Optimal Displacement Sensing with Spin-Dependent Squeezed States

TL;DR

This work introduces spin-dependent squeezed states as optimal references for quantum displacement sensing, achieving Heisenberg-limited performance for both amplitude and joint real-imaginary parameter estimation in many-body spin-boson systems. It provides explicit, experimentally feasible measurement protocols based on a time-reversal sequence and demonstrates a fast, scalable trapped-ion scheme to prepare these states using four-tone sideband driving, analyzed with Magnus and BCH formalisms. Numerics up to N=40 show strong performance: up to 8.7 dB of spin-dependent squeezing for N=20 achieved 15× faster preparation than conventional second-order sideband methods, highlighting practical advantages for sensing weak forces, dark matter searches, and photon-scattering measurements. The framework unifies single- and multi-parameter metrology in spin-boson platforms, showing HL saturation with finite resources and offering pathways to robust, scalable quantum-enhanced sensing in realistic devices.

Abstract

Displacement sensing is a fundamental task in metrology. However, the development of quantum-enhanced sensors that fully utilize the available degrees of freedom in many-body quantum systems remains an outstanding challenge. We propose novel many-body displacement sensing schemes that use spin-dependent squeezed (SDS) states -- hybrid spin-boson states whose bosonic squeezed quadrature is conditioned on an auxiliary spin. We prove that SDS states are \emph{optimal}, i.e. their quantum Cramér-Rao bound saturates the Heisenberg limit. We propose explicit measurement sequences that can be readily implemented in systems such as trapped ions. We also introduce a scalable state-preparation protocol and numerically demonstrate the preparation of $8.7$~dB of spin-dependent squeezing $15$ times faster than the standard approach using second-order sidebands in trapped ions. The potential applications of our sensing protocols range from measuring single-photon scattering to searches for dark matter.

Figures (8)

• Figure 1: Phase-insensitive sensing of a displacement amplitude using spin-dependent squeezed states. (a) Pulse sequence of the sensing protocol. Top (bottom) row shows for $N =1$ spin, schematic representations of the bosonic Wigner function associated with the spin $\ket{1/2}_s$ ($\ket{-1/2}_s$) state. After initialization to $\ket{0}_b \ket{\psi_0}_s \cong \ket{0}_b (\ket{-1/2}_s + \ket{1/2}_s)$, (panels (i) and (ii)), spin-dependent squeezing $\hat{S}\mathopen{(}-\zeta \hat{J}_z\mathclose{)}$ squeezes the bosonic mode conditioned on the spin state (panels (iii) and (iv)). The bosonic mode then undergoes an unknown displacement $\hat{D}(\beta)$ (panels (v) and (vi)). Finally, the spin-dependent squeezing is reversed (panels (vii) and (viii)). The signal's imaginary part, $\beta_{\rm im}$, is exponentially amplified in the $\ket{1/2}_s$ Hilbert space sector, while the real part, $\beta_{\rm re}$, is amplified in the $\ket{-1/2}_s$ sector. Unlike the phase-insensitive amplification protocol of Ref. burdExperimentalSpeedupQuantum2024, here the signal amplification is spin-dependent. Phase-insensitive information about the displacement is accessed via a spin measurement. Without spin-dependent squeezing, the final state is a purely bosonic displaced state (blue dashed circles). (b) Wigner function $W(x,p)$ of the bosonic states (i) $|\psi_{\rm GHZ}^{\rm (B)}\rangle \simeq \ket{-N\zeta/2}_b + \ket{N\zeta/2}_b$ and (ii) $|\psi_{\rm COH}^{\rm (B)}\rangle \simeq \sum_{m-J}^J \sqrt{\binom{N}{m+N/2}} \ket{\zeta m}_b$, with squeezing parameter $\zeta = 0.3$ and $N = 10$ spins. Both Wigner functions feature simultaneous squeezing along orthogonal quadratures, sub-Planck structure, and $Z_4$ symmetry. There is more phase space interference in panel (ii) due to the presence of additional states in the superposition.
• Figure 2: Metrological performance of spin-dependent squeezed coherent spin states, for phase-insensitive estimation of the displacement's magnitude, $|\beta|$. The classical Cramér-Rao bound (CCRB) for the time-reversal protocol and collective spin measurement (described in the main text) is plotted for increasing numbers of spins, $N$ (solid colored lines). The CCRB is below the standard quantum limit (SQL, dashed black line) for all $N$ at $\langle \hat{n} \rangle \gtrsim 1$. Moreover, the CCRB for $N=1$ is $22$ dB below the SQL at $\langle \hat{n} \rangle \approx 10$. The SQL is defined as the QCRB for bosonic coherent states, which for sensing of $|\beta|$ is $\rm{SQL} = 1/4$supp. The CCRB is further compared to the quantum Cramér-Rao bound (QCRB, solid black line). At $\langle \hat{n} \rangle \gtrsim 1$, the CCRB for all $N$ follows Heisenberg scaling, $\propto 1/\langle \hat{n} \rangle$. In this same regime, the CCRB for $N=1$ spin is a factor of two larger than the QCRB. The prefactor decreases to one in the $N \rightarrow \infty$ limit.
• Figure 3: Joint estimation of the real and imaginary components of a displacement signal using spin-dependent squeezed states. (a) Example measurement protocol for joint parameter displacement estimation using spin-dependent squeezing, which uses two additional ancillary spins initialized in $\ket{+}_\mathrm{1}$ and $\ket{+}_\mathrm{2}$, respectively. Spin-dependent displacements conditioned on each ancilla map information about the real and imaginary components of the displacement signal to the ancillas, which are then individually measured. (b) Metrological performance of the measurement scheme for joint displacement parameter estimation. The classical Cramér-Rao bound (CCRB, colored lines) is shown for $N = 1$ spin at various spin-dependent displacement strengths $g$. For each value of $g$, we vary the squeezing parameter in the range $\zeta \in [0.2,1.4]$ in steps of $0.2$. Solid lines are obtained from numerics, dashed lines are analytics of Eq. \ref{['eq:CCRB_MP']}. The CCRB is compared to the standard quantum limit (SQL, dashed black line), defined by the QCRB for bosonic coherent states, $\rm{SQL}=1/2$supp. For $g = 0.6$, the CCRB is below the SQL for mode occupations $\langle \hat{n} \rangle \gtrsim 1$. For fixed $\langle \hat{n} \rangle$, the CCRB approaches the quantum Cramér-Rao Bound (QCRB, solid black line) as $g$ increases.
• Figure 4: Experimental protocol for engineering fast and scalable spin-dependent squeezing in a trapped ion system. (a) We consider a system of $N$ trapped ions coupled via laser interactions to their collective motional mode. (b) Energy diagram for $N = 1$ spin, without loss of generality. Multiple laser tones are tuned close to the first-order sidebands (1SB) that are $\pm \omega_{\rm COM}$ from the carrier transition $\omega_q$. A first bichromatic field driving interactions in the $\hat{J}_x$ basis is detuned by $\pm (\omega_{\rm COM} + \Delta)$, while a second bichromatic field driving interactions in the $\hat{J}_y$ basis is detuned by $\pm (\omega_\mathrm{COM} - \Delta)$ (grey lines). The resulting effective spin-dependent squeezing has an interaction strength $\propto \Omega^2 \eta^2/\Delta^2$, offering a speed-up for appropriately chosen $\Delta$ over alternative approaches that use second-order sideband (2SB) interactions with a weaker coupling strength $\propto \Omega \eta^2$. (c) The laser detuning, phase and interaction strengths are dynamically modulated to obtain the desired spin-dependent squeezing interaction and to minimize unwanted errors from higher-order terms.
• Figure 5: Performance of first-order sideband protocol to engineer spin-dependent squeezing in a crystal of $N$ ions. (a) Minimum time $t_{\rm min}$ to prepare a spin-dependent squeezed state $\ket{\psi_t} = \hat{S}(\zeta \hat{J}_z) \ket{\psi_{\rm GHZ}}$ with fidelity $\mathcal{F} > 0.99$ versus squeezing parameter $z = \zeta N/2$. We use a typical coupling strength $g = 2\pi {\times} 5/\sqrt{N} \text{ kHz}$ and set the phase space direction $\phi_1 = \pi$ and the number of phase loops per segment $\ell = 1$. The minimum time is numerically optimized for increasing numbers of ions, $N$, and is compared to the theoretical speed limit $t_f^*$ (grey line). We further compare against the minimum time using second-order sidebands (2SB, black dashed line) using $\eta = 0.05$. In the inset, we fix the squeezing to $z=2.5$ and plot the minimum time $t_{\rm min}$ for increasing $N$, up to $N = 40$ spins. Solid black line is a least-squares fit to $\propto 1/\sqrt{N}$. (b) Detuning $\Delta$ versus the target squeezing $z$ for increasing number of spins, $N$. The inset plots the number of repetitions, $P$, of the stroboscopic modulation for varying squeezing $z$.