Super-radiance reveals infinite-range dipole interactions through a nanofiber

TL;DR

This paper demonstrates infinite-range dipole interactions mediated by a nanofiber, evidenced by super-radiance among atoms separated by hundreds of wavelengths, opening new avenues for quantum information and many-body physics.

Contribution

First experimental proof of infinite-range atom-atom interactions via a nanofiber, enabling collective quantum states over macroscopic distances.

Findings

Observation of super-radiance among atoms separated by hundreds of wavelengths

Measurement of sub-radiance in a nanofiber system

Evidence of collective radiative decay mediated by a waveguide

Abstract

Atoms interact with each other through the electromagnetic field, creating collective states that can radiate faster or slower than a single atom, i.e. super- and sub-radiance. The generation and control of such states by engineering the dipolar interactions between atoms can enable new tools for atomic-based technologies. Atom-atom interactions in free space are limited in range, since the amplitude of the radiated field decreases inversely with distance. When the field is confined to one dimension it enables infinite-range interactions. This has been observed for atoms in an optical cavity, but remains to be proven in one-dimensional waveguides, where the extent of the interactions is not limited by the cavity size. Here we present the first report of infinite-range interactions between macroscopically separated atomic dipoles mediated by an optical waveguide. This is evidenced by the collective radiative decay of a single photon distributed between distant atoms. We use cold $^{87}$Rb atoms in the vicinity of a single-mode optical nanofiber (ONF) that coherently exchange evanescently coupled photons through the ONF mode. In particular, we observe super-radiance of a few atoms separated by hundreds of resonant wavelengths. This effect is not possible for atoms separated by more than a wavelength interacting through free space. The same platform allows us to measure sub-radiance, a rarely observed effect, presenting a novel tool for quantum optics. This result constitutes a proof-of-principle for collective behavior of macroscopically delocalized atomic states, a crucial element for new proposals in quantum information and many-body physics. Given the application of one-dimensional waveguides in photonic-based quantum technologies, we envision infinite-range interactions as the natural next step towards interconnecting quantum systems on scales suitable for practical applications.

Figures

Equations

$$|{\Psi }_{\alpha }(t)⟩\propto e^{-\frac{1}{2}\left({\gamma }_{\alpha }+i{\Omega }_{\alpha }\right)t}\sum _{j=1}^N{c}_{\alpha j}|g_1{g}_2\cdot \cdot \cdot e_j\cdot \cdot \cdot g_N⟩$$

$$\dot{\rho }=-i\left[H_{\text{eff}},\rho \right]+\mathcal{L}[\rho ].$$

$$H_{\text{eff}}$$

$$\mathcal{L}[\rho ]$$

$${\hat{H}}_{int}=-\sum _k\sum _{j=1}^N\hslash G_{kj}\left[{\hat{\sigma }}_j{\hat{a}}_{\mathbf{k}}^{†}{e}^{\text{i}(\omega -{\omega }_0)t}+h.c.\right]$$

$$G_{\mu j}$$

$$G_{\nu j}$$

$${\gamma }_{ij}^{\text{(1D)}}$$

$${\gamma }_{ij}^{\text{(rad)}}$$

$${\Omega }_{ij}^{\text{(1D)}}$$

$$|\Psi ⟩$$

$$\dot{\text{B}}(t)=-\Gamma \text{B}(t)$$

$$|\Psi ⟩$$

$${\hat{\mathbf{E}}}_{\text{guided}}^{(+)}=i\sum _{fp}{\int }_0^{\infty }d\omega \sqrt{\frac{\hslash \omega {\beta }^{\prime }}{4\pi {ϵ}_0}}{\hat{a}}_{\mu }{\mathbf{e}}^{(\mu )}{e}^{-i(\omega t-f\beta z-p\varphi )}.$$

$${\hat{a}}_{\mu }(t)={\hat{a}}_{\mu }(t_0)+2\pi \sum _j{G}_{\mu j}^{\ast }\delta (\omega -{\omega }_0){\hat{\sigma }}_j(t),$$

$$⟨{\hat{\mathbf{E}}}_{\text{guided}}^{(-)}{\hat{\mathbf{E}}}_{\text{guided}}^{(+)}⟩=|\mathcal{E}(r){|}^2{\left|d(t)\right|}^2,$$

$$d(t)$$

$$|\mathcal{E}(r){|}^2$$

Full text

A new class of quantum technologies exploits the interfaces between propagating photons and cold atoms. Recent realizations using optical nanofibers (ONFs) platforms include optical isolators, switches, memories, and reflectorsSolano2017 . These devices guide the electromagnetic field, a feature that could allow us to engineer and control a collective time evolution of macroscopically separated subsystems. States that evolve as a whole with dynamics different to that of the independent subsystems are called collective states. These states emerge from atoms interacting via a common mode of the electromagnetic field.

For an ensemble of $N$ two level atoms, in the single excitation limit,

[TABLE]

represents the $\alpha$-th collective state of the system, where $\gamma_{\alpha}$ and $\Omega_{\alpha}$ are its collective decay and frequency shift respectively, and $\sum_{j=1}^{N}|c_{\alpha j}|^{2}e^{-\gamma_{\alpha}t}$ is the probability of having an excitation in the atoms. When $\gamma_{\alpha}$ is larger (shorter) than the natural radiative decay time $\gamma_{0}$, the system is super- (sub-) radiantdicke_coherence_1954 . For free-space coupling, collective states emerge when all the atoms lie within a wavelengthdevoe_observation_1996 . By externally exciting the atoms, super-radiant states are readily observed but because sub-radiant states are decoupled from the electromagnetic vacuum field, they are challenging to produceguerin_subradiance_2016 .

The master equation that describes the dynamics of an ensemble of atomic dipoles, of density matrix $\rho$, coupled through the electromagnetic field is given byLeKien2005

[TABLE]

The effective Hamiltonian $H_{\text{eff}}$ of the dipolar interaction between atoms and the Lindbland super operator $\mathcal{L}$ in Eq. (2) modify two atomic properties: the resonance frequency and the spontaneous decay rate respectively. They are given by

[TABLE]

with $\sigma_{i}$ ($\sigma^{{\dagger}}_{i}$) being the atomic lowering (raising) operator for an excitation of the $i$-th atom. $\Omega_{ij}$ is the rate of photons exchanged between atoms and $\gamma_{ij}$ is the term responsible for collective radiative decays, where $\gamma_{ii}$ is the single atom decay rate. The decay of an excitation in such a system, that leads to a collective state as in Eq. (1), depends on the coupling amplitudes and relative phase between the atoms given by $\gamma_{ij}$ .

When atoms are far apart in free space their interaction is mediated by a propagating field with an expanding wavefront, and a separation of few wavelengths is enough to make the interaction negligible. As atoms get closer together, $\Omega_{ij}$ in Eq. (3) diverges, reducing the coherence of a system with more than two atoms. These constraints can be circumvented by using longer wavelengths with larger atomic-dipole moments, such as Rydberg atomsbendkowsky_observation_2009 , or long-range phonon modes, implemented with trapped ionsricherme_2014 ; Bohnet1297 . However, these techniques are limited to subwavelength distances.

Waveguides offer an alternative by confining the mediating field. The guided field propagates unaltered, facilitating the coupling of atoms separated by many wavelengths (see Fig. 1). Dipole-dipole interactions, given by $\Omega_{ij}$, are finite for atoms along the waveguide, removing a practical limit for creating super-radiant states of a large number of atoms. Super-radiance of atoms around a waveguide has been observedgoban_superradiance_2015 , but its long-range interaction feature has not been proven or explored. Such effect has been implemented with superconducting waveguides and two artificial atoms one wavelength apartvanLoo1494 , but has not been realized for many atoms at multi-wavelength distances in the optical regime.

We present the implementation of collective atomic states through infinite-range interactions via a one-dimensional nanophotonic waveguide. We use a few atoms evanescently coupled to a single-mode ONF, observing super- and sub-radiant radiative decays of a single excitation in the system, evidence of collective behavior.

Atoms around the ONF interact at short and long distances (see Fig 1 (a)), the latter mediated by the ONF guided mode. The dipolar interaction that leads to a collective decay, is separated into two contributions of the electromagnetic field: from modes radiating outside the ONF, $\gamma_{12}^{(\text{rad})}$, and from the guided mode, $\gamma_{12}^{(\text{1D})}$LeKien2005 (see Fig. 1 (b) and (c)).

We overlap a cold atomic cloud of 87Rb atoms from a magneto-optical trap (MOT) with a 240 nm radius ONF. This ONF is single mode at the D2 resonant wavelength of 780 nm. After the MOT is turned off, the atoms form a cold thermal gas around the ONF. They are prepared in the $F=1$ ground level by an external free propagating beam. A repumper beam driving the $F=1\rightarrow F=2$ transition propagates through the nanofiber, leaving in the $F=2$ ground state only atoms that interact with the ONF guided mode. By detuning the repumper below resonance we address atoms near the nanofiber (whose levels have been shifted by van der Waals interactions) such that the atomic density distribution peaks at $\sim 30$ nm away from the surface. A weak free space probe pulse, propagating perpendicular to the fiber, excites atoms for 50 ns using the $F=2\rightarrow F^{\prime}=3$ transition. After the probe turns off (extinction ratio better than $1:2\times 10^{3}$ in one atomic natural lifetime), we collect photons spontaneously emitted into the ONF mode to measure the decay time using time-correlated single photon counting.

Collective states can be tailored by positioning the atoms in a particular arrangement. This kind of control has been challenging to implement for atoms trapped close enough to the ONF (tens of nanometers) to ensure significant mode coupling. However, collective states are still observed when atoms from a MOT are free to go near the ONF. Their random positioning leads to probabilistic super- or sub-radiant states on each experimental realization. Sub-radiant states have lifetimes much longer than most other processes, favoring their observation. Super-radiance can be measured as an enhanced decay rate at short times. Both effects can provide quantitative experimental evidence of collective states.

Figure 2 shows a typical signal of the atomic decay as measured through the ONF. Its time dependence can be described by two distinct exponential decays. The slow decay (green dashed line in Fig. 2 (a)) corresponds to an average of sub-radiant decays due to pairs of atoms located within a wavelength, i.e. free space interaction (Fig. 1 (b)). Infinite-range interactions also produce sub-radiant decay rates. However, these events are obscured by the dominant signal of slower decays produced from free space interactions. In our case $\gamma_{\text{1D}}\approx 0.13\gamma_{0}$, so sub-radiance from infinite-range interactions is limited to $\gamma_{0}-\gamma_{\text{1D}}\approx 0.87\gamma_{0}$. This is a factor of six faster than the observed sub-radiant rates. Sub-radiance of atoms interacting in free space has been observed in a very optically dense cloud of atomsguerin_subradiance_2016 , but we can observe it even for optical densities ($\mathit{OD}$) as small as $0.3$. The fast decay rate (red dashed line in Fig. 2 (a)) is larger than the natural decay rate, showing the presence of super-radiant initial states.

A full description of the temporal evolution of the entire data sample requires numerical (Monte Carlo) methods, as the solid black line in Fig. 2 shows. We use the average number of atoms ($N$) as the only free parameter for this simulation, allowing for variations of the background up to one-sigma. The two sigma deviation between simulation and data (see Fig. 2 (b) from 7 to 15 $\tau_{0}$) could come from otherwise a longer living subradiant state that gets prematurely destroyed because atoms fall onto the ONF, emitting the excitation into the guided mode. The initial state preparation – the polarization of the incoming pulse that produces the collective one-photon state – can favor super- or sub-radiant states, as the inset of Fig. 2 shows. In general the free space atom-atom coupling is larger for dipoles driven along the ONF, favoring sub-radiance, and the ONF mediated coupling is larger for dipoles driven perpendicular to the ONF, favoring super-radiance.

An important difference between sub- and super-radiant decay rates in ONF is that the latter increases as a function of $N$. We can vary $N$ from one to six by changing the MOT density, and quantify it through the $\mathit{OD}$ of the ONF mode. $n_{\text{eff}}OD=N\gamma_{\text{1D}}/\gamma_{0}$, where $n_{\text{eff}}$ is the mode effective refractive index, and in our case $n_{\text{eff}}\approx 1.15$. We measure the transmission spectrum through the ONF to extract the $OD$. The decay rate increases with $N$, as shown by the blue circles in Fig. 3, indicating super-radiance. The gray region represents the one sigma confidence bands of a linear fit to the data showing a linear dependence of the super-radiant decay rate for increasing $N$. The theoretical model implemented for the fit shown in Fig. 2 (solid black line) also predicts a linear dependence on $N$ of the decay rate $\gamma$ at short times. The red dashed line in Fig. 3 shows this prediction, corroborating the theory with the experiment.

The average spacing between atoms is larger than a wavelength for most of the realizations, meaning that infinite-range interactions are always present. However, to provide an unambiguous proof of infinite-range interactions, we split the atomic cloud in two (see inset of Fig. 3). We see that two atomic clouds separated by 400 wavelengths present the same super-radiant collective behavior as a function of the $\mathit{OD}$ as a single atomic cloud. This shows that the relevant parameter is the total $\mathit{OD}$ (or $N$) along the ONF mode, regardless the separation between atoms.

Optically guided modes can be used to mediate atom-atom interactions, creating macroscopically delocalized collective atomic states. We use the super-radiant behavior of distant atoms as evidence of infinite-range interaction, but other interesting collective quantum properties remain to be tested. The practical limits of infinite-range interactions are an open question, since in principle optical fibers can be easily connected and rerouted along several meters. An intriguing next step is the study of quantum systems beyond the Markov approximation, coupling atoms at distance greater than what light travels in an atomic lifetime. Moreover, by achieving fine control on the positioning of the interacting particles, and/or using the directional coupling produced by chiral atom-light interactionLodahl2016 , one can engineer desired states tailored to address specific applications. The implementation of infinite-range interactions opens new possibilities for quantum technologies and many-body physics.

Acknowledgments

We are grateful to A. Asenjo-Garcia, H. J. Carmichael, D. E. Chang, J. P. Clemens, M. Foss-Feig, M. Hafezi, B. D. Patterson, W. D. Phillips, and P. R. Rice for the useful discussions. We specially thank P. Zoller for helping us improve the manuscript. This research is supported by the National Science Foundation of the United States (NSF) (PHY-1307416); NSF Physics Frontier Center at the Joint Quantum Institute (PHY-1430094); and the USDOC, NIST, Joint Quantum Institute (70NANB16H168).

Methods

.1 Experimental methods

A tapered single mode ONF, with waist of $240\pm 20$ nm radius and $7$ mm length, is inside an ultrahigh vacuum (UHV) chamber, where it overlaps with a cloud of cold 87Rb atoms (less than half a millimeter width) created from a MOT. The MOT is loaded from a background gas produced by a 87Rb dispenser. Acousto optic modulators (AOMs) control the amplitude and frequencies of the MOT beams. After the atomic cloud loading reaches steady state, the MOT beams are extinguished. A free space propagating depump beam, resonant with the $F=2\rightarrow F^{\prime}=2$ transition (150 $\mu$s duration) prepares all atoms in the cloud in the $F=1$ ground state. A 0.4 nW fiber-repump beam, detuned below resonance by $15$ MHz to the $F=1\rightarrow F^{\prime}=2$ transition, propagates through the ONF during the entire cycle. It pumps back to the $F=2$ ground state only those atoms close enough to the ONF to interact with the guided mode. This detuning repumps only those atoms close enough to the ONF surface to experience an energy shift due to the van der Waals interaction with the dielectric body. This produces a narrow density distribution of atoms of $5$ nm width centered around $30$ nm away from the surface. We wait 300 $\mu$s until the AOMs reach maximum extinction. The atomic cloud free falls and expands around the ONF for $2.5$ ms creating a cold thermal gas (approx. $150$ $\mu$K), where each atom interacts with the nanofiber mode for approximately $1.5$ $\mu$sPhysRevA.92.013850 . The atomic density reduction due to the cloud expansion limits the probing time of the cycle. The atoms are excited by pulses of a weak probe beam incident perpendicularly to the nanofiber (see Fig. 1 (a)) and linearly polarized along the ONF for the data set shown in Fig. 3 . The pulses are resonant with the $F=2\rightarrow F^{\prime}=3$ transition of the $D2$ line and created with a double-passed Pockels cell (Conoptics 350-160), with a pulse extinction ratio better than 1:2000 in one atomic natural lifetime that remains at least an order of magnitude below the atomic decay signal for more than 20 lifetimes. The on-off stage of the light pulses is controlled with an electronic pulse generator (Stanford Research Systems DG645). The probe power is kept low, i.e. saturation parameter $s<0.1$, to ensure a single photon excitation while staying in the limit of low-excitation and avoiding photon pileup effects. Only those atoms that interact with the ONF guided mode are in the $F=2$ ground state and will be excited by the probe beam. During the probing time we send a train of 50 ns probe pulses every 1 $\mu$s. The probe is a 7 mm $1/e^{2}$ diameter collimated beam. After 2 ms of probing (approx. 2000 pulses) the probe beam is turned off and the MOT beams are turned back on. During the probing time the atomic density remains constant. We wait 20 ms after the MOT reloads and repeat the cycle. The average acquisition time for an experimental realization is around 5 hours, giving a total of about $1\times 10^{9}$ probe pulses. The photons emitted into the nanofiber and those emitted into free space are independently collected with avalanche photodiodes (APDs, Laser Components COUNT-250C-FC, with less than 250 dark counts per second). The TTL pulses created from photons detected by APD are processed with a PC time-stamp card (Becker and Hickl DPC-230) and time stamped relative to a trigger signal coming from the pulse generator. We use time-correlated single photon countingTCSPC to extract the decay rate of a single excitation in the system, eliminating after-pulsing events from the record.

When atoms are around the nanofiber, they tend to adhere due to van der Waals forces. After a few seconds of having the ONF exposed to rubidium atoms it gets coated, suppressing light propagation. To prevent this, we use $500$ $\mu$W of 750 nm blue-detuned light (Coherent Ti:Sapph 899) during the MOT-on stage to create a repulsive potential that keeps the atoms away from the ONF surface. This is intense enough to heat the ONF and accelerate the atomic desorption from the surface. The blue-detuned beam is turned off at the same time as the MOT beams, so the probed atoms are free to get close to the nanofiber.

Photons from the probe beam can be scattered multiple times by the atoms producing a signal that looks like a long decay, an effect known as radiation trapping. This effect can obscure sub-radiant signals. However, the small ODs involved in the experiment allow us to neglect contributions from radiation trapping. We confirm this assumption by observing the same temporal evolution of the signal at constant OD for several detunings of the probe beam in a range of $\pm 3$ linewidthsguerin_subradiance_2016 .

The atomic lifetime can also be altered by modification of the electromagnetic environment of the atoms in the presence of a ONF, i.e. the Purcell effect. However this effect is characterized separatelySolano2017b and well understood. More importantly, it does not depend on the number of atoms, in contrast with the super-radiant behavior.

Further evidence of collective states can be found in the resonance spectrum of the system (see Eqs. (2) and (3)). The dispersive part of the interaction modifies the resonance frequencies of the system, due to avoiding crossing of otherwise degenerate levels. This effect is in principle visible in the transmission spectrum. In our particular case the frequency splitting is a small percentage of the linewidth. Broadening mechanisms and other systematic errors prevent us from clearly observing such signal. However, a line-shape dependence on $N$ can be inferred from the statistical analysis of the fit of the spectrum to a Lorentzian. This effect might enable the exploration of features of collective states in the spectral domain.

ONFs can provide chiral atom-light couplingLodahl2016 . Even though this is a promising feature of the platform, it requires a particular positioning of the atoms and a preparation of their internal-state. This first exploration of infinite-range interactions involves detecting only on one end of the ONF and azimuthally averaging the atomic position, preventing studies of chiral effects that we do not consider crucial to our measurements.

.2 Theoretical model

We follow the work of Svidzinsky and ChangPhysRevA.77.043833 to implement the theoretical simulations of the experiment. Consider the Hamiltonian of $N$ atoms interacting with an electromagnetic field in the rotating-wave approximation

[TABLE]

where $\hat{\sigma}_{j}$ is the lowering operator for atom $j$; $\hat{a}^{{\dagger}}_{\mathbf{k}}$ is the photon creation operator in the mode $k$-th; $\omega_{0}$ and $\omega$ are the frequencies of atomic resonance and $k$-th mode of the field respectively. This is a general expression for the Hamiltonian, which leads to the master equation in Eq. (2) after some approximations. The sum on $j$ is done over the atoms and the sum on $k$ goes over the electromagnetic field modes, guided into the nanofiber and radiated outside. These modes can be found in the work of Le Kien et al.LeKien2005 The sum over the guided modes is $\sum_{\mu}=\sum_{f,p}\int_{0}^{\infty}d\omega$, where $f$ and $p$ are the propagation direction and polarization in the circular basis (plus or minus) of the guided mode respectively, and $\mu$ stands for modes with different parameters $(\omega,f,p)$. The sum over the radiated modes is $\sum_{\nu}=\sum_{m,p}\int_{0}^{\infty}d\omega\int_{-k}^{k}d\beta$; where $m$ is the mode order, $k$ is the wavenumber, $\beta$ is the projection of the wave vector along the fiber or propagation constant, and $\nu$ stands for modes with different parameters $(\omega,\beta,m,p)$. Then the total sum is $\sum_{k}=\sum_{\mu}+\sum_{\nu}$. The electromagnetic field modes and their relative coupling strength have been previously studiedLeKien2005 . The coupling frequencies $G_{kj}$ for the guided and radiated modes can be written as

[TABLE]

where $\beta^{\prime}=d\beta/d\omega$, $\mathbf{d}_{j}$ is the dipole moment of the $j$-th atom, and $\mathbf{e}^{(\mu,\nu)}$ are the electric field profile function (or spatial dependence of the amplitude) of the guided and radiated modes ($\mu$ and $\nu$).

Atoms interact with each other mediated by the electromagnetic field. The interaction between the atomic dipoles is proportional to the product of the atom-light coupling frequencies of the form $G_{ki}G_{kj}$, were $k$ labels the mediating field mode (the repetition of the letter implies summation if there is more than one mode) and $i$ and $j$ label the $i$-th and $j$-th atom. It is possible to identify two contributions from the coupling of atoms to the dynamics of the system, a dispersive and a dissipative one, as shown in Eq. (2). The dispersive part contributes to the unitary evolution of the system (see Eq. (3)), and it can be decomposed as $\Omega_{ij}=\Omega^{(\text{rad})}_{ij}+\Omega^{(\text{1D})}_{ij}$, were $\Omega^{(\text{rad})}_{ij}$ and $\Omega^{(\text{1D})}_{ij}$ come from the interaction of the $i$-th and $j$-th atoms mediated by the radiated and guided modes respectively. $\Omega_{ij}$ is usually called the dipole-dipole coupling frequency. The dissipative part contributes to the decay of the system (see Eq. (4)), and it can be decomposed as $\gamma_{ij}=\gamma^{(\text{rad})}_{ij}+\gamma^{(\text{1D})}_{ij}$, were $\gamma^{(\text{rad})}_{ij}$ and $\gamma^{(\text{1D})}_{ij}$ come from the interaction of the $i$-th and $j$-th atoms mediated by the radiated and guided modes respectively. For simplicity, here we consider the case of atomic dipoles oriented along the ONF ($z$-axis) placed in the position $\mathbf{r_{i}}=(r_{i},\phi_{i},z_{i})$ with reduced dipole moment $d_{i}$, obtaining

[TABLE]

where $\mu_{0}$ parametrizes the guided modes on resonance. The dispersive component of the interaction given by the radiated modes as $\Omega^{(\text{rad})}_{ij}$ is a complicated expression and hard to solve even numerically. We follow the work of Le Kien et al.kien_nanofiber-mediated_2016 and use the free space value of $\Omega^{(\text{rad})}_{ij}$ throughout the calculation as a reasonable approximation. $\gamma_{ii}=\gamma_{0}$ with $\gamma_{0}$ the single atom natural decay rate. $\gamma^{(\text{rad})}_{12}$ and $\gamma^{(\text{1D})}_{12}$ are plotted in Fig. 1 (b) and (c) respectively for an atom fixed at $\mathbf{r_{1}}=(30\text{ nm}+240\text{ nm},0,0)$ (240 nm being the ONF radius and 30 nm the distance of the atom to the surface). When atoms are too close to each other, the radiated terms $\Omega^{(\text{rad})}_{ij}$ and $\gamma^{(\text{rad})}_{ij}$ dominate over the guided ones ($\Omega^{(\text{1D})}_{ij}$ and $\gamma^{(\text{1D})}_{ij}$), with $\Omega^{(\text{rad})}_{ij}$ diverging and $\gamma^{(\text{rad})}_{ij}$ approaching the total decay rate. With a low number of atoms randomly distributed along the ONF the effects of short-range interaction are small but still observable.

For simplicity we are interested in the decay of only one excitation in a system of two level atoms, however generalizations to multilevel atoms can be found in the literatureHebenstreit2017 . Such system is represented by the state

[TABLE]

where $\mathbf{k_{\mu(\nu)}}$ represents the sum over the guided (radiated) modes, $b^{(g)}_{\mathbf{k}}$ is the probability amplitude of all the atoms being in the ground state and one excitation in the $\mathbf{k}$-th mode of the field, and $b^{(e)}_{j}$ is the probability amplitude of having zero excitation in the field and an excitation in the $i$-th atom. Assuming that we start the cycle with the excitation in the atoms, i.e. $b^{(g)}_{\mathbf{k}}(0)=0$, we can write the Schrödinger equation in the Markov approximation for the coefficients $b^{(e)}_{i}(t)$ in a matrix form asPhysRevA.77.043833

[TABLE]

where $\text{\bf{B}}(t)$ is a vector with entries given by the $b^{(e)}_{i}(t)$, and $\Gamma$ is a non-hermitian symmetric matrix with entries $2\Gamma_{ij}=\gamma_{ij}+\text{\bf{i}}\Omega_{ij}$, representing the couplings between the $i$-th and $j$-th atoms calculated from the optical nanofiber modes, radiated and guided. The eigenvalues $\eta_{\alpha}$ of Eq. (12) give the possible decay rates of the system. This are the collective sates mentioned in Eq. (1). The eigenvectors form a basis $\{|B_{\alpha}\rangle\}$ that allows us to write the state of the system as

[TABLE]

where the coefficients $c_{\alpha}$ are given by the initial state. In contrast with Eq. (1), here we have also included the states with one excitation in the field.

Following this approach the many-body problem, of calculating the decay of one excitation distributed among $N$ interacting atoms, becomes an eigenvalue problem in a Hilbert space of dimension $N^{2}$ instead of $2^{2N}$. This speeds the calculations, allowing us to compute the decay rate of the system with Monte Carlos simulations for a large $N$ in random positions.

The electromagnetic field operator for the guided modes is LeKien2005

[TABLE]

The formal solution of the Heisenberg equation for $\hat{a}_{\mu}(t)$ in the Markov and rotating wave approximation is

[TABLE]

The substitution of this expression into Eq. (14) gives the guided field operator as a function of the dipole operators.

Assuming that the guided modes are initially empty and that all the dipoles are oriented in the $z$ direction and at the same distance from the ONF, the intensity of the guided field as a function of the atomic dipole operators is

[TABLE]

where

[TABLE]

considering $\gamma_{\text{1D}}(r)=\gamma_{ii}^{\text{(1D)}}(r)$ from Eq. (8) and $A_{\text{eff}(z)}(r)=|n_{\text{eff}}e_{z}^{(\mu_{0})}(r)|^{-2}$ to be the effective mode area of the $z$ component of the electric field LeKien2005 . Eq. (18) relates the total radiated power into the waveguide with the energy radiated per unit time, i.e. $I(r)A_{\text{eff}(z)}(r)=\hbar\omega_{0}\gamma_{\text{1D}}(r)$, where $I(r)$ is the intensity of the radiated field.

Equation (16) shows that the measured intensity corresponds to the one produced by $N$ classical dipoles with different phases, different positions, and amplitudes given by the probability of being in the excited state $b^{(e)}_{j}$Araujo2017 .

.3 Theoretical methods

We use Monte Carlo simulations, randomly positioning $N$ atoms around the ONF. The position of each atom is given in cylindrical coordinates by $\mathbf{r_{i}}=(r_{0},\phi_{i},z_{i})$, where $r_{0}=240\text{ nm}+30\text{ nm}$, $\phi_{i}\in[0,2\pi]$, and $z_{i}$ is obtained from a Gaussian distribution with a FWHM of 200 $\mu$m, determined by the atomic cloud size. The radial position of the atoms is fixed, determined by the experimental procedure of repumping the atoms close to the nanofiber surface. In our case all the atoms are at a constant radial position of $30$ nm away from the surface of an ONF of 240 nm radius, with $\gamma_{\text{1D}}/\gamma_{0}\approx 0.13$. This is a good approximation given the narrow radial distribution of the atoms ($\sim 5$ nm), as explained in the experimental methods.

The initial state will depend on the amplitude and phase of the excitation beam. We assume that the initial state corresponds to a superposition of all the atoms in the ground state except one with an induced atomic dipole. The initial phase between the atoms depends on their position; assuming an excitation pulse with a wave vector perpendicular to the fiber, each atom initial phase can be calculated from its coordinates. For each random realization we solve Eq. (12) and calculate the intensity of the guided field, Eq. (16). We use these results to take the mean of the intensity of the guided field as a function of time. Typically, 100,000 realizations are required to converge to a level of precision higher than what it is visible in Figs. 2 and 3. If the mean of the intensity guided field is normalized, there is no dependence on the amplitude of the initial induced dipole in the weak excitation limit.

There is a correspondence between super-radiance (sub-radiance) configurations and constructive (destructive) interference of the field emitted by the dipoles into the ONF (see Eq. (17)); meaning that super-radiant configurations contributes more than sub-radiant configurations when taking the mean over all the realizations for an electric field detected through the ONF (Eq. (16)).

The theoretical model prediction for different dipole moment orientations relative to the ONFLeKien2005 qualitatively agrees with the observed experimental behavior: The long term sub-radiance disappears on our signal-to-background-ratio window when exciting with vertically polarized light (see inset of Fig. 2). A sensitivity analysis to the ONF radius shows no significant changes in the predictions up to a $\pm$10 nm variation.