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Engineering tunable decoherence-free subspaces with collective atom-cavity interactions

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Pith's one-line read Collective atom-cavity dissipation can be engineered into a tunable, multi-dimensional decoherence-free subspace, with shortcut driving preparing pure states in times far shorter than standard adiabatic or dissipative preparation.

desk verdict Useful analytic DFS construction for the SU(3) atom-cavity model, but the 'no purity loss' claim needs the strong-cooperativity condition stated; worth refereeing with revisions. read the letter →

arxiv 2412.02921 v2 pith:HO5UILV3 submitted 2024-12-04 quant-ph

Lyryl H. C. Vaecairn , Jarrod T. Reilly , John Drew Wilson , Simon B. Jaeger , Murray Holland This is my paper · ORCID
classification quant-ph PACS 03.65.Yz42.50.Pq03.67.-a
keywords decoherence-freesubspacecollectiveatom-cavityinteractionsnon-Hermitianjumpoperatoradiabaticshortcutsthree-levelatomsdissipativestatepreparationeffectivePaulioperatorsquantummetrology
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 argues that a dissipative atom-cavity system — $N$ three-level atoms collectively coupled to a lossy cavity and driven by tunable lasers — can be reduced, after eliminating the excited states and the cavity, to an effective master equation with a single collective jump operator $\hat{L}=\sqrt{\Gamma_c}(\hat{J}_-+\mu^2\hat{J}_++\chi)$, and that the kernel of this operator is a decoherence-free subspace whose dimension and orientation can be tuned in time. The paper derives the full structure of every such subspace analytically, shows that typical states inside them are highly entangled, and constructs transitionless shortcut Hamiltonians that keep the system in the instantaneous DFS at all times. The shortcuts prepare pure states in $t_f=1/\Gamma_c$, compared with $137/\Gamma_c$ for an adiabatic ramp and $318/\Gamma_c$ for a dissipative quench (for the edge DFS at $N=5$), with purity and fidelity above $0.9995$. If correct, this provides a fast, experimentally grounded route to protected state preparation and to a logical qubit whose gates commute with the dominant dissipation.

What carries the argument

The load-bearing object is the exact diagonalization of the non-Hermitian jump operator $\hat{L}$ through $L=VDV^{-1}$, which defines two non-canonical sets of Schwinger-boson operators $\vec{c}=V^\dagger\vec{b}$ and $\vec{d}=V^{-1}\vec{b}$ and turns $\hat{L}$ into $\sqrt{\Gamma_c}(\vec{c}^\dagger D\vec{d}+\chi)$. Because $\hat{L}$ is not Hermitian unless $\mu=1$, the eigenstates and the complementary space are built from $\hat{c}^\dagger$ and $\hat{d}^\dagger$ respectively; their biorthogonal overlap $\langle\psi^\perp_{\vec{n}}|\psi_{\vec{k}}\rangle\propto \delta_{\vec{n}\vec{k}}$ is what makes the complementary subspace analytically tractable. This diagonalization reduces the search for a DFS to a counting problem on the integer $C=k_3-k_1$ and gives the explicit dimension formula, and it provides the matrix elements $\langle\psi^\perp_{\vec{n}}|\partial_t\psi_{\vec{k}}\rangle$ that the shortcut Hamiltonians must match. The two shortcut mechanisms are the cavity-drive modification $\chi\to\chi+\chi_s$ for the edge DFSs and an additional laser coupling between the $|-1_g\rangle$ and $|1_g\rangle$ states for the central DFS; both are quadratic in single-particle operators and therefore reach only unentangled target states.

What would settle it

Numerically simulate the shortcut protocol for $|\psi_{N,0,0}\rangle$ with $N=5$ in the full model that keeps the excited states, adding a spontaneous-emission jump operator at rate $\gamma$ from each excited state, and scan $\gamma/\Gamma_c$ from 0.01 upward while holding all other parameters at the values used in Table I. If the final purity $P(t_f)$ or the target overlap $F(1,t_f)$ falls below 0.99 at a $\gamma$ small enough that the excited-state elimination should still be valid, the single-jump-operator shortcut is falsified; the same quantity can be checked experimentally by state tomography after a $1/\Gamma_c$ ramp and by looking for population in the DFS's orthogonal complement.

Watch

Extended reading notes

Core claim

At the center of the paper is the observation that the single non-Hermitian jump operator $\hat{L}=\sqrt{\Gamma_c}(\hat{J}_-+\mu^2\hat{J}_++\chi)$ can be diagonalized exactly by a non-unitary transformation in Schwinger-boson space, $\hat{L}=\sqrt{\Gamma_c}[\sqrt{2}\mu(\hat{c}_1^\dagger\hat{d}_1-\hat{c}_3^\dagger\hat{d}_3)+\chi]$. Its eigenstates are $|\psi_{\vec{k}}\rangle\propto(\hat{c}_1^\dagger)^{k_1}(\hat{c}_2^\dagger)^{k_2}(\hat{c}_3^\dagger)^{k_3}|0\rangle$ with eigenvalues $\sqrt{\Gamma_c}[\sqrt{2}\mu(k_1-k_3)+\chi]$, so the DFS is exactly the span of all eigenstates with $k_3-k_1=C=\chi/(\sqrt{2}\mu)$, a subspace of dimension $\lceil(N+1-|C|)/2\rceil$ whose states move as $\mu$ and $\chi$ are ramped. The paper then constructs shortcut Hamiltonians satisfying $\langle\psi^\perp_{\vec{n}}|\hat{H}_s|\psi_{\vec{k}}\rangle=i\hbar\langle\psi^\perp_{\vec{n}}|\partial_t\psi_{\vec{k}}\rangle$ to cancel all leakage into the complementary subspace; this yields the numerically demonstrated preparation of $|\psi_{N,0,0}\rangle$ and $|\psi_{0,N,0}\rangle$ in $t_f=1/\Gamma_c$. At $\mu=1$ the jump operator becomes Hermitian and the DFS hosts effective Pauli operators $\hat{\sigma}_x,\hat{\sigma}_y,\hat{\sigma}_z$ that drive a protected qubit while commuting with $\hat{L}$.

Load-bearing premise

The entire construction rests on the assumption that the fast degrees of freedom (cavity photons and excited atomic states) can be eliminated so that exactly one dissipation channel remains, described by the single jump operator $\hat{L}$; if spontaneous emission from the excited states is not negligible, or if the detuning and timescale hierarchies $\Delta_e\gg g,|\Omega_i|$, $g\gg|\Omega_i|$, $\kappa\gg|\sqrt{N}g\Omega_i/\Delta_e|$, $\kappa\gg \partial_t\eta/\eta,\partial_t\Omega_i/\Omega_i$ are violated, additional jump operators appear and the kernel of $\hat{L}$ is no longer the protected subspace.

Editorial extensions

If this is right

  • A pure target state can be prepared in a time $t_f=1/\Gamma_c$, roughly two orders of magnitude faster than the best adiabatic ramp ($137/\Gamma_c$) or quench ($318/\Gamma_c$) reported for the same $N=5$ system, so the state spends far less time exposed to noise the DFS does not protect against.
  • The DFS dimension can be dialed from 1 to $\lceil(N+1)/2\rceil$ by choosing the integer $C=\chi/(\sqrt{2}\mu)$, so a single setup can store a protected qubit, a qutrit, or a higher-dimensional qudit without changing the apparatus.
  • Because all states in the DFS are simultaneous eigenstates of $\hat{L}$, any operator that commutes with $\hat{L}$ (such as the effective Pauli operators of Sec. VI) can implement gates inside the subspace without causing decoherence; the qubit is therefore naturally robust against single-particle errors.
  • The DFS contains states with quantum Fisher information scaling as $N^2/2+N$ at $\mu=1$, so the protected subspace doubles as a source of Heisenberg-limited metrological sensitivity.
  • The shortcut to the central DFS state $|\psi_{0,N,0}\rangle$ is exact in simulation ($P=F=1$ up to numerical error), while the edge shortcuts carry only a small cutoff-induced error ($P=0.999992$, $F=0.9996$), meaning the main practical limitation is laser power near $\mu=1$, not the shortcut principle itself.

Reading between the lines

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

  • The shortcut Hamiltonians are quadratic in single-particle operators, so they can only reach unentangled product states; combining that with the paper's commuting Pauli operators suggests a two-stage recipe — shortcut to $|\psi_{0,N,0}\rangle$, then drive inside the DFS — for preparing the highly entangled, metrologically useful states, a concatenation the paper leaves implicit.
  • Because the edge-DFS shortcut requires $\chi_s\propto(\mu^4-1)^{-1}$, the 'not limited by any error rate' claim is in practice bounded by available laser power near $\mu=1$; a natural extension would be to optimize the pulse shape $\mu(t)$ or splice the two shortcut types to minimize peak power for a fixed total time.
  • If spontaneous emission is added perturbatively, it introduces jump operators that do not share the kernel of $\hat{L}$; computing the resulting leakage in first-order perturbation theory would give a quantitative bound on when the DFS protection degrades, which the paper does not provide.
  • The effective Pauli operators require four-operator (two-atom) interaction terms, so realizing the protected qubit gates will require a physical mechanism beyond the single-particle drives used for the shortcuts; identifying such a mechanism and testing gate fidelities under the same parameter hierarchy is a direct next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies N three-level atoms collectively coupled to a single cavity mode and derives an effective master equation with one jump operator L = √Γc(J_− + μ²J_+ + χ). Through a non-unitary diagonalization of L, the authors classify the decoherence-free subspaces by an integer C = χ/(√2μ), derive the dimension formula D(N,C)=⌈(N+1−|C|)/2⌉, and extract the structure of the dark states. They then compare three preparation protocols for the edge DFS H^{(−N)}_{DFS} and the central DFS H^{(0)}_{DFS}: an instantaneous quench, an adiabatic ramp, and two transitionless shortcuts to adiabaticity implemented by a modified cavity drive or by additional lasers. For N=5, numerical simulations report that the shortcut prepares |ψ_{N,0,0}⟩ with P=0.999992 and F=0.9996 in a time tf=1/Γc, whereas the quench and ramp require hundreds of 1/Γc to reach the 0.99 thresholds. The paper also defines effective Pauli operators that act on two-dimensional DFSs at μ=1.

Significance. If the effective single-jump model is justified, the paper makes a valuable contribution: it gives an explicit analytic diagonalization of a non-Hermitian collective jump operator, a closed-form DFS dimension formula, and shortcut Hamiltonians that are derived from the counterdiabatic condition rather than fitted. These are concrete, reproducible results, and the numerical simulations act as consistency checks on the analytic construction. The paper also identifies entangled dark states with Heisenberg-limit quantum Fisher information, which is a useful observation. The main gap is that the validity domain of the effective model, in particular the neglect of excited-state spontaneous emission, is stated only qualitatively, and the headline claim of purity-preserving preparation is therefore contingent on an unquantified strong-cooperativity condition.

major comments (2)
  1. [Section III and Appendix A] The single-jump effective master equation (6) is the foundation for the DFS and shortcut claims, and its derivation in Appendix A begins with the assumption that spontaneous decay from the excited states is negligible, but no quantitative condition is given. If an excited-state decay rate γ is included, elimination of the excited states produces additional Lindblad operators; the kernel of the single L in Eq. (7) is then no longer the protected subspace, and the shortcut condition (36) cannot guarantee purity. For the detuning regime used in the simulations (Δ′c/κ ≈ 0.1), estimating the effective Raman scattering rate as γ_eff ∼ γ(Ω/Δ_e)^2 and comparing it with Γ_c ∼ κg²Ω²/[Δ_e²(Δ_c′²+κ²)] gives γ_eff/Γ_c ∼ κγ/g², i.e., the single-jump model requires single-atom cooperativity g²/(κγ) ≫ 1. This condition should be derived and checked against the 87Rb parameters referenced in Refs. [44,45]; without it, the abstract's 'without any loss of purity' is contingent on an unstated strong-cooperativity assumption.
  2. [Section V D3, Eq. (43)] The statement that the final state for both the quench and the ramp 'must be' |j=N, k3−k1=C⟩ is not literally correct for the quench protocols, because in Tables I and II the quench ends at μ(tf)=μq<1 and the system damps into H^{(C)}_{DFS}(μq), not into H^{(C)}_{DFS}(1). The state |j=N, k3−k1=C⟩ is the unique state in the μ=1 target sector, and the actual quench final state has overlap F(1,tf)≈0.99 rather than exactly 1. The text should either restrict Eq. (43) to the ramp and to the μ=1 component of the quench state, or explain how the j=N conservation plus the C constraint uniquely fixes the quench final state inside H(μq); as written, the claim is inconsistent with the reported fidelities.
minor comments (5)
  1. [Abstract and Section V C] The abstract's phrase 'without any loss of purity' is stronger than the implemented protocol: the cavity-drive shortcut uses a cutoff |χ+χs|≤5√2N and the reported values are P=0.999992 and F=0.9996, with the deviation dominated by the cutoff rather than by numerical error. Please qualify the claim as applying to the ideal shortcut in the effective model.
  2. [Table I caption] The caption says the shortcut has 'unity purity and overlap up to numerical error,' but the F=0.9996 value is limited by the cutoff scheme described in Appendix E 1 and in Section V C; the caption should attribute this small error to the cutoff, not to numerical precision alone.
  3. [Section V C, after Eq. (37)] The text states that 'it can be proven that we can only shortcut to states that do not possess interparticle entanglement,' but immediately adds that 'the proof does not appear in this work.' Since this restriction is used to select the shortcut targets, please provide a proof, a sketch, or a precise reference; the current statement is an unsupported claim.
  4. [Section IV A and Appendix B] The linear independence of the states |ψ_k⟩ in Eq. (23), and hence the claim that they form a basis of the symmetric manifold of dimension (N+2)(N+1)/2, is asserted but not proved. A short argument using invertibility of V in Eq. (18) would make the dimensionality statement self-contained.
  5. [Section VI, Eq. (47)] The effective Pauli operators are introduced with the sufficient condition that they commute with L, but the verification is only shown on the two DFS basis states. Please state explicitly that [σ_i,L]=0 holds on the full Hilbert space (or on the full symmetric manifold), since invariance of the DFS under arbitrary superpositions is what guarantees decoherence-free driving.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the DFS and shortcut are derived from the effective master equation, and the numerics are consistency checks, not fitted predictions.

full rationale

The paper's central derivation is self-contained. The effective master equation (6) with the single jump operator (7) is obtained in Appendix A from a microscopic Hamiltonian under explicitly stated limits (|Δe|≫g,|Ω1|,|Ω2|; κ large compared with collective couplings and drive derivatives). The DFS is then defined as the kernel of this jump operator, and the Lidar–Chuang–Whaley criteria are applied directly; this is a mathematical construction, not a circular one. The eigenstates of the non-Hermitian jump operator are derived explicitly through the VDV⁻¹ diagonalization and the resulting states, normalizations, and overlaps are computed in the paper itself, rather than imported as black boxes. The shortcut Hamiltonian is obtained by solving the standard counterdiabatic condition of Eq. (36), with the matrix elements ⟨ψ⊥_n|H_s|ψ_k⟩ = iℏ⟨ψ⊥_n|∂_tψ_k⟩ computed in Appendix E. The reported purities and fidelities in Tables I and II are the results of integrating the same effective master equation with the analytically derived shortcut, and the deviation from unity is explicitly attributed to the chosen cutoff |χ+χs|≤5√2N, which the paper states is arbitrary and minimally influential. These are consistency checks of the analytic construction, not predictions obtained by fitting parameters to data. The paper does refer to the authors' earlier Ref. [27] for methodology (e.g., 'We now follow the methodology of Ref. [27]'), but the relevant formulas are re-derived in the appendices, so this citation is not load-bearing. The main caveat—that the no-purity-loss result holds only within the single-jump effective model and requires the assumptions of Appendix A, including negligible spontaneous emission from excited states—is a modeling limitation and a correctness risk, not a circularity. Because no step of the derivation reduces to its own input by definition or by fitted-parameter renaming, the circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The core DFS and shortcut derivations are analytic and rely on the standard adiabatic-elimination regime rather than on fitted constants. Three protocol parameters (μ_q, β, cutoff coefficient) are hand-tuned in the numerical comparisons, and the restriction to quadratic shortcuts rests on an unproved statement.

free parameters (3)
  • quench ratio μ_q = 0.96 (C=-N), 0.98 (C=0)
    Chosen by numerical search so that P(tf)≥0.99 and F(1,tf)≥0.99 (Eq. 35); it controls the dissipative gap and overlap with the target DFS.
  • adiabatic ramp rate β = 1/tf, tf=137/Γc (C=-N), 330/Γc (C=0)
    Largest β satisfying the threshold Eq. (35); determines the ramp speed and final fidelity.
  • shortcut cutoff coefficient = 5 in |χ+χ_s| ≤ 5√2N
    Introduced to truncate the diverging χ_s near μ=1; described as arbitrary, and it causes reported errors (F error ~1e-4, P error ~1e-6).
assumptions (5)
  • domain assumption The effective single-jump master equation (Eq. 6) is valid after eliminating excited states and the cavity mode.
    Requires the stated parameter hierarchy in Sec. III: |Δe|≫g,|Ω1|,|Ω2|; g≫|Ω1|,|Ω2|; κ≫|√N gΩ_i/Δe|; κ≫∂tη/η, ∂tΩ_i/Ω_i; and negligible spontaneous emission from excited states (Appendix A).
  • domain assumption The dynamics is confined to the permutationally symmetric SU(3) manifold.
    The initial state |−1g>^⊗N and all operators are symmetric, so Eq. (14) gives the exact Hilbert-space dimension for the accessible sector (Sec. IV).
  • domain assumption The transitionless condition of Ref. [24], Eq. (36), correctly characterizes shortcut Hamiltonians for an instantaneous DFS.
    The paper applies the standard counterdiabatic condition without re-derivation (Sec. V C, Appendix E).
  • ad hoc to paper Quadratic, single-particle shortcut Hamiltonians can only reach unentangled target states.
    Stated in Sec. V C with the note 'the proof does not appear in this work'; no citation is given, so this is an unsupported assumption that limits the shortcut construction.
  • standard math The SU(2) subalgebra J^2 and Clebsch-Gordan decomposition apply to the three-level collective states.
    Used in Appendix D to derive the conserved j=N final state |j=N, mk=C> for quench and ramp protocols.

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Pith. "Pith review of Engineering tunable decoherence-free subspaces with collective atom-cavity interactions." pith.science (2026). https://pith.science/paper/HO5UILV3

@misc{pith2026241202921,
  author       = {Pith},
  title        = {Pith review of: Engineering tunable decoherence-free subspaces with collective atom-cavity interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HO5UILV3}},
  note         = {Machine review of arXiv:2412.02921}
}
read the original abstract

We propose schemes to design and control a time-dependent decoherence-free subspace (DFS) in a dissipative atom-cavity system. These schemes use atoms with three internal energy levels, which allows for the DFS to be multi-dimensional--a condition important for quantum sensing, simulation, and computation. We consider the use of tunable external driving lasers to transfer the system from a coherent spin state to a highly degenerate DFS. We find that the typical state in the DFS is highly entangled. Throughout evolution the state is kept in an instantaneous DFS, thereby allowing for pure states to be prepared. We develop adiabatic shortcuts to carry out this evolution with higher purity and fidelity than standard adiabatic and dissipative methods.

Figures

Figures reproduced from arXiv: 2412.02921 by the authors.

Figure 1
Figure 1. we have highlighted this by sketching that Hˆ eff can￾not couple any decoherence-free states outside of HDFS. The system will typically damp into the DFS over time as the jump operators introduce stable decoherence-free population, whereas population in all other states will decay away. The Hamiltonian cannot affect this behavior as it cannot move states into or out of the DFS. We now manipulate the DFS via dynamica… view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The DFS structure for [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Time evolution of [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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Reference graph

Works this paper leans on

92 extracted references · 48 canonical work pages

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    S. Jin, H. Bao, J. Duan, X. Lu, M. Wang, K.-F. Zhao, H. Shen, and Y. Xiao, Adiabaticity in state preparation for spin squeezing of large atom ensembles, Photon. Res. 9, 2296 (2021)

    2021

  2. [1]

    This Hamiltonian can be created by changing the driving amplitude of the injected cavity drive η to η + ηs so that χ transforms to χ + χs

    Cavity drive modification We start by providing the shortcut Hamiltonian that is used for the preparation of |ψN,0,0⟩ and |ψ0,0,N ⟩, which are both in one-dimensional DFSs. This Hamiltonian can be created by changing the driving amplitude of the injected cavity drive η to η + ηs so that χ transforms to χ + χs. Remarkably, this does not require adding any ...

  3. [2]

    This subspace has the maximal dimension

    Additional lasers We now discuss how to shortcut to the state |ψ0,N,0⟩ in the central DFS where C = χ = 0. This subspace has the maximal dimension. To achieve this, consider a new shortcut Hamiltonian given by ˆHs = iαˆb† −1ˆb1 − iαˆb† 1ˆb−1 (40) which could be implemented experimentally by addi- tional lasers coupling between the |1g⟩ and |-1g⟩ inter- na...

  4. [3]

    the edge DFS H(−N ) DFS (µ)

    Edge decoherence-free subspace First, we consider the extremal value of C = −N , i.e. the edge DFS H(−N ) DFS (µ). This time-evolution is shown in Fig. 4. In Tab. I we show the numerical results for the evolution time tf , the final purity P(tf ), and the final overlap F (1, tf ) with the µ = 1 target subspace. For the quench, we also provide the final ov...

  5. [4]

    The inset shows behavior near t = tf

    (b) Purity of ˆρat(t). The inset shows behavior near t = tf . (c) Overlap of ˆρat(t) with H(−N ) DFS (µ(t)) which measures how well the state matches the current DFS. The inset shows behavior near t = tf . (d) Overlap of ˆρat(t) with H(−N ) DFS (1) which measures how well the state matches the target DFS. Note that tf is different for the different protoc...

  6. [5]

    Central decoherence-free subspace Now, we consider time dynamics where we set C = 0, corresponding to H(0) DFS(µ), as shown in Fig. 5. In Tab. II we provide a similar set of numerical results for the previous state in Tab. I. Here, the quench value for µ is µ(tf ) = µq = 0.98. (a) (b) (c) FIG. 5. Time evolution of H(0) DFS(µ) in units of tf via short- cut...

  7. [6]

    As the final purity P(tf ) is near 1 for all protocols, we can map ˆρat(tf ) to a pure state |ψ(tf )⟩

    Final state As H(0) DFS(µ) is multi-dimensional, determining the fi- nal state of the evolution analytically is nontrivial. As the final purity P(tf ) is near 1 for all protocols, we can map ˆρat(tf ) to a pure state |ψ(tf )⟩. For the shortcut, we have |ψ(tf )⟩ = |ψ0,5,0⟩ because our shortcuts are de- signed intentionally to drive between these states. Fo...

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    + ℏ NX j=1 −∆s (|1g⟩j⟨1g| − |-1g⟩j⟨-1g|) − ∆e|0e⟩j⟨0e| − (∆e − ∆s) |-1e⟩j⟨-1e| −(∆e + ∆s) |1e⟩j⟨1e| + gˆa† (|-1g⟩j⟨-1e| + |0g⟩j⟨0e| + |1g⟩j⟨1e|) + H.c

    Interaction picture In the first step we transform the Hamiltonian into the interaction picture using ˆ ρ → ˆU ˆρ ˆU † with ˆU = exp[i ˆH ′t/ℏ] with ˆH ′ =ℏ ω1 + ω2 2 ˆa†ˆa + NX j=1 −ω1 + ω2 2 ℏ(|1g⟩j⟨1g| − |-1g⟩j⟨-1g|) + ℏ ω1 + ω2 2 |0e⟩j⟨0e| + ℏω1|-1e⟩j⟨-1e| + ℏω2|1e⟩j⟨1e| (A2) so that our Hamiltonian becomes ˆHI = − ℏ∆cˆa†ˆa + ℏ ηˆa†ei∆dt + H.c. + ℏ NX...

Show all 92 references
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    This explicitly assumes the highly-detuned limit, |∆e| ≫ g, |∆s| , |Ω1| , |Ω2|

    Excited-state elimination We now eliminate the cavity degrees of freedom. This explicitly assumes the highly-detuned limit, |∆e| ≫ g, |∆s| , |Ω1| , |Ω2|. This allows us to adiabatically elim- inate the excited states |±1e⟩, |0e⟩ which evolve on a faster timescale than the grou...

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    For this we assume that the cavity photon lifetime is much shorter than collectively-enhanced processes, κ/ √ N ≫ |gΩ1/∆e|, |gΩ2/∆e|

    Cavity-mode elimination We now eliminate the cavity field to derive a time evolution that only involves the ground states. For this we assume that the cavity photon lifetime is much shorter than collectively-enhanced processes, κ/ √ N ≫ |gΩ1/∆e|, |gΩ2/∆e|. This allows us to ad...

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    For this we use the form of Eq

    Eigenvalue In this first part we discuss some technical details on the derivation of the eigenvalues. For this we use the form of Eq. (22) and apply it onto the eigenstate |ψ⃗k⟩ given by Eq. (23). We will commute the ladder operators in ˆL through |ψ⃗k⟩ which means the goal is...

  4. [11]

    orthogonalize

    Eigenstate overlap We now calculate the overlap between ˆL eigenstates to determine the normalization N⃗k. For µ = 1, the ˆ ci operators satisfy canonical com- mutation relations [ˆ ci, ˆc† j] = δi j, so their overlap is ⟨ψ⃗k′|ψ⃗k⟩ = δ⃗k′ ⃗k, and their normalization takes the ...

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    The overlap and normalization calculations for com- plementary states |ψ⊥ ⃗ n⟩ = N ⊥ ⃗ n( ˆd† 1)n1 ( ˆd† 2)n2 ( ˆd† 3)n3 |0⟩

    Complementary states We now discuss properties of the orthogonal comple- ment of the eigenspaces. The overlap and normalization calculations for com- plementary states |ψ⊥ ⃗ n⟩ = N ⊥ ⃗ n( ˆd† 1)n1 ( ˆd† 2)n2 ( ˆd† 3)n3 |0⟩. (B14) are similar to those of ˆL eigenstates, but we ...

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    In the µ = 1 limit, finding the eigenstates of ˆKz also finds the eigenstates of ˆL

    = √Γc √ 2 ˆKz + χ . In the µ = 1 limit, finding the eigenstates of ˆKz also finds the eigenstates of ˆL. Now, we rephrase the time evolution of the system as rotating the states from a simultaneous ˆJ 2, ˆJz eigenstate into a simultaneous ˆJ 2, ˆKz, eigenstate. The degeneracy ...

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    J. G. Bohnet, B. C. Sawyer, J. W. Britton, M. L. Wall, A. M. Rey, M. Foss-Feig, and J. J. Bollinger, Quantum spin dynamics and entanglement generation with hun- dreds of trapped ions, Science 352, 1297 (2016)

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    This shortcut is achieved by modifying the amplitude and phase of the cavity-mode drive η

    Cavity-drive modification We now discuss details for the implementation of the shortcut Hamiltonian for ⃗k ∈ {(0, 0, N), (N, 0, 0)}. This shortcut is achieved by modifying the amplitude and phase of the cavity-mode drive η. Keeping ∆e, g, Ω1 constant, we can write this as shif...

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    Additional lasers In this appendix we provide additional details for the shortcut to the ⃗k = (0, N,0) state. The shortcut Hamil- tonian that we use is ˆHs = iα ˆb† −1ˆb1 − ˆb† 1ˆb−1 (E11) for some real scalar α, which would be implemented by additional lasers coupling between...

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