Cavity-aided nondemolition measurements for atom counting and spin squeezing

Probing the collective spin state of an ensemble of atoms may provide a means to reduce heating via the photon recoil associated with the measurement and provide a robust, scalable route for preparing highly entangled states with spectroscopic sensitivity below the standard quantum limit for coherent spin states. The collective probing relies on obtaining a very large optical depth that can be effectively increased by placing the ensemble within an optical cavity such that the probe light passes many times through the ensemble. Here we provide expressions for measurement resolution and spectroscopic enhancement in such cavity-aided nondemolition measurements as a function of the cavity detuning. In particular, fundamental limits on spectroscopic enhancements in ${}^{87}\mathrm{Rb}$ are considered.

Figure 1

(a) Relevant energy levels. The pseudo-spin-1/2 system typically comprises two metastable states, $|\uparrow \rangle$ and $|\downarrow \rangle$, which are utilized in atomic sensors and clocks. The number of atoms $N_{\uparrow }$ in $|\uparrow \rangle$ modifies the cavity resonance frequency. Initially, we assume that the optically excited state $|e\rangle$ does not couple to $|\downarrow \rangle$ because of dipole selection rules or because the coupling is highly nonresonant, ${\delta }_c\ll {\omega }_{\mathrm{hf}}$, where ${\omega }_{\mathrm{hf}}$ is the frequency separation between $|\uparrow \rangle$ and $|\downarrow \rangle$. (b) A transmitted and/or reflected probe field is monitored to determine the cavity resonance frequencies and hence the number of atoms in $|\uparrow \rangle$. An applied NMR-like microwave rotation can swap the populations between ground states so that a subsequent measurement can also determine the $|\downarrow \rangle$ population $N_{\downarrow }$. The ensemble of atoms is trapped in an intracavity optical lattice (blue-green). The total cavity power decay rate is $\kappa$. Single-atom spontaneous decay from $|e\rangle$ at rate $\Gamma$ leads to photon recoil heating and single-atom wave function collapse. The goal then is to extract collective information from the probe mode more rapidly than the undesired single-atom photon scattering into free space.

Figure 2

Visualization of a nondemolition (ND) (upper right) and a quantum nondemolition (QND) (lower right) measurement. A coherent spin state (CSS) for an ensemble of $N$ spin-1/2 atoms prior to measurement is represented as a Bloch vector (red arrow) of length $N/2$. The quantum noise in the orientation of the Bloch vector is shown as a quasiprobability distribution (red-yellow region) perpendicular to the Bloch vector, with an rms opening angle at the standard quantum limit, $\Delta {\theta }_{\mathrm{SQL}}=1/\sqrt{N}$. An ND measurement in this paper refers to a measurement of the state's $\widehat{z}$ spin projection $J_z=(N_{\uparrow }-N_{\downarrow })/2$ with an rms imprecision, $\Delta J_z<\Delta J_{z,\mathrm{CSS}}=\sqrt{N}/2$, and with the majority of the atoms remaining trapped after the measurement. For example, in the ND measurement shown at the right, the measurement imprecision is $\Delta J_z=0$, but after the measurement the atoms are described by a product state of atoms in spin-up and spin-down due to single-atom state information gained by the environment via free-space scattering of light. We define a QND measurement via the additional requirement that a sufficiently large number of atoms remains in a coherent superposition of spin-up and spin-down such that the resulting state, conditioned on the measurement outcome, has a polar angle uncertainty $\Delta \theta <\Delta {\theta }_{\mathrm{SQL}}$. The definition of QND employed here is related to, but less restrictive than, that in Ref. [23]. The lower-right QND example shows the conditional state as a product state of a squeezed state and the atoms that have been collapsed into spin-up and spin-down.

Figure 3

(a) Graphical representation of the linearization achieved via the Holstein-Primakoff approximation. The atomic subsystem $|\uparrow \rangle$ and $|e\rangle$ is described by a Bloch vector of length $(N_{\uparrow }+N_{\mathrm{e}})/2$. In the regime in which the probe light only weakly excites the atoms such that $N_{\mathrm{e}}\ll N_{\uparrow }$, the two-level system can be described by its projection onto a 2D plane equivalent to that describing a light field. (b) With this approximation, the collective atomic mode may be treated as an equivalent cavity mode (lower, purple mode) whose coupling to the actual cavity (upper, red mode) is governed by a partially reflecting mirror (center) described by a field coupling rate constant $\left(\sqrt{N_{\uparrow }}g\right)$ that depends on the number of atoms in state $|\uparrow \rangle$. The physical mirrors have transmission coefficients described by the field coupling rates $\sqrt{{\kappa }_1}$ and $\sqrt{{\kappa }_2}$, and internal cavity losses are described by $\sqrt{{\kappa }_L}$, such that the total power decay rate is $\kappa ={\kappa }_1+{\kappa }_2+{\kappa }_L$. Decay of the atoms by emission of a photon into free space is described by the field transmission coefficient $\sqrt{\Gamma }$, which is the sum of the field scattering into free-space modes by the atoms in $|\uparrow \rangle$. Because the scattered modes are distinguishable (expanded view of free-space scattering port), it is possible to tell which atoms are in $|\uparrow \rangle$ from the free-space-scattered photons, destroying any coherent superposition between $|\uparrow \rangle$ and another state $|\downarrow \rangle$ (not shown) that may have been prepared for sensing a quantum phase.

Figure 4

Transmitted and reflected probe electric fields from driving the atoms-cavity system through the cavity mode. (a) The electric field phasors trace out circles in the $I,Q$-quadrature plane as the probe detuning ${\delta }_p$ from the dressed-cavity resonance varies from far below to far above resonance. The normalization is chosen such that the reflected electric field goes to 1 when far off resonance. The quantum noise of the probe normalized to the incident electric field is represented as a fuzzy blob with rms diameter $1/|c_i|$. In this illustration, a symmetric cavity ${\kappa }_1={\kappa }_2$ is assumed, so that the circles have the same diameter $\beta$. (b) Corresponding power transmission and reflection signals. (c, d) Typical experimental data (points) and least-squares fits to the $I,Q$-quadrature data (solid and dashed circles) from Ref. [9].

Figure 5

Theoretical scaling of key quantities with cavity detuning ${\delta }_c$ expressed in units of the the collective vacuum Rabi frequency ${\Omega }_{\uparrow }$. (a) The dressed-cavity linewidth ${\kappa }^{\prime }$ in units of the atomic excited state linewidth $\Gamma$. In (a) and (b), the scaling for different cavity finesses is shown for $\kappa /\Gamma =$ 0.01, 1, and 100 (blue, red, and green, respectively). (b) Right (black curve): The rms fluctuation of the dressed-cavity-mode frequency due to projection noise fluctuations $\Delta {\omega }_{+}^{\mathrm{proj}}$ decreases as $1/{\delta }_c$ above ${\delta }_c/{\Omega }_{\uparrow }=1$. The normalization is chosen such that one should multiply by $g/2\sqrt{2}$. Left (red, blue, and green curves): The ratio of the projection noise fluctuation of the cavity mode to the dressed-cavity linewidth $\Delta {\omega }^{\mathrm{proj}}/{\kappa }^{\prime }$ is shown normalized such that the plotted values should be multiplied by $g/\left(2\sqrt{2}\Gamma \right)$. A high ratio is desirable because technical noise may limit the ability to split the probed resonance by more than a fractional amount.

Figure 6

Theoretical scaling of key quantities with cavity detuning ${\delta }_c$ expressed in units of the the collective vacuum Rabi frequency ${\Omega }_{\uparrow }$. In (a) and (b), the scaling for different cavity finesses is shown, giving $\kappa /\Gamma =0.01$, 1, and 100 (blue, red, and green, respectively). (a) Left (blue, red, green): The average number of detected photons needed to resolve the projection noise fluctuations $M_d^{\mathrm{proj}}$ normalized such that the plotted values should be multiplied by ${\Gamma }^2/\left({\eta }_d{g}^2\right)$. Right (black): The ratio of the number of free-space-scattered photons for every detected photon $R_s$, normalized such that the plotted values should be multiplied by $\Gamma /\left(q_d\kappa \right)$. (b) The crucial average number of photons scattered into free space per atom $m_s^{\mathrm{proj}}$ when the atomic population measurement precision is equal to the projection noise fluctuations. The normalization is such that the plotted values should be multiplied by ${\Gamma }^2/\left(16q{N}_{\uparrow }{g}^2\right)$. In the bad-cavity limit of $\kappa \gg \Gamma$ (green curves), there is little fundamental advantage to operating away from resonance ${\delta }_c=0$. The technical requirements are simply increased as a result of detuning. As the finesse of the cavity $F$ is increased, the amount of free-space scattering falls roughly as $1/F$ until the good-cavity regime is reached when $\kappa \ll \Gamma$ (blue curves). Here, one must detune by roughly the critical detuning ${\delta }_c^{\circ }$ in order to realize the full advantage of having increased the cavity finesse. Importantly, note that $m_s^{\mathrm{proj}}$ does not significantly decrease above ${\delta }_c^{\circ }$ owing to cancellation in this regime of the scaling of $R_s\sim 1/{\delta }_c^2$ with the scaling of $M_d^{\mathrm{proj}}\sim {\delta }_c^2$.

Figure 7

The crucial average number of photons scattered into free space per atom such that the detected probe light allows resolution of projection noise fluctuations $m_s^{\mathrm{proj}}$ versus the ratio of the cavity power to atomic population decay rates $\kappa /\Gamma$. The normalization is chosen such that the plotted values should be multiplied by ${\Gamma }^2/\left(16q{N}_{\uparrow }{g}^2\right)$. Here, we assume that $\kappa$ is varied by changing the cavity finesse while holding the cavity length fixed (such that $g$ is constant). Each trace represents a fixed ratio of the bare-cavity detuning to the collective vacuum Rabi frequency ${\delta }_c/{\Omega }_{\uparrow }$, with values labeled on the traces and blue curves denoting ${\delta }_c/{\Omega }_{\uparrow }>0$, green curves ${\delta }_c/{\Omega }_{\uparrow }=0$, and dashed yellow curves ${\delta }_c/{\Omega }_{\uparrow }<0$. Here we consider probing the dressed mode at frequency ${\omega }_{+}$. At a fixed detuning (and therefore fixed projection-noise-driven frequency fluctuation size) a minimum is reached below which a better cavity is detrimental due to the dressed-cavity linewidth ${\kappa }^{\prime }$ becoming clamped while the ratio of free-space scattering to detected photon number $R_s$ continues to rise. In the units above, the locus of the minimum is just $4\kappa /\Gamma$ [diagonal (red) line]. Including the normalization, the locus of the minimum reduces to $m_s^{\mathrm{proj}}=1/\left(q{N}_{\uparrow }C\right)$. When including the dependence of $g^2$ and $\kappa \sim 1/l$ on the cavity length $l$, the minimum value of $m_s^{\mathrm{proj}}$ is not changed by just shortening the cavity at a fixed finesse. Only shortening the cavity length while simultaneously increasing the cavity finesse (such that $\kappa$ is constant) leads to a net fundamental reduction in $m_s^{\mathrm{proj}}$. Finally, the vertical black line indicates the cavity linewidth for a finesse $F={10}^6$ cavity, near the highest currently achievable, for ${}^{87}\mathrm{Rb}$ and $l=2$ cm.

Figure 8

Normalized spin noise variance versus fractional free-space scattering $m_s/m_s^{\mathrm{proj}}$. The curves assume no prior knowledge, i.e., ${\zeta }_{\mathrm{prep}}\to \infty$, and $NC={10}^3$. The measurement variance decreases as $1/m_s$ from averaging down photon shot noise (dotted black line) until noise due to random spin flips (dashed red and black lines) takes over at large $m_s$. The minimum noise variance is $4\sqrt{p{m}_s^{\mathrm{proj}}}\to \sqrt{8p/NC}$ assuming perfect quantum efficiency and probing in the far-detuned limit. This minimum is reached when two noise contributions are equal at $m_s^{\mathrm{opt}}=\sqrt{m_s^{\mathrm{proj}}/4p}\to 1/\sqrt{8pNC}$ assuming perfect quantum efficiency and probing in the far-detuned limit. A larger collective cooperativity $NC$ or lower spin-flip probability $p$ allows photon shot noise to be averaged down more before spin-flip noise takes over, as illustrated by the two curves for $p/NC={10}^{-5}$ (solid black) and $p/NC={10}^{-3}$ (solid red). The locus of the minimum variance is $2m_s/m_s^{\mathrm{proj}}$ (solid green line).

Figure 9

Optimal squeezing ${\xi }_m^{\mathrm{opt}}$ versus spin-flip probability $p$ for $NC=1000$ (solid black curve) and $NC=10$ (solid red curve) for probing in the far-detuned limit. Both curves take into account spin-flip diffusion noise and decoherence. The curves assume a perfect quantum efficiency $q=1$. The decoherence-limited squeezing ${\xi }_m^{\mathrm{opt}}=NC/e$ (dotted lines) is approached for $p\ll e^2/8NC$, and the spin-flip-limited squeezing ${\xi }_m^{\mathrm{opt}}=\sqrt{NC/8p}$ (dashed lines) is approached for $p\gg e^2/8NC$. The locus of optimal squeezing at the crossover point $p=e^2/8NC$ is $0.193/p$ (solid green line).

Figure 10

The fundamental optimum spectroscopic enhancement for differential probing of the ${}^{87}\mathrm{Rb}$ clock transition, as performed in Ref. [6]. The calculations assume a net quantum efficiency of $q=1$, a cavity-mode waist $w_0=71\mu$m, and a cavity length $l=1.91$ cm used for our experiments [9]. Purple, light-blue, green, orange, and red curves correspond to cavity finesses of $F={10}^2,{10}^3,{10}^4,{10}^5$, and ${10}^6$, respectively. All these finesses are experimentally feasible. The atom number $N$ at which the spectroscopic enhancement is maximized scales with the cavity-mode waist $w_0$ and cavity finesse $F$ as $w_0^2/F$.

Figure 11

Frequency resolution with which the relative frequency of the probe and dressed cavity mode must be measured to obtain the spectroscopic enhancements shown in (and under the same conditions as in) Fig. 10. Purple, light-blue, green, orange, and red curves correspond to cavity finesses of $F={10}^2,{10}^3,{10}^4,{10}^5$, and ${10}^6$, respectively.

Figure 12

Phase resolution with which the transmitted probe light must be measured to obtain the spectroscopic enhancements shown in (and under the same conditions as in) Fig. 10. Purple, light-blue, green, orange, and red curves correspond to cavity finesses of $F={10}^2,{10}^3,{10}^4,{10}^5$, and ${10}^6$, respectively.

Figure 13

Probing scheme for the cycling transition in ${}^{87}\mathrm{Rb}$. The ground hyperfine states $F=2$ and $F=1$ are split by ${\omega }_{\mathrm{hf}}=2\pi \times \left(6834\mathrm{MHz}\right)$. The values of $m_F$ are labeled across the top. The relevant $F^{\prime }=3$ and $F^{\prime }=2$ excited D2 transitions at wavelength 780 nm are also shown with the excited-state splitting ${\omega }_{\mathrm{ehf}}=2\pi \times \left(267\mathrm{MHz}\right)$. The pseudo-spin-1/2 system is here composed of $|\uparrow \rangle =|F=2,m_F=2\rangle$ and $|\downarrow \rangle =|F=1,m_F=1\rangle$. Ideally, the ${\sigma }^{+}$-polarized probing laser couples only $|\uparrow \rangle$ to the optically excited state $|e\rangle$ with dipole matrix element $M_{\uparrow e}$ at a frequency detuning ${\delta }_e$ that is approximately equal to the dressed-cavity-mode frequency ${\omega }_{-}$. In this illustration, ${\delta }_e$ is negative. By dipole selection rules, $|e\rangle$ can only decay back to $|\uparrow \rangle$. However, the same probe laser also couples $|\downarrow \rangle$ to the single excited state $|e^{\prime }\rangle$ with dipole matrix element $M_{\downarrow e^{\prime }}$ and larger detuning from resonance ${\delta }_{e^{\prime }}={\delta }_e-{\omega }_{\mathrm{hf}}+{\omega }_{\mathrm{ehf}}$. The ratio of matrix elements is $M_{\downarrow e^{\prime }}/M_{\uparrow e}=1/\sqrt{2}$. Finally, state $|e^{\prime }\rangle$ decays to states $|\downarrow \rangle$, $|\uparrow \rangle$, and $|3\rangle$ with fractional branching ratios $B_{\downarrow e^{\prime }}=1/2$, $B_{\uparrow e^{\prime }}=1/3$, and $B_{3e^{\prime }}=1/6$, respectively.

Figure 14

Theoretical optimal spectroscopic enhancement ${\xi }_m^{\mathrm{opt}}$ (in dB; solid curves) relative to the SQL versus effective ${}^{87}$Rb atom number $N$ when probing near the D2 cycling transition as shown in Fig. 13. Calculations are for the cavity geometry in Ref. [9] (waist $w_0=71\mu$m and cavity length $l=1.91$ cm) and assuming a net probe quantum efficiency $q=1$. The optimization of the spectroscopic enhancement is performed for a range of technically realizable cavity finesses, $F={10}^2$, ${10}^3$, ${10}^4$, ${10}^5$, and ${10}^6$ (blue, cyan, green, yellow, and red curves respectively). Note that spectroscopic enhancements below the Heisenberg limit is unphysical (disallowed gray region). A region of saturation of ${\xi }_m^{\mathrm{opt}}$ versus $N$ occurs near ${\xi }_m^{\mathrm{sat}}\approx {\omega }_{\mathrm{hf}}/\Gamma =30.5$ dB, shown by the horizontal dashed line. The physical origin of the saturation region arises from competition between the scaling of the off-resonance Raman scattering probabilities and the dressed-cavity-mode broadening, as described in the text. Points of 3-dB deviation from $qNC\propto qNF/w_0^2$ scaling at low effective atom numbers (filled circles) and $\sqrt{qNC}\propto \sqrt{qNF}/w_0$ scaling at large effective atom numbers (open circles) occur when the solid curves cross ${\xi }_m^{\mathrm{opt}}=27$ and 33 dB, respectively. For different cavity waist sizes $w_0^{\prime }$ and finite quantum efficiencies $q$, the saturation level is given by ${\xi }_m^{\mathrm{sat}}\approx \sqrt{q}{\omega }_{\mathrm{hf}}/\Gamma$. The lower saturation atom number $N_{\mathrm{lower}}$ (value of $N$ at filled circles) scales as $N_{\mathrm{lower}}\propto q^{-1/2}\times {(w_0^{\prime }/w_0)}^{-2}\times F^{-1}$, and the upper saturation atom number $N_{\mathrm{upper}}$ (value of $N$ at open circles) scales as $N_{\mathrm{upper}}\propto q^0\times {(w_0^{\prime }/w_0)}^{-2}\times F^{-1}$. Note that the cavity length $l$ is largely irrelevant here (although important for technical reasons) as long as one operates in the good-cavity limit $\kappa \ll \Gamma$. This is the case for all curves except for the finesse $F={10}^2$ curve, where $\kappa \lesssim \Gamma$.

Figure 15

The dressed mode frequency ${\omega }_{-}$ that optimizes the spectroscopic enhancement ${\xi }_m$ for the conditions described in the caption to Fig. 14, where, again, the cavity finesses $F={10}^2,{10}^3,{10}^4,{10}^5,$ and ${10}^6$ correspond to the blue, cyan, green, yellow, and red curves, respectively. Open and filled circles indicate the locations of the 3-dB points in Fig. 14. The mode frequency is normalized to the hyperfine splitting ${\omega }_{\mathrm{hf}}/2\pi =6.834$ GHz. Note that ${\omega }_{-}\approx \Gamma$ at ${\omega }_{-}/{\omega }_{\mathrm{hf}}\approx {10}^{-3}$. Comparing to Fig. 14, one finds that the transition to spectroscopic enhancement scaling as ${\xi }_m^{\mathrm{opt}}\propto \sqrt{qNC}$ occurs near ${\omega }_{-}\sim 1.8\times {\omega }_{\mathrm{hf}}$. Above this point, the spin-flip probabilities $p_{\uparrow ,3}$ change little with ${\omega }_{-}$.

Figure 16

The scattering $m_s$ that optimizes the spectroscopic enhancement ${\xi }_m$ shown in Fig. 14. Again, the cavity finesses $F={10}^2,{10}^3,{10}^4,{10}^5,$ and ${10}^6$ correspond to the solid blue, cyan, green, yellow, and red curves, respectively. Open and filled circles indicate the locations of the 3-dB points in Fig. 14. Comparing to Fig. 14, one finds that the transition to spectroscopic enhancement scaling as ${\xi }_m^{\mathrm{opt}}\propto qNC$ occurs when $m_s\approx 0.25$, indicating that at larger $N$, far-off-resonance Raman scattering begins to create significant diffusion of $J_z$, which limits the precision of the estimation of $J_z$ as discussed in the text.