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arXiv:2302.09829 · Physical review. B./Physical review. B

Spin squeezing in open Heisenberg spin chains

Tanausú Hernández Yanes, Giedrius Žlabys, Marcin Płodzień, D. Burba, Mažena Mackoit-Sinkevičienė, Emilia Witkowska, Gediminas Juzeliūnas

Published Sep 5, 2023• 16 citations
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Abstract

Spin squeezing protocols successfully generate entangled many-body quantum states, the key pillars of the second quantum revolution. In our recent work [Phys. Rev. Lett. 129, 090403 (2022)] we showed that spin squeezing described by the one-axis twisting model can be generated in the Heisenberg spin-$1/2$ chain with periodic boundary conditions when accompanied by a position-dependent spin-flip coupling induced by a single laser field. In this work, we show analytically that the change in boundary conditions from the periodic to the open ones significantly modifies spin squeezing dynamics. A broad family of twisting models can be simulated by the system in the weak-coupling regime, including one- and two-axis twisting under specific conditions, providing the Heisenberg level of squeezing and acceleration of the dynamics. Our analytical findings are confirmed by full numerical simulations.

DOI
10.1103/physrevb.108.104301

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Spin squeezing protocols successfully generate entangled many-body quantum states, the key pillars of the second quantum revolution. In our recent work [Phys. Rev. Lett. 129, 090403 (2022)] we showed that spin squeezing described by the one-axis twisting model can be generated in the Heisenberg spin-1/21/21/2 chain with periodic boundary conditions when accompanied by a position-dependent spin-flip coupling induced by a single laser field. In this work, we show analytically that the change in boundary conditions from the periodic to the open ones significantly modifies spin squeezing dynamics. A broad family of twisting models can be simulated by the system in the weak-coupling regime, including one- and two-axis twisting under specific conditions, providing the Heisenberg level of squeezing and acceleration of the dynamics. Our analytical findings are confirmed by full numerical simulations.
We propose an approach to produce spin squeezed states of a large number of nitrogen-vacancy centers in diamond nanostructures coupled to an optical cavity. Unlike the previous squeezing method proposed by Bennett et al. [Phys. Rev. Lett. 110, 156402 (2013)], which is limited by phonon number fluctuations due to the existence of phonon-spin entanglement, our proposal can completely erase the entanglement between spins and hybrid phonon-photon mode mediating the effective spin-spin interaction, and thus achieves unitary one-axis-twisting interactions between nitrogen-vacancy centres, yielding a squeezing scaling J−2/3J^{-2/3}J−2/3, where J is the total angular momentum. We found that, under certain conditions, our method has the potential to enhance the spin-spin nonlinear interactions. We also proposed a scheme utilizing repeatedly applying the one-axis-twisting evolution to two orthogonal spin directions, which enables the transformation of the one-axis-twisting interactions into two-axis-twisting type, and therefore leads to the spin squeezing with Heisenberg-limited scaling J−1J^{-1}J−1. Taking into account the noise effects of spin dephasing and relaxtion, we found that the proposed approaches are robust against imperfections.
It is well established that the optimal spin squeezing under a one-axis-twisting Hamiltonian follows a scaling law of ${J}^{\ensuremath{-}2/3}$ for JJJ interacting atoms after a quench dynamics. Here, we prove analytically and numerically that the spin squeezing of the ground state of the one-axis-twisting Hamiltonian actually reaches the Heisenberg limit ${J}^{\ensuremath{-}1}$. By constructing a bilinear Bogoliubov Hamiltonian with the raising and lowering spin operators, we exactly diagonalize the spin Bogoliubov Hamiltonian, which includes the one-axis twisting Hamiltonian as a limiting case. The ground state of the spin Bogoliubov Hamiltonian exhibits excellent spin squeezing, which approaches the Heisenberg limit in the case of the one-axis-twisting Hamiltonian. It is possible to realize experimentally the spin-squeezed ground state of the one-axis-twisting Hamiltonian in dipolar spinor condensates, ultracold atoms in optical lattices, spins in a cavity, or alkali atoms in a vapor cell.
It is well established that the optimal spin squeezing under a one-axis-twisting Hamiltonian follows a scaling law of J−2/3J^{-2/3}J−2/3 for JJJ interacting atoms after a quench dynamics. Here we prove analytically and numerically that the spin squeezing of the ground state of the one-axis-twisting Hamiltonian actually reaches the Heisenberg limit J−1J^{-1}J−1. By constructing a bilinear Bogoliubov Hamiltonian with the raising and lowering spin operators, we exactly diagonalize the spin Bogoliubov Hamiltonian, which includes the one-axis twisting Hamiltonian as a limiting case. The ground state of the spin Bogoliubov Hamiltonian exhibits wonderful spin squeezing, which approaches to the Heisenberg limit in the case of the one-axis twisting Hamiltonian. It is possible to realize experimentally the spin squeezed ground state of the one-axis-twisting Hamiltonian in dipolar spinor condensates, ultracold atoms in optical lattices, spins in a cavity, or alkali atoms in a vapor cell.
The creation and manipulation of quantum entanglement is central to improving precision measurements. A principal method of generating entanglement for use in atom interferometry is the process of spin squeezing whereupon the states become more sensitive to SU(2)\text{SU}(2)SU(2) rotations. One possibility to generate this entanglement is provided by one-axis twisting (OAT), where a many-particle entangled state of one degree of freedom is generated by a nonlinear Hamiltonian. We introduce a method which goes beyond OAT to create squeezing and entanglement across two distinct degrees of freedom. We present our work in the specific physical context of a system consisting of collective atomic energy levels and discrete collective momentum states, but also consider other possible realizations. Our system uses a nonlinear Hamiltonian to generate dynamics in SU(4)\text{SU}(4)SU(4), thereby creating the opportunity for dynamics not possible in typical SU(2)\text{SU}(2)SU(2) one-axis twisting. This leads to three axes undergoing twisting due to the two degrees of freedom and their entanglement, with the resulting potential for a more rich context of quantum entanglement. The states prepared in this system are potentially more versatile for use in multiparameter or auxiliary measurement schemes than those prepared by standard spin squeezing.
Squeezed spin states have important applications in quantum metrology and sensing. It has been shown by S\o{}rensen and M\o{}lmer [Phys. Rev. A 66, 022314 (2002)] that an effective one-axis-twisting interaction can be realized in a cavity setup via a double off-resonance stimulated Raman scattering, resulting in a noise reduction scaling $\ensuremath{\propto}1/{N}^{2/3}$ with NNN being the atom number. Here, we show that, by making an appropriate change of the initial input spin state, it is possible to produce a one-axis-twisting spin squeezing via a single off-resonance stimulated Raman scattering, which thus can greatly simplify the realistic implementation. We also show that the one-axis-twisting interaction can be transformed into a more efficient two-axis-twisting interaction by rotating the collective spin while coupling to the cavity, yielding a Heisenberg limited noise reduction $\ensuremath{\propto}1/N$. Considering the noise effects due to atomic decoherence and cavity decay, we find that substantial squeezing is still attainable with current laboratory techniques.
Spin squeezing in collective atomic ensembles enables quantum-enhanced metrology by reducing noise below the standard quantum limit through nonlinear interactions. Extending the one-axis and two-axis twisting paradigms of Kitagawa and Ueda, we introduce a general class of three-axis spin squeezed states within the anisotropic Lipkin-Meshkov-Glick model. The model features direction-dependent quadratic couplings that interpolate between uniaxial and biaxial regimes and can be interpreted as an asymmetric quantum rotor. Using semiclassical dynamics, Majorana representations, and Husimi-Q distributions, we analyze the structure and metrological properties of the resulting states. The three-axis framework reproduces the known N^(-2/3) scaling of one-axis twisting and the Heisenberg-limited N^(-1) scaling of two-axis twisting, while allowing additional tunability and enhanced entanglement generation in low-spin systems. We further show that tuning the anisotropy parameters induces ground-state and excited-state quantum phase transitions, including a second-order transition associated with level clustering and critical dynamics. These results unify spin squeezing, quantum criticality, and rotor analogies, and suggest implementations in Rydberg arrays and cavity-QED platforms for precision sensing and quantum simulation.