Gain engineering and atom lasing in a topological edge state in synthetic dimensions

Domain: physics
Published: 2025-12-13
Editorial outcome: minor_revision

Abstract
Recent advances in quantum technology have highlighted the importance of controlling quantum states, especially in open quantum systems, where the system interacts with the environment. Non-Hermitian quantum mechanics describes these systems. Photonic systems are a key platform for studying non-Hermitian quantum mechanics owing to their ability to engineer gain and loss. Ultracold atomic gases also have been used to study non-Hermitian quantum mechanics; however, unlike photonics, gain control is challenging, limiting exploration to control of loss. In this paper, we report engineering of effective gain through evaporative cooling of judiciously selected initial thermal atoms, leading to Bose–Einstein condensation (BEC) in the excited eigenstates of a synthetic lattice. We achieve BEC formation in a topological edge state of the Su–Schrieffer–Heeger lattice in the synthetic hyperfine lattice, akin to atomic laser oscillations at a topological edge mode, that is, a topological atom laser.

Introduction
Controlling quantum states has become crucial in quantum technology, particularly for open quantum systems coupled to the environment1. One important framework in describing open quantum systems is the non-Hermitian quantum mechanics2,3. Dating back to the study of nuclear decay4, non-Hermitian quantum mechanics has been rediscovered and studied in various contexts. One turning point is the discovery of parity-time reversal (PT)-symmetric Hamiltonians5,6, which, despite being non-Hermitian, give rise to real energy spectrum. Another very recent development came from understanding of the topological properties of non-Hermitian Hamiltonians7,8. Since non-Hermitian Hamiltonians generally have complex eigenenergies, modes with positive imaginary part can lead to amplification and lasing. Topological laser, which is a laser whose lasing mode is a topological edge mode, has been realized9–15, and its potential application in integrated photonics has been actively pursued16. In quantum many-body systems coupled to the environment, an effective description in terms of non-Hermitian Hamiltonians is possible in the quantum trajectory approach between quantum jumps17,18, and the effect of non-Hermiticity in many-body quantum phases have been actively studied19–33.

Since its experimental realization of PT-symmetric optics in 200934–36, non-Hermitian quantum mechanics has been actively studied in optics37–39. Despite the huge success, optical systems have a limitation that the interaction is typically classical nonlinearity, making it difficult to study quantum many-body physics in non-Hermitian setups. An emerging platform to study non-Hermitian quantum mechanics is ultracold atomic gases, where one can study quantum many-body physics in a controlled manner40–42. Recently, there have been experiments in which non-Hermitian Hamiltonians are realized by controlling dissipation in ultracold atomic gases43,44. However, ultracold atomic gases have their own limitation that the non-Hermiticity that can be introduced has been limited to the control of dissipation. It has thus not been possible to study non-Hermitian physics related to controlled gain in ultracold atomic gases. Without gain, it is not possible to study phenomena related to amplification and lasing.

Results
Experimental results

We achieve gain engineering in a lattice made of a synthetic dimension using ground-state hyperfine sublevels of 87Rb. Here, we utilize long-lived nature of the 87Rb hyperfine states in the electronic ground state to observe the formation of a BEC in the synthetic lattice. The scattering length of 87Rb exhibits only a small dependence on hyperfine or magnetic sublevels59,60. As a result, the spin-exchange collisions are significantly suppressed. By coupling hyperfine sublevels ∣F(=1,2),mF in the ground 5s2S1/2 state of 87Rb with microwaves, we prepare 3-site and 5-site lattices in a synthetic dimension, as shown in Fig. 1a, b.

Our experiments start with 87Rb atoms above the critical temperature Tc for BEC trapped in the crossed optical dipole trap at about 1064 nm61. The subsequent microwave sequence consists of two stages: First, we apply time-dependent microwaves to prepare thermal atoms in a superposition of the synthetic lattice sites close to one edge. Then, further evaporative cooling is performed with microwave couplings kept constant (see Fig. 1c, d). Resulting formation of a BEC corresponds to lasing of atoms at a specific eigenmode in the lattice. We properly choose initial thermal atoms so that the resulting BEC is formed not at the lowest-energy mode or a single hyperfine state but rather at an excited mode made of a coherent superposition of multiple hyperfine states. The resulting site distributions are measured by applying Stern–Gerlach separation immediately after switching off the trap, as shown in Fig. 1e.

We first perform an experiment with a 3-site lattice to demonstrate our ability to control the formation of BEC into a dark state, which is not the ground state of the 3-site lattice. Here, we exploit Λ-type coupling scheme with temporal evolution known as STIRAP. Three levels ∣1,1i, ∣2,0i and ∣1,−1i are coupled by microwaves and construct a 3-site synthetic lattice (site 4, 3 and 2). The site indices are labeled to be consistent with the 5-site lattice we introduce later. The Hamiltonian in the rotating frame, in the matrix form, is

H = ℏ/2 * [[0, Ω1, 0], [Ω1, 2δ, Ω2], [0, Ω2, -2δ]] (1)

where Ω1, Ω2 are the Rabi frequencies of microwave couplings, and δ is the detuning due to a magnetic field fluctuation. If the two-photon resonance δ=0 holds, a dark state, given by cosθ∣1,1i + sinθ∣1,−1i with tanθ=Ω1/Ω2, becomes one of the eigenstates of Hamiltonian (1). By applying two microwaves in counter-intuitive order so that θ changes from 0 to π/2, the dark state can adiabatically evolve from ∣1,1i (site 4) into ∣1,−1i (site 2). This process is well-known as STIRAP. We utilize this STIRAP process to prepare thermal atoms in a coherent superposition of synthetic lattice sites. Initially, atoms are evaporatively cooled in the state ∣1,1i (site 4). At T/Tc~1.1, evaporation is paused, and the STIRAP sequence is applied. The STIRAP with full sweep (θ=0→π/2) takes 10 ms. In our experiment, the actual sweep time Tprep is dependent on the final value of θ: e.g., Tprep=5 ms leads to the almost equal superposition of site 4 and 2 (tanθ=Ω1/Ω2=1.345). After the sweep, we keep the microwave coupling and restart evaporation to form BEC, followed by the site occupation measurement. As we will show, the BEC is predominantly formed in the dark state where the thermal atoms are prepared.

Figure 2 shows the site occupancies N4, N3 and N2 as well as the number of Bose-condensed atoms N4c, N3c, and N2c, measured at three different Ω1/Ω2 (see Methods). Figure 2a–c shows the case of Ω1/Ω2=19.27 where atoms almost localize in the site 4. By choosing different values of Ω1/Ω2, the atoms can be transferred to the site 2 (Fig. 2g–i) or in between the two sites as in Fig. 2d–f. The success of the STIRAP between the sites 4 and 2 indicate the coherent superposition is maintained during the process. In Fig. 2b, e, h, we plot the number of atoms (Bose-condensed atoms) in sites 2 and 4 versus site 3 normalized by the total number of atoms (Bose-condensed atoms) for each performed experiment. The absence of particles in the intermediate state N3 is the indication that the atoms are in the dark state. Small but nonzero N3 is caused by the deviation of the microwave frequencies from the resonances.

We note that the dark state where the condensate is formed is not the ground state of the lattice. This formation of BEC at an excited state can be understood by the difference of the critical temperatures; among the three dressed states, the dark state had most thermal atoms and thus has the highest BEC transition temperature, resulting in preferred formation of the BEC.

We can alternatively describe this process in terms of different effective gain for each lattice site. Although evaporative cooling acts on all sites equally, the initial nonequal distribution of thermal atoms results in different effective gain for each site. Sites with more thermal atoms imply larger effective gain, and since the thermal atoms are initially distributed to the sites forming the dark state, the preferred lasing, namely BEC, takes place at the dark state. This perspective on effective gain is more quantitatively discussed later for the case of 5-site SSH model, comparing the experiment with theoretical modeling involving effective gain. We note that formation of BEC in a dark state of momentum states was recently reported62 through a kind of mode-conversion by coupling atoms already in a BEC to an optical cavity. Our approach, instead, provides a direct means to control gain in ultracold atomic setups.

Next, we realize a 5-site SSH lattice57 to obtain the topological edge-state lasing, or the topological atom laser. The SSH model is a one-dimensional lattice model with alternating hopping strengths, whose Hamiltonian in the rotating frame is

H_SSH = ℏ/2 * sum_i (ΩA ∣2iih2i-1∣ + ΩB ∣2i+1ih2i∣ + H.c.) (2)

where the sum of i runs over integer values. The energy band gap closes at ΩA=ΩB, and the regions ΩA>ΩB and ΩA<ΩB are topologically distinct. Due to the bulk-edge correspondence, when a lattice is terminated at i=1 by restricting the above sum to i≥1, there exists a zero-energy edge state localized around the edge if and only if ΩA < ΩB. The energy of the zero-energy edge state is topologically robust; small spatial inhomogeneity in ΩA and ΩB will not destroy the edge state. As we see in Methods, such inhomogeneity is naturally present in our setup and thus the topological atom laser we realize is directly benefited from the topological robustness.

In our experiment, five levels ∣2,−2i, ∣1,−1i, ∣2,0i, ∣1,1i and ∣2,2i are coupled by microwaves and construct a 5-site synthetic lattice (site 1-5, respectively) for which the Hamiltonian in the rotating frame can be written in the matrix form

H = ℏ/2 * [[-4δ, ΩW(1), 0, 0, 0], [ΩW(1), 2δ, ΩS(1), 0, 0], [0, ΩS(1), 0, ΩW(2), 0], [0, 0, ΩW(2), -2δ, ΩS(2)], [0, 0, 0, ΩS(2), 4δ]] (3)

where ΩW(1,2) and ΩS(1,2) are the Rabi frequencies in microwave couplings, and δ is the detuning due to a magnetic field fluctuation. The ideal SSH model is obtained if Ωi(1) = Ωi(2) for i=S,W, and the edge state localized around site 5 exists when ΩW < ΩS; such an edge state is a superposition of only three sites, (sites 5, 3 and 1) if δ=0. In our setup, Ωi(1) and Ωi(2) are not exactly equal because of the different Clebsch-Gordan coefficients. As we noted above, the existence of a topological edge state is robust against such inhomogeneity or disorder in hopping amplitudes. Below, we use the averaged values 〈Ωi〉=(Ωi(1)+Ωi(2))/2 for these couplings to characterize the strong and weak couplings. (See Methods for more details).

To achieve BEC in the topological edge state, thermal atoms initially populated in the site 4 are first transferred to the site 5 by adiabatic rapid passage (ARP). Four microwaves ΩW(1,2), ΩS(1,2) are then applied as shown in Fig. 1b. Thermal atoms will then adiabatically populate the topological edge mode extending over sites 5, 3, and 1. Finally, we keep the microwave coupling and restart evaporation to accomplish BEC, followed by the site occupation measurement. We performed experiments with three values of the final ratios 〈ΩW〉/〈ΩS〉 (<1) to confirm the behavior of the topological edge-state lasing.

Figure 3a, d, f shows the time-of-flight signals after the Stern–Gerlach separation, revealing site occupancies Ni (i=1,…,5) at three different values of 〈ΩW〉/〈ΩS〉. We see that in all three values of 〈ΩW〉/〈ΩS〉, the BEC phase transition, namely, atom lasing is successfully observed, and the formed BEC has the largest population in site 5, with certain atom numbers in 3 and 1 with almost no occupation in 2 and 4, which is consistent with the wavefunction profile of the topological edge state of the SSH model. In the Fig. 3c, f, i, we plot the relative atom number of odd sites versus atom number of even sites. The histograms indicate that the population in sites 2 and 4 are small as expected. If the value of the diagonal components in the Hamiltonian shown in Eq. (1) is all zero, chiral symmetry is preserved and therefore, population in sites 2 and 4 should be zero for the topological edge state.

In our system, the stability of the state preparation is limited by the existence of a magnetic field fluctuation and the uncertainty in determining microwave resonances. Both of them can cause a photon detuning, which eliminates the edge state (See Methods for the stabilization of magnetic field).

Theoretical model

In order to verify that the formation of the BEC in the edge state is due to the effective gain in the system, we construct a theoretical model including gains, losses and magnetic fluctuation for a 5-site model. We simulate the formation of the condensate ∣ψ by numerically solving the nonlinear non-Hermitian Schrödinger equation

i d/dt ∣ψi = (H^0 + H^gain(nT) + H^loss) ∣ψi (4)

Here, H^0 represents the Hermitian Hamiltonian dynamics of Eq. (3) with the magnetic field fluctuation as well as the time dependence of the Rabi couplings. H^gain = i g nT ∣TihT∣ represents the gain term with the coefficient g being the effective strength of the gain per unit thermal component. We denote the (coherent) spin components of the thermal atoms as ∣Ti. The strength of the gain is multiplied by the number of atoms in the thermal components, which we denote nT. Note that the gain is related with the microscopic process of bosonic stimulation through atom collisions and thus, the gain coefficient g, introduced as a phenomenological parameter, should be multiplied by the number of thermal atoms, which is how we constructed the theoretical model. H^loss represents the state-dependent two-body inelastic loss of 87Rb atoms. Details of the theoretical modeling is also given in the Methods.

Figure 4 shows the numerically obtained dynamical formation of the BEC, which corresponds to the parameters of Fig. 3d–f (〈ΩW〉/〈ΩS〉=0.23). Figure 4a, b shows the time evolution of the site distribution of a condensed atoms and b thermal atoms, respectively. Figure 4c, d shows the relative population so that the sum of the relative population is 1 at each time. Here, the origin of time (0 ms) is when the gain is introduced, and the previous time (t < 0 ms) corresponds to the time when the adiabatic preparation of the system takes place. After the site preparation in 10 ms, the thermal populations (Fig. 4d) are nearly consistent with the expected edge states obtained by diagonalization of the Hamiltonian (Fig. 4e), indicating that the edge state are adiabatically prepared. The small oscillation of population indicate the influence of an external oscillating magnetic field of 0.1 mG.

After gain is introduced at t=0 ms, the site distribution of the condensed atomic cluster gradually approaches the edge state. Figure 4f plots (Nc,1+Nc,3+Nc,5)/Nc as a parameter indicating how close to the edge state it is. As expected, atoms are present only at odd-numbered sites, consistent with the zero-energy edge mode originated from the chiral symmetry. Figure 4g shows the time dependence of the fidelity ∣⟨Ψ∣ψ(t)⟩∣ = ∣⟨ψ(t)∣ψ(t)⟩ ⟨Ψ∣Ψ⟩∣, where ∣ψ(t) is obtained from numerical calculation, ∣Ψi is the expected edge states shown in e. We can see that the fidelity approaches one, which indicates that the final state is the expected edge state. The relative populations and fidelity for the BEC shown in Fig. 4c, f, g are only important when the BEC fraction is increased to a certain amount beyond the initial seed necessary in the numerical calculation. Thus, our experimental observation of edge-state formation of BEC can be reproduced and explained by the theoretical model with an effective gain, taking into account the realistic magnetic field fluctuations and two-body loss.

Confirming the coherence of the topological atom laser

We have moreover performed an additional experiment to confirm coherence of the superposition of spin states in the created BEC as shown in Fig. 5. After loading atoms into the lattice with largest ratio of 〈ΩW〉/〈ΩS〉, we apply the reverse process back to the 〈ΩW〉/〈ΩS〉=0. The resulting site population returns to the initial site 5, proving that the coherence persisted throughout the entire process; the system keeps adiabaticity even in the presence of random fluctuations in δ.

We can estimate a quantity I=∣⟨Ψfinal∣ψfinal⟩∣ from the relative population of site 5 in the final state ∣ψfinal, because ideally other sites are not occupied. The results shown in Fig. 5 indicate the quantity of I=0.90(1) for the condensates, consistent with 0.82(2) in terms of population, shown in Fig. 5c. This value of I close to 1 indicates that we successfully produce condensates in the edge states by gain engineering technique.

Methods
Experimental setup

Our experiments start with 87Rb atoms above the critical temperature for BEC trapped in the crossed optical dipole trap at about 1064 nm (see Fig. 6). The crossed optical dipole trap is created with two horizontal beams which are crossing at slight angle to provide a large trap volume and a vertical beam (see Fig. 6). Then the 87Rb atoms are pumped to a hyperfine state ∣F=1,mF=1 of the 5S1/2 electronic ground state by optical pumping. For 5-site experiment, we further transfer atoms into the ∣F=2,mF=2 state by ARP with microwave irradiation. For efficient evaporative cooling, we turn off a deeper horizontal beam in the middle of evaporative cooling. In order to induce coupling between the hyperfine states, or neighboring sites in the synthetic dimension, we apply resonant microwave fields. The subsequent sequence of the microwave irradiation consists of two stages: First, we apply time-dependent microwaves to prepare thermal atoms in a desired superposition of the synthetic lattice sites. Then, further evaporative cooling is performed with microwave couplings kept constant. Resulting formation of a BEC can be regarded as lasing of atoms at a specific eigenmode in the lattice.

Generation of 4-color microwave

We simultaneously apply 4-color microwaves around the clock frequency ~6.8 GHz. We use an output from a microwave synthesizer at a frequency fRF1 for the ∣2,−2i-∣1,−1i coupling and generate three sidebands. An image rejection mixer can create a microwave sideband only at higher frequency side at a frequency f0+fRF1, and is set on resonance to the ∣2,0i-∣1,1i transition. A series connection of an additional image rejection mixer further generates sidebands at f0+fRF1 and f0+fRF1+fRF2, and are set to the ∣2,0i-∣1,−1i and ∣2,2i-∣1,1i resonances (Fig. 7). In this configuration, ΩW can be controlled by the strength of the second RF. Measured Rabi frequencies for the 3-site and the 5-site experiment are shown in Tables 1 and 2, respectively. Numerical solutions to the 5-site SSH Hamiltonian using the measured Rabi frequencies are shown in Fig. 8.

Suppression of magnetic field fluctuation

One of the major sources of the short-term magnetic field fluctuations comes from electric devices around the experimental chamber. Transducers in power supply emit strong noises at alternate current (AC) line frequency (60 Hz) and its harmonics which amounts to ~0.8 mG at the chamber. We construct a feed-forward system to suppress the AC field fluctuation. We monitor field fluctuations by a flux gate sensor placed near the magnetic coil on the quantization axis. A function generator with a reference clock of 10 MHz locked to the AC line (DS technology) generates 60 Hz harmonics up to 7th order to be added to the analog control of the coil. The amplitudes and phases of the harmonics is tuned to minimize the fluctuation of sensor output. This enables us to suppress the 60 Hz line noise down to ~0.1 mG at all times during the experiment.

Numerical simulation: gain term

The thermal atoms present in the system provide the gain through the evaporative cooling. In order to model this process, we assume that the thermal atoms are kinetically thermal but made of coherent superposition of spin states. We denote the (coherent) spin components of the thermal atoms as ∣Ti, which we take to be normalized. The strength of the gain is proportional to the number of atoms in the thermal components, which we denote nT. We note that nT is a time-dependent quantity. The gain is then modeled by including the following term in the Hamiltonian

H^gain(nT) = i g nT ∣TihT∣, (7)

with the coefficient g being the effective strength of the gain per unit thermal component. We note that this Hamiltonian, which is a function of the thermal atom density nT, acts on the condensate wavefunction ∣ψ(t) and increases the condensate density ⟨ψ(t)∣ψ(t)⟩ but does not directly change the number of thermal atoms nT. As we discuss later, we decrease nT in an ad hoc manner to account for the increase of the condensate density.

The spin component ∣Ti of the thermal atoms we use in our simulation is determined by following realistic simulation of state preparation in the actual experiment. We start from the initial condition that all of the population is in site 5. Then, we simulate the time evolution under a linear increase of the Rabi coupling strength from zero to the final value Ωj,j+1,final as described in the state initialization part of Fig. 1:

Ωj,j+1(t) = (t+10)/10 * Ωj,j+1,final (8)

where t is time in millisecond. We note that we change from t=−10 to 0. The state realized at t=0 is what we use for ∣Ti.

Numerical simulation: loss term

Two-body inelastic collisions are the dominant source of atom loss of Bose condensates in our system. Two-body inelastic loss of condensate 87Rb have been studied67–70. For simplicity, we set the two-body loss coefficient γ to be γ=1×10−13 cm3/s for the atom pairs of

• (F=2,mF=2) and (F=2,mF=0)
• (F=2,mF=2) and (F=2,mF=−2)
• (F=2,mF=0) and (F=2,mF=−2)

We take the loss coefficient to be 2γ for the pair of

• (F=2,mF=0) and (F=2,mF=0)

and zero for other combinations.

This is equivalent to considering the following (nonlinear) Hamiltonian

H^loss = -i γ [[(n3+n5)/3, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, n1+2n3+n5, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, n1+n3]] (9)

where ni ≡ ∣ψi∣2 is the condensate density of ith site. The choice of this specific form of loss Hamiltonian of Eq. (9) is based on the reported measurement results, and does not have a direct relation to the appearance or suppression of the populations in the even sites where the loss is set to zero. In addition, we use the typical atom density of our experimental setup as 1014 cm−3.

Numerical simulation: procedure

We start from the random initial state ∣ψ with its norm equals to 10−6. This random initial seed of condensate is necessary to obtain the growth of condensate density in our numerical simulation. We then time evolve according to the nonlinear and non-Hermitian Schrödinger equation

i d/dt ∣ψi = (H^0 + H^gain(nT) + H^loss) ∣ψi (10)

In our numerical simulation, we discretize the time with the step of Δt and calculate

∣ψ(t+Δt)i = e−iH^lossΔt/ℏ e−iH^gain(nT)Δt/ℏ e−iH^0Δt/ℏ ∣ψ(t)i (11)

After every step we calculate the multiplication by e−iH^gain(nT)Δt/ℏ, we evaluate how much the norm of the wavefunction ⟨ψ(t)∣ψ(t)⟩ is increased, and subtract the same amount from nT to update the value of nT.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In the manuscript entitled “Gain engineering and topological atom laser in synthetic dimensions”, authors consider a multi-level bosonic Rb-87 system in which quantum states can be coherently controlled through two-photon processes. In particular, two hyperfine manifolds (F=1,2) are chosen to introduce 3 or 5 synthetic site coupled by microwave pulses. In this well-controlled platform, authors experimentally examine how thermal atoms cool down into different states – different sites in the synthetic dimension manner – when evaporative cooling is introduced. A dark state with the STRIAP sequence – which is well-known technique in quantum optics/AMO – and the SSH model with alternating hopping strengths are realized in this system.\n\nFor both models, authors observe the emergence of Bose-Einestein condensation at a specific site – usually at the edge – and this is interpreted as the lasing mode induced by gain engineering. This is of particular importance for pushing the limit of non-Hermitian dynamics, which is one of the frontier directions at the moment. The Authors point out that this gain engineering remains challenging in quantum systems in contrast to phonics, which is indeed true.\n\nOverall, the performed experiments seem very interesting with rock solid data sets. As noted, it is nice to see how topological edge mode (or a dark state) allow the control of thermal population. To me, the “observation” appears interesting on its own.\n\nI have some concerns about interpreting the experimental observations in terms of \"gain engineering.\" While the introduction connects gain engineering with non-Hermitian dynamics, this connection needs more clarity. In the presented experiment, the gain term is site-specific. Could the authors provide examples of how non-Hermitian dynamics can be realized through the demonstrated gain engineering and atom loss? It's not immediately clear how the proposed gain engineering helps implement a genuine PT-symmetric Hamiltonian with balanced gain-loss.\n\nAdditionally, what is the sub-system being considered when envisioning a non-Hermitian Hamiltonian? Since the authors discuss non-Hermitian (or open quantum) systems, readers would benefit from understanding how atoms flow into the subsystem as a gain term. Can the 3-site (or 5-site) system be described by non-Hermitian Hamiltonian? Would the non-Hermitian skin effect be observable with this gain term? I feel the date set presented here seems interesting, but it would be beneficial if authors put this in the right context.\n\nAnother question author may clarify is the quantum statistics. Like photons in photonics, it is not surprising that gain engineering properly works for bosonic particles, especially when BEC emerges, thanks to the bose enhancement. Can authors comment how proposed gain engineering works for fermions?\n\nIn summary, while I have several concerns and suggestions outlined above that the authors should address before publication in Nature Communications, I believe this work demonstrates significant breakthroughs in multiple areas—including quantum system gain engineering, topological lasing with bosonic atoms, and potential developments in non-Hermitian engineering. I look forward to reviewing the revised manuscript before making my final recommendation.", "reviewer_2": "The authors realise Bose-Einstein condensation in a micro-wave dressing of the hyperfine levels of 87Rb atoms. They perform evaporative cooling in the dressed states after carefully preparing the initial thermal states. In this approach, the thermal atoms can be seen as a reservoir, which provides “gain” for a “laser” formed of condensate atoms. They demonstrate this experimental approach in two configurations with 3 sites and 5 sites respectively, where the second engineers a SSH model, one of the simplest topological models in 1d. There, condensation occurs in a dressed state, which is a topological edge state of the 5-site system. Condensation is observed and quantified by observing the momentum distribution in time-of-flight with a magnetic gradient ; fitting the bimodal distribution identifies the condensed atoms and the thermal atoms.\n\nA novel aspect is the ability to engineer gain in a setup with ultracold atoms, where it is usually difficult to obtain, contrary to losses, which opens a new path to study non-Hermitian physics with both gain and losses. A theoretical analysis is provided to support the claims and the interpretation of the experimental data, which is based on solving the non-Hermitian Schrödinger equation for the condensate wavefunction, taking the thermal atoms as a gain medium (which empties as the condensate grows), in the presence of losses for the condensate.\n\nOverall I appreciated the manuscript and the work, but I have several questions and remarks for the authors, that would need to be addressed before supporting publication :\n\n1) I find that the denomination “topological atom laser” is a bit overselling and could be simplified to : Bose-Einstein condensation / Lasing in a topological edge state.\n\n2) While similar experiments have been performed in photonic systems, such as in lattices of polariton micropillars realising a SSH model [ref 9 of the authors], this is the first observation in cold atoms.\n\nIn the polariton experiment, the system is pumped incoherently and above a critical threshold, lasing suddenly dominates in an edge mode. However, here, the authors explain lasing in the edge state on page 2, right column, by a careful preparation of the reservoir of thermal atoms. As the atoms are initially in the dark state due to the time-dependent micro-waves, the population in the other states is indeed negligible, and thus we both only expect the dark state to be populated and the dark state to have a significant condensate fraction. This is obvious if there is no mechanism that would connect atoms in the dark state to a lower-energy state. In my opinion, this differs from the lasing demonstrated in other systems, where all modes are pumped and one wins above a critical threshold ; here, only the dark state is occupied by the thermal atoms which naturally forces Bose-Einstein condensation in this mode.\n\nFrom the sentence “We see that in all three values of r, the BEC phase transition, namely, atom lasing is successfully observed, and the formed BEC has the largest population in site 5, with certain atom numbers in 3 and 1 with almost no occupation in 2 and 4, which is consistent with the wavefunction profile of the topological edge state of the SSH model.”, I do not see what else I should expect.\n\nIt would be interesting to explain what would happen if the thermal atoms are prepared in a state which is not dark for example and discuss the robustness of the scheme with the preparation of the thermal atoms in a particular state. A clearer explanation of the role of the dark state is necessary for me.\n\n3) The most natural description of such a system for me is the one of a spinor condensate. In the dressed picture, the eigenstates are decomposed over the hyperfine states which can interact at low temperature with s-wave collisions with interaction parameters that depend on the magnetic field and on the states. I did not see any discussion or comment on interactions parameters, which I would appreciate.\n\n4) Related to my 2nd comment, I would find instructive to see what are the populations in the different hyperfine states at the end of the partial STIRAP procedure and before evaporation, to be able to compare them with the observed populations after evaporation. For example, in figure 2b, I would like to understand if the population in the state 3 originates purely from imperfections in the STIRAP procedure. This could be compared to the results of the theoretical model and figure 4d-e, which indicate that “that the edge state are adiabatically prepared”.\n\n5) In the 5-site SSH configuration, the condensate fraction are much smaller than in the 3-site configuration. Why is that ? The total condensate fraction that is reached is not written in any case.\n\n6) On page 5, equation 4 : the coefficient g is not explained. How can it be deduced from a microscopic analysis ?\n\n7) On page 6, figure 4g, can the fidelity be accurately extracted at times t < 30ms, where the population in the BEC state is actually very close to zero ? It seems to me that the fidelity if above 0.95 as soon as the population is non negligible.\n\nRelated to my 4th comment, what is the expected fidelity for the thermal component, especially at t=0 at the end of the MW preparation ? What is the sensitivity of the lasing in the dark state on the MW preparation based on the theoretical model ?\n\n8) On page 7, the part on “Confirming the coherence of the topological atom laser”, along with figure 5, is important in my opinion and I appreciated it. How is the fidelity 0.95(5) estimated ? It is not obvious for me based on the data of figure 5 and the relative population in state 5 below 0.8.\n\nAdditional smaller comments :\n\n1) In the figures 2 and 3, are the fourths rows commented in the main text ? I did not see their use. It seems to me that the sentence beginning by “In the third (fourth) row of Fig. 2,” on page 2, right column, does not exactly matches the content of the figure.\n\n2) On page 3, right column : typo “As we see in in Methods”\n\n3) On page 3, equation 3 : the indices (1,2) are not explicited for the Rabi frequencies in microwave coupling\n\n4) On page 4, right column, the reasons for the “not a simple ascending order” could be explicitly explained in a short way, based on the difference in sign of the magnetic moments of states of same projection m_F between the F=1 and F=2 states.\n\n5) The sentence with the definition of H_gain on page 5 left column does not have a verb.", "reviewer_3": "Summary\nThe authors report the experimental observation of BEC formation in excited eigenstates of a synthetic lattice of hyperfine states of 87Rb atoms. Their gain engineering scheme is original and well described, and supported by a theoretical model. They further employ their scheme to engineer a 5-site synthetic SSH chain, in which they promote lasing from topological edge modes. The technique presented in the article can catalyze the exploration of driven-dissipative phenomena in ultracold atomic gases.\n\nGeneral Remarks\nThe article is well detailed, both in terms of the experimental protocol and the theoretical modeling and simulations. I think that, overall, it warrants publication in Nature Communications. I do have, however, several remarks that I would like the authors to address.\n\n1. The information in the article is quite scattered throughout the pages, and early pages and statements become clear only later on. The reader, therefore, has to hold on to several bits of information before they are eventually clarified, which in my opinion leads to a confusing reading experience. There are also several repetitions, for example the abstract and the first lines of the introduction, or in the “Discussions” section.\n\n2. What is the rationale behind making an atom laser topological, especially an SSH (or 1D, in general) one? What would be the advantages (and disadvantages) over a conventional atom laser?\n\n3. Also in connection with my previous question, the authors show BEC formation, but what about lasing characteristics? What’s the threshold behavior? What are the spectral properties? What’s the second order coherence?\n\n4. This is more of a curiosity, not a request for review; would it be possible to directly measure the Zak phase in this particular synthetic lattice?\n\nSpecific Remarks\n1. Just before eq. 1, the authors write “λ-type”, although later they use “Λ-type” and “W-type”. Please use a consistent notation.\n\n2. At the beginning of the right column of page 2, they authors write “τ/τ leads to the equal superposition of site 4 and 2”. full\nWhile this seems to be clear from their previous expression for the dark state (at δ=0) and from the experimental results of Fig. 2b, third row, the “Theory” panel on the same row seems to predict a non-symmetric distribution. Why is that? Does it also account for a non-zero δ? And if so, why the population on site 3 is still zero?\n\n3. Again in Fig. 2, the legend text in the bottom row is way too small, and the remaining text could also be made a little bigger to improve legibility. If necessary, the authors can make more space by collapsing repeated x- and y-labels. This also applies to Fig. 3 and 1.\n\n4. Continuing the discussion of Fig. 2, why the a) and c) columns are show for τ/τ = 0.3 and 0.8? What happens at τ/τ = 0 full full\nand 1? Aren’t the “Theory” distributions predicted for those values?\n\n5. Let’s go back to Fig. 1. Why the site ordering 1-4-3-2-5 in column c? In the same column, are Ω and Ω the averages S W\ndefined later in the text? Shouldn’t the 5-site frequencies in the left panel be marked as Ω (1,2) and Ω (1,2)? S W\n\n6. Why is there a large mismatch wrt the theoretical prediction in the rightmost panel of the third row in Fig. 3?\n\n7. In the “Discussions” section, it’s stated again that a small occupation in sites 2 and 4 is to be expected due to the non-zero magnetic field fluctuations δ that break the chiral symmetry. While this seems to be incorporated in the theory, the “Theory” panels throughout the text have identically zero occupation in the even sites. Why is that?\n\n8. In relation to my previous answer, why do the authors choose the loss coefficient they way they do just before equation 9? There doesn’t seem to be any justification for their choice. This leads to the loss Hamiltonian in equation 9, which is 0 on sites 2 and 4. Could this be the source of the identically zero population on sites 2 and 4 in the “Theory” panels throughout the text? Wouldn’t this choice make the non-zero condensate density solution in sites 2 and 4 locally (dynamically) unstable, and thus forbidden in the numerical solutions?\n\n9. The legends in the last row of Fig 5 are again too small."}, "author_response": "We thank all the reviewers for their careful reading of our manuscript and their positive feedback. Their comments were very useful to us, and their questions brought up interesting discussions with new insights. The reviewers raised several points that gain profit from further clarification. This has led to several improvements in the manuscript. Numbers of equations, figures, etc. are those in the revised manuscript unless otherwise stated. The summary of changes are included in the end of this document.\n\nReviewer #1 (Remarks to the Author):\nIn the manuscript entitled “Gain engineering and topological atom laser in synthetic dimensions”, authors consider a multi-level bosonic Rb-87 system in which quantum states can be coherently controlled through two-photon processes. In particular, two hyperfine manifolds (F=1,2) are chosen to introduce 3 or 5 synthetic site coupled by microwave pulses. In this well-controlled platform, authors experimentally examine how thermal atoms cool down into different states – different sites in the synthetic dimension manner – when evaporative cooling is introduced. A dark state with the STRIAP sequence – which is well-known technique in quantum optics/AMO – and the SSH model with alternating hopping strengths are realized in this system.\n\nFor both models, authors observe the emergence of Bose-Einestein condensation at a specific site – usually at the edge – and this is interpreted as the lasing mode induced by gain engineering. This is of particular importance for pushing the limit of non-Hermitian dynamics, which is one of the frontier directions at the moment. The Authors point out that this gain engineering remains challenging in quantum systems in contrast to phonics, which is indeed true.\n\nOverall, the performed experiments seem very interesting with rock solid data sets. As noted, it is nice to see how topological edge mode (or a dark state) allow the control of thermal population. To me, the “observation” appears interesting on its own.\n\nI have some concerns about interpreting the experimental observations in terms of \"gain engineering.\" While the introduction connects gain engineering with non-Hermitian dynamics, this connection needs more clarity. In the presented experiment, the gain term is site-specific. Could the authors provide examples of how non-Hermitian dynamics can be realized through the demonstrated gain engineering and atom loss? It's not immediately clear how the proposed gain engineering helps implement a genuine PT-symmetric Hamiltonian with balanced gain-loss.\n\nWe appreciate your inquiries regarding gain engineering and non-Hermitian physics with a particular focus on PT-symmetric physics. As demonstrated in this study, the gain at each site was determined by the state of the thermal atoms. The loss arises from the intrinsic inelastic scattering between two atoms. In addition, various losses can be introduced artificially. Because we used an artificial dimension technique in this study, controlled site-dependent dissipation can be introduced by applying spin-dependent atomic loss. Thus, for example, if we prepare thermal atoms populated at sites 1, 3, and 5, and introduce dissipation at sites 2 and 4, we should be able to realize a PT-symmetric non-Hermitian system. Such models should feature PT-symmetric/broken phase transition, which should be observable within current experimental techniques. We have included information regarding the feasibility of realizing PT-symmetric non-Hermitian systems in the Discussion section.\n\nThe newly added sentence reads\n“Note that non-Hermitian dynamics can be realized through the demonstrated gain engineering and atom loss. For example, if we prepare thermal atoms populated at sites 1, 3, and 5, and introduce dissipation at sites 2 and 4, we should be able to realize a PT-symmetric non-Hermitian system. Such models should feature PT-symmetric/broken phase transition, which should be observable within current experimental techniques.”\n\nAdditionally, what is the sub-system being considered when envisioning a non-Hermitian Hamiltonian? Since the authors discuss non-Hermitian (or open quantum) systems, readers would benefit from understanding how atoms flow into the subsystem as a gain term. Can the 3-site (or 5-site) system be described by non-Hermitian Hamiltonian? Would the non-Hermitian skin effect be observable with this gain term? I feel the date set presented here seems interesting, but it would be beneficial if authors put this in the right context.\n\nWe thank the reviewer for the question. Here, the sub-system which can be described in terms of the non-Hermitian Hamiltonian is the Bose-Einstein condensate part of the atoms. The “external” bath which acts to provide the gain is the thermal (non-condensed) atoms.\nIn order to observe the non-Hermitian skin effect, we need additional ingredients. For example, by adding spin-dependent gain and loss in the spin-orbit coupled ultracold gases [Ren et al., Nat. Phys. 18, 385-389 (2022); Guo et al., Phys. Rev. A 106, L061302 (2022)], one should have the non-Hermitian skin effect. We have added these discussions in the revised manuscript.\n\nThe newly added sentence reads\n“Here, the subsystem which can be described in terms of the non-Hermitian Hamiltonian is the Bose-Einstein condensate part of the atoms. The “external” bath which acts to provide the gain is the thermal (non-condensed) atoms. By adding spin-dependent gain and loss in the spin-orbit coupled ultracold gases [Ren et al., Nat. Phys. 18, 385-389 (2022); Guo et al., Phys. Rev. A 106, L061302 (2022)], for example, one should also have the non-Hermitian skin effect, characteristic of a non-Hermitian system.”\n\nAnother question author may clarify is the quantum statistics. Like photons in photonics, it is not surprising that gain engineering properly works for bosonic particles, especially when BEC emerges, thanks to the bose enhancement. Can authors comment how proposed gain engineering works for fermions?\n\nWe thank the reviewer for the question. Since there is no clear difference between degenerate fermionic atoms and thermal atoms or, in other words, there is no Bose-Einstein condensation for fermions, our work cannot be directly applied to ultracold fermions. Lasing is a phenomenon specific to bosons. However, gain engineering should work for fermionic condensates like the BCS or molecular BEC states.\n\nThe newly added sentence reads\n“Since there is no Bose-Einstein condensation for fermions, our work cannot be directly applied to ultracold fermions. Lasing is a phenomenon specific to bosons. Gain engineering should, however, work for fermionic condensates like Bardeen-Cooper-Schriefer or molecular BEC states.”\n\nIn summary, while I have several concerns and suggestions outlined above that the authors should address before publication in Nature Communications, I believe this work demonstrates significant breakthroughs in multiple areas—including quantum system gain engineering, topological lasing with bosonic atoms, and potential developments in non-Hermitian engineering. I look forward to reviewing the revised manuscript before making my final recommendation.\n\nWe greatly appreciate your positive feedback and insightful comments. We believe that this response has properly addressed all your questions.\n\nReviewer #2 (Remarks to the Author):\nThe authors realise Bose-Einstein condensation in a micro-wave dressing of the hyperfine levels of 87Rb atoms. They perform evaporative cooling in the dressed states after carefully preparing the initial thermal states. In this approach, the thermal atoms can be seen as a reservoir, which provides “gain” for a “laser” formed of condensate atoms. They demonstrate this experimental approach in two configurations with 3 sites and 5 sites respectively, where the second engineers a SSH model, one of the simplest topological models in 1d. There, condensation occurs in a dressed state, which is a topological edge state of the 5-site system. Condensation is observed and quantified by observing the momentum distribution in time-of-flight with a magnetic gradient ; fitting the bimodal distribution identifies the condensed atoms and the thermal atoms.\n\nA novel aspect is the ability to engineer gain in a setup with ultracold atoms, where it is usually difficult to obtain, contrary to losses, which opens a new path to study non-Hermitian physics with both gain and losses. A theoretical analysis is provided to support the claims and the interpretation of the experimental data, which is based on solving the non-Hermitian Schrödinger equation for the condensate wavefunction, taking the thermal atoms as a gain medium (which empties as the condensate grows), in the presence of losses for the condensate.\n\nWe thank the accurate description and positive assessment of our paper.\n\nOverall I appreciated the manuscript and the work, but I have several questions and remarks for the authors, that would need to be addressed before supporting publication :\n\n1) I find that the denomination “topological atom laser” is a bit overselling and could be simplified to : Bose-Einstein condensation / Lasing in a topological edge state.\n\nWe appreciate your comment on the title and acknowledge your suggestion. We will update the title from “Gain engineering and topological atom laser in synthetic dimensions” to “Gain engineering and atom lasing in a topological edge state in synthetic dimensions”.\n\n2) While similar experiments have been performed in photonic systems, such as in lattices of polariton micropillars realising a SSH model [ref 9 of the authors], this is the first observation in cold atoms.\n\nIn the polariton experiment, the system is pumped incoherently and above a critical threshold, lasing suddenly dominates in an edge mode. However, here, the authors explain lasing in the edge state on page 2, right column, by a careful preparation of the reservoir of thermal atoms. As the atoms are initially in the dark state due to the time-dependent micro-waves, the population in the other states is indeed negligible, and thus we both only expect the dark state to be populated and the dark state to have a significant condensate fraction. This is obvious if there is no mechanism that would connect atoms in the dark state to a lower-energy state. In my opinion, this differs from the lasing demonstrated in other systems, where all modes are pumped and one wins above a critical threshold ; here, only the dark state is occupied by the thermal atoms which naturally forces Bose-Einstein condensation in this mode.\n\nFrom the sentence “We see that in all three values of r, the BEC phase transition, namely, atom lasing is successfully observed, and the formed BEC has the largest population in site 5, with certain atom numbers in 3 and 1 with almost no occupation in 2 and 4, which is consistent with the wavefunction profile of the topological edge state of the SSH model.”, I do not see what else I should expect.\n\nIt would be interesting to explain what would happen if the thermal atoms are prepared in a state which is not dark for example and discuss the robustness of the scheme with the preparation of the thermal atoms in a particular state. A clearer explanation of the role of the dark state is necessary for me.\n\nWe fully agree with the reviewer that what happens if the thermal atoms are not prepared only in the dark state is an interesting problem. For the photonic systems, it is true that the pumped states do not precisely correspond to the edge states, but, during the lasing only the edge state experiences growth. This behavior is also confirmed in our numerical simulations of our atom experiment. For instance, in Fig. R1, we show the result of our simulation where the initial state population for the site 5 is 0.8, while those of the sites 1, 2, 3, and 4 are randomly distributed, which is not apparently the pure edge state, corresponding to the edge-state fidelity of approximately 0.9. All other simulation conditions remain essentially the same as those outlined in the main text. Although, at t=0, the edge state fidelity does not reach 1 but 0.9, upon introducing gain for t>0, the BEC emerges with the fidelity approaching 1, surpassing that of the thermal state.\n\nFigure R1: Numerical simulation for the case of non-ideal population in the 5-site model.\nNote that the values of the gain, loss, and magnetic field fluctuations are the same as those shown in Figure 4, and the only difference is the initial state of the thermal atoms. When the total number of atoms is N , a fraction pN is allocated to site 5, whereas the remaining total total\n(1-p)N atoms are randomly distributed among sites 1-4 at the initial stage. Time t = 0 total\n(t=50 ms) marks the completion of state preparation (evaporative cooling). Φ and Φ sim edge\ndenote the state obtained through time evolution in a numerical simulation and the expected edge state derived by diagonalizing equation (3) in the main text with δ=0, respectively.\nFor a sufficiently large p (approximately p > 0.8), the BEC evolves into the same state as the edge state irrespective of the initial state.\n\nIt is important to note that this behavior is presumed to occur solely under the ideal conditions for numerical calculations. In actual experiments, significant magnetic field fluctuations were present, preventing the observation of situations predicted by numerical calculations. These discussions are elaborated upon in the Discussion section.\n\nThe newly added sentence reads\n“What happens if the initial thermal atoms are not prepared only in the edge state, like the photonic systems, is an interesting problem. Our simulation where the initial state is not the pure edge state shows, although the fidelity does not reach 1 at t=0, upon introducing gain for t>0, the BEC emerges with the fidelity approaching 1, surpassing that of the thermal state. This is similar to the observation in the photonic systems. In our actual experiments, however, significant magnetic field fluctuations were present, preventing the observation of behaviors predicted by numerical calculations.”\n\n3) The most natural description of such a system for me is the one of a spinor condensate. In the dressed picture, the eigenstates are decomposed over the hyperfine states which can interact at low temperature with s-wave collisions with interaction parameters that depend on the magnetic field and on the states. I did not see any discussion or comment on interactions parameters, which I would appreciate.\n\nWe thank the reviewer for the question. The scattering length of 87Rb exhibits only a small dependence on hyperfine or magnetic sublevels [Widera et al, New J. Phys. 8, 152 (2006); Kaufman et al, Phys. Rev. A 80, 050701 (2009)]. As a result, the spin-exchange collisions are significantly suppressed. We have added this sentence in the main text.\n\nThe newly added sentence reads,\n“The scattering length of 87Rb exhibits only a small dependence on hyperfine or magnetic sublevels [Widera et al, New J. Phys. 8, 152 (2006); Kaufman et al, Phys. Rev. A 80, 050701 (2009)]. As a result, the spin-exchange collisions are significantly suppressed.”\n\n4) Related to my 2nd comment, I would find instructive to see what are the populations in the different hyperfine states at the end of the partial STIRAP procedure and before evaporation, to be able to compare them with the observed populations after evaporation. For example, in figure 2b, I would like to understand if the population in the state 3 originates purely from imperfections in the STIRAP procedure. This could be compared to the results of the theoretical model and figure 4d-e, which indicate that “that the edge state are adiabatically prepared”.\n\nWe thank the reviewer for this important question. Note that “Theory” in Figs. 2, 3, and 5(b) in the original manuscript corresponds to the numerical simulation in the idealized case of no detuning (δ=0), as well as the data in Fig. 4(e). While the numerical simulations shown in Fig. 4(a-d, f, g) were done in the presence of the detuning oscillating at 60 Hz, here we performed the numerical simulation in the presence of a static detuning due to the slow drift of a magnetic field, which breaks the chiral symmetry for the 5 site system and dark-state condition for the 3-site system. The case of the 3-site system in the presence of 0.25 mG static magnetic field is shown in Fig. R2 where the finite population in the state 3 at the end of the partial STIRAP procedure and before evaporation is clearly found. Similar finite populations in the even sites are found in the case of 5-site system in the presence of 0.5 mG, as shown in Fig. R3. Note that, in these conditions, dark state and edge state fidelities of the subsequent BEC do no surpass those for the thermal atoms at t=0, contrary to the case of no static detuning but with the finite populations in the site 3 (3-site system) and even sites(5-site system), indicating the importance of the chiral symmetry and dark-state condition in the behavior of atom lasing.\n\nThe newly added sentence reads,\n“We also performed the numerical simulation in the presence of a static detuning due to the slow drift of a magnetic field, which breaks the chiral symmetry for the 5 site system and dark-state condition for the 3-site system. As expected, we found the finite populations in the state 3 and even sites in the case of 3- and 5-site systems, respectively, at the end of the partial STIRAP procedure and before evaporation.”\n\nIn the revised manuscript, we also explicitly note that “Theory” in Figs. 2, 3, and 5(b) in the original manuscript corresponds to the numerical simulation in the idealized case of no detuning (δ=0), as well as the data in Fig. 4(e), in the corresponding Figure Captions.\n\nFigure R2: Numerical simulation in the presence of a static detuning due to a 0.25 mG static magnetic field in the 3-site model. The values of the gain g=17.5 and loss gamma=10 are the same as those shown in Figure 2. The finite population in the state 3 is clearly found.\nThe region t<0 corresponds to the state preparation, that is, the Rabi coupling is gradually introduced to the thermal atoms. The region t>0 corresponds to evaporatively cooling with gain for the growth of BEC. Note that the adiabaticity is slightly violated, causing an oscillatory behavior at t>0.\n\nFigure R3: Numerical simulation in the presence of a static detuning due to a 0.5 mG static magnetic field in the 5-site model. The values of the gain and loss are the same as those shown in Figure 4. The finite population in the even states of 2 and 4 is clearly found. The region t<0 corresponds to the state preparation, that is, the Rabi coupling is gradually introduced to the thermal atoms. The region t>0 corresponds to evaporatively cooling with gain for the growth of BEC. The adiabaticity is slightly violated, causing a slightly oscillatory behavior at t>0.\n\n5) In the 5-site SSH configuration, the condensate fraction are much smaller than in the 3-site configuration. Why is that ? The total condensate fraction that is reached is not written in any case.\n\nWe thank the reviewer for the question. Compared to the 3-site system which mainly populates the lower hyperfine levels with negligible inelastic loss, the 5-site system mainly populates the upper hyperfine levels which suffer from larger inelastic collisions [63-66 in the revised manuscript] with more complex microwave couplings, making evaporative cooling more challenging. Information pertaining to the BEC fraction has been incorporated in Figures 2, 3 and 5.\n\nThe newly added sentence reads,\n“Compared to the 3-site system which mainly populates the lower hyperfine levels with negligible inelastic loss, the 5-site system mainly populates the upper hyperfine levels which suffer from larger inelastic collisions [63-66] with more complex microwave couplings, making evaporative cooling more challenging.”\n\n6) On page 5, equation 4 : the coefficient g is not explained. How can it be deduced from a microscopic analysis ?\n\nWe thank the reviewer for the question. As mentioned in the main text, the gain coefficient g is a phenomenological parameter. While we do not know at the moment how to microscopically derive this parameter, what we can say is that the gain is related with the microscopic process of bosonic stimulation through atom collisions and thus should be proportional to the number of thermal atoms, which is how we constructed the theoretical model.\n\nThe newly added sentence reads,\n“Note that the gain is related with the microscopic process of bosonic stimulation through atom collisions and thus, the gain coefficient g, introduced as a phenomenological parameter, should be multiplied by the number of thermal atoms, which is how we constructed the theoretical model. “\n\n7) On page 6, figure 4g, can the fidelity be accurately extracted at times t < 30ms, where the population in the BEC state is actually very close to zero ? It seems to me that the fidelity if above 0.95 as soon as the population is non negligible.\n\nRelated to my 4th comment, what is the expected fidelity for the thermal component, especially at t=0 at the end of the MW preparation ? What is the sensitivity of the lasing in the dark state on the MW preparation based on the theoretical model ?\n\nWe thank the reviewer for this important question. We here explain the details of behavior shown in Fig. 4 (g). As one can see in Fig. 4(a), (c), and (f), we set very small but finite “BEC” populations in the initial state as a “seed” for BEC. As the gain is introduced for t>0, the population of BEC in an edge state increases, approaching unity. We agree that the relative populations and “fidelity” for the BEC is only important when the BEC fraction is increased to a certain amount beyond the initial “seed” necessary in the numerical calculation.\n\nThe relative populations of the thermal atoms at t=0 depend on how adiabatic the state preparation, or the microwave sweep, is. The results of the numerical simulations showed that the sweep time of 10 ms is sufficiently adiabatic. As was also mentioned in the reply for the 4th comment, the numerical simulation in the presence of a static detuning due to the slow drift of a magnetic field of 0.5 mG, which breaks the dark-state condition for the 3-site system, shows that the finite population in the state 3 at t=0 is clearly found.\n\nThe newly added sentence reads,\n“The relative populations and fidelity for the BEC shown in Fig. 4 (c), (f), and (g) are only important when the BEC fraction is increased to a certain amount beyond the initial “seed” necessary in the numerical calculation.”\n\n8) On page 7, the part on “Confirming the coherence of the topological atom laser”, along with figure 5, is important in my opinion and I appreciated it. How is the fidelity 0.95(5) estimated ? It is not obvious for me based on the data of figure 5 and the relative population in state 5 below 0.8.\n\nWe apologize for the unclear explanation. First, the definition of fidelity used here is the first power of the inner product and not the second power. In addition, there are two methods to perform the required calculations. Originally we estimated the value of fidelity using a series of absorption images. The first method is to calculate the fidelity of each image and then averages the values. The second method is to average the number of atoms at site i (i=1…5) and then calculate the fidelity. The first method gave a result of 0.95(5), and the second method gave 0.90(1). In the revised manuscript, we chose the second method and used 0.90(1) for more appropriateness and clarity. We would like to note that the occupancy ratio of site 5 was 0.82(2), which is about the square of Fidelity of 0.90(1).\nTo avoid the possible confusion, we newly added the following sentence,\n“…the quantity of I=0.90(1) for the condensates, consistent with 0.82(2) in terms of population, shown in Fig. 5 c. This value of I close to 1 indicates …”\n\nAdditional smaller comments :\n\n1) In the figures 2 and 3, are the fourths rows commented in the main text ? I did not see their use. It seems to me that the sentence beginning by “In the third (fourth) row of Fig. 2,” on page 2, right column, does not exactly matches the content of the figure.\n\nWe apologize for the unclear wordings and missing connection with the figures. We have revised the text in the manner in which the figures were properly referenced.\nSee revised Fig. & Fig. caption\n\n2) On page 3, right column : typo “As we see in in Methods”\n\nWe thank the reviewer for your pointing out this mistake. We have made necessary corrections.\n\n3) On page 3, equation 3 : the indices (1,2) are not explicited for the Rabi frequencies in microwave coupling\n\nWe are sorry that we have not explained these in detail. Thank you for pointing these out to us. We have added an explanation in Figure 1.\n\nThe newly added sentence reads,\n“Rabi frequencies for the microwave coupling are indicated by … and Ω(i) (i=S,W, j=1,2) i\nfor 5-site lattice”\n\n4) On page 4, right column, the reasons for the “not a simple ascending order” could be explicitly explained in a short way, based on the difference in sign of the magnetic moments of states of same projection m_F between the F=1 and F=2 states.\n\nWe thank the reviewer for the comment. We have added a sentence to explain why the atom positions in the images after time-of-flight are not in a simple ascending order.\n\nThe revised sentence now reads,\n“The order of sites 1, 4, 3, 2, 5 in the absorption image is not a simple ascending order because of the difference in the sign of the magnetic moments of the states between the F=1 and F=2 states.”\n\n5) The sentence with the definition of H_gain on page 5 left column does not have a verb.\n\nThank you for your pointing out this mistake. We have made necessary corrections.\n\nThe revised sentence now reads,\nHˆgain =ignT |T ⟩ ⟨T | represents the gain term with the coefficient g being the effective strength of the gain per unit thermal component.\n\nReviewer #3 (Remarks to the Author):\nSummary\nThe authors report the experimental observation of BEC formation in excited eigenstates of a synthetic lattice of hyperfine states of 87Rb atoms. Their gain engineering scheme is original and well described, and supported by a theoretical model. They further employ their scheme to engineer a 5-site synthetic SSH chain, in which they promote lasing from topological edge modes. The technique presented in the article can catalyze the exploration of driven-dissipative phenomena in ultracold atomic gases.\n\nGeneral Remarks\nThe article is well detailed, both in terms of the experimental protocol and the theoretical modeling and simulations. I think that, overall, it warrants publication in Nature Communications. I do have, however, several remarks that I would like the authors to address.\n\nWe greatly appreciate the positive assessment of the reviewer.\n\n1. The information in the article is quite scattered throughout the pages, and early pages and statements become clear only later on. The reader, therefore, has to hold on to several bits of information before they are eventually clarified, which in my opinion leads to a confusing reading experience. There are also several repetitions, for example the abstract and the first lines of the introduction, or in the “Discussions” section.\n\nWe are sorry for the improper structure of our manuscript. We have revised the manuscript in a manner that the provided information should not be scattered. In addition, we removed the repetitions in the Abstract, Introduction, and Discussion sections as possible as we can.\nBecause of the numerous sections that have been updated, we will detail them in “Summary of Changes”.\n\n2. What is the rationale behind making an atom laser topological, especially an SSH (or 1D, in general) one? What would be the advantages (and disadvantages) over a conventional atom laser?\n\nWe thank the reviewer for the question. The purpose of this paper is proof of principle, but the topological atom laser should share the merits the topological photonic lasers have, such as the robust single-mode lasing (condensation) in the presence of disorder. However, we want to stress that the main technological implication of our paper is the ability to introduce gain in ultracold gases, which allows one to explore a variety of non-Hermitian Hamiltonians in ultracold settings.\n\nThe newly added sentence reads,\n“While the main technological implication of our work is the ability to introduce gain in ultracold gases, which allows one to explore a variety of non-Hermitian Hamiltonians in ultracold settings, the topological atom laser should share the merits the topological photonic lasers have, such as the robust single-mode lasing (condensation) in the presence of disorder.”\n\n3. Also in connection with my previous question, the authors show BEC formation, but what about lasing characteristics? What’s the threshold behavior? What are the spectral properties? What’s the second order coherence?\n\nWe thank the reviewer for this interesting question. What we mean by the atom laser is, following existing literatures, the Bose-Einstein condensation. It is difficult to find exact correspondence of lasing characteristics discussed in optical lasers, although we believe that they are promising subjects for future research. Second order coherence perhaps corresponds to two-point correlation function of Bose-Einstein condensation, but we believe there is nothing special about topological atom laser in terms of these characteristics compared to the conventional Bose-Einstein condensation.\n\nThe newly added sentence reads,\n“Note that we believe there is nothing special about topological atom laser in terms of characteristics such as threshold behavior, spectral properties, and the second order coherence compared to the conventional Bose-Einstein condensation, although they are subjects for future research.”\n\n4. This is more of a curiosity, not a request for review; would it be possible to directly measure the Zak phase in this particular synthetic lattice?\n\nWe thank the reviewer for this interesting question. Zak phase is a property of bulk, and in our experiment there are only 2.5 unit cells, which makes it difficult to probe bulk properties. In order to explore bulk topological physics we need to use a longer lattice, which is an interesting future work.\n\nSpecific Remarks\n1. Just before eq. 1, the authors write “λ-type”, although later they use “Λ-type” and “W-type”. Please use a consistent notation.\n\nThank you for your pointing out this inconsistency. In the revised manuscript, we use “Λ-type”, and “W-type” for the 3-site and 5-site systems, respectively.\n\n2. At the beginning of the right column of page 2, they authors write “τ/τ leads to the equal superposition of site 4 and 2”. full\nWhile this seems to be clear from their previous expression for the dark state (at δ=0) and from the experimental results of Fig. 2b, third row, the “Theory” panel on the same row seems to predict a non-symmetric distribution. Why is that? Does it also account for a non-zero δ? And if so, why the population on site 3 is still zero?\n\nWe thank the Reviewer for the questions. We apologize for these potentially misleading expressions. As shown in Table I, Ω and Ω are not equal whenτ/τ = 0.5. In the 3-site system, we believe that labeling with τ is not appropriate. In the revised manuscript, we have labeled using the Ω to Ω ratio. The time over which the values change adiabatically 1 2\nis newly presented in Table I. In this revised manuscript, we show three cases where Ω /Ω 1 2\nis much less than 1, approximately equal to 1, or much greater than 1, and we believe that it is a more reader-friendly way of presenting the experimental parameters.\n\nWe again note, as in the reply of 4th comment of reviewer#2, that the “Theory” in Figs. 2, 3, and 5(b) in the original manuscript corresponds to the numerical simulation in the idealized case of no detuning (δ=0), as well as the data in Fig. 4(e).\n\n3. Again in Fig. 2, the legend text in the bottom row is way too small, and the remaining text could also be made a little bigger to improve legibility. If necessary, the authors can make more space by collapsing repeated x- and y-labels. This also applies to Fig. 3 and 1.\n\nThank you for your pointing out these issues. We have made necessary corrections.\n\n4. Continuing the discussion of Fig. 2, why the a) and c) columns are show for τ/τ = 0.3 and 0.8? What happens at τ/τ = 0 full full\nand 1? Aren’t the “Theory” distributions predicted for those values?\n\nWe thank the Reviewer for the questions. We did not have experimental data corresponding toτ/τ =0 orτ/τ =1 in the original manuscript.τ/τ =0 and τ/τ =1 are almost full full full ful\nthe same asτ/τ =0.3 and τ/τ =0.8, respectively, and the differences are only trivial.\nfull full\nAs mentioned in our answer to Specific Remark 2), we believe that labeling with τ/τ is full\nnot appropriate. Therefore, in the revised manuscript, we used Ω /Ω for labeling. We 1 2\nbelieve this is a less misleading way to express it.\n\n5. Let’s go back to Fig. 1. Why the site ordering 1-4-3-2-5 in column c? In the same column, are Ω and Ω the averages S W\ndefined later in the text? Shouldn’t the 5-site frequencies in the left panel be marked as Ω (1,2) and Ω (1,2)? S W\n\nWe thank the reviewer for the questions. The first question is the same as the question (4) by Reviewer 1. We have added a sentence to explain why the atom positions in the images after time-of-flight are not in a simple ascending order.\n\nThe revised sentence now reads,\n“The order of sites 1, 4, 3, 2, 5 in the absorption image is not a simple ascending order because of the difference in the sign of the magnetic moments of the states between the F = 1 and F = 2 states.”\n\nAs for the second question, we revised Fig. 1 so that the 5-site frequencies are marked as Ω (1,2) and Ω (1,2), according to the suggestion. The averages of Ω (1,2) and Ω (1,2), marked S W S W\nas <Ω > and <Ω > are only defined later in the text (“Experimental results”). S W,\n\n6. Why is there a large mismatch wrt the theoretical prediction in the rightmost panel of the third row in Fig. 3?\n\nWe thank the reviewer for the question. The rightmost panel of the third row in Fig. 3 in the original manuscript corresponds to the case of the largest microwave power to construct the edge state, beyond the regime of Ω (1,2) > Ω (1,2), and thus the largest change of S W\npopulations, possibly suffering from imperfections such as a magnetic field fluctuation.\n\nThe newly added sentence now reads,\n“One can see a relatively large mismatch between the theoretical prediction and the data in the case of the largest microwave power, beyond the regime of Ω (1,2) > Ω (1,2). Since this S W\ncase corresponds to the largest change of populations, and thus possibly suffers relatively much severely from imperfections such as magnetic field fluctuation.”\n\n7. In the “Discussions” section, it’s stated again that a small occupation in sites 2 and 4 is to be expected due to the non-zero magnetic field fluctuations δ that break the chiral symmetry. While this seems to be incorporated in the theory, the “Theory” panels throughout the text have identically zero occupation in the even sites. Why is that?\n\nWe thank the reviewer for this important question. Please see the answer for the Reviewer #2 ‘s question (4). “Theory” in Figs. 2, 3, and 5(b) in the original manuscript corresponds to the numerical simulation in the idealized case of no detuning (δ=0), as well as the data in Fig. 4(e). The numerical simulations shown in Fig. 4(a-d, f, g) were done in the presence of the small detuning oscillating at 60 Hz. Here we performed the numerical simulation in the presence of a static detuning due to the slow drift of a magnetic field, which breaks the chiral symmetry for the 5 site system. The case of 0.5 mG static magnetic field is shown in Fig. R3 where the finite population in the even states at the end of the partial STIRAP procedure and before evaporation is clearly found.\n\nAs is already noted in the reply for Reviewer #2’s Question (4), the corresponding newly added sentence reads,\n“We also performed the numerical simulation in the presence of a static detuning due to the slow drift of a magnetic field, which breaks the chiral symmetry for the 5 site system and dark-state condition for the 3-site system. As expected, we found the finite populations in the state 3 and even sites in the case of 3- and 5-site systems, respectively, at the end of the partial STIRAP procedure and before evaporation.”\n\nWe also explicitly note that “Theory” in Figs. 2, 3, and 5(b) in the original manuscript corresponds to the numerical simulation in the idealized case of no detuning (δ=0), as well as the data in Fig. 4(e), in the corresponding Figure Captions.\n\n8. In relation to my previous answer, why do the authors choose the loss coefficient they way they do just before equation 9? There doesn’t seem to be any justification for their choice. This leads to the loss Hamiltonian in equation 9, which is 0 on sites 2 and 4. Could this be the source of the identically zero population on sites 2 and 4 in the “Theory” panels throughout the text? Wouldn’t this choice make the non-zero condensate density solution in sites 2 and 4 locally (dynamically) unstable, and thus forbidden in the numerical solutions?\n\nWe thank the reviewer for this important question. We do not know all the loss coefficients.\nThe loss Hamiltonian in Eq. (9) based on the reported measurement results is adopted in this work.\n\nAs was explained in the reply for the previous question as well as that for the reviewer #2’s question (4), “Theory” in Figs. 2, 3, and 5(b) in the original manuscript corresponds to the numerical simulation in the idealized case of no detuning (δ=0), as well as the data in Fig. 4(e), which is the source of the identically zero population on sites 2 and 4 in the “Theory” panels of Fig. 3, as well as the site 3 of Fig. 2.\n\nAs for the numerical simulation including the loss and gain terms, as shown in Fig. 4(a-d, f, g), only when we introduce a static detuning, we find a finite population in the site 3(3-site system) and even-sites (5-site system), which illustrates the importance of the chiral symmetry and dark-state condition to suppress the unwanted populations. In this respect, a specific form of loss Hamiltonian does not play an important role in the current work.\n\nThe newly added sentence reads,\n“The choice of this specific form of loss Hamiltonian of Eq. (9) is based on the reported measurement results, and does not have a direct relation to the appearance or suppression of the populations in the even sites where the loss is set to zero.”\n\n9. The legends in the last row of Fig 5 are again too small.\n\nWe thank the reviewer for this pointing out. We deleted the legend from the image and explained in the figure caption because of limited available space.\n\nSummary of Changes:\n1. We changed the title according to the advice of the Reviewer #2. The new title is “Gain engineering and atom lasing in a topological edge state in synthetic dimensions”.\n2. We modified the Abstract section in order to remove repetition with respect to the “INTRODUCTION” and “DISCUSSION” sections.\n3. We added the sentence “The scattering length of 87Rb exhibits only a small dependence on hyperfine or magnetic sublevels [Widera et al, New J. Phys. 8, 152 (2006); Kaufman et al, Phys. Rev. A 80, 050701 (2009)]. As a result, the spin-exchange collisions are significantly suppressed.” in the “EXPERIMENTAL RESULTS” section.\n4. We added the sentence “Note that the gain is related with the microscopic process of bosonic stimulation through atom collisions and thus, the gain coefficient g, introduced as a phenomenological parameter, should be multiplied by the number of thermal atoms, which is how we constructed the theoretical model. “ in the “THEORETICAL MODEL” section.\n5. We added the sentence “The relative populations and fidelity for the BEC shown in Fig. 4 (c), (f), and (g) are only important when the BEC fraction is increased to a certain amount beyond the initial “seed” necessary in the numerical calculation.” in the “THEORETICAL MODEL” section.\n6. We added the sentence ““…the quantity of I=0.90(1) for the condensates, consistent with 0.82(2) in terms of population, shown in Fig. 5 c. This value of I close to 1 indicates …” in the “CONFIRMING THE COHERENCE OF THE TOPOLOGICAL ATOM LASER” section.\n7. We added the sentence “We also performed the numerical simulation in the presence of a static detuning due to the slow drift of a magnetic field, which breaks the chiral symmetry for the 5 site system and dark-state condition for the 3-site system. As expected, we found the finite populations in the state 3 and even sites in the case of 3- and 5-site systems, respectively, at the end of the partial STIRAP procedure and before evaporation.” in the “DISCUSION” section.\n8. We added the sentence “What happens if the initial thermal atoms are not prepared only in the edge state, like the photonic systems, is an interesting problem. Our simulation where the initial state is not the pure edge state shows, although the fidelity does not reach 1 at t=0, upon introducing gain for t>0, the BEC emerges with the fidelity approaching 1, surpassing that of the thermal state. This is similar to the observation in the photonic systems. In our actual experiments, however, significant magnetic field fluctuations were present, preventing the observation of behaviors predicted by numerical calculations.” in the “DISCUSION” section.\n9. We added the sentence “Note that non-Hermitian dynamics can be realized through the demonstrated gain engineering and atom loss. For example, if we prepare thermal atoms populated at sites 1, 3, and 5, and introduce dissipation at sites 2 and 4, we should be able to realize a PT-symmetric non-Hermitian system. Such models should feature PT-symmetric/broken phase transition, which should be observable within current experimental techniques.” in the “DISCUSION” section.\n10. We added the sentence “Here, the subsystem which can be described in terms of the non-Hermitian Hamiltonian is the Bose-Einstein condensate part of the atoms. The “external” bath which acts to provide the gain is the thermal (non-condensed) atoms. By adding spin-dependent gain and loss in the spin-orbit coupled ultracold gases [Ren et al., Nat. Phys. 18, 385-389 (2022); Guo et al., Phys. Rev. A 106, L061302 (2022)], for example, one should also have the non-Hermitian skin effect, characteristic of a non-Hermitian system.” in the “DISCUSION” section.\n11. We added the sentence “Note that we believe there is nothing special about topological atom laser in terms of characteristics such as threshold behavior, spectral properties, and the second order coherence compared to the conventional Bose-Einstein condensation, although they are subjects for future research.” in the “CONCLUSION AND PROSPECTS” section.\n12. We added the sentence “Since there is no Bose-Einstein condensation for fermions, our work cannot be directly applied to ultracold fermions. Lasing is a phenomenon specific to bosons. Gain engineering should, however, work for fermionic condensates like Bardeen-Cooper-Schriefer or molecular BEC states.” in the “CONCLUSION AND PROSPECTS” section.\n13. We added the sentence “While the main technological implication of our work is the ability to introduce gain in ultracold gases, which allows one to explore a variety of non-Hermitian Hamiltonians in ultracold settings, the topological atom laser should share the merits the topological photonic lasers have, such as the robust single-mode lasing (condensation) in the presence of disorder.” in the “CONCLUSION AND PROSPECTS” section.\n14. We added the sentence “The choice of this specific form of loss Hamiltonian of Eq. (9) is based on the reported measurement results, and does not have a direct relation to the appearance or suppression of the populations in the even sites where the loss is set to zero.” in the “METHOD” section.\n15. We added the sentence “Rabi frequencies for the microwave coupling are indicated by … and Ω(i) (i=S,W, j=1,2) for 5-site lattice” in the caption of Figure 1. i\n16. We added the sentence “The order of sites 1, 4, 3, 2, 5 in the absorption image is not a simple ascending order because of the difference in the sign of the magnetic moments of the states between the F = 1 and F = 2 states.” in the caption of Figure 1.\n17. We added the sentence “Compared to the 3-site system which mainly populates the lower hyperfine levels with negligible inelastic loss, the 5-site system mainly populates the upper hyperfine levels which suffer from larger inelastic collisions [63-66] with more complex microwave couplings, making evaporative cooling more challenging.” in the caption of Figure 3.\n18. We added the sentence “One can see a relatively large mismatch between the theoretical prediction and the data in the case of the largest microwave power, beyond the regime of Ω (1,2) > Ω (1,2). Since this case corresponds to the largest change of populations, and thus possibly suffers relatively much severely from imperfections such as magnetic field fluctuation.” in the caption of Figure 3.\n19. We changed to use Ω /Ω to denote the experimental conditions for the 3-site system, 1 2\nrather than τ.\n20. We changed to use <Ωs>/<Ω > to denote the experimental conditions for the 5-site W\nsystem, rather than r.\n21. We revised the manuscript to improve readability, such as increasing the font size and avoiding the use of expressions such as \"Top\" and \"Bottom.\" for Figures 1, 2, 3 and 5.\n22. We added the information of BEC fractions for Figures 2, 3, and 5.\nIn addition, we corrected as many errors and typos as possible in the manuscript. For details, please refer to the attached file, starting from the next page. Items highlighted in red indicate deletions and those in blue indicate additions."}, {"round": 2, "reviewer_comments": {"reviewer_1": "I have thoroughly reviewed the revised manuscript with considerable interest and find that the authors have addressed all referee comments and suggestions comprehensively and thoughtfully. The authors have not only implemented specific technical corrections but have also substantially improved the clarity throughout the paper. Particularly noteworthy is how they have addressed the common concerns raised by multiple referees, providing additional data, analyses, and explanations that significantly strengthen their conclusions.\n\nThis innovative work represents a significant advancement in the quantum simulation field, with potential implications for both theoretical understanding and practical applications. The data presentation is clear and accessible.\n\nBased on my assessment, this manuscript fully satisfies the criteria of Nature Communications for novelty, significance, and technical quality. I am therefore happy to strongly support its publication in its current form.", "reviewer_2": "The authors thoroughly answered my questions with additional numerical data to support their claims, as well as the questions from the other reviewers. In particular, they have clarified the role of the dark state and of the initial preparation of the thermal gas in the dark state.\n\nThey have clarified and improved the manuscript, and I do not have any concerns left and I advise publication.", "reviewer_3": "I thank the authors for their exhaustive reply; they have addressed all of my concerns. In particular, I find the new labeling with Ω /Ω and explicit δ=0 to be definitely clearer than before. I also appreciate the additional simulations prepared in 1 2\nresponse to Reviewer #2, which have also resolved some of my doubts.\n\nAs a result, I recommend the revised manuscript for publication."}, "author_response": ""}]

Decision letter