It is well established that the optimal spin squeezing under a one-axis-twisting Hamiltonian follows a scaling law of
J−2/3 for
J 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−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.
Squeezed spin states possess unique quantum correlation or entanglement and are significantly promising for advancing quantum information processing and quantum metrology. In recent back-to-back publications [C. Gross et al., Nature (London) 464, 1165 (2010) and Max F. Riedel et al., Nature (London) 464, 1170 (2010)], reduced spin fluctuations are observed leading to spin squeezing at -8.2 and -2.5 dB, respectively, in two-component atomic condensates exhibiting one-axis-twisting interactions. The noise reduction limit for the one-axis twisting scales as ∝1/N(2/3), which for a condensate with N∼10(3) atoms is about 100 times below the standard quantum limit. We present a scheme using repeated Rabi pulses capable of transforming the one-axis-twisting spin squeezing into the two-axis-twisting type, leading to Heisenberg limited noise reduction ∝1/N or an extra tenfold improvement for N∼10(3).
In two-component Bose-Einstein condensates, the one-axis twisting Hamiltonian leads to spin squeezing with a limitation that scales with the number of atoms as ${N}^{\ensuremath{-}2/3}$. We propose a scheme to transform the one-axis twisting Hamiltonian into a two-axis twisting Hamiltonian, resulting in enhanced spin squeezing proportional to ${N}^{\ensuremath{-}1}$ approaching the Heisenberg limit. Instead of pulse sequences, only one continuous driving field is required to realize such a transformation, thus the scheme is promising for experimental realization of a two-axis twisting Hamiltonian. Quantum information processing and quantum metrology may benefit from this method in the future.
Ultracold atoms in an ultrahigh-finesse optical cavity are a powerful platform to produce spin squeezing since photon of cavity mode can induce nonlinear spin-spin interaction and thus generate a one-axis twisting Hamiltonian
HOAT=qJx2, whose corresponding maximal squeezing factor scales as
N−2/3, where
N is the atomic number. On the contrary, for the other two-axis twisting Hamiltonian
HTAT=q(Jx2−Jy2), the maximal squeezing factor scales as
N−1, approaching the Heisenberg limit. In this paper, inspired by recent experiments of cavity-assisted Raman transitions, we propose a scheme, in which an ensemble of ultracold six-level atoms interacts with two quantized cavity fields and two pairs of Raman lasers, to realize a tunable two-axis spin Hamiltonian
. For proper parameters, the above one- and two- axis twisting Hamiltonians are recovered, and the scaling of
N−1 of the maximal squeezing factor can occur naturally. On the other hand, in the two-axis twisting Hamiltonian, spin squeezing is usually reduced when increasing the effective atomic resonant frequency
ω0. Surprisingly, we find that by combined with the dimensionless parameter
χ(>−1), the effective atomic resonant frequency
ω0 can enhance spin squeezing largely. These results are benefit for achieving the required spin squeezing in experiments.
Spin squeezed states are a class of entangled states of spins that have practical applications to precision measurements. In recent years spin squeezing with one-axis twisting (OAT) has been demonstrated experimentally with spinor Bose-Einstein condensates (BECs) with more than
103 atoms. Although the noise is below the standard quantum limit, the OAT scheme cannot reduce the noise down to the ultimate Heisenberg limit. Here we propose an experimentally feasible scheme based on optimized quantum control to greatly enhance the performance of OAT to approach the Heisenberg limit, requiring only an OAT Hamiltonian and the use of several coherent driving pulses. The scheme is robust against technical noise and can be readily implemented for spinor BECs or trapped ions with current technology.
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.