P Hilaire

C Millet

J C Loredo

C Antón

A Harouri

A Lemaître

I Sagnes

N Somaschi

O Krebs

P Senellart

L Lanco

Deterministic assembly of a charged quantum dot -micropillar cavity device

Quantum-dot based spin-photon interfaces are highly sought systems to implement deterministic photon-photon gates as well as to generate photonic cluster states. This requires mastering the technological challenge of fully controlling the coupling of a charged quantum dot to a cavity mode. Here, we report on a set of technological and experimental developments that allows doing so. The first ingredient consists in combining the in-situ lithography technique, that allows a deterministic spatial and spectral coupling of the emitter to a pillar cavity mode, with a pre-identification of the quantum dot charge states. The second ingredient relies on the design of an asymmetric tunneling barrier to inject the carrier in the quantum dot and an optical control of the charge occupation probability. We show that we can ensure both a high occupation probability of the charge in the quantum dot and an optimal coupling to the cavity mode. This is demonstrated through second order auto-correlation measurements and by measuring the performance of the device as a bright source of indistinguishable single photons.

The efficient interfacing of single photons to natural or artificial atoms is central to the development of efficient quantum light sources [START_REF] Varnava | [END_REF]2 , quantum repeaters 3,4 , photonphoton gates 5,6 as well as for the generation of photonic cluster states 7,8 . Strong atom-photon interactions are commonly obtained by inserting a single atom in an optical resonator to control the spontaneous emission into a well designed optical mode 9,10 . The atom-photon interface can act as an efficient emitter of indistinguishable, single photons [11][12][13] , but also as an efficient single-photon receiver. In the latter case, the atom can be used as a quantum memory to store the photonic state, or to realize various atom-photon, atom-atom or photon-photon gates 5,14,15 .

Semiconductor quantum dots (QDs) are excellent artificial atoms, able to emit highly-pure single-photons and to route light at the single-photon level when inserted in cavities 6,16 . Exploiting the spin state of a single charge in the QD, either an electron or a hole, is essential to demonstrate deterministic photon-photon gates and generate highly sought photonic cluster states 17,18 , two central features for the scalability of optical quantum technologies.

However, the deterministic fabrication of such singlycharged cavity quantum electrodynamics (cQED) devices is very challenging as it requires fulfilling multiple stringent requirements at the same time. The first one is the optimal spatial and spectral coupling of a singly-charged QD state to a cavity mode. A second challenge is to controllably prepare the QD in the desired charge state, consisting of either an electron or a hole in excess. Finally, the cavity must allow the photons to be efficiently injected and collected after having interacted with the QD. In this respect, pillar microcavities allow for both efficient injection [19][20][21] and collection of photons [11][12][13] .

Here, we report on a set of experimental and tech-nological methods that result in a high control on the injection of a single charge in a quantum dot, which is at the same time optimally coupled to a micropillar cavity mode. To do so, we make use of an engineered asymmetric band structure that hinders the tunneling of an optically-injected hole out of the QD. An in-plane magnetic field photoluminescence study of the various QD charge states allows us to identify the positively-charged state within the pattern of various emission lines. We define electrically connected micropillar cavities centered on chosen quantum dots, using a cryogenic in-situ lithography step 22,23 , during which we measure the positive trion energy transition and tune the cavity frequency accordingly. The quality of the fabricated charged quantum dot-cavity interfaces is evaluated via intensity correlations measurements and by showing the excellent performance of the device as a single-photon source. This paper is organized as follows. In Sec. I, we present the sample structure and the technological procedure. The scheme used to trap a single carrier in the QD is presented in Sec. II. We then discuss the deterministic coupling of the cavity to the trion transition in Sec. IV and evaluate its performances in Sec. V. Finally, the properties of the devices operating as single-photon sources are discussed in Sec. VI.

I. SAMPLE STRUCTURE AND TECHNOLOGICAL PROCEDURE.

The micropillars are fabricated from a planar sample embedding a λ-cavity, surrounded by two distributed Bragg reflectors (GaAs and Al 0.9 Ga 0.1 As, with 14 and 28 pairs for the top and bottom mirrors respectively). The reduced number of layer pairs in the top mirror allows for a higher output coupling efficiency of the photons emit-ted by the QD towards the top. Fig. 1(a) displays the electromagnetic field intensity calculated by a standard transfer matrix method as a function of the position in the structure (due to its small width, the QD layer was neglected in these simulations). The micropillar structure confines the electromagnetic field with a maximum intensity in its center where the quantum dot is located.

The vertical structure comprises a p-i-n junction 19,24 with a doping density that gradually decreases while approaching the cavity (negative doping for the bottom mirror and positive doping for the top mirror). Such an approach allows minimizing optical losses due to doping in the cavity layer. The structure is connected to an electrically-contacted diode through four ridges and a circular frame as shown in Fig. 1

(b).

A well-known technique to control the charge state of QDs is via the bias voltage, which brings the Fermi energy close to a charged QD state 25,26 . Here, we adopt the approach proposed in Ref. 27, based on an asymmetric confinement of an optically-injected electron-hole pair. The λ-cavity consists of a GaAs layer that embeds a single InGaAs QD and a 20-nm thick tunneling barrier of Al 0.1 Ga 0.9 As, positioned 10nm above the QD layer.

The energy levels of the QD and its near environment are schematized in Fig. 1(c), which sketches the energy bands as a function of the vertical position in the cavity layer. The doped regions tilt the bands, allowing the tunneling of carriers out of the quantum dot. Yet the Al 0.1 Ga 0.9 As barrier, being placed above the QD layer, is used to strongly increase the tunneling time for a hole trapped in the quantum dot, typically by three orders of magnitude 27 , while keeping the electron tunneling time unchanged. The resulting difference in the electron-hole tunneling time allows the trapping of a single hole when optically creating an electron-hole pair.

The micropillar cavities are fabricated using the in-situ lithography technique to center the chosen QDs within the pillar mode 22,23 . This is obtained by measuring the QD position with nanometer-scale accuracy through photoluminescence mapping and by defining a connected pillar cavity centered on the QD during the same step. The cavity diameter is adjusted so that the trion transition is matched to the cavity mode. It is thus essential to be able to identify the QD emitting charge state (neutral exciton, charged exciton, etc) during the in-situ lithography procedure. To do so, we detail the experimental conditions used here to optically trap a hole in Sec. II and discuss how we identify the lines through magnetic field spectroscopy in Sec. III.

II. OPTICAL INJECTION OF A SINGLE-HOLE.

We describe here the optical method used to inject a hole in the quantum dot. The stable QD state is the crystal ground state, here denoted as neutral QD, whose energy levels are represented in the left part of Fig. 2 (a).

An electron-hole pair can be generated by resonantly pumping the exciton (ω X ) or other discrete transitions (such as "p-shell" transitions or LO-phonon-assisted transitions) here denoted as quasi-resonant (QR) transitions with energy ω QR > ω X 28,29 . Here, this is done using a CW laser at a wavelength around λ QR = 901nm as depicted in the left panel of Fig. 2(b).

When an exciton is quasi-resonantly created in the QD, the electron and the hole excess energy non-radiatively decays (in typically less than 100ps 30 ) and the system reaches the lower exciton level, as shown in the middle panel of Fig. 2(b). If the exciton radiatively recombines by emitting a photon, the QD then returns to its ground state. This generation-recombination cycle keeps going until an optically-created electron eventually tunnels out towards the n-doped electrical contact before the radiative recombination occurs. Due to the Al 0.1 Ga 0.9 As barrier, the remaining hole is confined in the QD for a longer time, during which the QD is in the desired positivelycharged state.

The addition of a hole in the QD induces a strong modification of the electric environment by Coulomb interaction, modifying all the discrete energy levels of the QD, as illustrated in the right-hand side of Fig. 2(a).

Electron hole pair creation

Hole state generation

Hole state excitation The positively-charged QD thus presents a different energy structure than the neutral one, a feature that applies to LO-phonon assisted and p-shell optical transitions as well. Thus, the incoming frequency ω QR does not induce any optical excitation when a hole is confined. This allows trapping only a single hole in the quantum dot as shown later on.

ω QR ω X e _ h + + ω X (

III. CHARGE STATE IDENTIFICATION.

In order to define the pillar cavity diameter to match the trion transition of a positively-charged QD, we have developed spectroscopy tools to identify the QD energy levels in the experimental conditions of the in-situ lithography -i.e. non-resonant excitation around 850 nm. With above-band excitation, the emission spectrum of the QD presents emission lines corresponding to both the neutral and charged energy levels. To be able to distinguish these lines, we have performed a systematic study of the QD emission under an in-plane magnetic field.

The trion transitions with and without magnetic field are sketched respectively in the top-right and top-left panels of Fig. 3(a). At zero external magnetic field (topleft panel of Fig. 3(a)), a trion radiatively decays into the hole state by emitting a circularly-polarized photon with either a right-handed or a left-handed helicity. There are therefore two trion transitions that are energy degenerate. When an in-plane magnetic field is applied, it induces a Zeeman splitting between the hole states and the trion states, and it also modifies the system eigenstates and thus the optical selection rules, as illustrated in the top-right panel of Fig. 3(a). For a high in-plane magnetic field (B x > 1T typically), a trion decays into a hole by emitting a single photon with 4 combinations of linear polarizations and energies.

The neutral exciton transitions with and without magnetic field are sketched respectively in the bottom-right and bottom-left panels of Fig. 3(a). At zero external magnetic field, the two exciton states |e X and |e Y radiatively decay into the same ground state |g (the neutral state) by emitting a photon with linear polarization X and Y, respectively. An in-plane magnetic field increases the energy splitting between the two excited states and modifies the polarization of the emitted photons to either horizontal or vertical. Note that even in the case of a strong in-plane magnetic field, an exciton decays by emitting a single photon with only two combinations of linear polarizations and energies and thus, can be discriminated from the trion transition.

The bottom panel of Fig. 3(b) shows a typical photoluminescence spectrum obtained on a QD in a planar cavity sample, under non resonant excitation (λ N R = 850nm) and without any applied magnetic field. There are three dominant transitions observed, at 925.1nm, 925.3nm and 925.7nm; the lower intensity transition (at 925.5nm) is related to another QD spatially close to the one under study.

When an in-plane magnetic field is applied, the QD transitions split as seen in the middle panel of Fig 3(b), which displays the evolution of the photoluminescence spectrum under an increasing in-plane magnetic field intensity. At B x = 4T and above, all the observed peaks are split into two transitions. At high magnetic field, all the observed QD transitions are also blue-shifted, due to the diamagnetic shift 31 . The trion transition splits into four different transitions; however, the limited spectrometer resolution impedes the visualization of such effect which is revealed by polarization analysis instead.

The top panel of Fig. 3(b) displays the polarizationresolved photoluminescence for the same QD at B x = 4T. The red (green) curve corresponds to the horizontallypolarized (vertically-polarized) photoluminescence. The horizontal (vertical) polarization peak centers are highlighted by red (green) vertical lines.

Let us first consider the higher wavelength transi- The magnetic field scan and the polarization analysis show that it is split in two transitions with orthogonal linear polarizations. This is the expected behavior for an exciton 32 .

V H H V R L (b) (c) B x (a) X + X X + X X B x Trion Exciton Y V H ↑ ⇑ ⇑ ⇑ ⇑ ⇑ ⇑ ↑ ⇑ ⇑ ↑ ⇑ ⇑ ↑ ⇑ ⇑
As can be seen in the middle and top panels of Fig. 3(b), the lower wavelength transition (925.1nm at B x = 0T) analysis shows that it is split in four transitions with different energies and polarizations: the highest and lowest have the same horizontal polarizations and are orthogonal to the two intermediate vertically-polarized transitions. This state is therefore identified as a trion, as this is the behavior expected from the polarization selection rules of Fig. 3(a).

We note that the center transition (925.3nm at B x = 0T) analysis is also split in four transitions with similar polarizations as the QD trion. However, there is a clear asymmetry in the photoluminescence intensity for these four transitions. This feature may be explained by the X 2+ states, corresponding to a QD charged with two holes 33,34 .

IV. POSITIVE TRION-OPTICAL MODE COUPLING

We have conducted the above emission analysis on a large number of QDs on the planar cavity sample used to fabricate the pillars. Such a systematic study allows us to identify the various QD transitions in a fingerprint-type analysis 35 . During the in-situ lithography step, while no magnetic field is applied, we can thus precisely identify the lines corresponding to the charged-QD and tune the pillar cavity mode energy to the right transition. After the in-situ lithography procedure, and the following etching and electrical contacting steps 22,23 , we return to magneto-optics measurements to demonstrate the coupling of the positively charged trion transition to the cavity mode.

Fig. 3(c) presents the emission spectrum of a QD inserted in a micropillar under above-band excitation. The bottom panel represents the photoluminescence at B x = 0T, displaying discrete peaks: the most intense peak corresponds to the QD transition that is coupled to the fundamental cavity mode. The middle panel of Fig. 3(c) shows the evolution of the photoluminescence when increasing the magnetic field. The top panel of Fig. 3(c) represents the photoluminescence collected for an external transverse magnetic field of approximately 6T in H and in V polarizations. The QD transition that is in resonance with the fundamental cavity mode is characterized by four Zeeman-split transitions as expected for a positive trion transition.

V. PERFORMANCES OF THE HOLE TRAPPING

In the following, the hole trapping scheme performances, namely the single-hole occupation probability and the single-hole trapping time, are evaluated by observing the resonance fluorescence signal on multiple QDcavity devices. In addition to the quasi-resonant laser used for the hole trapping, we use a second laser in res-onance with the trion transition, i.e. at the energy ω X + ( = ω X ) with typical power P X + ≈ 0.1-3nW, to measure the resonance fluorecence of the charge QD.

To suppress the resonant laser and only collect the emitted single photons, we use a cross-polarized setup where the trion transition is resonantly excited with a horizontally-polarized resonant laser. The resonance fluorescence is collected through a vertical polarizer that filters out the resonant laser (the quasi-resonant laser is spectrally filtered). The 15-ps pulse laser in resonance with the trion transition induces Rabi oscillations between the hole and the trion state [36][37][38] depending on the pulse area P X + . Note that single photons are emitted only if the hole is trapped in the QD. Therefore, the detection of a single photon ensures that a single hole is confined in the QD. Fig. 4(a) represents the time trace of the crosspolarized resonance fluorescence intensity emitted by one QD-pillar device, observed with a quasi-resonant power of P QR = 50µW, a π-pulse resonant excitation and a time bin of ∆t = 4µs. It evidences a clear blinking of the QD occupation levels between charged and neutral states. This is further shown in the histogram of Fig. 4(b) representing the distribution of the number of detected events per time bin ∆t. The histogram shows the presence of two states for the QD, a bright and dark state, the latter corresponding to any case where the QD does not trap a single hole, leading to the absence of detected photons and thus N = 0 photon per time bin. The signal corresponding to the bright state shows a typical Poisson distribution, centered on N = 7.9 photons per time bin, with a width given by the square root of the average value.

From this histogram, comparing the area of the bright and dark state events, we can deduce a hole occupancy of around 60%. Note that there is another signal at N = 1 photon per time bin which prevents an accurate determination of the hole occupation probability using this method. We identify this noise to a background signal arising from the imperfect laser rejection and the detector dark counts. To precisely determine the hole occupation probability together with the hole trapping time, we now turn to second-order correlations of the crosspolarized resonance fluorescence as pioneered by Ref. 39. A typical auto-correlation measurement is displayed in Fig. 5(a), where the two photon coincidence histogram is plotted for different detection delays. The peaks are separated by a delay T R = 1/f , where f = 82MHz is the laser repetition rate. In Fig. 5(a), the continuous background between two peaks is due partly to the detector dark counts, but mainly to imperfect filtering of the quasi-resonant laser. (2) (t) of the same set of data (QD1) and for two other devices (QD2 and QD3), for even longer timescale and with time bin ∆t = 10TR. The decay is fitted by an exponential curve (in red). The inset illustrates how the hole occupation probability P h and the hole trapping time T h can be deduced from these correlation measurements.

Fig. 5(b) shows the same set of data for longer delays, at which we can observe that the envelope of the peaks is decaying. Fig. 5(c) shows the intensity auto-correlations g (2) (t) at even longer times (integrated with a time bin ∆t = 10T R ) with the same set of data (QD1 in red) and for two other devices (QD2 and QD3 respectively in purple and blue). Here, the large binning blurs out the peaks which were visible in Fig. 5(a) and (b), as well as the zero-delay antibunching. The correlation measurements evidence an exponential decay for the three devices, originating from the on/off fluctuations of the hole state in the QD. This effect is used to extract the hole confinement characteristics, as is shown in the following.

The auto-correlation function g (2) (t) can be interpreted as the probability to detect a photon in a detector (denoted APD1) at time t, conditioned to the detection at time t = 0 of a photon in the other detector (denoted APD0) normalized by the uncorrelated probability of detecting a photon at any time:

g (t) = P (APD1, t|APD0, 0) P (APD1) (1)
Immediately after a photon detection in APD0, the probability that a hole is trapped in the QD is P h (0) = 1 and the detection of a photon in APD1 is thus more probable shortly after this first detection.

To interpret the dynamics of the charge state, we develop a model using two states: "0" (the crystal ground state, with zero excess charge) and "h" (the hole state) 39 . The CW quasi-resonant laser transfers the quantum dot state from "0" to "h" at a rate γ (which directly depends on the quasi-resonant power P QR , used to populate the single hole state). Conversely, the hole can tunnel out from the quantum dot with a tunneling time T h . The occupation probability of the empty state and the charged states are denoted P 0 (t) and P h (t) respectively, with P 0 (t) + P h (t) = 1. Their rate equations are:

dP h (t) dt = γP 0 (t) - 1 T h P h (t) dP 0 (t) dt = -γP 0 (t) + 1 T h P h (t) (2)
We now assume that at time t = 0, a single photon emitted by the QD is detected. A hole is thus trapped inside the quantum dot with a probability P h (0) = 1 and the hole occupation probability evolution is given by:

P h (t) = (1 -P h )e -t τ ef f + P h (3)
with P h = P h (∞) = γ/(γ + 1 T h ), the average hole occupation probability, and

τ ef f = γ + 1 T h -1
, the effective time characterizing the charge dynamics.

In this model, the enveloppe of the auto-correlation function is given by: g (2) (t) = P h (t)

P h = 1 P h -1 e -t/τ ef f + 1 (4)
Therefore, it is possible to extract the hole occupation probability and the hole tunneling time through these auto-correlation measurements, as illustrated in the inset of Fig. 5(c): the enveloppe of the g (2) (t) varies from 1/ P h at zero delay to 1 in a characteristic time τ ef f . In addition, its tangent at zero-delay crosses the x-axis at t = T h .

We include the effect of small background noise and obtain the real auto-correlation function of the quantum dot light source g (2) (t) deduced from the experimental one g

(2) exp (t). Let P QD be the probability that a photon detected is originated from the quantum dot and 1 -P QD , the probability that it is originated from bad laser filtering or dark counts. The relation between g exp (t) and g (2) (t) is given by: g (2) (t) = g

(2)

exp (t) -2(1 -P QD ) + (1 -P QD ) 2 P 2 QD (5)
In Fig. 5(c) and 6(a,b), the corrected g (2) (t) is displayed, from which the occupation probability and the hole trapping time can be directly extracted using the fit with Eq. 4.

The dependence of the hole trapping time and hole occupancy is now studied as a function of the quasi-resonant (P QR ) and resonant (P X + ) laser powers. Fig. 6(a) shows the dependence of correlation measurements with respect to P QR , with P X + fixed to π-pulse. Similarly, Fig. 6(b) shows the dependence of correlation measurements with P X + , with P QR fixed to 50µW. These two figures show that the correlation timescales and the zero-delay value both depend on the resonant and quasi-resonant laser powers.

For all P QR and P X + , the resulting correlations are fitted with an exponential decay, to extract the hole occupation probability P h and the hole trapping time T h , that are displayed in Fig. 6(c) and (d), respectively. As expected, P h is increasing with P QR : it reaches P h = 85 ± 1% for QD1. The hole trapping time is higher than 20µs except for P QR > 100µW. This shows that T h is always higher than the typical hole spin lifetime at zero magnetic field (which is generally around 1µs) 40 .

We note that T h and P h should not vary with the resonant power according to the simple model considering only the zero and one hole states. In addition, the hole trapping time T h should be constant, equal to the hole tunneling time. The observed deviation from our model can be explained if we consider the possibility of generating a two-hole state: due to the pumping by the two lasers, it is possible to generate a second electron-hole pair when a single hole is already present in the QD and, if the electron tunnels out of the quantum dot, to obtain a two-hole state. This limits the hole occupation probability and reduces the hole trapping time in the strong pumping regime.

Finally, Fig. 6(e) shows the evolution of the polarized brightness B p as a function of the measured hole occupation probability. B p is defined as the probability per excitation pulse to detect a polarized single photon after the first lens 2 . It is measured by dividing the measured count rate C, by the laser repetition rate f , the setup transmission T (evaluated by independant component transmission measurements), and the detector efficiency η det :

B p = C f T η det (6)
As expected, the measured brightness is proportional to the hole occupation probability : the measured B p corresponding to π, π/2, π/3 pulses are fitted altogether by 3 linear functions with relative slopes s π = 26.2%, s π/2 = s π cos 2 (π/2) and s π/3 = s π cos 2 (π/3) (the cos 2 takes into account the partial population inversion to the trion state for π/2 and π/3 pulses).

The probability that an intracavity photon escapes the cavity through the top mirror is called the output coupling and is measured by independent reflectivity measurements and shows values around η top = 85 ± 5%. For QD1, the maximum observed brightness is 21% and corresponds to a 85 ± 1% occupation probability.

These experiments have been repeated on two other devices, QD2 and QD3. In these cases, a remarkably high polarized brightness has been measured: B p = 28 ± 5% for QD2 and B p = 33 ± 5% for QD3. They both show a similarly high hole occupation probability with P h = 85 ± 1% for QD2 and with P h = 91 ± 1% for QD3.

VI. PERFORMANCES OPERATING AS SINGLE-PHOTON SOURCES.

Finally, the quantum properties of the cQED devices are investigated by assessing performances as single photon sources. The single-photon purity is evaluated by the short timescale zero-delay intensity correlations, measuring the g (2) (0) with a Hanbury-Brown and Twiss (HBT) experiment 41 . A spectral filter (30pm bandwidth) is inserted in the collection setup to further suppress the spectrally-wide excitation laser and phonon side band emission. The results obtained on QD1 show a good .

single-photon purity g (2) (0) = 1.6 ± 0.4% as displayed in Fig. 7(a).

The single-photon indistinguishability is evaluated by coalescence measurements via the Hong-Ou-Mandel (HOM) effect 42 . This is performed experimentally using a path-unbalanced Mach-Zender interferometer 43 where the difference of delay between the two arms is set to be equal to the laser period T R . Therefore, two singlephotons generated by two immediately separated laser pulses can interfere. Fig. 7(b) displays the experimental results obtained on QD1: the measured photon indistinguishability is at the state of the art with a raw HOM visibility V = 97 ± 0.4%.

VII. CONCLUSION

We have provided a set of experimental and technological tools for the controlled realization of singly-charged QD-photon interfaces. The key points of our work are: facilitating the hole confinement with a tunneling barrier; using in-plane magnetic-field spectroscopy to identify the trion transition under non-resonant excitation; defining the cavities with the proper geometry to match the targeted trion transition using the in-situ lithography technique; and optically injecting a single hole with a quasi-resonant pumping scheme. The resulting charged QD-photon interfaces have then been used to monitor in real time the jumping of the hole in and out of the quantum dot on a time scale of tens of microseconds. Autocorrelation measurements give access to the hole tunnelling time, and to the hole occupation probability P h which has been shown to reach large values, between 85% and 91%, for multiple devices. The high-quality of the spin-photon interfacing devices was evidenced by stateof-the-art performances of the devices operated as bright sources of pure and indistinguishable photons.

Figure 1 .

 1 Figure 1. (a) Simulated electromagnetic field intensity inside the micropillar structure. The electromagnetic field is confined at the cavity layer position (green: GaAs layers, yellow: Al0.9Ga0.1As layers, purple: Al0.1Ga0.9As tunneling barrier). (b) Scanning electron microscopy image of a sample. (c) QD + doping structure: the n-doping and p-doping region tilt the forbidden band so that fluctuating charges remain far from the QD, therefore stabilizing the electrical fluctuations. The tunneling barrier reduces the hole tunneling rate.

Figure 2 .

 2 Figure 2. (a) Sketch comparing the energy levels of a neutral QD and a singly-charged QD. The resonant trion transition energy ω X + is different from the exciton transition energy ωX (ω X + = ωX ) and the p-shell transition energies of a charged QD are also different from a neutral QD: ω QR = ωQR. (b) Hole trapping scheme using quasi-resonant excitation. Left panel: the QD stable state is the crystal ground state. A quasi-resonant laser creates an electron-hole pair by exciting a p-shell transition (ωQR). Middle panel: If the electron tunnels out the QD before radiative recombination with the hole, it generates of a single hole QD state. Right panel: the QD can then be resonantly excited by a laser with energy ω X + corresponding to the trion transition.

Figure 3 .

 3 Figure 3. (a) Optical selection rules of a trion (top) and an exciton (bottom), with (right) and without (left) external transverse magnetic field. (b, c) QD photoluminescence under an 850-nm non-resonant laser in (b) a planar cavity sample before etching and (c) after the pillar etching ((b) and (c) represent photoluminescence from different quantum dots.). Bottom panel: without any applied external magnetic field and polarization selection. Middle panel: under varying in-plane magnetic field. Top panel: with strong in-plane magnetic field ((b) Bx = 4T and (c) Bx = 6T) measured with horizontal (red) and vertical (green) polarizations. (In the middle panel of (c), a waveplate is also rotated before the spectrometer, during the magnetic field scan, to obtain polarization resolution.)

Figure 4 .

 4 Figure 4. (a) Time trace recording the number of singlephotons detected per time bin ∆t = 4µs (with PQR = 50µW and P X + corresponding to π-pulse). (b) Histogram of the number of time bins corresponding to a given number of detected photons, during an acquisition time of approximately 10s. It represents the intensity distribution of the singlephoton source and displays the addition of a high zero detected photon probability and a gaussian distribution (with N = 7.9 photons /∆t and a width of σ = 2.8(= N )).

Figure 5 .

 5 Figure 5. (a,b) Coincidence observed at (a) short and (b) intermediate timescales. (c) Intensity autocorrelations g(2) (t) of the same set of data (QD1) and for two other devices (QD2 and QD3), for even longer timescale and with time bin ∆t = 10TR. The decay is fitted by an exponential curve (in red). The inset illustrates how the hole occupation probability P h and the hole trapping time T h can be deduced from these correlation measurements.

Figure 6 .

 6 Figure 6. (a, b) Auto-correlation measurements for (a) varying non resonant powers PQR (with fixed P X + corresponding to π-pulse) and (b) varying resonant power P X + (with fixed PQR = 50µW). Red curves are exponential decay fits. (c,d) Extracted (c) hole occupation probability P h and (d) trapping time T h as a function of the non resonant PQR and resonant P X + powers. (e) Polarized brightness dependence on the hole occupation probability. Points are experimental data that are fitted with solid linear lines. The brightness measurement error bars are not represented on this panel but are typically of ±5%.

Figure 7 .

 7 Figure 7. Quantum performances of the cQED devices. Single photon purity and photon indistinguishability are estimated by HBT (a) and HOM (b) experiments for device QD1 (with a Fabry-Pérot etalon to better suppress the reflected laser)..

ACKNOWLEDGMENTS

This work was partially supported by the ERC PoC PHOW, the French Agence Nationale pour la Recherche (grant ANR SPIQE and ASTRID LIGHT), the Quan-tERA network (project HIPHOP), the French RENAT-ECH network and a public grant overseen by the French National Research Agency (ANR) as part of the Investissements d'Avenir programme (Labex NanoSaclay, reference ANR-10-LABX-0035). J.C.L. and C.A. acknowledge support from Marie Skłodowska-Curie Individual Fellowships SMUPHOS and SQUAPH, respectively.


Zakariya Ghalmane

Stephany Rajeh

Chantal Cherifi

Hocine Cherifi

Mohammed El Hassouni

Finding Influential Nodes in Networks with Community Structure 1 Introduction

Identifying influential nodes is a fundamental issue in complex networks. Several centrality measures take advantage of various network topological properties to target the top spreaders. However, the vast majority of works ignore its community structure while it is one of the main properties of many real-world networks. In our previous work 4 , we show that the centrality of a node in a network with non-overlapping communities depends on two features: Its local influence on the nodes belonging to its community, and its global influence on nodes belonging to the other communities. For this end, we introduced a framework to adapt all the classical centrality measures proposed for networks with no community structure to non-overlapping modular networks. We proposed a two-dimensional vector (the so-called "Modular centrality"), where each dimension accounts for a different type of influence that the nodes can exert in the network. Its first component is measured by computing the classical centrality on the Local network. This network is formed only from the intra-community links of the original network. Additionally, its second component is quantified by computing the classical centrality on the Global network. This network is formed only from the inter-community links of the original network. Depending of the strength of the community structure, these two components are more or less influential. In a recent study5 , we extended this framework to networks with overlapping modules. Indeed, it is a frequent scenario in real-world networks where nodes usually belong to several communities, especially for social networks. The "Overlapping Modular Centrality" is a two-dimensional measure that quantifies the local and global influence of overlapping and non-overlapping nodes. The global component of this vector is defined in the same way as the Modular centrality.

It is computed on the global network obtained by removing all the intra-community links from the original network. However, the local component computation depends on the nature of nodes. For a non-overlapping node, as previously, only members of its community are considered. For an overlapping node, all the communities that the node belongs to are merged in a single community. Extensive experiments have been performed on synthetic and real-world data using the Susceptible-Infected-Recovered (SIR) epidemic model. Results show that the proposed framework (the Modular centrality for networks with non-overlapping modules and the Overlapping Modular Centrality for networks with overlapping modules) outperforms its standard centralities designed for non-modular networks. These investigations provide better knowledge on the influence of the various parameters setting the community structure on the nodes' centrality. Moreover, a straightforward combination of both components (modulus of the twodimensional vector) has been evaluated. Comparative analysis with competing methods shows that they produce more efficient centrality scores.

Proposed approach

We restrict our attention to non-overlapping community structure. We only present the algorithm used to compute the Modular centrality. It can be summarized as follows:

Step 1. Choose a standard centrality measure β .

Step 2. Remove all the inter-community edges from the original network G to obtain the set of communities C forming the local network G l .

Step 3. Compute the local measure β L for each node in its own community.

Step 4. Remove all the intra-community edges from the original network to reveal the set of connected components S formed by the inter-community links.

Step 5. Form the global network G g based on the union of all the connected component.

Step 6. Compute the global measure β G of the nodes linking the communities based on each component of the global network.

Step 7. Add β L and β G to the Modular centrality vector B M .

Results

In Figure 1, we report the results of the SIR simulations using the Degree centrality. The relative difference of the outbreak size ∆ r between the various measures (Local component, Global component, and the modulus of either the Modular Degree centrality or the Overlapping Modular Degree centrality) and the classical Degree centrality have been computed while varying both the fraction of initial spreaders and the mixing proportion parameter µ. The relative difference of the outbreak size represents the difference between the final number of recovered nodes for the centrality measure under test and the standard centrality. It is positive if the centrality under test is more efficient than standard centrality. Moreover, the mixing parameter µ allows to control the strength of the community structure. For low values of µ, the communities are well separated (few inter-community links), while high values of µ indicates a weak community structure.

Results for networks with strong community structure are illustrated in the left panel of Figure 1. We notice that in both type of networks, ∆ r is always positive for the local component , while it is always negative for the global component. Indeed, in networks with strong community structure the local influence predominates. The same type of results are shown in the right panel of Figure 1 for networks with weak community structure strength. This time, one can notice that ∆ r is always positive for the global component, while it is negative for the local component. Indeed, in this situation, the global influence takes over the local influence. Furthermore, we notice that the measure that combines both components is always the most efficient in any case. b) is for networks with overlapping community structure, where the number of overlapping nodes on is fixed to 5% of the size of the network, and the community membership parameter om is equal to 5% of the total number of communities.

To summarize, in order to evaluate efficiently the centrality of nodes in networks with community structure (overlapping communities and non-overlapping communities), we cannot rely on the traditional centrality measures that do not take into account this topological property. It is necessary to consider and combine properly the two types of influence (local, global) occurring in such networks. In this study, we ignored the positional hierarchy of nodes in the original network. In our future work, we intend to take account this property.

Fig. 1 .

 1 Fig.1. Relative difference of the outbreak size ∆ r as a function of the portion of the initial spreaders f 0 . Where ∆ r = (R c -R s /R s ), R c and R s are the final number of recovered nodes for the centrality measure under test and the standard Degree centrality. Each value is the average of 100 SIR simulations per method and fraction of initially infected nodes. Positive value of ∆ r means that the centrality under test is more efficient than the standard centrality. Synthetic networks generated with the LFR algorithm are used with various community structure strength µ. (a) is for networks with non-overlapping community structure. (b) is for networks with overlapping community structure, where the number of overlapping nodes on is fixed to 5% of the size of the network, and the community membership parameter om is equal to 5% of the total number of communities.

Ghalmane, Z., El Hassouni, M., Cherifi, C., & Cherifi, H. (2019). Centrality in modular networks. EPJ Data Science, 8(1), 15.

Ghalmane, Z., Cherifi, C., Cherifi, H., & El Hassouni, M. (2019). Centrality in complex networks with overlapping community structure. Scientific reports, 9(1), 1-29.


domain: phys chim

text:
Rajarshi Sinha-Roy
email: sinharoy@cinam.univ-mrs.fr

Jérôme Hurst

Giovanni Manfredi
email: giovanni.manfredi@ipcms.unistra.fr

Paul-Antoine Hervieux

Driving Orbital Magnetism in Metallic Nanoparticles Through Circularly Polarized Light: A Real-Time TDDFT Study

Keywords: Magneto-optical response, Inverse Faraday effect, Time-Dependent Density-Functional Theory, Plasmon, Orbital angular momentum, Laser induced magnetism

Transfer of angular momentum from helicity-controlled laser fields to non-magnetic electronic system can lead to the creation of magnetization. The underlying mechanism in metallic nanoparticles have been studied using different theoretical approaches.

However, an understanding of the dynamics using orbital-based quantum-mechanical method within the many-body theoretical framework is still due. To this end, the realtime formulation of time-dependent density-functional theory is used to study induced orbital magnetism in metallic nanoparticles (clusters) excited by circularly polarized light. The nanoparticles are described by a spherical jellium model on a real-space grid. The polarized laser field gives rise to an angular momentum, and hence a magnetic moment, which is maximum at the surface plasmon frequency of the nanoparticle, revealing that this is a resonant plasmonic effect. The primary contribution to the magnetic moment comes from surface currents generated by the plasmonic field, although some bulk contributions due to the quantum-mechanical nature of the system (Friedel oscillations) still persist. We compare several nanoparticles of K, Na, and Au having the same size and excited at their respective plasmon frequencies and show that the generated magnetic moment per energy pumped into the system is maximum for K and minimum for Au. A similar trend is observed for nanoparticles of the same chemical species but different sizes.

Introduction. Inducing magnetism in non-magnetic materials by means of the controlled helicity of laser fields is currently unraveling new paths for all-optical magnetic switching. The resonant interaction between an electromagnetic field and the conduction electrons in a metal, giving rise to collective oscillations, is one of the key controlling parameters of the underlying mechanism. A typical example of this interaction is the localized surface plasmon (LSP) in a gold nanoparticle excited by a short laser pulse, which corresponds to the coherent oscillations of the electron cloud, leading to a strong local enhancement of the electric field. [START_REF] Kreibig | Optical properties of metal clusters[END_REF] The plasmonic properties can be used to enhance and tailor the magnetic properties of matter at nanoscale thanks to the material-specific magneto-optical response. This has already been applied to all-optical magnetic switching in ferri-or ferromagnetic materials and graphene. [START_REF] Stanciu | All-optical magnetic recording with circularly polarized light[END_REF][START_REF] Alebrand | Light-induced magnetization reversal of highanisotropy TbCo alloy films[END_REF][START_REF] Mangin | Engineered materials for all-optical helicity-dependent magnetic switching[END_REF][START_REF] Lambert | All-optical control of ferromagnetic thin films and nanostructures[END_REF][START_REF] El Hadri | Electrical characterization of all-optical helicity-dependent switching in ferromagnetic Hall crosses[END_REF][START_REF] John | Magnetisation switching of FePt nanoparticle recording medium by femtosecond laser pulses[END_REF][START_REF] Vomir | Single shot all optical switching of intrinsic micron size magnetic domains of a Pt/Co/Pt ferromagnetic stack[END_REF][START_REF] Bychkov | Plasmon mediated inverse Faraday effect in a graphene-dielectric-metal structure[END_REF][START_REF] Chekhov | Surface Plasmon-Mediated Nanoscale Localization of Laser-Driven sub-Terahertz Spin Dynamics in Magnetic Dielectrics[END_REF] Surface plasmon polaritons were used for improving light-assisted magnetic switching mechanisms [START_REF] Bychkov | Plasmon mediated inverse Faraday effect in a graphene-dielectric-metal structure[END_REF][START_REF] Chekhov | Surface Plasmon-Mediated Nanoscale Localization of Laser-Driven sub-Terahertz Spin Dynamics in Magnetic Dielectrics[END_REF] giving rise to the field of magneto-plasmonics. It has been shown that one of the possible pathways to enable fast all-optical switching is the inverse Faraday effect (IFE), [START_REF] Stanciu | All-optical magnetic recording with circularly polarized light[END_REF][START_REF] Mangin | Engineered materials for all-optical helicity-dependent magnetic switching[END_REF][START_REF] Lambert | All-optical control of ferromagnetic thin films and nanostructures[END_REF][START_REF] John | Magnetisation switching of FePt nanoparticle recording medium by femtosecond laser pulses[END_REF] which corresponds to a transfer of angular momentum from a circularly polarized laser field to the electronic system, leading to the creation of a static magnetization. [START_REF] Pershan | Theoretical Discussion of the Inverse Faraday Effect, Raman Scattering, and Related Phenomena[END_REF] For instance, the Faraday rotation in gold-coated maghemite nanoparticles can be increased thanks to the plasmonic properties of gold. [START_REF] Jain | Surface Plasmon Resonance Enhanced Magneto-Optics (SuPREMO): Faraday Rotation Enhancement in Gold-Coated Iron Oxide Nanocrystals[END_REF] Similarly, the magneto-optical response of nickel nano-disks can be adjusted by controlling the phase of the localized surface plasmons. [START_REF] Maccaferri | Tuning the Magneto-Optical Response of Nanosize Ferromagnetic Ni Disks Using the Phase of Localized Plasmons[END_REF] In this context, very recently light-induced magnetism in plasmonic gold nanoparticles has also been reported. [START_REF] Cheng | Light-induced magnetism in plasmonic gold nanoparticles[END_REF] The inverse Faraday effect has been investigated using different theoretical approaches. [START_REF] Battiato | Quantum theory of the inverse Faraday effect[END_REF][START_REF] Kundu | Effect of laser induced orbital momentum on magnetization switching[END_REF][START_REF] Scheid | Ab initio theory of magnetization induced by light absorption in ferromagnets[END_REF][START_REF] Scholl | Off-resonant all-optical switching dynamics in a ferromagnetic model system[END_REF] This effect was also studied for different bulk materials using density functional theory (DFT), [START_REF] Berritta | Ab Initio Theory of Coherent Laser-Induced Magnetization in Metals[END_REF] and the effect of spin-orbit coupling was also recently explored. [START_REF] Scheid | Ab initio theory of magnetization induced by light absorption in ferromagnets[END_REF] The plasmonic properties of the inverse Faraday effect in metal nanoparticles were also investigated by several groups. [START_REF] Nadarajah | Optoelectronic phenomena in gold metal nanostructures due to the inverse Faraday effect[END_REF][START_REF] Koshelev | Resonant inverse Faraday effect in nanorings[END_REF] In a recent work, [START_REF] Hurst | Magnetic moment generation in small gold nanoparticles via the plasmonic inverse Faraday effect[END_REF] some of the authors of the present study used a quantum hydrodynamic (QHD) model [START_REF] Manfredi | How to model quantum plasmas[END_REF][START_REF] Vladimirov | On description of a collisionless quantum plasma[END_REF][START_REF] Ciracì | Quantum hydrodynamic theory for plasmonics: Impact of the electron density tail[END_REF] to describe the interaction between the surface plasmons in gold nanoparticles and circularly polarized light. It was shown that, under the action of a circularly polarized laser field at the LSP frequency, the gold nanoparticles can acquire a static magnetic moment whose direction is reversed when changing the light polarization.

The model stressed the importance of the finite size of the electronic system: LSP generates electronic currents localized on the surface of the nanoparticle, which in turn create a quasi-static orbital magnetic moment. However, in this model, the electrons are treated as a charged fluid [START_REF] Manfredi | How to model quantum plasmas[END_REF][START_REF] Ciracì | Quantum hydrodynamic theory for plasmonics: Impact of the electron density tail[END_REF][START_REF] Crouseilles | Quantum hydrodynamic model for the nonlinear electron dynamics in thin metal films[END_REF][START_REF] Toscano | Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics[END_REF] and lack the orbital resolution of a fully quantum-mechanical description of an electronic system. Therefore, although the QHD model and previous theoretical approaches describe the magneto-optical responses in nanoparticles reasonably well, a thorough understanding of the chiro-optical dynamics of the generation of orbital magnetic moment in metallic nanoparticles is still missing from a fully quantum many-body perspective.

In the present work, we investigate the orbital magnetism using a quantum mechanical method for many-body systems, namely the real-time approach of the time-dependent density-functional theory (RT-TDDFT). The calculations are performed for jellium spherical nanoparticles (size 6 nm) containing different numbers of electrons, in order to study the effect of size on the generated orbital angular momentum. The generation of an orbital angular momentum and the associated magnetic moment through the IFE is fully confirmed and its magnitude is consistent with results based on the QHD 22 model. We discuss the effects of the energy and polarization angle of the incident laser field on the magnetic moment generation. Finally, we study the effect of chemical species (K, Na, and Au) by comparing results for different Wigner-Seitz radii for two different cases: nanoparticles having the same number of electrons and nanoparticles having the same jellium radius.

The present study confirms that orbital magnetism can be successfully triggered through the IFE by irradiating non-magnetic nanoparticles with circularly polarized light. The resulting magnetic moment is significant (about 0.04 µ B per electron in the best case for K) and robust over a wide range of parameters, including atomic species, laser intensity, and size of the nanoparticle. The crucial point is that the material should support a strong plasmonic response at the driving laser frequency. The generated magnetic moment could be tailored to trigger ultrafast all-optical switching in nanometric systems such as core/shell nanoparticles. [START_REF] Mullins | Chiral symmetry breaking yields the I-Au60 perfect golden shell of singular rigidity[END_REF][START_REF] Sinha-Roy | How metallic are noble-metal clusters? Static screening and polarizability in quantum-sized silver and gold nanoparticles[END_REF] Theoretical background. The real-time approach of the time-dependent densityfunctional theory (RT-TDDFT) is used to study the optical response in metallic systems.

The evolution of the many-electron system under an external perturbation is obtained from the solution of the time-dependent Kohn-Sham equations, under the action of an effective time-dependent potential which is a functional of the time-dependent electron density n(r, t).

The effective potential contains both the Hartree and the exchange-correlation contributions. The latter is not known exactly and in practice, requires to be approximated. In this work, we have employed the adiabatic local-density approximation (ALDA), which suffices to describe energy absorption in metallic jellium systems, where collective excitations dominate the spectrum.

The ground state electron density of the jellium system is calculated in static DFT. The absorption spectrum is obtained in RT-TDDFT using the the time-propagation formalism of Yabana and Bertsch. [START_REF] Yabana | Time-dependent local-density approximation in real time[END_REF][START_REF] Yabana | Real-time, real-space implementation of the linear response time-dependent density-functional theory[END_REF] The system is excited by an electric field with strength K 0 , having a temporal form of a dirac-delta function, and linearly polarized along a given direction, say

z, δE(r, t) = K 0 δ(t)e z . (1)
From the time-dependent induced density δn(r, t) corresponding to this excitation, the frequency-dependent absorption cross-section is obtained through Fourier transformation as,

σ abs (ω) = - 4πω cK 0 Im ˆ∞ 0 dt e iωt ˆV dr z δn(r, t), ( 2
)
where c is the speed of light and V denotes the volume of integration. For a spherically symmetric jellium system, Eq.( 2) gives the total absorption spectrum. The strength K 0 in Eq.( 1) is usually chosen such that the perturbation remains within the linear response regime.

In the present work, we use a homogeneous circularly (or, more generally, elliptically) polarized field to trigger the generation of an orbital angular momentum in the nanoparticle.

The homogeneous electric field can be written as: E = (E x , E y , 0), where

E x (t) =        E 0 cos(ωt) sin πt T , 0 ≤ t ≤ T 0 , otherwise (3)
E y (t) =        E 0 cos(ωt -φ) sin πt T , 0 ≤ t ≤ T 0 , otherwise. (4)
Here, φ is the polarization angle (φ = ±90 • for left or right circularly-polarized light). The second sinusoidal term in Eqs. ( 3)-( 4) constitutes the temporal envelope of the pulse, which has a duration equal to T . As shown in Fig. S2, a Gaussian functional form could also be chosen for the temporal envelope, without changing significantly the results.

The time-dependent current density j(r, t) is obtained in the length gauge from the timedependent Kohn-Sham wave functions ϕ k (r, t) as

j(r, t) = -ie 2m e N k=1 ϕ * k (r, t)∇ϕ k (r, t) -ϕ k (r, t)∇ϕ * k (r, t), ( 5
)
where N is the total number of Kohn-Sham states, m e and e are respectively the mass and charge of the electron, and is Planck's constant. The total orbital magnetic moment generated in the system is then obtained from the time-dependent current density as

M(t) = 1 2 ˆV dr r × j(r, t). (6)
We will also use, as a diagnostic tool, the dipole moment, which is defined as

p(t) = ˆV dr r δn(r, t), (7)
where δn(r, t) is the time-dependent induced density corresponding to the excitation by the laser field.

Systems and simulation details. We study jellium spheres with valence electron densities corresponding to three different chemical species: K, Na, and Au, with values of the Wigner-Seitz radius being r s = 2.572 Å, 2.083 Å and 1.593 Å respectively. The radius of the jellium sphere is related to the total number of electrons N e through the expression

R = N 1/3 e r s .
For Au, we have considered only the s electrons and no polarizable jellium background is used in the model to mimic the dynamical screening by the d electrons. [START_REF] Lermé | Introduction of quantum finite-size effects in the Mie's theory for a multilayered metal sphere in the dipolar approximation: Application to free and matrixembedded noble metal clusters[END_REF][START_REF] Hervieux | Electronic temperature effects onthe optical response of silver nanoparticles[END_REF] All the calculations are performed using the real-space code octopus. [START_REF] Marques | octopus a first-principles tool for excited electron-ion dynamics[END_REF] The calculation domain is taken to be a sphere centered at the the center of the nanoparticle. The radius of the calculation domain is chosen to be 5.0 Å larger than the radius of the jellium nanoparticle.

The real-space grid spacing was set to 0.5 Å for simple metals (K and Na) and 0.25 Å for the noble metal. For the calculation of the ground-state electron density, a stopping criterion of 10 -7 Å-3 for the relative electron density is chosen for the convergence of the self-consistent loop. For the time evolution, a time-step of 0.00658 fs (≈ 0.01 /eV ≈ 0.272 /Hartree) is used for all the systems. The absorption spectra are obtained from a total evolution time of ≈ 25 fs.

In the calculations using a circularly polarized electric field, the amplitude of the field was set to E 0 = 0.01 V/ Å = 10 8 V/m, corresponding to a peak laser intensity (I 0 ) of 1.327 × 10 9 W/cm 2 . This choice is made so that the perturbation remains weak enough for the response to be within the linear regime and for avoiding photo ionization. The duration of the laser pulse was set to T ≈ 13.16 fs for all the calculations. We have studied the optical response for different values of the polarization angle, changing the polarization gradually from circular (φ = 90 • ) to elliptical (φ = 60 • and 30 • ) and finally to linear (φ = 0 • ) polarization. The response of the system is calculated for a time duration equal to twice the duration of the laser pulse.

Magnetic moment generation. In this section, we describe the effect of the energy and the polarization of the incident laser field on the generation of angular momentum and present the spatio-temporal dynamics of the electron density and current density at the plasmon energy. The absorption spectra calculated using the adiabatic local-density approximation (TD-ALDA) for potassium clusters of different sizes are shown in Fig. 1.

They show the well-known red shift [START_REF] Bréchignac | Optical response of large lithium clusters: Evolution toward the bulk[END_REF][START_REF] De Heer | The physics of simple metal clusters: experimental aspects and simple models[END_REF] of the LSP peak in a simple metal as the size decreases. This red shift is attributed to the spill-out 37 of the ground state electron density, which becomes more and more significant with decreasing jellium radius, cf Fig. S1 in the Supporting Information. Moreover, we note that other spectral features than the plasmonic peak appear, due to the fragmentation of the collective excitation in the spectrum of the smallest system considered here (K 58 ).

Figure 1: Absorption spectra for potassium jellium clusters of different sizes: K 58 , K 561 , K 923 , K 2057 . The spectra are normalized to the total number of electrons contained in the system.

The absorption spectrum of jellium K 561 cluster displays a single prominent plasmon peak at 2.39 eV with a full-width at half maximum of ≈ 0.18 eV, as shown in Fig. 1. In order to study the effect of the laser frequency on the generation of a magnetic moment, we calculated the response of this system for quasi-monochomatic circularly polarized pulses with frequencies 2.39 eV (at the plasmon resonance) and 1.55 eV (off-resonance). The results of the TDDFT simulations are presented in Fig. 2, for the on-resonance (right panels) and off-resonance (left panels) cases. This figure shows several features of the laser excitation (laser electric field) and the electronic response of the system (dipole moment, absorbed energy, magnetic moment). The differences between the resonant and off-resonant response of the system are clearly captured in the magnitudes of both the dipole moment (panels (b) and (f)) and the absorbed energy (panels (c) and (g)).

As it can be seen in panel (d) and (h) of Fig. 2, the generated magnetic moment (M z ) is two orders of magnitude larger at resonance as compared to the off-resonance case. The

x and y components of the magnetic moment (Fig. 2(h)) remain negligibly small, as they should be, both during and after the laser excitation. In contrast, M z increases in time and closely follows the time dependence of the energy absorbed by the system. After the laser is turned-off, both the total energy and M z reach saturation values, as they are conserved in the simulation. These results are in line with earlier findings using a QHD approach, 22 thus confirming that the observed magnetic moment generation is indeed a resonant plasmonic effect. At the end of the laser pulse, the total energy given to the system remains constant, cf. Fig. 2(g). The energy does not decay as there is no decay channel (radiation losses, etc.) present in the simulations. However, Landau damping, which accounts for the phase mixing between the electronic states, is taken into account and can be observed in the timedependent dipole moments, which slowly decay in time -see Fig. 2(f). In Fig. 3, we present the time dependence of the normalized dipole moment (top panel) and the normalized z component of the magnetic moment M z (t) (bottom panel) generated in a K 561 jellium sphere, for different values of the peak amplitude E 0 of the circularly polarized laser field. The dipole moments are divided by the corresponding E 0 , whereas M z (t) is divided by E 2 0 (this is because the IFE is proportional to the intensity of the laser, hence to the square of the electric field). The figure shows that below E 0 ≈ 0.01 V/ Å the corresponding normalized dipole moments do not differ from each other, signaling that in this range the electron response is linear. The transition to the nonlinear response regime takes place between E 0 = 0.01 V/ Å and E 0 = 0.1 V/ Å, as for E 0 = 0.1 V/ Å the amplitude of both the dipole and the magnetic moment is no longer proportional to the excitation. For even larger amplitudes (E 0 = 1 V/ Å, not shown in the figure), the systems is very strongly perturbed and goes into the ionization regime. Figure 3 ensures that by choosing E 0 = 0.01

V/ Å we remain indeed within the linear response regime. Incidentally, it also confirms that the induced magnetization is due to the IFE in the linear response regime, as it follows the same scaling with the field intensity as that of IFE. Induced Density. As shown in Fig. 2, the generation of a magnetic moment in a K 561 jellium sphere is strongly favored at the LSP frequency. In order to better understand the influence of the LSP on the magnetic moment generation, we have studied the induced time-dependent density, from the corresponding TDDFT simulation. The induced density is defined as

δn(r, t) = n(r, t) -n GS (r), (8)
where n GS is the ground-state electron density. The result is presented in Fig. 5. The contents of panel (a) of Fig. 5 is the same as in panel (f) of Fig. 2. It shows the time dependence of the dipole moment in K 561 due to the excitation by right-circularly polarized light at the LSP frequency. The shaded grey area corresponds to the duration of the laser pulse. In Fig. 5(b), we show an iso-surface of the induced density δn(r, t) at time t = 15.96 fs i.e., after the laser is switched off. The induced density shows accumulation of opposite charges at the surface of the jellium sphere along the y direction. This confirms that the LSP mode is dipolar and also that at time t = 15.96 fs the y component of the dipole moment reaches a maximum, as indicated by the blue dot (iii) in panel (a). We note that, at this instant, the x component of the dipole moment is close to zero. However, the induced density also shows other contributions from the interior of the sphere, which are concentric and not fully symmetric along the y direction.

In order to better understand the time-dependent evolution of the electron density, we have taken snapshots of the induced density for five different instants, indicated by the red and blue dots respectively on the x and y components of the dipole moment in Fig. 5(a).

All five snapshots at times (i) 15.14 fs, (ii) 15.47 fs, (iii) 15.96 fs, (iv) 16.45 fs, and (v) 16.78 fs together represent a full period of the circular rotation of the total dipole moment, and hence of the induced electron density of the system, after the laser is switched off. The snapshots of these induced densities are shown in five different panels in Fig. 5(c), which represent the electron density in the (x, y) plane, passing through the center of the jellium sphere and parallel to the plane of polarization (z = 0). These snapshots confirm that, at the LSP frequency, the principal contribution responsible for the circular motion of the induced density lies at the surface of the cluster. As we move from the surface to the center of the cluster, we notice that dipolar contributions, of smaller magnitude than at the surface, form concentric shells of different radii. These contributions correspond to those captured at the interior of the system in the isosurface representation of the induced density in panel (b).

The origin of these contributions is fully quantum mechanical and can be traced back to the Friedel-like radially oscillating nature of the ground-state electron density [START_REF] Lang | Theory of Metal Surfaces: Charge Density and Surface Energy[END_REF] (cf. Fig. S1 of the Supporting Information). A careful observation reveals that, although these shells are undergoing the same circular motion as the surface contribution, there exists a lag of phase between them. An animation presenting the time-dependent circular motion of the induced density is given in Fig. S3 of the Supporting Information. Fig. 5 shows that, when the cluster is excited by a circularly polarized wave at the LSP frequency, the surface of the system plays the leading role in the circular motion of the induced electron density and hence also in the generation of the magnetic moment. Current density. The current density directly concerns the generation of the magnetic moment through Eq. ( 6). Consequently, we have studied the spatial behavior of j(r, t) after switching off the laser (t > T ). Figure 6 shows the spatial distribution of the current density j(x, y, 0) in the z = 0 plane, which is parallel to the polarization plane of the driving field.

The current density is shown at every third grid point (shown in red dots) of the real-space grid on which it was computed (using the octopus code). The top panel of Fig. 6 shows j(x, y, 0) at the time 15.96 fs for which the induced densities are shown in Fig. 5(b) and also in the middle panel of Fig. 5(c). At a given instant of time after the laser is switched off, the current density within the cluster is not directed towards a preferred direction. The origin of these fluctuations of the instantaneous current is quantum-mechanical and is related to the size-confinement effects that give rise to Friedel-like oscillations of the electron density.

This effect was not captured in the earlier QHD model. An animation that displays the time-dependent current corresponding to the full period of the dipole oscillation between instants (i) and (v) of Fig. 5(a) is given in Fig. S3 of the Supporting Information.

The bottom panel of Fig. 6 shows the current density for the same z = 0 plane as in the top panel, but averaged over one period of oscillation of the dipole moment. In order to produce a magnetic moment, the current must form a loop. In the case of a metal cluster, this is achieved by a full rotation of its free electron density. Therefore, the spatial distribution of the average current density in Fig. 6 shows the region of space where current loops are formed to generate the saturated magnetic moment in the cluster. As expected at the LSP frequency, there is a non-vanishing current at the surface of the cluster. However, at the interior of the cluster there are also two different concentric loops of current: one of them is rotating in the sense of the current at the surface, and the other in the opposite sense.

From the figure, it is qualitatively clear that the resultant of the contributions of all these different loops creates a magnetic moment along the z direction. This result, along with the observations on the induced density in Fig. 5, confirms that, although the surface takes a leading role in the generation of the magnetic moment through IFE, contributions from the core of the cluster are of comparable importance.

Effect of cluster size and chemical species. In order to study the effect of size and chemical species on the generation of angular momentum, we have performed RT-TDDFT calculations for jellium clusters of K, Na, and Au having different numbers of electrons. Here the different chemical species differ from each other only by their respective values of the Wigner-Seitz radius r s . The LSP resonant frequencies for these systems are obtained from the absorption spectra calculated in TD-ALDA. These values of N e have been chosen so that the jellium systems contain the same number of electrons as the number of atoms in icosahedral clusters (except for K 58 ). The jellium radii and the TD-ALDA calculated LSP frequencies are listed in Table 1. The absorption spectra were shown in Fig. 1.

The systems are excited by circularly polarized electric fields at their respective plasmon frequencies. The results are displayed in Fig. 7. For all cases, the evolution of the M z component of the magnetic moment displays an initial growth, followed by saturation when the laser is switched off (left panel). The value reached by M z at saturation scales linearly with the number of electrons N e in the system as shown in the right panel of Fig. 7. This indicates that the per-electron magnetic moment generated through IFE remains constant for clusters with a diameter 6.5 nm, such as those considered here (see Table 1).

Effect of chemical species for the same radius. In order to study the effect of the chemical species on clusters of the same size, we consider jellium spheres of sodium (Na 106 ) and gold (Au 225 ) which have almost the same jellium radius as that of K 58 . The absorption spectra calculated in TD-ALDA for these three systems are shown in Fig. S4 of the Supporting Information. The Wigner-Seitz radii r s , jellium radii, and the LSP frequencies are summarized in Table 1. The systems are then excited by circularly polarized light at their respective plasmon frequencies. The results of the simulation are shown in Fig. 8.

The top most panel shows M z (t) for the three systems. Although they are excited at different respective LSP frequencies, the generated magnetic moments are very similar to each other. On the contrary, the induced dipole moments are rather different. This can be attributed to the fact that Landau damping is different for different chemical species (cf.

Fig. S3 in the Supporting Information). Nevertheless, the dipole moments display similar maximum amplitudes for all cases (≈ 5.5 Å).

The pulse duration of the lasers used for the three different systems is the same. Therefore, as the frequencies are different, the number of oscillations of the electric field within the pulse duration is not the same. Hence, during the laser excitation, the three clusters did not absorb the same amount of energy (as listed in Table 2). The total energy absorbed by the three clusters as a function of time is shown in Fig. S4 of the Supporting 1). Lower panels: Time dependence of the x (red) and y (blue) components of the induced dipole moment for each system. Information. In order to better understand the effect of the chemical species, the generated magnetic moment normalized to the total absorbed energy M z (t)/E T ot is shown in Fig. 9, left panel. This quantity is maximum for K 58 and minimum for Au 254 , and appears to be directly proportional to the the Wigner-Seitz radius r s of each species as shown in Fig. 9, right panel.

Effect of chemical species for same N e . We consider jellium clusters of potassium, sodium, and gold, all having 58 electrons (their characteristics are summarized in Table 1) and excited by circularly polarized laser at their respective LSP frequencies. The TD-ALDA absorption spectra of K 58 , Na 58 , and Au 58 are provided in Fig. S5 of the Supporting Information. Unlike in Fig. 8 (clusters of same size, but different N e ), here the three systems with same N e do not display a similar M z at saturation (cf. Fig. 10). The dipole moments are also rather very different from each other in this case.

The total energy pumped by the laser pulse into these 58-electron jellium spheres is given in Table 2 and in Fig. S5 the Supporting Information. In Fig. 11, we plot the magnetic moment M z normalized to such energy, which shows that M z /E T ot is maximum for K 58 and minimum for Au 58 . Thus, also in this case, M z /E T ot appears to be proportional to the Wigner-Seitz radius r s of the material.

Classical Model. A simple illustration of our findings can be obtained by making Considering electric field propagation in the z direction, the electronic dipoles d x (t) and d y (t) in the transverse plane obey the linearized equations of motion:

dx = -ω 2 Mie d x -γ ḋx + e m e E x (t), (9) dy
= -ω 2 Mie d y -γ ḋy + e m e E y (t). (10)
Here, e and m e are respectively the charge and mass of an electron, E x (t) and E y (t) are respectively the x and y component of the electric field, ω Mie is the Mie frequency, which is a good approximation of the surface plasmon frequency for relatively large nanoparticles, and γ is a phenomenological damping rate that incorporates the effect of all incoherent interactions between the collective electron modes (Landau damping) and between the electrons and the environment. The dipoles are excited by an elliptically-polarized monochromatic laser pulse, with frequency ω L and phase shift φ L :

E x (t) = E 0 cos (ω L t) , E y (t) = E 0 cos (ω L t + φ L ) . (11)
For simplicity, we have neglected the envelope of the laser pulse.

We consider solutions of equations ( 9)- (10) in the stationary regime, where the dipoles are assumed to oscillate at a single frequency, equal to the laser frequency ω L . In this case, the general solutions for the two dipole modes can be written as:

d x (t) = d 0 cos (ω L t + ϕ) , (12)
d y (t) = d 0 cos (ω L t + ϕ ) , (13)
where d 0 denotes the amplitude of the dipole modes. Injecting Eqs. ( 12)-( 13) in Eqs. ( 9)-( 10), we get:

tan (ϕ) = γω L ω 2 L -ω 2 Mie (14)
and ϕ = ϕ + φ L , i.e., the phase shift between the two dipoles is the same as the one between the two corresponding components of the electric field. We also obtain the relation between the amplitude of the dipole modes and the amplitude of the electric field:

d 0 = eE 0 /m e (ω 2 L -ω 2 Mie ) 2 + γ 2 ω 2 L 1/2 (15)
As expected, Eq. ( 15) corroborates that the dipoles are strongly excited at the surface plasmon resonance, where ω L = ω Mie .

Considering a homogeneous electron density for a spherical metallic nanoparticle, the magnetic moment associated with the displacement of the center of mass of the electron gas can be written as: [START_REF] Battiato | Quantum theory of the inverse Faraday effect[END_REF][START_REF] Hurst | Magnetic moment generation in small gold nanoparticles via the plasmonic inverse Faraday effect[END_REF]

M = 1 2 ˆr × j dr = - N e 2 d(t) × ḋ(t), ( 16
)
where N is the total number of electrons in the nanoparticle. Injecting the expressions of the two dipoles ( 12)-( 13), together with Eq. ( 15), into Eq. ( 16), we obtain:

M z = N 2 e 3 m 2 e ω L E 2 0 sin (φ L ) (ω 2 L -ω 2 Mie ) 2 + γ 2 ω 2 L . (17)
The above expression corresponds to the classical inverse Faraday effect in a nanoparticle containing N delocalized electrons. Indeed, the magnetization is proportional to the square of the electric field, vanishes for linearly polarized light (φ L = 0), and is maximum for circularly polarized light (φ L = ±π/2). For small damping γ, the generated magnetic moment can reach very high values if the laser pulse is tuned at the surface plasmon frequency (ω L = ω Mie ).

This simple result demonstrates that the magnetic moment generation is indeed due to a plasmonic resonance.

Further, one can also show that the current density contribution relevant for the magnetic moment generation is localized on the nanoparticle surface. Following Ref. 39, the electron density can be written as

n = n + δn, (18)
where n is the time-averaged value of the density (which is constant or slowly varying in time) and δn represents the rapidly oscillating fluctuation driven by the external electric field. In a finite-size object, such as a metal nanoparticle, δn is localized on the surface. The same decomposition can be made on the electric current density: j = j + δj. As δn and δj vary with the oscillating external electric field, they vanish when averaged over a period of oscillation, i.e.:

δn = δj = 0. ( 19
)
Making use of the fact that δn n (linear response) and separating the slow and fast terms in the continuity equation, we obtain that:

δj = e n δu, (20)
where δu is the fluctuating part of the velocity.

In the Drude approximation, δj = σE (note that the electric field E only has an oscillating component), where σ = i n e 2 /(mω L ) is the electric conductivity. Inserting the Drude expression into Eq. ( 20), we get the fluctuating part of the velocity

δu = σ e n E. (21)
Finally, using the time-averaged continuity equation and assuming u = 0, the slowly varying part of the current becomes j = e δn δu = ie 2 m e ω L δn E .

From Eq. ( 22), it is clear that the time-averaged current j is localized on the surface of the nanoparticle, because it is proportional to the localized density fluctuation δn. By contrast, the fluctuating current δj depends on the equilibrium density n [cf. Eq. ( 20)]

and not on δn. Thus, it is not localized on the surface. Moreover, it vanishes when averaged over a full period of oscillation [cf. Eq. ( 19)]. These conclusions are in line with our earlier results obtained using a QHD model [START_REF] Hurst | Magnetic moment generation in small gold nanoparticles via the plasmonic inverse Faraday effect[END_REF] and with the present simulations, as shown in Fig. 6.

Our RT-TDDFT simulations provide strong proof that the laser-induced generation of the magnetization is a robust effect, which persists even when the fully quantum, self-consistent, and nonlinear many-electron dynamics is taken into account. In addition, in contrast to the intuitive classical approach sketched above, the RT-TDDFT simulations also explore the quantum-mechanical effects due to the finite size of the nanoparticle. The resulting confinement of the electron gas gives rise to Friedel-like oscillations in the electron density (Fig. 5), yielding in turn the spatial fluctuations of the instantaneous current density observed in Fig. 6. Consequently, concentric contributions with opposed directions of rotation appear in the current density averaged over one period of oscillation (bottom panel of Fig. 6). These interior contributions are of purely quantum-mechanical origin and cannot be captured by simple classical models.

Discussion. In the simulations presented in Figs. 8 and 10, the induced dipole moment at LSP frequency decays differently for different chemical species and for different sizes of the clusters. As no decay channel other than the Landau damping is incorporated, the observed decay of the dipole moments in the presented results is necessarily due to this effect, and corresponds to the electronic dephasing, i.e., a decay of the phase correlations between the electron-hole excitations that form the plasmon. It is worth mentioning that, although the dipole moments are observed here to decay exponentially, due to the absence of other decay channels they would revive again later in time. This revival of dipole oscillations forms beat-like features as can be seen for the cases of Au 254 (cf. Fig. 8) and Na 58 (cf. Fig. 10). All these results together confirm that, in systems belonging to the size range considered here (see Table 1), Landau damping is important. An exception may be Au 58 , where the plasmon peak is most likely due to a single highly-renormalized electron-hole excitation which remains well separated from any other neighboring excitation in the spectrum (cf. However, Figs. 8 and 10 also indicate that Landau damping does not affect the magnetic moment generated in those systems. It also does not affect the total energy of the system, as can be seen from Fig. 2. This is an intriguing result, as it suggests that the life-time of the magnetic moment may be much larger than a few tens of femtoseconds, which was the duration of the simulations presented here. The absence of decay for M z (t) is linked to the conservation of the total angular momentum for the highly symmetric systems considered here. For systems possessing fewer symmetries, it may no longer be true, but further studies are needed to clarify this important issue. Of course, other energy-dissipating mechanisms, not included here, may also be responsible for the decay of M z in metal clusters.

Another important issue concerning the generated orbital magnetic moment is the effect of the chemical species. Figures 9 and11 shows that the magnetic moment generated per unit energy absorbed, and at LSP frequency is maximum in K and minimum in Au. To this end, we emphasize that we define chemical species through their r s values only. In our simplified description, the Au clusters do not have d electrons, nor do they have a polarizable background that would mimic the effect of the d electrons on the delocalized electrons of the system. Using a proper description of strongly localized d electrons, the dynamical response in the Au systems would be screened by their presence, which might diminish the generation of orbital magnetic moment observed here. We would also like to mention that in our simulations we did not consider the spin degrees of freedom, which in principle can couple to the orbital magnetic moment. Depending on the species-specific spin-orbit coupling, the overall induced magnetic moment can thus be modified. However, in this work we focused solely on the orbital magnetic moment and discussed its generation and dynamics. Spin effects are left for further studies.

Finally, for Au clusters we have compared the per-atom magnetic moment at saturation normalized with respect to pulse duration and peak intensity (i.e., M z /(N e T I 0 ) and considering one electron per atom for Au), with the recent experimental results of Cheng et al. [START_REF] Cheng | Light-induced magnetism in plasmonic gold nanoparticles[END_REF] For Au 254 and Au 58 we obtained respectively 4.91 and 7.06 in units of µ B m 2 fs -1 W -1 × 10 -17 , which should be compared to 2.82 obtained from the experimental results. [START_REF] Cheng | Light-induced magnetism in plasmonic gold nanoparticles[END_REF] The comparison is shown in Table T1 of the Supporting Information. One of the reasons for getting higher values in calculations is that unlike in the experiment, 14 the presented calculations do not take into account the effects of chemical environment around the Au clusters in solution.

Conclusion.

In this work, we have presented real-time TDDFT calculations for the optical response of jellium clusters excited by circularly polarized laser fields in the linearresponse regime. We have considered the effects of different physical parameters, both pertaining to the laser (frequency, polarization, intensity) and to the clusters (size and chemical species).

We showed that irradiating small metal clusters with circularly polarized laser light induces an orbital magnetic moment M z through the inverse Faraday effect. Similarly to results

obtained in an earlier study using a simpler QHD model, [START_REF] Hurst | Magnetic moment generation in small gold nanoparticles via the plasmonic inverse Faraday effect[END_REF] we showed that orbital magnetic moment generation is a resonant effect and only occurs at or near the surface plasmon frequency of the cluster. In the linear-response regime, M z scales as the square of the peak amplitude of the electric field, E 2 0 . The transition to the nonlinear regime appears to occur for 10 8 < E 0 < 10 9 V/m. Crucially, by changing the polarization φ of the laser field, we can tune M z at saturation within a wide range of values: maximum for φ = ±90 • (circular polarization) and zero for φ = 0 • (linear polarization). One can also flip the direction of M z by changing the polarization by an angle of 180 • . After the perturbation is switched-off, the induced magnetic moment does not decay in our calculations owing to the high symmetry of the jellium systems, and the absence of defects and phonon scattering.

A detailed spatial analysis of the induced electron density and the current density shows that, the primary contribution in building up orbital magnetic moment comes from the surface. However, significant contributions from the core of the cluster are also present.

These core contributions are a direct consequences of the Friedel-like oscillations of the ground-state electron density due to quantum-mechanical size-confinement effects. They are not captured in the QHD model, [START_REF] Hurst | Magnetic moment generation in small gold nanoparticles via the plasmonic inverse Faraday effect[END_REF] which lacks the orbital resolution of the electronic system.

We also presented the effect of size and of chemical species of the system (i.e., r s ) on the laser-generated magnetic dipole. For a given chemical species (i.e., constant r s ), the magnetic moment per electron remains almost constant in the size range discussed in this work (up to about ∼ 6.5 nm in diameter, for K), i.e. M z varies linearly with the total number of electrons. The effect of chemical species was studied for K, Na, and Au using their respective r s values. For systems of different chemical nature having the same size or having the same number of electrons, we showed that at LSP frequency the generated magnetic moment per energy absorbed increases with the Wigner-Seitz radius r s .

All in all, the present findings confirm the results obtained using a simpler QHD approach: 22 a sizeable orbital magnetic moment can be generated in metal clusters using circularly polarized laser pulses at or near the surface plasmon frequency. However, unlike in the QHD approach where the magnetic moment is entirely due to surface contributions, the present results show that both the surface and the core of the cluster contribute to the magnetic moment. The generated orbital magnetic moment is very stable and can in principle persist for many tens of femtoseconds. The effect is robust when testing for different chemical species and clusters sizes.

Finally, we mention that our simulations of the generation of orbital magnetic moment in gold clusters produced results having magnitudes consistent with recently reported exper-imental study. [START_REF] Cheng | Light-induced magnetism in plasmonic gold nanoparticles[END_REF] The decisive point for the magnetic moment generation is that the material supports a strong plasmonic response at the driving laser frequency. The robust confirmation of this magneto-plasmonic phenomenon presented here is believed to lead to novel techniques in achieving ultrafast plasmon-assisted optical switching in suitable systems such as, core/shell nanoparticles, supported gold discs, or gold rings with a magnetic core.

Associated Content

The following supporting information is available free of charge at http://pubs.acs.org.

Figure 2 :

 2 Figure 2: Quasi-monochromatic circularly-polarized laser excitation of a jellium K 561 cluster. Panels on the left and right columns represent respectively a non-resonant (1.55 eV) and a resonant (2.39 eV) excitation. From top to bottom, the panels (a) and (e) show the time dependence of the x (red) and y (blue) components of the laser electric field in V/ Å; panels (b) and (f) show the three components of the dipole moment in Å; panels (c) and (g) show the total energy absorbed by the electronic system in eV; and panels (d) and (h) show the three components of the magnetic moment, in units of the Bohr magneton µ B .

Figure 3 :

 3 Figure 3: Top panel: Time-dependent dipole moments (y components) divided by the peak amplitude E 0 of the laser field, for different values of E 0 ranging from 10 -5 to 10 -1 V/ Å. Bottom panel: Time evolution of the z component of the generated magnetic moment M z , divided by E 2 0 (in units of µ B Å2 /V 2 ). The simulations were performed for the K 561 spherical nanoparticle.

Figure 4 :

 4 Figure 4: Generated magnetic moment M z (t) (in units of µ B ) in a jellium K 561 cluster for different polarization of the perturbing laser field: circular |φ| = 90 • , elliptical |φ| = 60 • , 30 • , linear |φ| = 0 • polarizations.

Figure 5 :

 5 Figure 5: Response of a K 561 spherical nanoparticle excited by circularly polarized light at the plasmon frequency (2.39 eV). (a) Time evolution of the three components of the dipole moment. The shaded grey area represents the duration of the laser pulse. (b) Threedimensional iso-surface of the induced density at time t = 15.96 fs. (c) Snapshots of the induced electron density at five different times corresponding to the red and blue dots on the dipole moment curves in panel (a). The solid circle represents the jellium boundary and the induced density is shown as a color map, where the colors correspond to values of the color bar (on the right) in units of electrons per Å3 .

Figure 6 :

 6 Figure 6: The blue circles shows the jellium boundary. The red dots are grid points. Top panel: current density j(x, y, 0) at z = 0 at time 15.96 fs. The arrows' length and direction respectively correspond to the magnitude and the direction of the current. Bottom panel: j(x, y, 0) averaged over one period of oscillation of the dipole moment, and multiplied by 10.

Figure 7 :

 7 Figure 7: Left panel: Time dependence of the z component of the generated magnetic moment (in units of µ B ) for different potassium clusters. For K 58 the values are multiplied by a factor 5 in order to make them visible in the same scale. Right panel: saturated z components of the generated magnetic moment versus the number of electrons.

Figure 8 :

 8 Figure 8: Top panel: Magnetic moment M z (t) in K 58 (red), Na 106 (orange), and Au 225 (brown) clusters generated through excitation by circularly polarized light at the LSP frequency. The three jellium systems have almost the same radius (see Table1). Lower panels: Time dependence of the x (red) and y (blue) components of the induced dipole moment for each system.

Figure 9 :

 9 Figure 9: Time evolution of the z component of the generated magnetic moment M z (t), divided by the total absorbed energy, for K 58 (brown), Na 106 (red), and Au 225 (orange).

Figure 10 :

 10 Figure 10: Top panel: M z (t) for K 58 (red), Na 58 (green), and Au 58 (blue) clusters due to excitation by circularly polarized light at the LSP frequency (see Table1). The three jellium systems have the same number of electrons. Bottom panel: time dependence of the x (red) and y (blue) component of the corresponding induced dipole moments.

Figure 11 :

 11 Figure 11: Time dependence of the z component of the generated magnetic moment per unit energy absorbed by the system, for K 58 (red), Na 58 (green), and Au 58 (blue) clusters.

  Fig S5, Supporting Information), therefore resulting in almost no damping.

  Ground-state electron density for Potassium clusters of different size. Animation showing the dynamics of electrons (in K 561 jellium cluster at resonance) after laser is switched-off.Absorption spectra of jellium spheres with different electron density but having almost the same jellium radius, and the time-dependence of energy in them due to laser excitation at resonance. Absorption spectra of jellium spheres with different electron density but having the same number of electrons, and the time-dependence of energy in them due to laser excitation at resonance. Comparison of magnetic moment for gold with recent experimental findings.

Table 1 :

 1 Properties of the various jellium clusters.

	r s [ Å]	System	R [ Å]	ω p [eV]
		K 58	9.955	2.26
	2.572	K 561	21.211	2.39
		K 923	25.040	2.42
		K 2057	32.707	2.44
	2.083	Na 58	8.06	2.89
		Na 106	9.86	3.10
	1.593	Au 58	6.165	4.01
		Au 254	10.08	4.82
	Effect of size for the same chemical species. Four different jellium spheres of
	potassium containing respectively N e = 58, 561, 923, and 2057 electrons are considered.

Table 2 :

 2 Total energy absorbed by various types of clusters.

	Cluster	ω p [eV]	E T ot [eV]
	K 58	2.26	1.335
	Na 58	2.89	0.94442
	Na 106	3.10	1.8756
	Au 58	4.01	0.75508
	Au 254	4.82	2.7425

Acknowledgement

This work has used HPC resources from GENCI-IDRIS (Grant 2017-096829). R. S.-R. would like to acknowledge the contribution of the International Research Network IRN Nanoalloys (CNRS), and the in-house computational resources in CINaM (CNRS). P.-A. H. acknowledges fruitful discussions with M. Vomir and M. Rastei.