Pauli blocking of stimulated emission in a degenerate Fermi gas

The Pauli exclusion principle in quantum mechanics has a profound influence on the structure of matter and on interactions between fermions. Almost 30 years ago it was predicted that the Pauli exclusion principle could lead to a suppression of spontaneous emission, and only recently several experiments confirmed this phenomenon. Here we report that this so-called Pauli blockade not only affects incoherent processes but also, more generally, coherently driven systems. It manifests itself as an intriguing sub-Doppler narrowing of a doubly-forbidden transition profile in an optically trapped Fermi gas of $^3\mathrm{He}$. By actively pumping atoms out of the excited state, we break the coherence of the excitation and lift the narrowing effect, confirming the influence of Pauli blockade on the transition profile. This new insight into the interplay between quantum statistics and coherent driving is a promising development for future applications involving fermionic systems.

ing to electrical (semi-)conductivity. Another example is the Fermi pressure which stabilizes the densest observable matter in our universe like white dwarves and neutron stars against gravitational collapse. In the field of ultracold atomic physics, the Pauli exclusion principle has a direct impact on the collisional properties of identical fermions, due to the absence of s-wave (even parity) collisions, which has been observed experimentally 5,6 , and has cleared the way towards unprecedented fractional uncertainties at or below the 10 −18 level of state-of-the-art fermionic 1D lattice clocks [7][8][9][10][11] .
Since the idea was suggested almost three decades ago that spontaneous decay of an excited ultracold fermion confined in a Fermi sea would be suppressed due to quantum statistics, it has regularly attracted theoretical interest [12][13][14][15][16][17][18][19] . This suppression of spontaneous emission can become relevant in photon scattering events, where the absorption and subsequent spontaneous emission of a photon imparts a momentum transfer k = (k abs − k emi ) on the atom. If the imparted photon recoil is smaller than the Fermi momentum of the Fermi sea, these scattering events couple to states that are already occupied, and are thus strongly suppressed. This so-called Pauli blockade of spontaneous emission in ultracold degenerate Fermi gases has only recently been experimentally observed [1][2][3] .
In this work we demonstrate, for the first time to our knowledge, the Pauli blockade of stimulated emission, which leads to the intriguing phenomenon of narrowing the linewidth of the 2 3 S 1 → 2 1 S 0 transition that is studied in our experiment. This transition at 1557 nm connects the two metastable states of helium. Due to the very small Einstein coefficient of about 9×10 −8 s −1 , the upper state lifetime fully determines the natural linewidth of 8 Hz, making this an ideal candidate for precision spectroscopy in helium. Moreover, it is trivial to make the coherent driving Rabi frequency (2π×40 Hz for our experimental conditions) orders of magnitude larger than the Einstein coefficient, eliminating all influence of spontaneous decay back to the 2 3 S 1 state.
The excitation is performed in a degenerate Fermi gas of 3 He confined in a crossed optical dipole trap (ODT) at the magic wavelength (where the trapping potential is identical for both 2 3 S 1 and 2 1 S 0 states). If the linewidth of the excitation source is narrow enough to resolve the energy difference between the motional states induced by the trapping potential, only pairs of states with the same energy difference are coupled, as illustrated in Fig.1(a). We define this as carrier transitions, and the hole left in the lower state by excitation to the upper state can only be refilled by stimulated emission of the same atom back to the lower state again. The evolution of the system can be described as an ensemble of independent atoms each performing its own Rabi oscillation. The absorption profile will therefore simply reflect the momentum distribution of the atoms in the trap, in the form of a Doppler broadening.
If, on the other hand, the linewidth of the excitation laser is much broader than the energy spacing of the motional levels (which is the case in our experiment), each vibrational state couples to multiple others. This situation is depicted in Fig.1(b). The sideband transitions lead to an exchange of motional states, which is affected by the Pauli exclusion principle. Upon absorption of a photon, de-excitation is again only possible towards states which are not occupied, but now the laser bandwidth covers many more states. The excitation profile will then reflect both the ab-sorption profile, determined by the phase space density, and stimulated emission downward again, which depends on the distribution of holes in the Fermi gas. It ultimately leads to a narrowing of the measured transition, which counter-intuitively happens only if the laser is broad enough to couple different motional states. The model explaining this narrowing effect will be developed in the next paragraphs.
We can describe the excitation process by the following Hamiltonian in the interaction picture: whereĝ † n andê † m (ĝ n andê m ) represent the creation (annihilation) operators of a fermion in state |g, n and |e, m respectively, where n and m represent the motional quantum states for atoms in the lower (|g ) and upper (|e ) internal atomic states (see Fig.1), and Ω gn,em represents the Rabi frequencies coupling these states 20 (see the Supplementary material for more details). The weights a m−n in Eq.(1) describe the spectral intensity distribution of the excitation laser light, which is peaked at |m − n| = 0 . When e.g. only a 0 is non-zero, then only carrier transitions are driven, as shown in Fig.1(a). In order to capture the physics resulting from the excitation light, we compute the transition rates Γ gn→em and Γ em→gn using Fermi's golden rule (see the Supplementary material for details): with n gn and n em representing the occupation numbers of the different states and ω g (ω e ) corre-sponding to the trapping frequency felt by the internal state |g (|e ) (for a magic wavelength ODT, ω g = ω e ). These excitation rates reflect the Pauli exclusion principle since they both depend on the occupation of the initial state and the availability of holes in the final state. From this we obtain the shape of our spectroscopy signal (measured as depletion of the 2 3 S 1 population): in which 1/Γ 0 represents the lifetime of the |e state, f g = ω g /2π andΩ g k em = Ω g k em /Ω (see the Supplementary material for an elaborate derivation). The first term describes the contribution of absorption to the signal and the second one accounts for exchange of motional states within the |g manifold through stimulated emission, which is subject to Pauli blocking.
To get to a practically applicable model, the motional states are treated in a semi-classical approach and denoted with |r, k , where r represents the position and k the momentum of the atoms. The number density of the initial Fermi gas is given by the Fermi-Dirac distribution function 21,22 : with β = 1/k B T , where T and µ are the temperature and chemical potential of the gas respectively, and H g (r, k) is the Hamiltonian describing state |g .
Similarly to References 15, 17 , we use a local density approach to estimate the effective excitation rate between |g and |e . Expression (4) yields (see the Supplementary material for details): with and a modification factor M(ω), representing the Pauli-blocked stimulated emission, defined as: In the Supplementary material a full evaluation of equations (6)-(8) is given. In principle M(ω) depends also on the spectral profile of the excitation laser, but to include this effect would significantly complicate the calculation (see the Supplementary). As the linewidth of the excitation laser is much broader than the energy spacing of the motional states, we therefore only consider that the main contribution will come from the center frequency and thus neglect the small change in position or momentum. For those atoms almost no holes are available in a degenerate Fermi gas to go back to, so M(ω) goes to zero and stimulated emission is suppressed. On the other hand, atoms with p p F are excited at the wings of the spectral profile due to the Doppler effect, for these atoms the availability of holes goes to unity. Stimulated emission back is allowed, leading to a reduced effective excitation rate from |g to |e . As the Doppler effect is the only broadening mechanism for fermions 6 in a magic wavelength ODT, the spectral linewidth narrows compared to pure Doppler broadening. Moreover, the effect does not induce any additional shift on the center frequency since it is fully symmetrical in momentum space. The validity of equation (6)  We can effectively decrease the lifetime of the 2 1 S 0 state by exciting it to the short-lived 4 1 P 1 8 state with a 397 nm "depumper" laser 29, 30 . The relevant four-level system is depicted in Fig. 2(a).
Any excitation to the 4 1 P 1 state will result in a quick decay to the 1 1 S 0 level since the Einstein A coefficient for this transition is 35 times higher than the one leading back to 2 1 S 0 31 . Therefore, those atoms will be lost from the Fermi gas. By varying the intensity of the 397 nm depumper, we are able to adjust the effective lifetime of the 2 1 S 0 state (1/Γ 0 in equation (4)) and thus change the effect of the modification factor M(ω) on the linewidth.  (6) considering an effective lifetime for the excited state. In order to also predict the threshold intensity where the transition happens between the two regimes, one would have to include the full effect of the excitation laser linewidth in the model. As mentioned before, this would further complicate the calculations significantly, and was therefore not pursued.
By varying our experimental cooling and trapping conditions, we can change the parameters of the produced Fermi gases over a range of temperatures and chemical potentials. It is then possible to construct a universal curve displaying the linewidth reduction factor as a function of temperature. This is shown in Figure 5. In the limit of zero temperature, all states below the Fermi energy are occupied (except one hole from excitation), hence it is very improbable for excited atoms to be stimulated back into the lower electronic state, resulting in a Doppler excitation profile.
As the temperature of the Fermi gas increases, the availability of holes around the Fermi energy increases too and stimulated emission becomes possible, leading to an increased narrowing of the line profile. When T T F , the quantum statistical nature of the atoms does not play a role anymore in the expression of the rates (2) and (3), and the Doppler profile should be retrieved again (this is out of the validity range of the presented theoretical approach). As can be seen in Figure 5, the reduction of the actual FWHM compared to the Doppler linewidth becomes stronger as T /T F increases until it reaches a reduction by a factor ∼ 0.7 when the temperature is comparable to the Fermi temperature. Although it was not possible to achieve temperatures lower than T 0.25 T F to measure into the deeply degenerate regime where the availability of holes becomes negligible, the experimental values consistently show a significant narrowing compared to the Doppler width, with a good agreement to the theoretical curve over the full range of experimental parameters.
In contrast to what was observed by 1-3 , our study shows that Pauli blockade can affect coherent processes in specific situations where quantum exchange symmetry plays a role. Interestingly, a broader linewidth of the excitation laser results in this case in a narrowing of the spectroscopic linewidth. Even though we are experimentally limited to temperatures down to T /T F 0.25, our narrowed spectra mostly reflect the low momentum atoms, so we nevertheless achieve linewidths which are otherwise only reachable for highly degenerate gases. In a broader context, our observation of Pauli blockade of coherent processes could be of interest in the context of cooling of fermionic samples. Additionally, the mechanism shows similarities with conduction phenomena in semi-conductor materials and could be used to perform quantum simulations of such materials, or be of great interest for quantum logic and information processing with fermionic species.
In conclusion, we observed a signature of Pauli blockade in a coherently driven system, by the means of spectroscopy of a doubly-forbidden transition in an ultracold Fermi gas of metastable and laser parameters. In the context of precision spectroscopy, it proves to be a useful feature as it helps improving the accuracy of the determination of the transition frequency without inducing any additional shift. In a broader perspective, the narrowing effect showcases Pauli blockade of stimulated emission. This influence of quantum statistics on a coherently driven system shows great potential as a tool for applications with many-body physics and quantum information. For each set, we determine the temperature T /T F through the relation:

Methods
where Li 3 (−ζ) is the trilogarithm function and ζ = e βµ is the fugacity of the gas. The measurements are divided into two categories: below the depumping intensity threshold where the stimulated emission is relevant, and above this threshold where the stimulated emission is eliminated. We discard lifetime-broadened measurements from the analysis shown in Figure 5.
where the density matrix ρ is flattened into a 16 elements column vector and M is a 16 × 16 matrix containing the couplings. The solution is obtained by propagating the initial density matrix by numerically computing the expression: where initially the only non-zero element is the one describing a population in the metastable triplet state (equal to 1). The 2 3 S 1 population at time t = 3 s is evaluated over a range of 500 kHz around the resonance frequency to build up a spectrum, from which the FWHM is extracted. Since our modelling describes only single atom dynamics, the coupling between the metastable triplet and singlet states inserted in equation (11)

Calculation of the transition rates and derivation of the line profile
As explained in the main text, we consider a Fermi gas in an internal state |g confined in a harmonic potential. This trapping potential has an angular frequency ω g along its axial direction. State |g is coupled at time t = 0 to the excited state |e by resonant light with a wavenumber κ and angular frequency ω. The energy difference between these two levels is ω 0 . The excitation laser beam propagates along the axial direction of the trapping potential (which we denote whereas the radial direction is denoted ⊥). The atoms in state |e feel a harmonic potential with an axial angular frequency ω e , which in the particular case of a magic wavelength trap is ω e = ω g . The confinement potential is assumed to be weak compared to the recoil energy ω rec of the transition, such that the system is out of the Lamb-Dicke regime (η 1, with the Lamb-Dicke parameter defined as η = ω rec /ω g and the recoil energy ω rec = 2 κ 2 /2m). For the transition of metastable helium considered in this study, the Lamb-Dicke parameter is η 30.
Consider first that the excitation light has a narrower spectral linewidth than the splitting in energy of the motional states. It is assumed to be tuned to a central frequency ω = ω 0 + 0 ω g . The system can be described by the following Hamiltonian in the interaction picture: whereî † n andî n represent the fermionic creation and annihilation operators of an atom in internal state |i and motional state |n , and the Rabi angular frequencies are given by [1]: where n < (n > ) is the lesser (greater) of n and n + 0 . The evolution of the system is straightforward and leads to independent Rabi oscillations between pairs of |g, n and |e, n + 0 states, as depicted on Figure 1(a) from the main text. These sets of transitions constitute the carrier transitions of the system. Since all these transitions couple pairs of states independently, no collective many-body effect is expected and each fermion composing the Fermi sea can be seen as a single particle.
(1) LaserLaB, Department of Physics and Astronomy, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, the Netherlands where β = 1/(k B T ) is the reciprocal temperature, and k B Boltzmann's constant. The line profile is found by rewriting the expression (15) within the Thomas-Fermi approximation in the same spirit as [3]. We assume for simplicity that the function defined in expression (11) is peaked at j = n when its denominator is not dominated by f g Γ 0 /Ω 2 (thereby neglecting the small change in phase space induced by stimulated emission). We then simplify it as: such that we can rewrite it in the semiclassical approach as: with the resonance condition The lineshape thus reads: with the absorption profile given by [3]: and the modification factor expressed as: It can be rewritten as: in which the second term represents the Pauli blocking induced inhibition factor on stimulated emission.
We follow the derivation from [3] to obtain an expression for S Absorption (ω). Making use of the spherical symmetry over space and cylindrical one over momenta, we rewrite the integrals as: which yields after integration over the axial momenta k : Performing the integration over the transverse momenta k ⊥ , performing the change of variable R = rω g βm/2 and denoting the detuning ∆ = ω − ω 0 − ω rec , we can obtain an expression for S Absorption (∆): − βµ, where m ex = 1 − ω 2 e /ω 2 g characterizes the deviation from the magic wavelength case, and where R is only an integration variable.
We follow the exact same procedure for the calculation of d 3 r d 3 k ρ 2 g (r, k)δ(ω− ω r,k ), considering the magic wavelength case where ω g = ω e and m ex = 0, and we find : with F (R, ∆) which now reads: Integrating S Absorption (∆) and M(∆) then gives: where the fugacity is ζ = e βµ and Li n (z) denotes the n-th order polylogarithm function of z.
Hence, S(∆) becomes: In case the lifetime of the |e internal state τ becomes small compared to f g /Ω 2 (by applying sufficient intensity in the depump laser from the 2 1 S 0 to the 4 1 P 1 states as presented in the main text), the second term related to M(ω), vanishes and we retrieve the non Pauli-blocked profile as derived in [3] and experimentally verified in [4] (in that case, the condition τ → 0 was realized by the fact that the |e state was expelled from the dipole trap by a blue-detuned harmonic potential).

Few-body numerical analysis
In order to verify our modelling, we numerically solved the master equation describing both the coherent excitation and the decoherence due to the decay to the 1 1 S 0 state. The equation reads: where ρ(t) represents the density matrix of the system at time t, and with the Hamiltonian given in expression (3) and the collapse operators defined as: the internal state |0 standing for the 1 1 S 0 state and Γ 0 = 1/τ . Since we are not interested in collective effects due to the decay processes, we modelled them as conserving the motional state while flipping the internal state from |e to |0 . The solution ρ(t) is numerically calculated using either an exact solving method [5,6] or a quantum Monte-Carlo one [7]. From the density matrix, the population of each manifold, |g , |e , or |0 , is extracted according to: i standing for one of these three states and N being the total number of particles. Due to the computational time increasing very fast, we limited our analysis to few-atoms systems not exceeding N = 3. A first thing that we confirmed is that no many-body effects can be observed if only carrier transitions are allowed and the Fermi gas can be treated as an ensemble of independent particles. In contrast, the dynamics is modified by the presence of the sideband couplings enabling cross-talk between the particles. The coherent dynamics exhibits Pauli blocking and can not be obtained by considering independent single-particles. Figure 1 shows an example of such behaviour for different numbers of fermions in the system. Having more than a single atom results in an enhanced lower state depletion due to Pauli blockade.
We then simulated line profiles by modelling the Fermi gas as an ensemble of sets of N particles each of which is weighted by the Fermi-Dirac distribution. The remaining |g population of each of those is calculated and added to obtain the total remaining population. The procedure is repeated for different detunings to simulate a line profile. Figure 2.a. shows the results of these simulations with increasing N from 1 to 3. In our experiment, the Fermi gases are composed of 10 5 − 10 6 atoms, hence we are only interested in a trend for the linewidth as it is not possible to simulate the full system. The excitation strength increases with N and the profile narrows as shown in figure 2.b., displaying a similar behaviour as what we experimentally observe.

Averaging of the Rabi frequency
In order to estimate the average Rabi frequency that we use for solving the four level dynamics explained in the main text, we follow a similar approach as in Ref. [8]. We label the states with two indices, n and n ⊥ , which represent the axial and radial directions respectively. With this notation, excitation light of angular frequency ω = ω 0 + ω rec + ω couples only the sets of internal (electronic) and motional states |2 3 S 1 |n and |2 1 S 0 |n + with a Rabi frequency Ω n ,n + given by the equation (2). The averaged Rabi frequency is thus given by: with the normalization N = n ,n ⊥ f FD (n , n ⊥ ), and the Fermi-Dirac distribution given by: f FD (n , n ⊥ ) = 1 1 + e −βµ+β n ω +β n ⊥ ω ⊥ . (38)

Comparison of the behaviour of different datasets
As explained in the main text, we varied the intensity of the depumper beam for Fermi gases with different thermodynamical parameters. To achieve totally different T /T F values, we changed the intensity of the ODT beams. All the datasets we gathered this way exhibit the same behaviour as the one shown in the main text. Figure 3 shows a summary of two additional datasets obtained with different trap depth than Figure 3 of the main text. The intensity value for which the transition between the two regimes happens is changed (also compared to the dataset shown in the main text) because of the interplay between the laser linewidth, the effective lifetime due to the pump to the 4 1 P 1 state and the trap frequency. We did not take these data into account in the main text because it corresponds to an early implementation of the experimental procedure and the parameters were not wellcontrolled. SUPPLEMENTARY MATERIAL 9 Figure 3. Variation of the FWHM as a function of depumper laser intensity for deeper dipole traps than the data shown in the main text. Each of the two datasets were acquired with similar DFGs composed of 10 6 atoms at a temperature of 300 nK and exhibit the same behaviour as the dataset shown in the main text.