{\rm d}^{2}{\bf r}_{\perp}{\rm d}t=N\mathcal{E}_{j}.$ (33)

Here $v_{c}=\omega_{c}/\beta_{c}$ is the phase velocity of the ONF guided
control pulses ($\beta_{c}$ is the associated propagation constant) and
$\mathcal{N}({\bf r}_{\perp})$ is the transverse ONF refractive index profile.

We denote the pulse energy $\mathcal{E}_{j}$ from the estimation above as
$\mathcal{E}_{j,{\rm raw}}$. To simulate the experiments, in the simulations
we assume accurately programmed control pulse phase $\varphi_{j}$ and pulse
interval $\tau_{d}$ according to Eq. (14). By rescaling $\mathcal{E}_{j,{\rm
raw}}$ with a common factor $\eta_{\rm P}$, the experimentally observed
$\overline{\delta T}$ features can be reproduced as those in Fig. 2 3 4, 12
with $\Omega_{j}=\sqrt{\eta_{\rm P}}\Omega_{j,{\rm raw}}$. For each graph in
these figures, we typically find $\eta_{\rm P}\approx 1.3\sim 1.6$ which is
centered around $\eta_{\rm P}=1.45$. This level of systematic offset is
expected, considering the moderate accuracy in the optical power estimation.
On the other hand, the $\pm 10\%$ uncertainty around the mean $\eta_{P}$,
applied to different graphs of Figs. 2 3 4, 12, is illustrated in Fig. 3b with
the horizontal double-sided arrow.

Finally, while the simulated $\overline{\delta T}$ patterns are matched to the
measurements by the $\Omega_{j}$ adjustment, the simulation typically suggests
“actual” $\overline{\delta T}$ to be different slightly from the
$\overline{\delta T}_{\rm raw}$ “raw data”. The difference is highly likely
due to the aforementioned rubidium pressure drifts. We therefore also assign a
rescaling factor $\eta_{\delta T}$ to $\overline{\delta T}_{\rm raw}$, leading
to $\overline{\delta T}=\eta_{\delta T}\overline{\delta T}_{\rm raw}$ in the
figures to best match the numerical $\overline{\delta T}$ values. The
$\eta_{\delta T}$ for Fig. 2, Fig. 3, Fig. 4(j-m), Fig. 4(n) and Fig. 12 are
$\\{0.53,1.07,0.82,1.05,1.07\\}$ respectively (The Fig. 4(j-m) and Fig. 4(n)
data were taken on different days.). The adjustments are therefore within $\pm
20\%$ as suggested by the vertical double-sided arrow in Fig. 3d, except for
the Fig. 2e data which is likely related to its early measurement time within
the initial hour of the rubidium dispenser operation. We note that despite the
uncertainty associated with the slow drifts, the consistent match between
groups of measurements, such as the Fig. 3b single pulse data, and the Fig.
4(e-h) simulation and the uniformly down-scaled Fig. 4(j-m) experimental data,
all strongly suggest close-to-ideal performance of the composite picosecond
control in this work. In particular, an optimal absolute $\overline{\delta
T}\approx 70\%$ is likely reached in the Fig. 4(m) data, assuming
$\overline{\delta T}\approx 50\%$ at
$\Delta\varphi_{1,3}=\Delta\varphi_{2,3}=0$, as by Fig. 9, similar to the
single-pulse saturation effect discussed in Sec. III.4.

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