Tailored generation of quantum states in an entangled spinor interferometer to overcome detection noise

We theoretically investigate how entangled atomic states generated via spin-changing collisions in a spinor Bose-Einstein condensate can be designed and controllably prepared for atom interferometry that is robust against common technical issues, such as limited detector resolution. We use analytic and numerical treatments of the spin-changing collision process to demonstrate that triggering the entangling collisions with a small classical seed rather than vacuum fluctuations leads to a more robust and superior sensitivity when technical noise is accounted for, despite the generated atomic state ideally featuring less metrologically useful entanglement. Our results are relevant for understanding how entangled atomic states are best designed and generated for use in quantum-enhanced matter-wave interferometry.

Figure 1

(a) Interferometric sequence. (i) Illustration of a Mach-Zehnder scheme, where entangled pairs of $m_F=\pm 1$ atoms are the input state of the upper and lower paths. Beam splitters are realized by coherently mixing the $m_F=\pm 1$ modes [see (b)] and a relative phase shift $\phi$ is imprinted. Accruing a phase shift due to gravity requires combining state-dependent momentum kicks imparted by laser pulses (red curves) with the internal-state beam splitters. An estimate of $\phi$ is obtained by measuring populations ${\widehat{n}}_{\pm 1}$ at the outputs. (ii) In an equivalent Ramsey sequence, the beam-splitter and phase-shift operations correspond to rotations (axes indicated) of the quantum state on a collective Bloch sphere, illustrated using the Wigner phase-space distribution of an example squeezed state (see Sec. 3 for details). (b) Example internal state dynamics for $F=1$ sodium spinor BEC. Initialize: Coherent seeding is implemented via microwave pulses resonantly tuned to an ancillary $F=2$ manifold that transfer a fraction of the total population to the $m_F=\pm 1$ modes. Pulse 1 transfers $n_s$ atoms from $(F,m_F)=(1,0)\to (2,0)$ and pulse 2 completes the transfer to (1,1) [split equally to $(1,-1)$ for dual seeding]. Entangle: Subsequent spin-changing collisions between $m_F=0$ atoms produce entangled pairs in $m_F=\pm 1$ to realize $|{\psi }_t^{s,d}\rangle$. Beam splitter: Coherent mixing of the $m_F=\pm 1$ modes is also implemented via resonant microwaves. Pulse 1 transfers the entire population from $(1,-1)\to (2,0)$ before pulse 2 implements coherent 50:50 mixing of $(2,0)\leftrightarrow (1,1)$. Finally, pulse 3 returns the remaining population from $(2,0)\to (1,-1)$.

Figure 2

(a) $\mathrm{SU}(2$) Wigner distributions $W_{|\psi \rangle }^J\left(\mathbf{J}\right)$ for $|{\psi }_t^s\rangle$ projected into the $J=10$ sector for (i) $n_s=0.1$ and (ii) $n_s=4$ and fixed $N_{+}={10}^2$. The distributions are renormalized to account for the projection onto the $J=10$ sector. (b) QFI, $F_Q$, as a function of (i) collision duration $\tau$ and (ii) total population $N_{+}$ in $m_F=\pm 1$ modes. In panel (ii), we plot the normalized QFI per particle, where $F_Q/N_{+}>1$ indicates sub-SQL sensitivity when only the $m_F=\pm 1$ populations are considered. Predictions are from the undepleted pump approximation, Eq. (17), with initial seed ocupation: $n_s=0$ (blue solid lines), $n_s=10$ (red dashed lines), and $n_s=100$ (green dot-dashed lines).

Figure 3

(a) Example interferometric sensitivities as a function of phase shift $\phi$ for different initial conditions: (i) $n_s=0.1$ and (ii) $n_s=4$. The sensitivities are computed from measurements of ${\widehat{J}}_z$ (green lines) and ${\widehat{J}}_z^2$ (blue lines) as labeled. Line style indicates single (solid) and dual seed (dotted) initial conditions. We also plot the case of $n_s=0$ with a ${\widehat{J}}_z^2$ measurement (red dot-dashed line), which saturates the HL, as a reference in both panels. For all data, ${\theta }_s$ is optimally chosen (see main text) and $N_{+}={10}^2$ is fixed. (b) Optimal sensitivity as a function of initial seed occupation $n_s$ and $N_{+}={10}^2$. We indicate with arrows the results of measurements of ${\widehat{J}}_z$ (green lines) and ${\widehat{J}}_z^2$ (blue lines) against the CFI $F_C^{-1}$ [Eq. (27), narrow magenta line] that also saturates the QCRB [Eq. (17), thick black line underlying CFI]. The line styles indicate single (solid) and dual seed (dotted) initial conditions. For clarity, we also indicate the Heisenberg (HL) and standard quantum limits (SQL) for $N_{+}={10}^2$ in all panels.

Figure 4

(a) Optimal sensitivity as a function of seed occupation $n_s$ with fixed detection noise $\sigma =8$. We label results for measurements of ${\widehat{J}}_z$ (green lines) (green solid and dashed lines for single and dual seeds), ${\widehat{J}}_z^2$ (blue lines), and the CFI $F_C^{-1}$ [Eq. (27), magenta line]. Line style indicates single (solid line) and dual (dashed line) seed initial conditions. For reference, we also indicate the sensitivity achievable with a coherent spin state (CSS) including identical detection noise (faded gray line). Inset: Corresponding dynamic range (DNR) depending on seed occupation and measurement signal (same line styles as main panel). Results are indistinguishable for single or dual seed initial conditions. We compare to the ideal ($\sigma =0$) result (33) (black line). (b) Best sensitivity (optimized over both $\phi$ and $n_s$) as a function of detection noise $\sigma$. Line styles are the same as (a) with the additional comparison to the reference case of $n_s=0$ (red dot-dashed line) and measurement of ${\widehat{J}}_z^2$ (other signals are labeled in plot). Inset: Optimal seed occupation $n_s^{\mathrm{opt}}$ as a function of detection noise. The shaded grey regions in (a) and (b) indicates sensitivity below the SQL, ${\left(\mathrm{\Delta }\phi \right)}_{\mathrm{SQL}}^2=1/N_{+}$.

Figure 5

(a) Normalized sensitivity $N_{+}{\left(\mathrm{\Delta }\varphi \right)}^2$ obtained from full quantum dynamics [governed by $\widehat{H}$, Eq. (1)] of an initial BEC of $N={10}^4$ atoms as a function of initial seed occupation $n_s$ and interaction time $\tau$. Measurement signals are indicated on panels. Note that the plotted data saturates the color scale in lower left corner of panels (i.e., $N_{+}{\left(\mathrm{\Delta }\varphi \right)}^2>10\mathrm{dB}$). (b) Optimal sensitivity as a function of seed $n_s$, with $\tau$ chosen such that $N_{+}=1000$ or 2000 (indicated by arrows). Markers indicate results of full numerical simulations for a measurement of ${\widehat{J}}_z$ (circles) or ${\widehat{J}}_z^2$ (squares), while the tracking solid lines are equivalent analytic predictions from the undepleted pump approximation [e.g., Eq. (20)] with $\overline{n}$ and $n_s$ chosen to match numerical results. In all panels of (a) and (b), we include fixed detection noise $\sigma =8$.

Figure 6

Best achievable sensitivity as a function of seed $n_s$ after optimization over interaction time $\tau$. The sensitivity is rescaled relative to the absolute SQL ${\left(\mathrm{\Delta }\phi \right)}_{\mathrm{SQL}}^2=1/N$. We compare results of full numerical simulations for a measurement of ${\widehat{J}}_z$ (green circles) or ${\widehat{J}}_z^2$ (blue squares) with analytic expressions (corresponding blue and green lines overlaying the markers) derived from the undepleted pump approximation [e.g., optimization of Eq. (20)] with $\overline{n}$ matched to numerical simulations. We include fixed detection noise $\sigma =8$. Inset: Sensitivity as a function of $\sigma$ obtained from numerical calculations with optimally chosen initial seed $n_s$ (green squares) compared to $n_s=0$ prediction (red data). Lines (red dashed for $n_s=0$ and green solid for $n_s\ne 0$) are analytic predictions from undepleted pump approximation with matching $N_{+}$.