Spin squeezing generated by the anisotropic central spin model

arXiv:2311.11308 · quant-ph · Submitted 2023-11-19 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: Today's paper: "Spin squeezing generated by the anisotropic central spin model".

Mira: The gist The anisotropic central spin model can be mapped to the anisotropic Lipkin-Meshkov-Glick model in the limit where the ratio of transition frequencies between the central spin and the…

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, this paper is called "Spin squeezing generated by the anisotropic central spin model". It looks at how you can actually get high-precision parameter estimation schemes using spin squeezing.

Mira: Right, it investigates spin squeezing and a quantum phase transition in an anisotropic central spin system. The main claim they make is that these kinds of central spin systems can be mapped to the anisotropic Lipkin-Meshkov-Glick model when the ratio of transition frequencies between the central spin and the bath spins goes to infinity.

Lev: Mapping it to the LMG model sounds like a big simplification, but we need to know what that means for real hardware. If this mapping holds in that limit, it suggests a specific type of interaction emerges.

Kai: Exactly, and they say this property induces a one-axis twisting interaction which is what provides this new possibility for generating spin squeezing. They look at generating squeezed states both through the ground state and through the dynamic evolution of the central spin model.

Mira: When they look at the ground state, they find that as the anisotropy parameter decreases, you see that the spin squeezing parameter improves. And its value scales with system size as N to minus two over three.

Lev: Scaling with N to minus two over three is interesting for hardware because it suggests how fast we'd need to evolve a state just to get that level of squeezing on real systems.

Kai: And they also did this numerical simulation for the critical exponent of the quantum Fisher information around the critical point. They found that this value tends toward four over three when both the frequency ratio and system size approach infinity.

Mira: That's a key piece of information because it connects spin squeezing directly to quantum metrology resources near a phase transition, which is where you get high precision for sensing.

Lev: But they also noted that if the frequency ratio is finite, this critical exponent will decay to zero as N goes to infinity. So, they're saying the actual quantum phase transition only happens when both the frequency ratio and system size go toward infinity.

Kai: So, this paper provides a scheme for generating spin-squeezed states and it opens up a new pathway for potential advancements in quantum sensing. It's about how to use these specific spin systems to make better measurements.

Paper summary: Mira: What they do here is look at the anisotropic central spin model, which has coupling strengths where the X and Y directions have different interaction strengths, specifically setting A x equals one plus lambda times A, A y equals one minus lambda times A, and zero for Az.

Lev: So they're starting with a Hamiltonian that describes an ensemble of N identical spin-half particles interacting with a central spin-half particle. The transition frequencies are called Omega and omega, which are those of the central spin and bath spins induced by an external magnetic field.

Kai: When they apply a Schrieffer-Wolff transformation to this anisotropic Hamiltonian in Eq. two they get an effective Hamiltonian in the limit where eta equals Omega over omega tends toward infinity <ref:2311.11308#pg2>.

Mira: That limit is what makes the central spin system exactly equal to the Hamiltonian of the anisotropic Lipkin-Meshkov-Glick model. This mapping shows that when you have that ratio of transition frequencies going to infinity, your central spin system acts like it's inducing a one-axis twisting interaction.

Lev: That one-axis twisting interaction is what dynamically generates spin squeezing, so this transformation gives us a clear dynamic process to look at for squeezing generation.

Kai: They separately looked at generating spin-squeezed states using the ground state and also through the dynamic evolution of the central spin model itself. And in both cases, they found that the anisotropy parameter decreasing makes that spin squeezing parameter smaller, which means it gets better when lambda is small.

Mira: Specifically, for the isotropic case where lambda equals zero, they show that under certain conditions, like when g is less than tilde two over square root of delta times one plus lambda times one minus lambda, the ground state itself is a spin-squeezed state with parameters scaling as one to the power of two.

Lev: So for real hardware running this, you'd need to make sure your coupling strengths and transition frequencies are tuned such that that condition holds for the ground state to be squeezed.

Kai: And when the frequency ratio is large enough, those two spin squeezing parameters become essentially consistent with those of the LMG model. It means you can use both methods if you have a sufficiently large frequency ratio.

Paper summary: Mira: The paper also explored a quantum phase transition in this anisotropic central spin model, treating its quantum criticality as a resource for quantum metrology. They numerically calculated the critical exponent of the quantum Fisher information around that critical point by numerical simulation.

Lev: So they're looking at how you can use these transitions for better sensing, which is what we want to see in practical applications where precision matters.

Kai: The numerical simulation found that this critical exponent tends toward four over three as the frequency ratio and the system size approach infinity. This result ties the spin squeezing directly to how well you can estimate a parameter near that critical point.

Mira: That's a big result because it shows a direct link between the quantum phase transition physics and the quantum metrology you get from it.

Lev: But remember, they also said if the frequency ratio is finite, this critical exponent decays to zero as N goes to infinity, meaning that for the transition to actually happen in this context, both those ratios need to go toward infinity.

Kai: So what's left is the dynamic generation of squeezed states through an initial state of one down spin tensor product with a spin coherent state. They solved the exact solution for this dynamic process and got some scaling laws.

Mira: The optimal squeezing parameter in this dynamic process scales as N to minus two over three, and the corresponding squeezing time scales as N to minus two over three for the one-axis twisting interaction when that frequency ratio is sufficiently large.

Lev: If you're trying to build a circuit based on this, you need to keep in mind that those scaling laws for squeezing time are tied directly to the one-axis twisting case when eta is sufficiently large.

Kai: So overall, the paper shows that under fixed N, the most effective way to generate an optimal spin-squeezed state is through the dynamic evolution of the isotropic central spin model. And it really emphasizes making that frequency ratio between those two spins as large as possible.

Mira: This work offers a new potential pathway for generating spin-squeezed states in practical physical systems, and it also gives us a fresh perspective on the relationship between spin squeezing and quantum criticality.

Lev: It's a promising scheme because it suggests that we can create these states in physical systems, and it paves the way for potential advancements in quantum sensing.

Conclusion: Kai: So, we're wrapping up this paper on "Spin squeezing generated by the anisotropic central spin model". It's about how this specific type of central spin system can actually generate good quantum resources for sensing.

Mira: Right, it boils down to taking a complex central spin setup and showing it can be mapped to something simpler when you crank up the ratio of transition frequencies.

Lev: From my side, the main thing I'm looking at is that this mapping turns a complicated interaction into a one-axis twisting kind of thing, which is what we need for dynamic squeezing.

Kai: So, if we can engineer that high frequency ratio in our physical setup, it opens up a new path for creating these squeezed states dynamically.

Mira: The authors show two ways to get squeezing—using the ground state and using the time evolution of the central spin—and they both point toward that large frequency ratio being essential.

Lev: And they give us some hard numbers on that dynamic part, showing how fast you need to evolve, scaling with N to minus two over three for the best result in this one-axis twisting case.

Kai: That scaling is pretty concrete for hardware design; it tells you exactly what kind of evolution time your system needs to achieve optimal squeezing.

Mira: And they also found that the critical exponent for quantum Fisher information, which relates to how much precision you get near a phase transition, hits four over three when everything scales up infinitely.

Lev: But there's a catch there, and this is important for real hardware; if you keep the frequency ratio finite while making N really big, that critical exponent drops to zero.

Kai: So the authors are saying that to see that strong quantum metrology benefit tied to squeezing, you have to push both the system size and that frequency ratio toward infinity together.

Mira: It really frames spin squeezing not just as a static property but as something deeply connected to the quantum critical behavior of these systems.

Lev: It suggests we need very specific control over the bath and central spin interactions if we want to utilize this mechanism for error correction overhead or actual measurement precision.

Graduate School of China Academy of Engineering Physics

quant-ph

Submitted: 2023-11-19

Updated: 2023-11-28

Comments: 10 pages, 4 figures

Journal ref: Phys. Rev. A 109, 052618 (2024)

DOI: 10.1103/PhysRevA.109.052618

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 83/100

The gist: The gist The anisotropic central spin model can be mapped to the anisotropic Lipkin-Meshkov-Glick model in the limit where the ratio of transition frequencies between the central spin and the spin

Key concepts

Spin Squeezing: Spin squeezing is a quantum resource used to achieve high-precision measurements beyond the standard limits. It involves preparing a quantum state where the uncertainty (variance) of one spin component is reduced below what is possible for a classical system, allowing for better estimation of parameters.
Anisotropic Central Spin Model: This model describes a central spin interacting with an ensemble of N other spins, where the interaction strengths in different directions (X and Y) are unequal. This asymmetry allows the system to generate specific quantum dynamics, such as a one-axis twisting interaction when certain frequency ratios are high.
One-Axis Twisting (OAT) Interaction: The OAT interaction is a type of quantum evolution that drives the system towards states with reduced uncertainty in one direction. In this context, the mapping to the Lipkin-Meshkov-Glick model shows that this interaction is dynamically generated by increasing the ratio of transition frequencies between the central spin and its environment.

Terminology

Spin Squeezing — 93× in this paper · explained in 3 episodes
Phase Transition — 26× in this paper · explained in 2 episodes
Entanglement — 13× in this paper · explained in 4 episodes
Fisher Information — 12× in this paper · explained in 4 episodes
Anisotropy — 7× in this paper · explained in 4 episodes
Quantum sensing — 5× in this paper · explained in 2 episodes
Effective Hamiltonian — 4× in this paper · explained in 2 episodes
Phase Transitions — 4× in this paper · explained in 2 episodes
Fidelity — 3× in this paper · explained in 2 episodes
Core — 2× in this paper · explained in 2 episodes
Generator — 2× in this paper · explained in 2 episodes
Quantum Fisher Information (QFI) — 2× in this paper · explained in 16 episodes
Scaling Laws — 2× in this paper · explained in 4 episodes
Coherence — 1× in this paper · explained in 2 episodes
Hubbard Model — 1× in this paper · explained in 3 episodes
Superradiance — 1× in this paper · explained in 3 episodes

Summary

The gist The anisotropic central spin model can be mapped to the anisotropic Lipkin-Meshkov-Glick model in the limit where the ratio of transition frequencies between the central spin and the spin bath tends towards infinity, which induces a one-axis twisting interaction and provides a new possibility for generating spin squeezing<ref:2311.11308#pg8>.

Model Mapping and Interaction

The central spin model is described by a Hamiltonian involving an ensemble of N identical spin- 1/2 particles interacting with a central spin- 1/2 particle<ref:2311.11308#pg3>. The anisotropic version of this model is defined by coupling strengths where the X and Y directions have different interaction strengths, specifically setting A x = (1 + lambda) A, A y = (1 - lambda) A, and A z = 0<ref:2311.11308#pg3>. Through a Schrieffer-Wolff transformation applied to the anisotropic Hamiltonian in Eq. (2), the authors obtain an effective Hamiltonian in the limit of η = Ω/ω → ∞ that is exactly the Hamiltonian of the anisotropic Lipkin-Meshkov-Glick (LMG) model<ref:2311.11308#pg4>. This mapping demonstrates that when the ratio of transition frequencies tends to infinity, the central spin system can be regarded as inducing a one-axis twisting (OAT) interaction, which dynamically generates spin squeezing<ref:2311.11308#pg7>.

Spin Squeezing Analysis

The paper investigates spin squeezing using two definitions proposed by Kitagawa et al. and Wineland et al., defined as ξ 2 S=4min Δ 2 I n⊥/N and ξ 2 R=Nmin Δ 2 I n⊥/E<ref:2311.11308#pg4>. For the isotropic case (lambda = 0), the ground state is a spin-squeezed state under certain conditions, with parameters scaling as ξ 2 S=ξ 2 R=1 when g. For the anisotropic case, the spin squeezing parameter ξ 2 S is shown to be less than 1 when g <˜ 2/√Δ(1 + lambda)(1 - lambda), implying that a spin-squeezed state can be generated by preparing the ground state of the anisotropic central spin model with a large frequency ratio η<ref:2311.11308#pg6>. Furthermore, when the frequency ratio is sufficiently large, the two spin squeezing parameters become essentially consistent with those of the LMG model<ref:2311.11308#pg5>.

Quantum Phase Transitions and Critical Exponents

The study explores a quantum phase transition (QPT) in this anisotropic central spin model, viewing its quantum criticality as a resource for quantum metrology<ref:2311.11308#pg6>. The critical exponent of the quantum Fisher information (QFI) around the critical point is numerically calculated, finding that it tends to 4/3 as the frequency ratio and the system size approach infinity<ref:2311.11308#pg8>. However, when the frequency ratio is finite, this critical exponent will decay to zero as N tends to infinity, implying that the quantum phase transition will not occur unless both η and N tend towards infinity<ref:2311.11308#pg8>.

Dynamical Generation of Squeezed States

The paper also examines spin squeezing through a dynamical approach, starting with an initial state of the form ↓⟩ ⊗ ψcs⟩, where ψcs⟩ is a spin coherent state<ref:2311.11308#pg5>. The exact solution of this dynamical process yields the time evolution of the spin squeezing parameters, with the optimal squeezing parameter scaling as ξ 2 S,min ∼ N(-2/3) and the corresponding squeezing time scaling as tmin ∼ N(-2/3) for the OAT interaction when η is sufficiently large<ref:2311.11308#pg5>. This confirms that the power-law scalings of ξ 2 S,min and tmin are the same as those of the one-axis twisting case when η is sufficiently large<ref:2311.11308#pg5>.

Conclusion

The analysis concludes that under fixed N, the most effective method for generating the optimal spin-squeezed state is through the dynamic evolution of the isotropic central spin model, emphasizing that it is crucial to make the ratio η between the frequency of the central spin and that of the bath spin as large as possible<ref:2311.11308#pg7>. The work provides a new potential pathway for generating spin-squeezed states in practical physical systems, simultaneously offering a fresh perspective for our understanding of the relationship between spin squeezing and quantum criticality<ref:2311.11308#pg8>.

Acknowledgments

We acknowledge Dr. Zhucheng Zhang for valuable suggestions<ref:2311.11308#pg8>. This work was supported by the National Natural Science Foundation of China (Grants No. 12088101 and No. U2330401) and Science Challenge Project (Grant No. TZ2018005)<ref:2311.11308#pg8>.

Appendix A: Derivation of Mapping Between Central Spin Model and LMG Model

The derivation of the mapping involves rewriting the Hamiltonian as H = H0 + AV, where H0 is diagonal with respect to the spin subspace while V is off-diagonal<ref:2311.11308#pg8>. Applying a Schrieffer-Wolff transformation leads to an effective Hamiltonian in Eq. (7) which corresponds precisely to the Hamiltonian of the anisotropic LMG model in the limit of η → ∞<ref:2311.11308#pg8>.

Appendix B: Derivation of Spin Squeezing Parameter in Dynamical Process

The derivation for the spin squeezing parameter in the dynamical process involves approximations such as P n↓ ≃ e(iΩnt/2) and P n↑ ≃ 0 when η is sufficiently large<ref:2311.11308#pg8>. The analytical solution for the time evolution of the spin squeezing parameter ξ 2 S is obtained by substituting Eq. (B9), Eq. (B10), and Eq. (B11) into Eq. (19)<ref:2311.11308#pg8>. This substitution results in an analytical solution for the time evolution of the spin squeezing parameter ξ 2 S<ref:2311.11308#pg8>.

Appendix A: Derivation of Mapping Between Central Spin Model and LMG Model (Cont.)

The block-off-diagonal part of H' must be zero up to second order in A, leading to the relationship [H0, S] = -V<ref:2311.11308#pg8>. The generator S satisfying this condition is found to be S = -i(1 + λ)/Ω I x σ y + i(1 - λ)/Ω I y σ x<ref:2311.11308#pg8>. In the limit of η → ∞, the Hamiltonian in Eq. (A4) becomes H' = H0 + A 2/2 [V, S], which precisely corresponds to the Hamiltonian of the anisotropic LMG model<ref:2311.11308#pg8>.

Appendix B: Derivation of Spin Squeezing Parameter in Dynamical Process (Cont.)

For the ground state of the LMG model, it is established that ⟨I x⟩ = ⟨I y⟩ = 0<ref:2311.11308#pg8>. This leads to a minimum variance for the perpendicular spin components given by min Δ 2 I n⊥ = -1/2(n - N 2/2 + N 4/8 + N 4)<ref:2311.11308#pg8>. Utilizing this result, Eq. (B16) and Eq. (B17) are obtained<ref:2311.11308#pg8>. The derivation of the time evolution of spin squeezing parameters ξ 2 S and ξ 2 R follows from approximations such as Ω n ≃ Ω, P n↓ ≃ e(iΩnt/2), and P n↑ ≃ 0<ref:2311.11308#pg8>.

**Appendix B: Derivation of Spin Squeezing Parameter in Dynamical Process (Cont.

Improvements for AI systems

textbfIncreased Precision Parameter Estimation in Quantum Sensing Systems: The improved AI system can achieve high-precision parameter estimation schemes by leveraging spin squeezing generated by anisotropic central spin models, as this scheme can induce a one-axis twisting interaction and provides a new possibility for generating spin squeezing.
textbfReal-time Quantum Resource Optimization: The system can dynamically generate optimal states by utilizing the time evolution described in Eq. (18), which shows how the spin squeezing parameter improves as the anisotropy parameter decreases and its scaling with system size as N−2⁄3.
textbfEnhanced Quantum Phase Transition Detection: By calculating the critical exponent of the quantum Fisher information (QFI) around a critical point, the AI can detect phase transitions in anisotropic central spin systems, finding that the critical exponent of the quantum Fisher information around the critical point by numerical simulation, and find its value tends to 4/3 as the frequency ratio and the system size approach infinity.
textbfModel Mapping for Predictive Analysis: The system can rigorously map complex physical models, such as "the anisotropic central spin model can be mapped to the anisotropic Lipkin-Meshkov-Glick (LMG) model in the limit where the ratio of transition frequencies between the central spin and the spin bath tends towards infinity."
textbfAdaptive Squeezing Strategy: The AI can select a state generation strategy based on system parameters; for fixed N, it finds that the most effective method for generating the optimal spin-squeezed state is through the dynamic evolution of the isotropic central spin model when the ratio η between the frequency of the central spin and that of the bath spin as large as possible.