Observation of perfect absorption in hyperfine levels of molecular spins with hermitian subspaces

Domain: earth_science
Published: 2025-12-10
Editorial outcome: minor_revision

Abstract
We investigate Perfect Absorption (PA) of radiation, in which incoming energy is entirely dissipated, in a system consisting of molecular spin centers coherently coupled to a planar microwave resonator operated at milliKelvin temperature and in the single photon regime. This platform allows us to fine tune the spin-photon coupling and to control the effective dissipation of the two subsystems towards the environment, thus giving us the opportunity to span over a wide space of parameters. Our system can be effectively described by a non-Hermitian Hamiltonian exhibiting distinct Hermitian subspaces. We experimentally show that these subspaces, linked to the presence of PA, can be engineered through the resonator-spin detuning, which controls the composition of the polaritons in terms of photon and spin content. In such a way, the required balance between the feeding and the loss rates is effectively recovered even in the absence of PT-symmetry. We show that Hermitian subspaces influence the overall aspect of coherent spectra of cavity QED systems and enlarge the possibility to explore non-Hermitian effects in open quantum systems. We finally discuss how our results can be potentially exploited for applications, in particular as single-photon switches and modulators.

Introduction
Passive open light-matter quantum systems offer the opportunity to investigate non-Hermitian physics1–4. Non-Hermiticity provides an insightful theoretical framework to understand unconventional phenomena such as wave-scattering anomalies, e.g., coherent perfect absorption (CPA)5–7, perfect absorption (PA, often also referred to as Reflectionless Scattering Modes, RSM, in single-port configuration)8–10, and electromagnetically-induced transparency11,12, which have attracted a great deal of interest in the last decades for their potential applications in photonics13,14. These peculiar wave scattering effects are often associated with the presence of Exceptional Points (EP, i.e., degeneracy of the complex resonance frequencies of the system)7,9,15, Parity-time (PT) symmetry13,16–19, anti-PT-symmetry20–23, and Bound States in the Continuum (BIC)23–25. It is known that coherent scattering of electromagnetic radiation is fully described by the positions of the poles and zeros of the scattering matrix6,8,25–27. In particular, for the reflection element of the scattering matrix (e.g., S11), the relevant spectral features are determined by the eigenvalues of two closely related, non-Hermitian effective Hamiltonians8,25,27. Following the convention adopted in ref. 27, we will denote these as H^RZ and H^res, associated with the poles and zeros of the reflection coefficient, respectively (see Section “Theoretical modeling” and Section “Methods”). PA can occur only when one or more eigenvalues of H^RZ lie on the real axis, and the frequency ω of the input field matches such real eigenvalues. In ideal passive PT-symmetric systems (with respect to H^RZ), where the losses and effective gains introduced through the input port are balanced, both the two relevant eigenvalues of H^RZ are real in the symmetry-unbroken phase, thus enabling the realization of PA or CPA. The full absorption of an input signal, which is often regarded as an undesired effect, can actually improve the performance of operations, such as detection28,29, quantum state transfer and wavelength conversion. In this way, it may offer new opportunities in the development of optical modulators and switches or quantum sensing.

PA or CPA have been observed in optics using microdisks7, slab waveguides30, optical metamaterials31,32, slabs of conductive materials33, or nanostructured semiconductors6. These systems are constituted by passive components and are fed with one or more input signals. The necessary balance between incoming energy and losses is achieved by adjusting the coupling of the system with its feeding lines and/or the losses of the system. PA can also be realized at microwave frequencies using metamaterials34, conducting films35,36, microwave resonators1,2, or dielectrics37. Similar results have been obtained using microwave cavities coupled to ferromagnetic Yttrium-Iron-Garnet (YIG) spheres4,12,38. Here, the system typically has a two-port configuration, which is used to send microwave excitations and, due to the low damping of YIG, its magnetic coupling with the cavity is adjusted to obtain the necessary critical coupling condition.

Diluted spin centers in a non-magnetic matrix offer the possibility to play with multiple degrees of freedom, for instance, by choosing the crystal field or exploiting nuclear spins. These features can be tailored at the synthetic level in molecular spin systems39–42, leading to some extensive control of their coherence times and over genuine quantum features43–45, which, in turn, translates into large potential for applications in quantum computing46–49 and in quantum sensing50–52. Molecular spins and, more in general, spin qubits in solids inevitably operate in a dissipative environment and require a proper tailoring of the photon-spin qubit transduction. Thus, having a deep control of the key experimental parameters and a full description of the theoretical problem is highly desirable to design quantum computing or quantum sensing experiments. For instance, finding optimal conditions for qubit encoding typically requires tailoring the environmental bath/s53 or being insensitive to it and/or its fluctuations, as for atomic clock transitions54. Conversely, running molecular spins as quantum sensors requires finding conditions for which small perturbations of the external parameters lead to large signal variations, e.g., those obtained at singularities in the scattering parameters. Finally, since molecular spins can be embedded into hybrid spin-superconducting circuits55–58, they offer an ideal testbed to investigate open non-Hermitian systems at microwave frequency.

In this work, we theoretically and experimentally investigate an open passive system that is not PT-symmetric. Our geometry consists of a planar superconducting microwave resonator in the purely quantum regime (milliKelvin temperature and average single microwave photon number) overcoupled to its input-output line and magnetically coupled to molecular spin centers, as depicted in Fig. 1. We first consider a diluted α, γ-bisdiphenylene-β-phenylallyl (BDPA, for short) organic radical spin as a prototypical Two-level-System (TLS) and, then, a tetraphenylporphyrinatooxovandium(IV) complex (VOTPP, for short), with large (I=7/2) nuclear spin and anisotropic hyperfine tensor to enlarge the number of subsystems investigated (Fig. 1d). Tuning the position of the sample on the resonator allows us to change the coupling rate with the spins, gμ, thus crossing from the strong to the weak coupling regime.

Results
Theoretical modeling
We begin by presenting the model developed for our system, which is sketched in Fig. 1a, b. The system consists of a lumped element resonator coupled to an antenna acting as an input-output port with rate γr. The μ-th spin ensemble is coupled to the resonator through its collective coupling strength gμ, and has energy ωsμ=gμμBB0/ℏ (being gμμ the Landé factor and B0 the applied static magnetic field). BDPA can be effectively described using a single resonance frequency (μ=1), the VOTPP sample comprises multiple molecular spin levels due to hyperfine splitting (μ=1, …, 8), in Fig. 1d. These multiple-spin ensembles can be effectively described using the Holstein-Primakoff mapping with N non-interacting bosons. In the low-excitation regime, such mapping can be linearized, leading to the harmonic Hamiltonian

H^S = ℏω0a^ya^+ℏ∑Nμ=1ωsμb^μyb^μ+ℏ∑Nμ=1gμ(a^yb^μ+b^μya^), (1)

where a^ and b^μ denote the bosonic annihilation operators of the resonator and the μ-th spin ensemble, respectively. The coupling strengths gμ observed are sufficiently small with respect to ω0 to justify the use of the Rotating Wave Approximation59. Equation (1) can describe either BDPA for N=1 or VOTPP for N=8. The coupling with the external environment, which consists of the decay channels of the resonator (radiative and non-radiative) and of the spin ensembles (Fig. 1b), can be conveniently described by Heisenberg–Langevin equations, which take into account also the coherent feeding field through the antenna (see Supplementary Information). The complex reflection scattering parameter, S11(ω), can be shown to have poles (i.e., resonance states) corresponding to the complex eigenfrequencies, Ω, of the non-Hermitian Hamiltonian H^ = ℏα^y(A−iΓ/2)α^, where α^T=(a^, b^), A is the Hopfield matrix of Hamiltonian in Eq. (1) and Γ is the corresponding decay matrix. Moreover, we observe that the zeros of S11(ω) exhibit the analogous structure as the denominator, with the only difference being the reversal of the sign of the decay rate associated with the input-output port, γr (see Section “Methods”)8,26,27,60. Therefore, these reflection zeros can be interpreted as the complex eigenfrequencies, Ω~j, of the effective non-Hermitian Hamiltonian

H^RZ = ℏα^y(A−iΓ~ /2)α^, (2)

where the decay matrix Γ~ differs from Γ only for the inversion of the sign of the input-output port decay rate (notice that −γr corresponds to the feeding rate). Hence, the reflection scattering parameter reads:

S11(ω) = det(H^RZ−ω^I) / det(H^res−ω^I) = ∏Nj=1 (ω−Ω~j) / ∏Nj=1 (ω−Ωj), (3)

where I^ is the identity operator. Here we notice that, in Eq. (3), we have implicitly performed an analytic continuation of the reflection scattering parameter S11(ω) into the complex ω-plane. However, in the context of our experimental implementation, ω must be real-valued. Equation (3) explicitly highlights the connection between the PA condition (S11(ω)=0) and the eigenvalues of the effective non-Hermitian Hamiltonian H^RZ in Eq. (2). Indeed, when the imaginary part of one of the complex eigenfrequencies of Eq. (2) vanishes (ℑ(Ω~j)=0 for some j), it is always possible to nullify the numerator of S11 by appropriately tuning the frequency of the input probing field ω. Reflection zeros on the real axis are also referred to as Reflectionless Scattering Modes (RSM)8,27. To further elucidate this point, we first perform a rotation on the bare bosonic operators to transform them in the polariton basis (normal modes). Specifically, we introduce a suitable Hopfield transformation U, defined as P^=Uα^, such that the Hamiltonian in Eq. (1) takes the diagonal form H^S = ℏ∑Nj=1 ωjP^j yP^j. Subsequently, the effective non-Hermitian Hamiltonian describing the open system dynamics of the reflection zeros, in the strong coupling regime, is given by

H^RZ = ∑Nj=1 (ωj−iγ~j/2) P^jyP^j, (4)

where the effective (or dressed) loss rates γ~j are defined by the diagonal elements of UΓ~Uy (see Section “Methods”). Equation (4) provides a clear physical interpretation of the reflection zeros in terms of the polariton modes. In particular, the PA condition turns out to be γ~j=0 for some j. We remark that this effective Hamiltonian is derived under the hypothesis of PA and is valid in the strong coupling regime, as, in this regime, the mixed terms in the decay channels can be safely neglected due to the different resonance frequencies of the polariton peaks.

The single ensemble case
We begin by analyzing the BDPA organic radical. Figure 2a displays experimental reflection spectra maps as a function of the static magnetic field. Figure 2b illustrates the corresponding theoretical fit performed using Eq. (3) with N=1, showing excellent agreement. An anticrossing, which is typical of the strong spin–photon coupling regime, is clearly visible when ωs crosses ω0. Fits gives g/2π≈20.7 MHz and γs/2π=5 MHz, further corroborating the strong coupling regime (see Supplementary Table 1 for full fit parameters). We observe the presence of two dips (Fig. 2c), approaching zero reflection, corresponding to PA. Remarkably, these occur before (0.3443 T) and after (0.3459 T) the resonant magnetic field value, corresponding to symmetric resonator-spin detuning Δ=ωs−ω0 ≈±8 ⋅ 10−4 T. This clearly differs from what is typically observed in systems where the corresponding H^RZ is PT-symmetric, in which the dips occur at resonance4,7. The positions and the absolute values of the two reflection zeros are also predicted by our model (Fig. 2b). As a first important result, this demonstrates (both experimentally and theoretically) that PA can be realized on molecular spin centers by tuning their resonance frequency. More specifically, in the strong coupling regime, the imaginary part, ℑ(Ω~j), is approximately equal to the effective loss rates γ~j/2, as given by Eq. (4), which, in turn, can be expressed as a combination of the single-channel loss rates:

γ~j = (γr+γnr)|Uj1|2+γs|Uj2|2 = γ~cj+γ~sj, (5)

Here Ujk are the Hopfield coefficients defined by U, which determines the degree of photon (|Uj1|2) and spin (|Uj2|2) hybridization of the polaritons, and which can be tuned through the spin-resonator detuning. γ~cj=(γr+γnr)|Uj1|2 and γ~sj=γs|Uj2|2 represent the dressed cavity feeding rate and the dressed spin loss rate, respectively. In our experiments, we vary the detuning, Δ, through the externally applied magnetic field. Therefore, PA (defined by γ~j=0) occurs when the balance between dressed feeding and loss rates is achieved, analogously to the balance condition in PT-symmetric systems for the bare rates, even if the H^RZ Hamiltonian does not display such symmetry (see Section “Discussion” for a more detailed comparison with PT-symmetric systems). Equation (5) can be satisfied for either the lower or upper polariton, leading to the emergence of a Hermitian subspace within the Hilbert space of H^RZ. This subspace H(j) is effectively described by the Hamiltonian H^RZ(j) = ωjP^jyP^j, where j=1 or j=2 for the upper or lower polariton, respectively. Furthermore, it can be easily shown that if a Hermitian subspace exists for one of the polariton modes at a given detuning Δ0, a second one will exist at −Δ0 for the other polariton mode. This property holds even in the case of multiple spin resonances, as it will be shown for the VOTPP sample. Figure 2d shows the imaginary part of the complex eigenvalues of Eq. (2), ℑ(Ω~j), along with the corresponding contributions to the jth polariton mode of the dressed cavity feeding and spin loss rates. The balance between the two contributions (γ~cj = γ~sj), corresponding to the vanishing of ℑ(Ω~j), is observed for either of the polariton modes at symmetric detunings, as discussed above.

Effects of coupling strength
We now investigate theoretically the effect of the coupling strength on PA. Figure 3 shows reflection maps simulated with Eq. (8) by using the relaxation rates obtained from the fits of Fig. 2, along with different coupling strengths. Notably, as the coupling decreases, the dips progressively merge towards the resonant field value. We identify a threshold value, gth = (γr−γnr+γs)/4 (green line in Fig. 3c), marking the transition from the strong to the weak coupling regime. An intermediate region emerges, where no crossing of the imaginary parts occurs anymore but PA still persists (yellow region), until the coupling strength reaches a minimum value gmin = (γr−γnr)γs/2, at which ℑ(Ω~j) exhibits a double root at zero detuning (blue line) and the reflection dips coalesce into a single one. For couplings below gmin, the imaginary part ℑ(Ω~j) cannot cross the zero, preventing the observation of PA (red line). The simulations in Fig. 3 suggest that our model in Section “Theoretical modeling“, although developed under the assumption of a strong coupling regime, also holds in the weak coupling regime.

The multiple ensemble case
We now experimentally investigate an analogous phenomenology using the VOTPP sample, which, in contrast to BDPA, exhibits multiple resonance frequencies, ωsμ. As mentioned in the introduction, the overall coupling regime for the whole group of transitions is primarily determined by the position of the sample with respect to the resonator (see Fig. 1). In addition, the values of the hyperfine tensor of VOTPP (see Section “Methods”) give different thermal population of hyperfine levels at milliKelvin temperature, thus allowing us to span a large set of g values within a unique field scan.

Figure 4a, d shows experimental reflection maps, |S11|, measured as a function of the static magnetic field, for positions #A and #D of VOTPP (see Fig. 1). In position #A, the observed avoided level crossings clearly indicate that the high cooperativity regime is achieved on each line. This is also supported by the fitted values gμ/2π=13–16 MHz and γsμ/2π=7–13 MHz (depending on the line, see Supplementary Table 5). A pair of dips in reflection, approaching near-zero values and at non-zero symmetric detunings from resonance, is clearly visible for each line, indicating the presence of PA. Conversely, data for position #D show the weak coupling regime for all resonances and significantly lower coupling strengths (see Supplementary Table 8). Here, due to the different thermal population of each line, PA occurs only for a subset of the eight resonant lines (from left to right, the first four), and the coalescence of those dips is observed for increasing magnetic field values. The theoretical reflection maps in Fig. 4b, e, simulated through Eq. (3), are in excellent agreement with the data. For position #A, the PA points are perfectly predicted by the zeros of the imaginary parts of the corresponding effective eigenfrequencies (each line crosses the zero, see Fig. 4c), which are calculated by using the fit parameters obtained from Fig. 4a. Conversely, for position #D, the imaginary parts of the complex eigenvalues cross zero only for the first four lines. This can be explained by the minimal coupling gmin discussed in Section “The Single Ensemble Case”, since moving across the different resonances results in the coupling strengths g becoming smaller than gmin. These latter results further show that Eq. (3) holds for any coupling regime, demonstrating that the PA condition can be achieved even in the weak coupling regime, provided that a minimum coupling rate value is overcome for a given set of relaxation rates.

Methods
Derivation of the reflection scattering parameter
To derive the reflection spectrum, we first express the output fields in terms of the input fields using the quantum Langevin equations. This leads to the general relation (see Supplementary Information)

F^out(ω) = [1 − Γ(iωI−A)−1 Γ(iωI−A)−1] F^in(ω), (7)

where F^in(out)(ω) is the input (output) Langevin force vector. These vectors contain the input (output) operators associated to the different channels, i.e., the antenna (both the radiative and non-radiative channels, a^r,in(out)(ω) and a^nr,in(out)(ω), respectively) and the spin ensembles (b^μ,in(out)(ω)). Since we focus on coherent reflection spectra where the signal enters and is sampled through the radiative port of the antenna, we take a^r,in(ω) ≠ 0, while a^nr,in(ω) = 0, b^μ,in(ω) = 0, etc. Therefore, taking the input–output relation for the radiative port, a^r,out(ω) = a^r,in(ω) − √γr a^(ω), we derive the reflection coefficient

S11(ω) = a^r,out(ω) / a^r,in(ω) = 1 − γr / [γr+γnr−i(ω−ω0) + ∑Nμ=1 γsμ / 2 − i(gμ2 / (ω−ωsμ))]

This equation serves as a model for fitting experimental reflectivity spectra. Notably, as pointed out in Section “Theoretical modeling” and in refs. 8,27,60, the resonances of the reflection scattering coefficient S11(ω) corresponds to the zeros Ω of the characteristic polynomial of the non-Hermitian Hamiltonian H^RZ = ℏα^y(A−iΓ/2)α^. The numerator has the same structure as the denominator, differing only in the structure of Γ but with the sign of γr reversed, the numerator can be interpreted as the characteristic polynomial of the effective non-Hermitian Hamiltonian H^RZ = ℏα^y(A−iΓ~ /2)α^, whose eigenfrequencies are Ω~j.

Derivation of the effective Hamiltonian in the strong coupling regime
In this section, we present the derivation for a single spin ensemble of the effective non-Hermitian Hamiltonian in the strong coupling regime. Starting from the quantum Langevin equations for the photon and the collective spin operators presented in the Supplementary Information, we apply the unitary transformation U defined in Section “Theoretical modeling” to rotate in the polariton basis, i.e., P^=Uα^. Therefore, we obtain the following equations of motion for the polaritonic bosonic operators

∂P^j = −iωjP^j − 1/2 ∫ dωUj1 ∑k=r,nr γk c^k(ω) − 1/2 ∫ dωUj2 γs d^ (ω), (9)

where c^k and d^ are the baths’ bosonic operators, which are used to define the input (output) operators, a^k,in(out)(ω) and b^μ,in(out)(ω), respectively (see Supplementary Information). ωj and Ujk are the eigenfrequencies and Hopfield coefficients, respectively, as defined in the main text.

Following the approach outlined in refs. 67,68, we substitute the formal solutions of the equations of motion for c^k(ω) and d^ into Eq. (9). By taking the expectation value of the resulting expression, we obtain

∂p j(t) = −iωj p j(t) − γj/2 p j(t) − Uj1U*m1 (γr+γnr)/2 p m(t) − Uj2U*m2 γs/2 p m(t) + √γr Uj1 a^in(t), (10)

where m is the complementary index of j, i.e., m=2 when j=1 and vice versa. In addition, we introduced the definitions pj(t)=⟨P^j(t)⟩, γj=(γr+γnr)|Uj1|2+γs|Uj2|2 and a^in(t)=a^r,in(t), which represents the coherent input signal. Specifically, in the derivation of Eq. (10), we took into account that the system has a coherent feeding only through the antenna radiative channel, thus implying a^nr,in=⟨b^μi⟩=0. Eq. (10) can be rewritten in the frequency domain as

−iωp j(ω) = −iωj p j(ω) − γj/2 p j(ω) − Uj1U*m1 (γr+γnr)/2 p m(ω) − Uj2U*m2 γs/2 p m(ω) + √γr Uj1 a^in(ω): (11)

By imposing the PA condition, a^(ω)=0, we obtain from the input-output relations the explicit expression a^out(ω)=√γr (Uj1 p j(ω) + Um1 p m(ω)) − a^in(ω). Hence, the effective dynamics in the time domain can be written as

∂p j(t) = −iωj p j(t) − γ~j/2 p j(t), (12)

where γ~j = (γr+γnr)|Uj1|2+γs|Uj2|2, as in Eq. (5). These equations of motions can be derived by the effective non-Hermitian Hamiltonian

H^RZ = ∑j=1,2 (ωj − iγ~j/2) P^jyP^j, (13)

which coincides with Eq. (4). This procedure can be easily generalized in the case of multiple spin resonances, as in the strong coupling regime, each photon-spin anti-crossing is separated from the others.

Experimental setup
We use a planar superconducting lumped-element LC microwave resonator made of superconducting Niobium films (thickness, 50 nm) on sapphire substrate (thickness, 420 μm), as shown in Fig. 1. The resonator has a small inductive loop to enhance the generated microwave magnetic field, which is coupled to a large interdigitated capacitance. The resonator works in reflection mode, at a fundamental frequency ω0/2π ≈ 9.9 GHz, and it is coupled to the input-output line through an antenna, whose position can be adjusted at room temperature, before cooling the sample. The chip carrying the resonator is loaded into a cylindrical copper waveguide sample holder hosting the antenna in his bottom69,70. Preliminary characterization of the empty resonator is reported in Supplementary Information.

All experiments are carried out inside a QinuSion ludi dilution refrigerator at base temperature (20 mK) and with an average single microwave photon in the resonator69,70. The input signal is attenuated by 60 dB inside the cryostat before reaching the sample box with the antenna, while the output line hosts a cryogenic High Electron Mobility Transition (HEMT) amplifier (Low Noise Factory, 37 dB gain). The signal is further amplified at room temperature before acquisition. The complex reflection scattering parameter, S11, is measured with a Vector Network Analyzer (VNA) for different values of the static magnetic field applied, obtaining the 2D maps shown in Figs. 2 and 4. The input power at the position of the antenna is between −130 and −120 dBm, corresponding to an average single microwave photon in the resonator (see Supplementary Information). The cylindrical box and the resonator are aligned into the magnet in order to have the static magnetic field along the plane of the chip, i.e., in a in-plane configuration (Fig. 1a). All the measured reflection scattering parameters have been normalized over the average value of the signal baseline measured off-resonance.

Molecular spin samples
We use two samples of the molecular compounds shown in Fig. 1c. The first one is a diluted solid dispersion of α, γ-bisdiphenylene-β-phenylallyl (BDPA, for short) organic radical into a polystyrene matrix, with a spin density of ≈1 ⋅ 1015 spin/mm3. The sample was prepared as described in ref. 71 and then cut into a rectangular shape with a size ≈1.5×1 mm2. The sample is placed on the resonator as shown in Fig. 1a. Each molecule has electronic spin S=1/2, which is due to its single unpaired electron71,72. Pure BDPA samples typically have antiferromagnetic exchange interaction occuring below 10 K72–74, with a Curie-Weiss temperature between TC =−8 K and TC =−1 K, strongly dependent by the spin concentration and by the solvent or matrix used72–74. Although there are no reports for BDPA diluted in Polystyrene, due to the relatively high concentration, we can expect that a residual antiferromagnetic interaction still occurs among molecules. Therefore, our BDPA sample constitutes a prototypical TLS collection with essentially negligible magnetic anisotropy and no hyperfine splitting, which gives a single transition frequency71,72.

The other sample is a single crystal of VOTPP with 2% concentration in its isostructural diamagnetic analog, TiO(TPP). Each molecule has an electronic spin S = 1/2 ground state and an additional hyperfine splitting due to the interaction with the I=7/2 nuclear spin of the 51V ion (natural abundance: 99.75%). This results in a multiplet with eight {Sz, Iz} electronuclear transitions, which can be exploited as eight independent spin ensembles. The magnetic properties and the electron spin resonance spectroscopy of this molecule have been previously reported in ref. 40. In particular, the hyperfine tensor shows uniaxial anisotropy with parallel component A∥=477 MHz=23 mK and perpendicular component A⊥=68 MHz=8 mK40, respectively. These relatively large values, combined with the temperature of the experiments and the frequency of the resonator, give different thermal populations on the eight lines, thus decreasing the coupling rate gμ for increasing line number i (see Supplementary Information). The position of the VOTPP crystal (from #A to #D in Fig. 1a) is adjusted before each cooldown. Due to the experimental configuration used and the orientation of the molecules inside the unit cell, all molecules experience the same static magnetic field, which lies on the TPP plane and it is nearly perpendicular to the direction of the V=O double bound.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This work investigated the perfect absorption phenomenon based on a non-Hermitian framework in a system consisting of spin-ensembles coupled to a microwave resonator. The manuscript presents the formalism of non-Hermitian Hamiltonian that describes the various sources of dissipation, and transformed it to a diagonal form under the polariton eigenbasis. The experiments demonstrated perfect absorption in both weak and strong coupling regimes, i.e., zero reflectance at the eigenfrequencies, for the cases of one and multiple spin ensembles. The resonance frequencies of spins can be controlled through a magnetic field while the resonator-spin coupling can be adjusted by position tuning. The conditions of perfect absorption were verified through simulation of dressed cavity feeding rates and dressed spin loss rates versus the applied magnetic field. Potential application of a microwave photon switch is discussed.\n\nThe perfect absorption has been demonstrated in a plethora of platforms. As mentioned in the introduction it has been explored in PT-symmetric systems, and here the manuscript demonstrated a PA in absence of PT symmetry. However, there is no clear connection between those two concepts, since the PA is a generalized critical coupling phenomenon that can be found by balancing the coupling loss with intrinsic loss.\n\nWhile the theoretical framework built in this work clarifies how to find perfect absorption in a spin-resonator system, the demonstration is not essentially distinct from other previous works in other systems. The link of this work to PT symmetry is also not clear. Thus I do not recommend its publication in this form.\n\nDetailed comments:\n1) The demonstration of PA in the absence of PT-symmetry is natural, since PA is not related to PT-symmetry. The PT symmetry requires that whenever the zeros become real the poles are also real. Without PT symmetry, one can tailor the parameters to move zeros to the real axis which realizes PA, leaving the poles in the complex plane. Thus I do not see a clear necessity to highlight the connection between PA and PT symmetry.\n2) The text before Eq. (4) assumes that the effective Hamiltonian is diagonalized under a polariton basis. However, since it involves a generic non-Hermitian description, we cannot neglect the possibility that the system reaches an exceptional point with degenerate eigenstates. In this situation, the system Hamiltonian given should take Jordan normal form instead. Then what would be the form for Eq. (4)?\n3) Explain the PT symmetry condition mentioned in page 7, especially how this system satisfies PT symmetry.\n4) What is the physical meaning of having a Hermitian subspace, and how is it related to previous works, especially the PA in the PT symmetry case, as authors claim it to be a special case of this work?\n5) Regarding the application in microwave switch, though the modulation depth can reach 50dB, the on-state only leads to <0.2 transmission indicating a large loss. The limited bandwidth is another issue. These factors could hinder its practical use.", "reviewer_2": "This manuscript presents perfect absorption (PA) phenomena in passive open quantum systems based on molecular spins coupled to planar superconducting microwave resonators. The authors demonstrate that PA can be realized and understood in terms of Hermitian subspaces in the absence of parity-time (PT) symmetry. The theoretical framework is verified using two types of molecular spin systems: BDPA (a prototypical two-level system) and VOTPP (with multiple hyperfine transitions). The results seem to be new in the subfield of molecular spintronics, but may not be new in more general field such as non- Hermitian physics. Therefore, I cannot recommend the publication in the present form unless the authors address my serious concerns as in the following:\n(1) The theoretical framework including two non-hermitian hamiltonians (H(NH) and H(eff)) which correspond respectively to poles and zeros of S11 is not original, as earlier work (Nat.Phys. 20, 1904 (2024)) has been adopted. In fact, Eq.(3) in this manuscript is same as Eq.(1) in Nat.Phys. 20, 1904 (2024) where the physical meaning of two effective non-Hermitian Hamiltonians are explained explicitly. The authors should cite earlier work and specify the difference if any.\n(2) The authors observed PA and hence emphasize the importance of Hermitian subspaces in the absence of PT symmetry. However, this is rather obvious from previous works such as PRL 125, 147202 (2020) and Sci.Adv.9, eadg4730 (2023) using different teminology but essentially same underlying physics. And in the one-port configuration, this is sometimes termed as reflectionless scattering mode (RSM) which is a special case for complex zeros of S11. See for example Phys. Rev. A 102, 063511 (2020).\n(3) The authors claim in the discussion part that \"previous ffndings in PT-symmetric systems (e.g. Ref. [18]) represent a particular case of the more general framework introduced here\". I would say that \"the more general framework\" is already well-know in those previous works and they aimed at achieving the more strict and challenging PT-symmetric condition. I would recommend authors to rephrase so that readers will not be misled. Accordingly, I would encourage the authors to discuss whether the more strict PT-symmetry can be acchieved in present work or not, and if not, explain the reason.\n\nIn addition, the format of present manuscript seems not to satisfy the requirement of Nat.Comm.\n\nIn summary, the theorectical framework of present manuscript seems to be not new and I think that the main advance in this paper lies in the experimental demonstration of these non-hermitian physics in the cryogenci platform with molecular spins which is indeed impressive."}, "author_response": "We thank the Editor for sending our manuscript out for review. We also thank Reviewers #1 and #2 for their careful evaluation of our manuscript and for their constructive and helpful comments.\n\nIn the following we first give a brief introduction to our reply and we then provide a point-by-point reply (black text) to all the comments raised by the Reviewers (blue text). A summary of all changes made is given at the end of the present rebuttal. Changes are marked with red color in the revised version of the manuscript.\n\n*** INTRODUCTION TO OUR REPLY ***\nThe main concerns raised by both Reviewers are about the novelty of our proposed theoretical framework and the connections between the concepts of Perfect Absorption (PA), non-Hermitian systems and PT-symmetry. In the revised version we have done an extensive and careful effort in checking the suggested literature, in better clarifying their novelty and in the connections among the three above-mentioned concepts. In our opinion, this has brought to significant revisions and to a significant improvement of our manuscript.\n\nWe agree that PA can be studied in open quantum systems without the need to invoke non-Hermitian physics. Here we notice that in several works (see e.g., [1-3] and PRL 125, 147202 (2020), Sci.Adv.9, eadg4730 (2023) suggested by Reviewer #2) scattering parameters and PA has been studied in the context of open systems with non-Hermitian physics. This motivates our interest in linking PA to non-Hermitian physics. In particular, we remark that we show that: i) the detuning can give the proper balance between radiative and non-radiative coupling rates for the observation of PA and, ii) this balance corresponds to an effective non-Hermitian Hamiltonian with an Hermitian subspace.\n\nWe agree with the Reviewers that our formalism based on effective Hamiltonians is not a new tool introduced by us. However, as detailed in our reply, we believe that there are several important differences between our findings and previous works. These are related to the experimental realization and the experimental conditions used (e.g, YIG spheres in 3D cavities, temperature, absence of hyperfine interaction), to the observed coupling regime (e.g, presence or not of polaritons) and in the type of interaction coupling spins and resonator (e.g, dissipative coupling).\n\nNext, we remark that our system does not satisfy the conditions required for PT-symmetry. To better highlight the connection between our results (in particular, the proposed Hermitian subspaces) and PT-symmetry we have prepared the plot in Fig. R1 of the present reply, which has also been added in the manuscript (new Figure 5). This additional simulation better clarifies how Hermitian subspaces can predict the occurence of PT-symmetry conditions for a special choice of the parameters of our system.\n\nFinally, we would like to further stress that we demonstrate PA on cryogenic platform based on microwave resonator coupled to molecular spins at milliKelvin temperature and in the single-photon regime. To the best of our knowledge, this not only corresponds to the first experimental demonstration achieved by using molecular spin centers but also on ensembles of paramagnetic centers with large set of spin degrees of freedom (electronic and nuclear spins). In the revised version we have further improved the discussion concerning the link between our experimental results and applications.\n\nAll considerations above are discussed in detail in the following.\n\n*** REPLY TO REVIEWER #1 ***\n[...]"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors clarified the link between perfect absorption and PT symmetry under the non-Hermitian framework, and illustrated the Hermitian subspace by Fig. 5 with associated explanations. It is worth noting that PT symmetry discussed here is defined within the effective Hamiltonian that differs from the conventional one by the sign of the coupling term associated with the input port. The authors added the definition of the two Hamiltonians in the first paragraph in the introduction. To improve clarity further, it would be necessary to highlight the definition of PT symmetry in that paragraph since in most of previous works in non-Hermitian physics, PT symmetry was defined through the effective Hamiltonian of the system and described the behaviors of poles.\n\nThough the theoretical framework is inherited from previous works, the authors claim that this work is the first example of perfect absorption demonstration in a quantum system made of spin centers coupled with microwave resonators. The demonstration in such new platforms could be interesting. Further, some potential applications in microwave photon control and detection have been discussed. Nevertheless, in the introduction the authors mentioned quantum information processing and sensing, but didn’t explain how these applications are related to or motivate the engineering of PA in such a system. I would suggest further clarifying the motivation of this study in the introduction section.\n\nIn overall, the authors well addressed my questions related to physics and other issues along with appropriate revision. I would therefore recommend its publication, after making the necessary revision suggested above.", "reviewer_2": "The authors addressed most of my concerns and made revision to the manuscript. My additional suggestion is that it's better to specify which Hamiltonian (Hres or Hrz) authors refer to when they discuss the terms like PA, PT-symmetry, Hermitian subspace.\nPA is only related to Hrz and has nothing to do with Hres; but PT-symmetry is often referring to Hres in most of previous works and occassionally to Hrz (e.g., Ref.[27]). As a purely passive open system in present work, Hres must be PT-broken and hence it is trivial to say \"in the absence of PT-symmetry for a passive system. Also, for the same reason, Hres has no any Hermitian subspace at all. Therefore, I recommend authors specify clearly that the term of \"Hermitian subspace\" applies only to Hrz, not generally to the system (which may mislead most of readers to mistakenly think Hres has any Hermitian subspace)."}, "author_response": "We thank the Editor for overseeing the review process. We thank again Reviewers #1 and #2 for their additional comments and valuable suggestions.\n\nIn the following we provide a point-by-point reply (black text) to all the suggestions and comments raised by the Reviewers (blue text). A summary of all changes made is given at the end of the present document. Changes are marked with red color in the revised manuscript (pdf version with changes marked).\n\n*** REPLY TO REVIEWER #1 ***\n[...]"}]

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