Long-lifetime coherence in a quantum emitter induced by a metasurface

Brian Stout; David Wilkowski; Emmanuel Lassalle; Patrice Genevet; Philippe Lalanne; Syed Aljunid; Thomas Durt

REVIEW 2 major objections 6 minor 59 references

Long-lifetime coherence in a quantum emitter induced by a metasurface

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Spontaneous emission alone can create a long-lived quantum coherence

desk verdict A genuinely new Lambda-scheme coherence effect from an anisotropic vacuum, with a sound master-equation core but an unquantified degeneracy requirement and a rough device-efficiency estimate. read the letter →

arxiv 1909.02409 v2 pith:YUJD6FNI submitted 2019-09-05 quant-ph

Emmanuel Lassalle , Philippe Lalanne , Syed Aljunid , Patrice Genevet , Brian Stout , Thomas Durt , David Wilkowski This is my paper · ORCID

classification quant-ph

keywords spontaneousemissionanisotropicquantumvacuumLambdatransitionground-statecoherencemetasurfaceemitterGreentensorNVcenter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper predicts that a quantum emitter with a Λ-shaped level structure—one excited state and two nearly degenerate ground states—can settle into a coherent superposition of its two ground states purely by emitting a photon into an anisotropic vacuum. Ground-state coherence normally needs an external coherent laser; here spontaneous emission itself is the source, and the resulting coherence lives on the slow ground-state timescale rather than on the fast excited-state decay. Starting from a Born–Markov master equation, the paper derives the steady-state value $\rho_{12}(\infty)=\kappa_{12}/(\gamma_1+\gamma_2)$, factorizes it into an emitter term and a vacuum-anisotropy term, and shows the maximum is $\pm 1/2$, a pure ground-state superposition. It then designs two flat metasurfaces that make the vacuum anisotropic at distances much larger than the wavelength, and estimates that a realistic device with numerical aperture $NA=0.7$ would produce a coherence of about $0.05$, small but detectable with NV-center platforms.

What carries the argument

The load-bearing identity is the steady-state factorization $\rho_{12}(\infty)=R\,A$ with $R=d_{01}d_{02}/(d_{01}^2+d_{02}^2)\le 1/2$ and $A=\mathrm{Im}(G_{xx}-G_{yy})/\mathrm{Im}(G_{xx}+G_{yy})$ when $G_{xy}=0$; equivalently $A=\mathrm{Im}(G_{+-})/\mathrm{Im}(G_{++})$ in the circular basis. Here $G_{ij}$ are components of the Green tensor at the emitter position, related to the vacuum field correlations by the fluctuation–dissipation theorem. The coefficient $\kappa_{12}=\hbar^{-2}d_{01}^*\cdot\hat{C}\cdot d_{02}$ is the central object that carries the coherence. To make $A$ nonzero at remote distances, the paper designs a resonant-phase-delay metasurface acting as a polarization-selective spherical mirror and a geometric-phase (Pancharatnam–Berry) metasurface that flips circular polarization on reflection.

What would settle it

Prepare a Λ-emitter such as an NV center in its excited state with the two ground levels degenerate during the emission, in an environment where $\mathrm{Im}(G_{xx})\neq\mathrm{Im}(G_{yy})$ at the emitter, then wait several excited-state lifetimes and read $\rho_{12}$ via a phase-sensitive microwave transfer. If the measured off-diagonal element is zero while the two decay rates are measurably different, the central prediction fails; repeating with the degeneracy lifted during emission and seeing the coherence vanish would confirm the frequency-degeneracy condition.

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Extended reading notes

Core claim

The paper solves the Born–Markov master equation for a Λ-emitter (one excited state $|0\rangle$, two nearly degenerate ground states $|1\rangle,|2\rangle$ with orthogonal dipole transitions) initially prepared in $|0\rangle$. The excited population decays exponentially, and a ground-state coherence grows as $\rho_{12}(t)=\frac{\kappa_{12}}{\gamma_1+\gamma_2}\left[1-e^{-(\gamma_1+\gamma_2)t}\right]$. The coefficient $\kappa_{12}$ is the cross-coupling between the two decay channels through the vacuum field-correlation tensor, so it is zero in the isotropic vacuum and nonzero only when the vacuum is anisotropic. In the ideal anisotropic limit the atom ends in the pure superposition $(|1\rangle\pm|2\rangle)/\sqrt{2}$; in isotropic vacuum it ends in the mixture $\frac{1}{2}I$. The paper interprets this as the environment acting as a polarization filter that erases the atom–photon entanglement created during emission. For the realistic metasurface, the steady coherence is estimated as $\rho_{12}(\infty)\simeq \frac{1}{2}(\gamma_x-\gamma_y)/(\gamma_x+\gamma_y)\simeq 0.05$ at $NA=0.7$; the estimate uses the analytic spherical-mirror decay formula with numerically computed reflectance values, and the paper notes that a fully numerical calculation including all metasurface details was not performed.

Load-bearing premise

The two decay transitions must be nearly degenerate in frequency, because the cross-coupling term that builds the ground-state coherence oscillates as $e^{i(\omega_1-\omega_2)t}$ and averages away unless $\omega_1\simeq\omega_2$; a magnetic-field splitting of the ground states therefore has to be applied after the spontaneous-emission step.

Editorial extensions

If this is right

If the prediction holds, ground-state coherence can be prepared without any coherent laser drive: a single spontaneous photon in an engineered vacuum is enough.
Because the coherence lives in ground states, it survives milliseconds in NV centers and seconds in cold atoms, far longer than the sub-nanosecond excited-state decay that creates it.
The two proposed metasurfaces create the anisotropic vacuum at distances $d\simeq 10\lambda_0$, so the emitter and its environment need not be in the near field; a realistic design yields $|\rho_{12}|\simeq 0.05$ at $NA=0.7$.
Detecting this coherence would be a direct, macroscopic test of vacuum anisotropy acting on a quantum emitter, and would underwrite the use of metasurfaces for remote coherent coupling between emitters.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same formula implies that the effect is not tied to metasurfaces: any local environment with unequal imaginary Green-tensor components at the emitter—a planar interface, a nanodisk, a waveguide—should produce the same Λ ground-state coherence, so a near-field tabletop test could precede the harder far-field metasurface experiment.
One metasurface shared by two Λ-emitters could leave each in a correlated ground-state superposition, so remote entanglement generation may only require coherent local operations afterwards; the paper gestures at this but does not work it out.
Since the effect only accumulates while the ground states are degenerate, a detection scheme that uses a magnetic field to split them must switch the field on after emission; this timing constraint makes the effect directly testable with pump-probe sequencing and protects it against stray-field dephasing during the emission stage.
Absorption and discretization losses at large deflection angles are what cap the estimate near 0.05; inverse-designed metasurfaces that maintain high reflectance at oblique incidence could plausibly approach the ideal 1/2, making the effect a practical state-preparation tool.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear [Colleague],

The one thing you should know: this paper predicts a real new effect—long-lived ground-state coherence in a Lambda emitter generated purely by spontaneous emission in an anisotropic vacuum, without any external driving. The central result, ρ12(∞)=κ12/(γ1+γ2), is derived cleanly from a standard Born-Markov master equation and is exact in the degenerate limit ω1=ω2. That part is solid.

What's genuinely new: prior work (Agarwal 2000; Jha et al. 2015, 2018) considered a V-scheme, where the coherence lives in the excited states and dies with the excitation. Here the coherence is between two ground states, so it survives after photon emission. The dressed-state interpretation—where a polarization-filtering environment erases the atom-photon entanglement and leaves the atom in a pure superposition—is illuminating. The two metasurface designs are also concrete and technically credible, with RCWA simulations of phase and reflectance.

The soft spots, in order of importance. First, the near-degeneracy condition is not quantified. In going from Eq. (A19) to Eq. (A22) they drop exp[i(ω1−ω2)t] with a hand-wavy 'ω1≃ω2'. If δ=ω1−ω2 is not much smaller than γ1+γ2, the coherence never accumulates; the suppression is roughly Lorentzian in δ/(γ1+γ2). This matters because their proposed NV-center detection protocol uses a magnetic-field bias to split the |±1⟩ ground states. If the bias is on during the decay, the effect washes out. They need to state the bound on δ and explain how the detection scheme actually works in time (or accept that coherence must be generated before the bias is applied).

Second, the quantitative estimate of ρ12≈0.05 for a realistic metasurface leans on Eq. (23), a spherical-mirror result applied to a metasurface, with reflectances taken from periodic-grating simulations rather than a full device simulation. They are upfront about this compromise, and it's a reasonable first estimate, but there's no convergence or uncertainty analysis, so the number should be treated as an order-of-magnitude guide, not a prediction.

The paper deserves a serious referee. The central physics is likely correct in the ideal degenerate regime, and the metasurface part is a useful engineering contribution. But the unquantified degeneracy requirement is a real gap that should be fixed before publication.

Recommendation: send to peer review; request a quantitative criterion on δ and a clearer discussion of the detection sequence.

Best

Referee Report

2 major / 6 minor

Summary. The paper considers a three-level Lambda-type quantum emitter (one excited state |0>, two ground states |1>, |2>) with orthogonal dipole transitions, in an anisotropic electromagnetic vacuum. Starting from a Born-Markov master equation derived in the Appendix, it predicts that spontaneous emission from the initially excited state generates a steady-state coherence rho12(infinity)=kappa12/(gamma1+gamma2) between the two ground states, without any coherent external drive. For ideal vacuum anisotropy this reaches +/-1/2 and corresponds to a pure ground-state superposition. The paper gives a dressed-state interpretation of the effect as a quantum eraser that removes atom-photon entanglement. It then proposes two flat metasurface designs (resonant phase-delay and geometric-phase) that produce the required anisotropy at remote distances, using RCWA simulations to characterize their reflection efficiencies. Finally, using a spherical-mirror formula for the decay-rate modification, it estimates that a realistic metasurface with NA=0.7 yields |rho12| approximately equal to 0.05, about ten times smaller than the ideal case.

Significance. The central theoretical result is clean and physically interesting: it shows that spontaneous emission, normally a decoherence mechanism, can create long-lived atomic coherence in a degenerate Lambda-system when the vacuum is anisotropic. The master-equation derivation is standard and the steady-state formula Eq. (8) follows directly from the model with no fitted parameters. The metasurface designs are concrete and the numerical parameters are reported in sufficient detail to be reproduced. If the effect is confirmed experimentally, it would provide a new test of QED in an anisotropic vacuum and a potential route to remote coherent coupling. However, the practical significance is currently limited by two unquantified approximations: the near-degeneracy condition on the two ground states and the use of the spherical-mirror formula beyond its original context in the quantitative estimate. Both need to be addressed before the observability claim can be considered robust.

major comments (2)

[Section II A and Appendix, Eq. (A22)] The derivation of the master equation leading to Eq. (8) assumes omega1 approximately equal to omega2, denoted omega0, which removes the phase factors exp[i(omega1-omega2)t] in Eq. (A19). No quantitative condition is given for the allowed splitting delta = omega1 - omega2. For any nonzero delta, the coherence source term oscillates in time; in a Markovian treatment the steady-state coherence is suppressed by a Lorentzian factor (gamma1+gamma2)^2/[delta^2+(gamma1+gamma2)^2]. This is not a minor caveat: the proposed NV-center detection protocol applies a magnetic-field bias that splits the |+-1> ground states, so the spontaneous-emission generation step takes place under non-degenerate conditions if the field is always on. The manuscript must provide (i) an explicit bound on delta, relative to gamma1+gamma2, under which Eq. (8) is valid to a stated accuracy, or a modified calculation that includes delta, and (ii) a protocol showing that the coherence can be generated before the magnetic-field bias is applied, or that a small splitting is experimentally achievable. Without this, the claim that the effect is observable with current NV-center platforms is not supported.
[Section IV, Eq. (23)] The quantitative estimate of the induced coherence uses Eq. (23), a result derived for a two-level atom at the focus of an ideal spherical mirror, and applies it to a finite, discretized metasurface. The authors correctly state that this is 'out of its original context' and that no full numerical computation of the Green tensor was performed, but they do not quantify the resulting error. The actual metasurface differs from the spherical-mirror model in several ways: the phase profile is discretized and sampled by a finite number of super-cells, the local reflectance varies with incident angle, and the numerical aperture is limited by discretization losses. In addition, the assumption gamma_y = gamma_0 neglects the modification of the y-polarized decay rate by the planar-mirror response of the same metasurface at finite distance. Consequently, the quoted value |rho12| approximately equal to 0.05 should be regarded as an order-of-magnitude estimate with unquantified uncertainty. I request either a validation of Eq. (23) against a direct computation of Im(G) for at least the d=10 lambda0 design, or an explicit statement of the expected error bars and a softening of the observability claim in the abstract and conclusion.

minor comments (6)

[Section II B, Eq. (12)] The notation '- i2Im[Gxy]' is ambiguous; it should be written '-2i Im[Gxy]' for clarity.
[Section II C, Eq. (13)] The dressed-state expression is introduced without a derivation; a short justification or a reference to standard methods would help the reader follow the interpretation.
[Section IV, Fig. 9] The plotted quantity is the absolute coherence |rho12|, but for the first design rho12 is negative because gamma_x is less than gamma_y. Please state the sign explicitly in the caption or text.
[Section IV, after Eq. (22)] The statement 'gamma_y is not modified compared to its free space value' is an approximation; for a planar mirror at distance d=10 lambda0 the decay rate oscillates around gamma0 by a small amount. A brief justification, such as quoting the order of the correction, would remove a potential source of confusion.
[Abstract and Introduction] The condition of near-degeneracy of the two ground states is not mentioned in the abstract; given its importance, it should be stated explicitly.
[Section II B, Eq. (9)] Reference [11] is cited for the fluctuation-dissipation theorem; a more standard reference for the FDT in this context would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Lambda-coherence result is derived from an independent Born-Markov master equation with no fitted parameters.

full rationale

The central prediction (Eq. 8) is obtained by solving the Born-Markov master equation derived in the Appendix from the dipole interaction Hamiltonian (Eq. 1). The coefficients gamma_i and kappa_12 are defined independently through the vacuum correlation tensor (Eqs. 4-5) and are not fitted to the coherence being predicted; the steady-state value rho12(infinity)=kappa12/(gamma1+gamma2) is the explicit solution of drho12/dt=kappa12 rho00(t) (Eqs. A36-A46). The later Green-tensor form (Eqs. 10-12) only rewrites these coefficients via the standard fluctuation-dissipation theorem. No input quantity is defined in terms of rho12, and no target coherence value is used to determine any parameter. The metasurface estimate (Section IV) uses numerically computed reflectances (Table II) as inputs to the independent spherical-mirror result of Ref. [3] and explicitly labels the lack of a full simulation as 'an acceptable compromise, in-between a fully numerical treatment and an educated guess'; this is a numerical caveat, not a circular reduction. The near-degeneracy assumption omega1 approx omega2 = omega0 (Eq. A22) is a stated physical simplification that removes oscillatory factors; it is not an input fitted to produce the reported coherence. Existing self-citations (e.g., Ref. [46] for the two-level decay-rate formula, RETICOLO software, review articles) are not load-bearing: the same formulas appear in standard external references, and the coherence result does not rely on them. Therefore no circular step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on standard open-quantum-system approximations (Born-Markov, rotating-wave) and macroscopic QED (fluctuation-dissipation theorem). The key domain assumption is the near-degeneracy of the two ground states; if omega1 and omega2 are split, the coherence generation is suppressed. No free parameters are fitted to the target result, and no new physical entities are introduced.

assumptions (5)

domain assumption Born-Markov approximation and factorization of the system-environment density matrix: rho_T(t) = rho(t) tensor rho_e(0)
Introduced in Section II A and the Appendix (Eq. A8) to derive the master equation; requires weak atom-field coupling and short environment memory.
domain assumption Rotating-wave approximation and electric-dipole approximation for the interaction Hamiltonian
Used in Eq. (1) and in the Appendix (Eq. A1); valid for optical transitions with small Lamb shifts.
standard math Fluctuation-dissipation theorem at zero temperature: C(r,r',omega) = 2 hbar omega^2/(epsilon_0 c^2) Im G(r,r',omega)
Used in Eq. (9) to relate vacuum field correlations to the Green tensor; standard result in macroscopic quantum electrodynamics.
domain assumption Near-degeneracy of the two transition frequencies omega1 approximately equal to omega2
Stated in Section II A and used in Eq. A22 to drop fast oscillations in the cross-coupling term; required for stationary coherence generation.
domain assumption Orthogonality of the two dipole transitions: d01 dot d02 = 0
Defines the Lambda-configuration considered; guarantees zero coherence in isotropic vacuum.

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Pith. "Pith review of Long-lifetime coherence in a quantum emitter induced by a metasurface." pith.science (2026). https://pith.science/paper/YUJD6FNI

@misc{pith2026190902409,
  author       = {Pith},
  title        = {Pith review of: Long-lifetime coherence in a quantum emitter induced by a metasurface},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YUJD6FNI}},
  note         = {Machine review of arXiv:1909.02409}
}

abstract

An anisotropic quantum vacuum (AQV) has been predicted to induce quantum interferences during the spontaneous emission process in an atomic $V$-transition [G. S. Agarwal, Phys. Rev. Lett. 84, 5500 (2000)]. Nevertheless, the finite lifetime of the excited states is expected to strongly limit the observability of this phenomenon. In this paper, we predict that an AQV can induce a long-lifetime coherence in an atomic $\Lambda$-transition from the process of spontaneous emission, which has an additional advantage of removing the need for coherent laser excitation. We also carry out two metasurface designs and compare their respective efficiencies for creating an AQV over remote distances. The detection of this coherence induced by a metasurface, in addition to being yet another vindication of quantum electrodynamics, could pave the way towards the remote distance control of coherent coupling between quantum emitters, which is a key requirement to produce entanglement in quantum technology applications.

Figures

Figures reproduced from arXiv: 1909.02409 by the authors.

**Figure 2.** Unit-cell of a reflect-array metasurface made of: a [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Power reflectance (in purple) and phase-shift [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Illustration of the phase-mapping approach for the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Diffraction performances of a linear-phase gradi [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Geometric phase ϕ as a function of the rotation angle φ of the nanorods in the plane (~x, ~y) (see inset), computed for a 2-D grating (see main text) (green circles). The analytical expression [Eq. (19)] is also plotted (black dashed line). The number of Fourier mode…

**Figure 8.** Figure 8: Cross-polarization (CP) power reflectance, which [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: Relative decay rate modifications γx/γ0 (green circles) and absolute coherence |ρ12| (red triangles) as a function of the numerical aperture of the metasurface NA. For comparison, the relative decay rate [resp. coherence] for an ideal spherical mirror of power reflec…

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Short notations It will be convenient for the following calculations to rewrite ˆHI(t) of Eq. (1) in a more compact form: ˆHI(t) = ˆd†(t)· ˆE(+) v (t) + ˆd(t)· ˆE(−) v (t) (A1) where ˆd(t) and ˆd†(t) are deﬁned by: ˆd(t)≡− ( d∗ 01|1⟩⟨ 0|e−iω1t + d∗ 02|2⟩⟨ 0|e−iω2t) (A2) ˆd†(t)≡− ( d01|0⟩⟨ 1|eiω1t + d02|0⟩⟨ 2|eiω2t) (A3) Note that for clarity we dropped th...

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Short notations It will be convenient for the following calculations to rewrite ˆHI(t) of Eq. (1) in a more compact form: ˆHI(t) = ˆd†(t)· ˆE(+) v (t) + ˆd(t)· ˆE(−) v (t) (A1) where ˆd(t) and ˆd†(t) are deﬁned by: ˆd(t)≡− ( d∗ 01|1⟩⟨ 0|e−iω1t + d∗ 02|2⟩⟨ 0|e−iω2t) (A2) ˆd†(t)≡− ( d01|0⟩⟨ 1|eiω1t + d02|0⟩⟨ 2|eiω2t) (A3) Note that for clarity we dropped th...

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[2] [2]

anisotropy

We use the ⃗ zdirection as the quantization axis. This scheme appears naturally in NV-centers in diamond, using the magnetic sublevels|± 1⟩ as the ground states and |A2⟩ as the excited state [18]. It also can be found in atoms, using Zeeman manifold with|F,m =±1⟩ for the ground states and|F′,m = 0⟩ for the excited state, where m are the magnetic quantum n...

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[3] [3]

=γi/(γ1 +γ2) (fori = 1, 2), in agreement with Eq. (7). In isotropic vacuum, and when the two ground states are equally weighted ( d01 = d02, γ1 = γ2), it is well- known that the atom and the emitted photons are fully entangled [22]: at the end of the decay process, the atom- ﬁeld state is of the form: |ψ(∞)⟩ = 1√ 2 ( |1⟩⊗| σ+⟩ +|2⟩⊗| σ−⟩ ) . (14) The redu...

work page 2018

[4] [4]

Solution of the master equation From the Master Equation, given in Eq. (A22), we obtain the following equations for the atomic populations ρii(t) and atomic coherences ρij(t) with j⁄=i ˙ρii(t) =γiρ00(t) for i = 1, 2 (A36) ˙ρ00(t) =−(γ1 +γ2)ρ00(t) (A37) ˙ρi0(t) =− (γ1 +γ2 2 − iω0 ) ρi0(t) for i = 1, 2 (A38) ˙ρ12(t) =κ12ρ00(t) (A39) where we used the fact t...

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[5] [5]

(A4): ρT (t) =ρT (0) + 1 iℏ ∫ t 0 dt′ [ ˆHI(t′),ρT (t′)] (A6) and substitute this expression in Eq

Master equation framework In the interaction picture , the density matrix ρT (t) of the total system {atom+environment} obeys the Schr¨ odinger equation [19, 20]: ∂ρT (t) ∂t = 1 iℏ[ ˆHI(t),ρT (t)] (A4) The atomic density matrixρ(t) is obtained by taking the trace over the degrees of freedom of the environment: ρ(t) = Tre(ρT (t)), and therefore obeys: ∂ρ(t...

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[6] [6]

(A13) is the starting point to calculate the dynam- ical evolution of any multilevel atom

Master equation for an atomic Λ-transition Eq. (A13) is the starting point to calculate the dynam- ical evolution of any multilevel atom. Here, we proceed by writing explicitely the terms in the integrand using the expressions for ˆd(t) and ˆd†(t) from Eqs. (A2) and (A3), which corresponds to the Λ-conﬁguration with or- thogonal transitions: ˆd†(t)ˆd(t′)ρ...

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[7] [7]

P. K. Jha, X. Ni, C. Wu, Y. Wang, and X. Zhang, Phys- ical Review Letters 115, 025501 (2015)

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Raimond and S

J.-M. Raimond and S. Haroche, Exploring the quantum (Oxford University Press, Oxford, 2006)

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