REVIEW 2 major objections 4 minor 1 cited by
Hybrid sub- and superradiant states in emitter arrays with quantized motion
T0 review · 2 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash
Pith's one-line read Trapped atom arrays retain sub- and superradiance under quantized motion, with decay rates acquiring phonon-number corrections of order $\eta_0^2$.
desk verdict Solid theory paper with a real master-equation derivation and two genuinely new results; the main weakness is an unquantified secular approximation, which is fixable in revision. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the effective single-excitation non-Hermitian Hamiltonian (2), obtained from a Lindblad master equation derived under Born, Markov, and Lamb-Dicke approximations. Its four sets of jump operators, $\sigma_j$, $\sigma_j a_j$, $\sigma_j a_j^\dagger$, and $\sigma_j(1+2a_j^\dagger a_j)$, encode photon recoil: phonon loss, phonon gain, and a phonon-number-dependent renormalization of the spin transition. The motion corrections are carried by the second-derivative matrices $\Gamma''_{jj'}$ and $V''_{jj'}$, which replace the collective coupling by its curvature with respect to reduced interatomic distance. Because the effective Hamiltonian conserves total phonon number, the ansatz of product spin-phonon states can be checked algebraically, yielding the exact two-atom spectrum and the exact center-of-mass eigenstates in chains.
What would settle it
Measure the decay rate of the antisymmetric two-atom state with one phonon as a function of interatomic distance and trap frequency: the formulas predict a correction proportional to $\eta_0^2 (2n_{\rm a}+1)\Gamma''_{12}$ that vanishes at $d=d_0$, and independence of $\eta_0$ there; a numerical simulation that omits the secular approximation and keeps the phonon-number non-conserving $H_2^2$ terms should show deviations growing as the trap frequency approaches the collective decay scale, which would test the exactness of the product eigenstates.
Extended reading notes
Core claim
Within the linear-optics regime (at most one spin excitation) and up to order $\eta_0^2$ in the Lamb-Dicke parameter, the dynamics of a chain of trapped two-level emitters is governed by the non-Hermitian effective Hamiltonian (2), which conserves the total number of phonons. The collective decay matrix $\Gamma$ and dipole-dipole interaction matrix $V$ acquire motion corrections through their second derivatives $\Gamma''$ and $V''$. For two atoms every eigenstate is a separable product $|s/a\rangle |n_{\rm ph}, n_{\rm a}\rangle$ of a symmetric or antisymmetric spin state and a phonon Fock state, with decay rates $\gamma_{s/a}^{n_{\rm ph},n_{\rm a}} = \gamma \pm \Gamma_{12} \pm \eta_0^2 (2 n_{\rm a}+1) \Gamma''_{12}$. At the special separation $d_0$ where $\Gamma''_{12}=0$ these rates are independent of both phonon number and $\eta_0$; for perpendicular dipoles at $\kappa_0=2$ the energy shifts are likewise protected. In an infinite chain the center-of-mass phonon states $|q\rangle (\tilde{a}_0^\dagger)^{n_{\rm ph}} |0\rangle_{\rm ph}$ are exact eigenstates with decay rates $\tilde\Gamma_q = -2\,\mathrm{Im}\,(\tilde{M}_q + \eta_0^2(\tilde{M}''_q - M''_{11}))$, while in finite chains numerical diagonalization shows that the remaining eigenstates are genuinely entangled spin-phonon hybrids whose entanglement entropy can approach $\log N$ for dense arrays.
Load-bearing premise
The load-bearing premise, flagged in the Supplemental Material, is the secular approximation that drops phonon-number non-conserving terms oscillating at twice the trap frequency; if the trap frequency is not large compared with the collective decay rates, those terms do not average out, total phonon number is no longer conserved, and the separable product eigenstates are no longer exact.
Editorial extensions
If this is right
- For two atoms, the collective decay rates depend on the phonon state only through the occupancy of the antisymmetric phonon mode, so phonons in the symmetric center-of-mass mode leave the sub- and superradiant rates unchanged.
- At the special distance $d_0$ where $\Gamma''_{12}=0$, two-atom collective decay is completely insensitive to quantized motion to order $\eta_0^2$, regardless of phonon number or Lamb-Dicke parameter.
- In chains, the center-of-mass phonon states $|q\rangle(\tilde a_0^\dagger)^{n_{\rm ph}}|0\rangle_{\rm ph}$ are exact separable eigenstates, so the spin and motion degrees of freedom factorize for these states even though the decay rates are renormalized by $\eta_0^2(\tilde M''_q - M''_{11})$.
- For finite chains, the majority of eigenstates are entangled spin-phonon hybrids, and for dense chains some approach the maximum entanglement entropy $\log N$ while still exhibiting super- and subradiant rates.
- The derived master equation provides the starting point for going beyond the single-excitation regime to study many-photon superradiance, driven systems, higher-dimensional lattices, and couplings beyond the Lamb-Dicke limit.
Reading between the lines
- Going beyond the paper, the special distance $d_0$ where $\Gamma''_{12}=0$ is a natural experimental sweet spot: two atoms placed there should show collective emission almost unaffected by thermal phonon occupation, reducing the need for ground-state cooling.
- The exactness of center-of-mass phonon eigenstates suggests a test the authors leave implicit: a global oscillation of the whole trap should not degrade subradiance in the chain, since the center-of-mass phonon mode factorizes from the spin state.
- The secular approximation is expected to break down when the trap frequency is comparable to the collective decay rates, which can happen in dense subwavelength arrays; a numerical study retaining the phonon-number non-conserving $H_2^2$ terms would map where the hybrid eigenstates acquire entanglement and where the simple rate formulas fail.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops a first-principles Lindblad master equation for a one-dimensional chain of trapped two-level emitters coupled to the radiation field, including the quantized center-of-mass motion in the Lamb-Dicke regime to order η0^2. From this master equation the authors define an effective non-Hermitian Hamiltonian for the single-excitation sector and analyze its eigenstates. For two atoms they find fully separable product states with collective decay rates γ ± Γ12 ± η0^2(2n_a^ph+1)Γ''12 and identify a distance d0 where Γ''12 vanishes, so that the spin-motion coupling leaves the decay rates unchanged. For an infinite chain they show that COM-phonon states |q>(a†_0)^{nph}|0>_ph are exact eigenstates with renormalized rates, and for finite chains they numerically study the decay rates and spin-phonon entanglement, finding that most eigenstates are entangled hybrids while a subset remains separable.
Significance. If the central claims hold, this is a useful and fairly complete theory of collective atomic emission with quantized motion. The zero-order reduction to the well-established Lehmberg equation is a strong internal check, the analytic results for two atoms and for the infinite-chain COM states are explicit and falsifiable, and the identification of a distance at which motional corrections vanish is a concrete, experimentally testable prediction. The paper contains no fitted free parameters and clearly separates exact product states from numerically studied entangled hybrids. The main value lies in the framework and the explicit formulas, which should stimulate experimental work on hybrid sub- and superradiance in optical-lattice or microtrap settings.
major comments (2)
- [Supplemental Material, Sec. I (secular approximation; paragraph beginning 'Note, that we neglect directly the terms...')] The central exactness of the phonon-number-conserving eigenstates rests on dropping the phonon-number non-conserving Hamiltonian H2^2, whose terms carry e^{±2iωt}, and also on dropping the H0–H1 cross terms carrying e^{±iωt}. The paper states only that these terms 'drop out under the secular approximation', but no quantitative condition is given. The relevant requirement is that 2ω (and ω) be large compared with the dissipative timescales of the problem, i.e. the collective decay rates, which can scale as Nγ. Since the paper only assumes ω ≪ ω0, and in many optical-trap experiments ω can be comparable to or smaller than γ, this condition is not automatically satisfied. If 2ω is not large, H2^2 couples phonon sectors differing by two, the effective Hamiltonian no longer conserves total phonon number, and the product states of Eq. (3) as well as the chain states |q>(a†_0)^{nph}|0>_ph are not exact eigenstates; their rates receive corrections at the same order η0^2Γ as the quoted formulas. Please state the precise validity condition, discuss the parameter regimes where it holds (e.g. resolved-sideband condition ω ≫ γ_collective), and either include the H2^2 contributions or clearly restrict the claims to the regime where they are negligible.
- [Main text, Eqs. (1) and (2) and definition M_jj' = V_jj' - iΓ_jj'/2] As written, Eq. (1) and Eq. (2) are mutually inconsistent with respect to the sign of the coherent coupling. The second term of Eq. (1) is +i Σ Ṽ [J†_m J_{m'}, ρ]. For a single excitation, the no-jump part of this master equation is reproduced by an effective Hamiltonian whose real part is -Ṽ, because -i(H_eff ρ - ρ H†_eff) yields a coherent commutator -iṼ[J†J,ρ]. Equation (2), with M = V - iΓ/2, has real part +V and would instead correspond to a master equation with coherent term -iV[J†J,ρ]. Unless a nonstandard commutator sign convention is intended and explicitly stated, the energy shifts obtained from Eq. (2) — including the statement that at φ = π/2 and κ0 = 2 both Γ12 and V12 possess inflection points so that energy shifts are also unaffected — are opposite to those implied by Eq. (1). Please correct the sign convention in Eq. (1) or in the definition of M, and include the reduction from the Lindblad equation (1) to the single-excitation effective Hamiltonian (2), which is not shown in the paper or the Supplement.
minor comments (4)
- [Main text, Eq. (2)] The parentheses around the phonon-number operator combination are unbalanced; please rewrite the equation so that the bracket structure is unambiguous.
- [Main text, Fig. 3 caption and surrounding text] The sentence 'Sm = 0 for all values of η0 when η0' is incomplete; it should say 'for all values of η0 for the separable states' or similar.
- [Supplemental Material, Sec. I] There are several typographical errors, including 'the the spin ladder operators' and the definition 'η = (k·ẑ)z_ho/√2 =≪ 1'; these should be corrected.
- [Main text, Many atoms section] The Fourier sums are typeset in a garbled way ('π/dΣ_q=-π/d'); the sums over q and p should be written with explicit limits from -π/d to π/d.
Circularity Check
No significant circularity: the master equation and hybrid-state decay rates are derived from the microscopic Hamiltonian without fitted inputs or load-bearing self-citations.
full rationale
The derivation chain is self-contained. The paper starts from the full spin-phonon-photon Hamiltonian (main text and Supplement Sec. I), applies Born, Markov, and Lamb-Dicke expansions, and obtains the Lindblad equation (1) with explicit rate matrices Γ, V and their derivatives Γ'', V'' computed from the radiation-mode integrals. The η0=0 limit reproduces Lehmberg's well-established equation, providing an external benchmark; the η0² corrections come from explicit evaluation of [H1,[H1,ρ]] and [H0,[H0_2,ρ]] terms in the Supplement, not from an ansatz. No free parameter is fitted to any target result: the distance d0 with Γ''12=0 is a mathematical zero of the derived function Γ12(κ), and the two-atom rates γ^{nph,na}_{s/a}=γ±Γ12±η0²(2na+1)Γ''12 and the chain states |q⟩(a†0)^{nph}|0⟩_ph follow from exact diagonalization or Fourier transformation of the derived effective Hamiltonian (2). The self-citations (Refs. 16, 18, 22, 45) are background references and prior applications; they do not supply the load-bearing argument. The one concern raised by the Supplement, the secular approximation that drops the phonon-number non-conserving H2^2 terms, is a regime-of-validity question about whether 2ω exceeds the collective decay rates; it is not a circular identification of a prediction with an input. No circular step can be exhibited by quoting equations that reduce to themselves.
Assumptions & free parameters
assumptions (6)
- domain assumption Born-Markov approximation and factorized initial state ρ ⊗ ρ_E.
- domain assumption Rotating-wave / secular approximation neglecting terms oscillating at e^{±2iωt}, including the phonon-number non-conserving Hamiltonian H2^2.
- domain assumption Lamb-Dicke expansion to order η0^2 with z_ho ≪ λ0.
- domain assumption Zero-temperature photon bath (vacuum fluctuations only).
- domain assumption Single-excitation (linear optics) regime: at most one spin excitation in the system.
- domain assumption One-dimensional atomic chain with vibrations strictly along the chain axis, and uniform dipole orientation angle φ.
Cite this review
Pith. "Pith review of Hybrid sub- and superradiant states in emitter arrays with quantized motion." pith.science (2026). https://pith.science/paper/LJCOPC5Z
@misc{pith2026250201428,
author = {Pith},
title = {Pith review of: Hybrid sub- and superradiant states in emitter arrays with quantized motion},
year = {2026},
howpublished = {\url{https://pith.science/paper/LJCOPC5Z}},
note = {Machine review of arXiv:2502.01428}
}
read the original abstract
Ensembles of dipolar emitters which couple collectively to the radiation field display sub- and superradiance. These terms refer to a reduction or an enhancement of photon emission rates due to the interference of emission channels. Arrays of trapped neutral atoms constitute a promising platform for harnessing this phenomenon in technological applications, e.g. for excitation storage, single-photon switches and mirrors. However, vibrational motion of the atoms within their traps leads to position fluctuations that entangle the motion and the internal atomic degrees of freedom, which is expected to affect the collective photon emission. We develop here a theory for collective atom-light coupling in the presence of this quantized motion within the Lamb-Dicke limit. We show the existence of sub- and superradiant states, which are hybrids of electronic and vibrational excitations and explore their properties for analytically and numerically efficiently solvable cases.
Figures
Forward citations
Cited by 1 Pith paper
-
Effects of finite trapping on the decay, recoil, and decoherence of dark states of quantum emitter arrays
Finite trap strength makes subradiant atomic-array dark states decay faster over time, heat up, and lose fidelity; infidelity scales as (γ0η/ωt)^2 and is minimized with strong traps and perpendicular polarization.
Reference graph
Works this paper leans on
-
[1]
The dynamics is then governed by the Lindblad master equation ˙ρ = 4NX m,m′=1 ˜Γmm′ Jm′ρJ † m − 1 2 J † mJm′, ρ + i X m̸=m′ ˜Vmm′ J † mJm′, ρ . (1) This equation accounts for collective photon emission and dipole-dipole exchange interactions (first and sec- ond term, respectively). The two matrices ˜Γ and ˜V are 4N × 4N block matrices of the form ˜Γ = ...
-
[2]
to the dynamics due to the recoil of the emitters under photon emission. The processes that accompany these corrections correspond either to the loss or the gain of a phonon in each spin transition (see Fig. 1b). A further effect of the same order is the renormalization of the spin transition rates, which acquire a dependence on the number of phonons. 3 I...
-
[3]
This yields H eff =M11 + (b† sbs − b† aba) M12 + η2 0M ′′ 12 + 2η2 0(b† sbs − b† aba)M ′′ 12a† aaa, (3) where M11 = M22 = −iγ/2. This expression reveals that all eigenstates of the Hamiltonian are separable (product states) in the atomic and phonon degrees of freedom. The atomic eigenstates are the symmetric and antisymmetric superpositions |s/a⟩ = b† s/a...
-
[4]
The phonon eigenstates are |nph, na ph⟩ = a† s (nph−na ph) a† a na ph |0⟩ph, where nph is the total number of phonons and na ph is the number of phonons in the an- tisymmetric mode. The corresponding collective decay rates are given by γ nph,na ph s/a = γ ± Γ12 ± η2 0(2na ph + 1)Γ′′ 12. When η0 = 0 there exist only two degenerate rates, namely γ nph,na ph...
-
[5]
J. Pellegrino, R. Bourgain, S. Jennewein, Y. R. P. Sor- tais, A. Browaeys, S. D. Jenkins, and J. Ruostekoski, Observation of suppression of light scattering induced by dipole-dipole interactions in a cold-atom ensemble, Phys. Rev. Lett. 113, 133602 (2014)
work page 2014
-
[6]
R. H. Dicke, Coherence in spontaneous radiation pro- cesses, Phys. Rev. 93, 99 (1954)
1954
-
[7]
R. H. Lehmberg, Radiation from an n-atom system. I. General formalism, Phys. Rev. A 2, 883 (1970)
work page 1970
-
[8]
D. F. V. James, Frequency shifts in spontaneous emis- sion from two interacting atoms, Phys. Rev. A 47, 1336 (1993)
work page 1993
Show all 51 references
-
[9]
Bienaim´ e, N
T. Bienaim´ e, N. Piovella, and R. Kaiser, Controlled dicke subradiance from a large cloud of two-level systems, Phys. Rev. Lett. 108, 123602 (2012)
2012
-
[10]
Ferioli, A
G. Ferioli, A. Glicenstein, L. Henriet, I. Ferrier-Barbut, and A. Browaeys, Storage and release of subradiant exci- tations in a dense atomic cloud, Phys. Rev. X 11, 021031 (2021)
2021
-
[11]
S. D. Jenkins, J. Ruostekoski, J. Javanainen, S. Jen- newein, R. Bourgain, J. Pellegrino, Y. R. P. Sortais, and A. Browaeys, Collective resonance fluorescence in small and dense atom clouds: Comparison between theory and experiment, Phys. Rev. A 94, 023842 (2016)
2016
-
[12]
Note that, unlike in long-range interacting sys- tems, where the interactions decay monotonically with the distance, the behaviour of both Γ 12 and V12 with the reduced distance κ is non-monotonic (see Fig. 2a). Hence, here it is possible to find an interatomic distance d0 whe...
-
[13]
M. O. Ara´ ujo, I. Kreˇ si´ c, R. Kaiser, and W. Guerin, Su- perradiance in a large and dilute cloud of cold atoms in the linear-optics regime, Phys. Rev. Lett. 117, 073002 (2016)
2016
-
[14]
Guerin, M
W. Guerin, M. O. Ara´ ujo, and R. Kaiser, Subradiance in a large cloud of cold atoms, Phys. Rev. Lett. 116, 083601 (2016)
2016
-
[15]
S. L. Bromley, B. Zhu, M. Bishof, X. Zhang, T. Both- well, J. Schachenmayer, T. L. Nicholson, R. Kaiser, S. F. Yelin, M. D. Lukin, A. M. Rey, and J. Ye, Collective atomic scattering and motional effects in a dense coher- ent medium, Nat. Comm. 7, 11039 (2016)
2016
-
[16]
C. M. Lange, E. Daggett, V. Walther, L. Huang, and J. D. Hood, Superradiant and subradiant states in lifetime-limited organic molecules through laser-induced tuning, Nat. Phys. 20 (2024)
2024
-
[17]
Tiranov, V
A. Tiranov, V. Angelopoulou, C. J. van Diepen, B. Schrinski, O. A. D. Sandberg, Y. Wang, L. Midolo, S. Scholz, A. D. Wieck, A. Ludwig, A. S. Sørensen, and P. Lodahl, Collective super- and subradiant dynamics be- tween distant optical quantum emitters, Science379, 389 (2023)
2023
-
[18]
Ferioli, A
G. Ferioli, A. Glicenstein, I. Ferrier-Barbut, and A. Browaeys, A non-equilibrium superradiant phase tran- sition in free space, Nat. Phys. 19, 1345 (2023)
2023
-
[19]
selective radiance
A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht, H. J. Kimble, and D. E. Chang, Exponential improve- ment in photon storage fidelities using subradiance and “selective radiance” in atomic arrays, Phys. Rev. X 7, 031024 (2017)
2017
-
[20]
Sierra, S
E. Sierra, S. J. Masson, and A. Asenjo-Garcia, Dicke su- perradiance in ordered lattices: Dimensionality matters, Phys. Rev. Res. 4, 023207 (2022)
2022
-
[21]
M. Cech, I. Lesanovsky, and B. Olmos, Dispersionless subradiant photon storage in one-dimensional emitter chains, Phys. Rev. A 108, L051702 (2023)
2023
-
[22]
Holzinger, O
R. Holzinger, O. Rubies-Bigorda, S. F. Yelin, and H. Ritsch, Symmetry based efficient simulation of dissi- pative quantum many-body dynamics in subwavelength quantum emitter arrays (2024), arXiv:2409.02790 [quant- ph]
2024 arXiv
-
[23]
Olmos, D
B. Olmos, D. Yu, Y. Singh, F. Schreck, K. Bongs, and I. Lesanovsky, Long-range interacting many-body sys- tems with alkaline-earth-metal atoms, Phys. Rev. Lett. 110, 143602 (2013)
2013
-
[24]
H. H. Jen, M.-S. Chang, and Y.-C. Chen, Cooperative single-photon subradiant states, Phys. Rev. A94, 013803 (2016)
2016
-
[25]
Facchinetti, S
G. Facchinetti, S. D. Jenkins, and J. Ruostekoski, Stor- ing light with subradiant correlations in arrays of atoms, Phys. Rev. Lett. 117, 243601 (2016)
2016
-
[26]
M. T. Manzoni, M. Moreno-Cardoner, A. Asenjo-Garcia, J. V. Porto, A. V. Gorshkov, and D. E. Chang, Optimiza- tion of photon storage fidelity in ordered atomic arrays, New J. Phys. 20, 083048 (2018)
2018
-
[27]
J. A. Needham, I. Lesanovsky, and B. Olmos, Subradiance-protected excitation transport, New J. Phys. 21, 073061 (2019)
2019
-
[28]
K. E. Ballantine and J. Ruostekoski, Subradiance- protected excitation spreading in the generation of colli- mated photon emission from an atomic array, Phys. Rev. 6 Res. 2, 023086 (2020)
2020
-
[29]
K. E. Ballantine and J. Ruostekoski, Quantum single- photon control, storage, and entanglement generation with planar atomic arrays, PRX Quantum 2, 040362 (2021)
2021
-
[30]
K. E. Ballantine and J. Ruostekoski, Unidirectional ab- sorption, storage, and emission of single photons in a collectively responding bilayer atomic array, Phys. Rev. Res. 4, 033200 (2022)
2022
-
[31]
Ostermann, H
L. Ostermann, H. Ritsch, and C. Genes, Protected state enhanced quantum metrology with interacting two-level ensembles, Phys. Rev. Lett. 111, 123601 (2013)
2013
-
[32]
Reitz, C
M. Reitz, C. Sommer, and C. Genes, Cooperative quan- tum phenomena in light-matter platforms, PRX Quan- tum 3, 010201 (2022)
2022
-
[33]
Qu and A
C. Qu and A. M. Rey, Spin squeezing and many-body dipolar dynamics in optical lattice clocks, Phys. Rev. A 100, 041602 (2019)
2019
-
[34]
Pi˜ neiro Orioli and A
A. Pi˜ neiro Orioli and A. M. Rey, Subradiance of multi- level fermionic atoms in arrays with filling n ≥ 2, Phys. Rev. A 101, 043816 (2020)
2020
-
[35]
R. J. Bettles, S. A. Gardiner, and C. S. Adams, En- hanced optical cross section via collective coupling of atomic dipoles in a 2d array, Phys. Rev. Lett.116, 103602 (2016)
2016
-
[36]
J. Rui, D. Wei, A. Rubio-Abadal, S. Hollerith, J. Zeiher, D. M. Stamper-Kurn, C. Gross, and I. Bloch, A subradi- ant optical mirror formed by a single structured atomic layer, Nature 583, 369–374 (2020)
2020
-
[37]
Buckley-Bonanno, S
S. Buckley-Bonanno, S. Ostermann, O. Rubies-Bigorda, T. L. Patti, and S. F. Yelin, Optimized geometries for cooperative photon storage in an impurity coupled to a two-dimensional atomic array, Phys. Rev. A 106, 053706 (2022)
2022
-
[38]
Srakaew, P
K. Srakaew, P. Weckesser, S. Hollerith, D. Wei, D. Adler, I. Bloch, and J. Zeiher, A subwavelength atomic array switched by a single Rydberg atom, Nat. Phys. (2023)
2023
-
[39]
Ruostekoski, Cooperative quantum-optical planar ar- rays of atoms, Phys
J. Ruostekoski, Cooperative quantum-optical planar ar- rays of atoms, Phys. Rev. A 108, 030101 (2023)
2023
-
[40]
N. O. Gjonbalaj, S. Ostermann, and S. F. Yelin, Modify- ing cooperative decay via disorder in atom arrays, Phys. Rev. A 109, 013720 (2024)
2024
-
[41]
R. N. Palmer and A. Beige, Enhancing laser sideband cooling in one-dimensional optical lattices via the dipole interaction, Phys. Rev. A 81, 053411 (2010)
2010
-
[42]
F. m. c. Damanet, D. Braun, and J. Martin, Master equa- tion for collective spontaneous emission with quantized atomic motion, Phys. Rev. A 93, 022124 (2016)
2016
-
[43]
Guimond, A
P.-O. Guimond, A. Grankin, D. V. Vasilyev, B. Verm- ersch, and P. Zoller, Subradiant Bell States in distant atomic arrays, Phys. Rev. Lett. 122, 093601 (2019)
2019
-
[44]
C. C. Rusconi, T. Shi, and J. I. Cirac, Exploiting the photonic nonlinearity of free-space subwavelength arrays of atoms, Phys. Rev. A 104, 033718 (2021)
2021
-
[45]
Rubies-Bigorda, R
O. Rubies-Bigorda, R. Holzinger, A. Asenjo-Garcia, O. Romero-Isart, H. Ritsch, S. Ostermann, C. Gonzalez- Ballestero, S. F. Yelin, and C. C. Rusconi, Collectively enhanced ground-state cooling in subwavelength atomic arrays (2024), arXiv:2405.18482 [quant-ph]
2024 arXiv
-
[46]
Manzano, A short introduction to the Lindblad master equation, AIP Advances 10, 025106 (2020)
D. Manzano, A short introduction to the Lindblad master equation, AIP Advances 10, 025106 (2020)
2020
-
[47]
[1–3], for details on how to obtain the quantum master equation in the Lamb-Dicke regime
See Supplemental Material, which further contains Refs. [1–3], for details on how to obtain the quantum master equation in the Lamb-Dicke regime
-
[48]
S. J. Masson and A. Asenjo-Garcia, Universality of dicke superradiance in arrays of quantum emitters, Nat. Com- mun. 13 (2022)
2022
-
[49]
J. R. Ott, M. Wubs, P. Lodahl, N. A. Mortensen, and R. Kaiser, Cooperative fluorescence from a strongly driven dilute cloud of atoms, Phys. Rev. A 87, 061801 (2013)
2013
-
[50]
Olmos, D
B. Olmos, D. Yu, and I. Lesanovsky, Steady-state prop- erties of a driven atomic ensemble with nonlocal dissipa- tion, Phys. Rev. A 89, 023616 (2014). 7 SUPPLEMENTAL MATERIAL Hybrid sub- and superradiant states in emitter arrays with quantized motion Beatriz Olmos1 and Igor Le...
2014
-
[51]
H ′ = ¯hω X j a† jaj − ¯h X j,k,λ gλ k(eiω0tσ† j + e−iω0tσj) h ckλeiη(a† j +aj )ei(k·r0 j −νkt) + c† kλe−iη(a† j +aj )e−i(k·r0 j −νkt) i
We go now into the interaction picture with respect to the atomic and field frequencies, i.e. H ′ = ¯hω X j a† jaj − ¯h X j,k,λ gλ k(eiω0tσ† j + e−iω0tσj) h ckλeiη(a† j +aj )ei(k·r0 j −νkt) + c† kλe−iη(a† j +aj )e−i(k·r0 j −νkt) i . Now we need to go into the interaction pictu...
Reviewed August 9, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.