{"id":34290,"date":"2025-07-03T03:35:15","date_gmt":"2025-07-03T03:35:15","guid":{"rendered":"https:\/\/www.europesays.com\/us\/34290\/"},"modified":"2025-07-03T03:35:15","modified_gmt":"2025-07-03T03:35:15","slug":"hybrid-quantum-network-for-sensing-in-the-acoustic-frequency-range","status":"publish","type":"post","link":"https:\/\/www.europesays.com\/us\/34290\/","title":{"rendered":"Hybrid quantum network for sensing in the acoustic frequency range"},"content":{"rendered":"

A detailed layout of the experimental set-up is shown in Extended Data Fig. 1<\/a>. Below, we discuss the main components of the system and their characterization.<\/p>\n

EPR source<\/p>\n

As shown in Extended Data Fig. 1<\/a>, two lasers at 852\u2009nm (SolsTiS, M Squared) and 1,064\u2009nm (Mephisto, Innolight) drive the sum frequency generation producing the pump beam at 473\u2009nm for a NOPO, which generates the EPR state of light. The NOPO cavity design and the operation principle are reported in ref.\u200941<\/a>. The cavity has a bow-tie configuration with a periodically poled potassium titanyl phosphate nonlinear crystal. The resonance of the NOPO for the 852-nm and 1,064-nm downconverted modes is maintained by locking the cavity to the 1,064-nm laser and locking the 852-nm laser to the cavity, using beams counterpropagating to the EPR modes. To achieve quantum-noise-limited performance in the audio frequency band, we suppress the classical noise of the probe laser using an active noise eater and implement the robust control of the EPR state phases, as described in depth in V. Novikov et al., manuscript in preparation.<\/p>\n

The EPR output modes of the NOPO cavity are separated by a dichroic mirror. The signal beam at 1,064\u2009nm is mixed with the corresponding LOs on a 50:50 beam splitter and the canonical operators (xs,\u2009ps) are measured using a balanced homodyne detector (Extended Data Fig. 1<\/a>). The 852-nm idler beam is combined with an orthogonally polarized probe beam LOi on a polarizing beam splitter and polarization homodyning is performed.<\/p>\n

We observe 9\u2009dB of two-mode squeezing, corresponding to 6\u2009dB of conditional squeezing of the signal field when the two EPR beams are analysed directly (the idler bypasses the atoms)\u00a0(A. Grimaldi et al., manuscript in preparation). The electronic noise floor is more than 17\u2009dB below the shot noise level.<\/p>\n

Before entering the atomic ensemble, the combined idler beam and LOi are shaped into a square top-hat profile, enabling the optimal readout of atomic spins (see the \u2018Atomic spin oscillator\u2019 section below). To characterize the overall propagation losses, including those from the cell windows, beam shaper and other optical elements, a measurement with the Larmor frequency tuned to 1\u2009MHz, beyond the detection frequency band, has been performed.<\/p>\n

We observe roughly\u00a03.3\u2009dB conditional squeezing from 3 to 60\u2009kHz relative to the signal vacuum noise, as shown in Extended Data Fig. 2a<\/a>. Noise peaks at 26 and 36\u2009kHz are experimental artefacts caused by the intensity noise eater. The theory of the conditional squeezing, detailed in Supplementary Information Section\u00a0I<\/a>, allows us to extract a squeezing factor r\u2009=\u20091.42 and an unbalanced detection efficiency of \u03b7s\u2009\u2248\u20090.92 for the signal arm and \u03b7i\u2009=\u2009\u03b7i,out\u03b7i,in\u00a0\u2009\u2248\u20090.8 for the idler arm by fitting the measured power spectrum densities. See Extended Data Table 1<\/a> for parameter values.<\/p>\n

Atomic spin oscillator<\/p>\n

The atomic spin oscillator is implemented in a\u00a0133Cs gas of NCs\u00a0\u2009\u2248\u20091010 atoms in a 2\u2009\u00d7\u20092\u2009\u00d7\u200980-mm3 glass channel inside a vacuum-tight glass cell. The cell windows are anti-reflection coated and the inner walls are coated with an anti-relaxation paraffin material to minimize the decoherence from spin\u2013wall collisions. The atomic vapour density in the cell is defined by the temperature of the caesium droplet in the stem of the cell, which was set to \u00a0about\u00a040\u2009\u00b0C in the experiment. A multilayer magnetic shield around the cell provides isolation from the Earth\u2019s magnetic field and other high-frequency magnetic noise, whereas a set of coils running low-noise DC currents generates a highly homogeneous bias magnetic field within the cell. By adjusting the DC current, we can control the Larmor frequency |\u03a9a| from 1\u2009MHz down to 3\u2009kHz, maintaining quantum-dominated performance throughout this frequency spectrum36<\/a>.<\/p>\n

As depicted in Extended Data Fig. 1<\/a>, the vapour cell is illuminated by two optical fields. A circularly polarized optical repumping field, propagating transversely, prepares the spin ensemble in the hyperfine |F\u2009=\u20094,\u2009mF\u2009=\u20094\u27e9 or |F\u2009=\u20094,\u2009mF\u2009=\u2009\u22124\u27e9 ground state manifold with 82% efficiency, enabling it to function as a macroscopic spin oscillator with an adjustable sign of the effective mass.<\/p>\n

The probe beam is blue-detuned by 1.6\u2009GHz from the D2 line F\u2009=\u20094\u2009\u2192\u2009F\u2032\u2009=\u20095 transition to eliminate the absorption. The polarization of the probe entering the atomic ensemble is chosen to optimize the light\u2013spin interaction, as discussed in the main text20<\/a>,36<\/a>. The light\u2013atoms interaction strength is characterized by the readout rate \\({\\varGamma }_{{\\rm{a}}}\\propto {g}_{{\\rm{cs}}}^{2}{{\\mathcal{S}}}_{1}\\,{J}_{x}\\,\\propto d\\), in which gcs is the single photon\u2013atom coupling rate that depends on the probe detuning and \\({{\\mathcal{S}}}_{1}\\) is a Stokes parameter proportional to the power of the probe light. The optical depth of the spin ensemble d\u2009\u221d\u2009Jx\u2009\u221d\u2009NCs, in which Jx is the macroscopic component of the collective spin and NCs is the number of atoms36<\/a>.<\/p>\n

The spin oscillator experiences depolarization, primarily because of the spontaneous emission that occurs in the presence of the probe field. This leads to the spin thermal noise imprinted onto the output probe beam. In the regime of strong light\u2013atom coupling \u0393a\u2009\u226b\u2009\u03b3a, the QBA induced by the probe dominates over the atomic thermal noise. Atoms that maintain interaction with the probe by several passages across the probe beam during the coherent evolution time contribute to the narrow-band atomic response limited by the spin decoherence rate \u03b3a. The remaining atoms contribute to faster-decaying atomic modes, leading to further, broadband atomic noise with the bandwidth of \u03b3bb\u2009\u226b\u2009\u03b3a (refs.\u200920<\/a>,52<\/a>,53<\/a>). To minimize the broadband noise, the input field, comprising two orthogonally polarized fields (idler and LOi), is shaped into a collimated square top-hat beam with a seventh-order super-Gaussian waist of 1.7\u2009mm by the top-hat shaper, as shown in Extended Data Fig. 1<\/a>. It propagates through the cell with the filling factor of approximately\u00a080% without introducing extra losses. The effect of the remaining broadband noise is illustrated in Extended Data Fig. 2<\/a>.<\/p>\n

To characterize the spin oscillator, we block the idler EPR beam and drive the atoms with a coherent state of light. The Larmor frequency is set to \u03a9a\/2\u03c0\u2009\u2248\u200910.5\u2009kHz and the quadrature of the output probe light is adjusted by a quarter-wave plate and a half-wave plate. In Extended Data Fig. 2b<\/a>, the recorded quantum noise, dominated by the spin QBA noise is shown as the red area. Choosing an optimal polarization of detected light, we observe 5\u2009dB of ponderomotive squeezing, as shown in the left inset. By fitting the shown spin noise spectra, we can extract both the narrowband and the broadband spin readout and decay rates, along with the losses after the spin ensemble and the effective thermal occupancies (see Extended Data Table 1<\/a>). The total extracted occupancy number of \u00a0approximately 3.5 is the result of two factors: the imperfect spin polarization (measured using the magneto-optical resonances) contributes \u00a0about 1 and the technical noise sources contribute \u00a0about 2.5 (for details, see Supplementary Information Section\u00a0IIB<\/a>). The effective thermal occupancies for the narrowband and broadband spin responses are the same. The reconstructed atomic thermal noise (the light-blue area) and broadband noise (the purple area) are also presented in Extended Data Fig. 2b<\/a>. The impact of these distinct spectral features on the conditional frequency-dependent squeezing level is discussed below. The red area, representing the QBA noise, highlights the key quantum contribution that enables the frequency-dependent rotation of the squeezing phase. The same calibration procedure was applied to experimental data at 54\u2009kHz Larmor frequency, demonstrated in Extended Data Fig. 2<\/a>. The reduced classical noise at this frequency band allowed for better fitting.<\/p>\n

After characterizing the spin oscillator, we unblock the idler output of the NOPO and record the dynamics of the spin oscillator driven by the idler component of the EPR field. The virtual rigidity phase \u03b4\u03b8i is extracted by fitting the noise spectrum shown in the right inset of Extended Data Fig. 2b<\/a> (green trace). By combining the overall losses obtained from the EPR source calibration with the optical and detection losses after atoms evaluated from ponderomotive squeezing, we estimate the propagation efficiency between the NOPO and atoms to be \u03b7i,in\u2009\u2248\u200989%. The optical losses from the atoms to the detection in the idler arm have been measured and result in the efficiency \u03b7i,out\u2009\u2248\u200990%. Propagation efficiency from the NOPO and the detection efficiency in the signal arm result in \u03b7s\u2009\u2248\u200992%. On the basis of those numbers, the predicted degree of two-mode squeezing fits the observed degree of squeezing as shown in Extended Data Fig. 2<\/a>.<\/p>\n

Using the parameters extracted from these independent calibrations and the model presented in the next sections, equation (10<\/a>), we calculate the predicted frequency-dependent conditional squeezing as a function of the signal homodyne angle \u03b8s shown in Figs. 2<\/a> and 3<\/a>. The figures show good agreement between the predicted spectra of quantum noise and the experimental data.<\/p>\n

Phase control of the signal and idler fields<\/p>\n

A feature of the EPR source critical for the present work is phase control of the two-colour EPR state. Here we describe its underlying principles and a complete report is in preparation. The phases of the signal and idler beams of the NOPO are related to the pump phase by \u03b8p\u2009=\u2009\u03d5i\u2009+\u2009\u03d5s (refs.\u200935<\/a>,54<\/a>). To precisely track the phase of the signal and the idler, we inject a coherent beam \u03b1c co-propagating with the pump and frequency-shifted by \u03b4\u03c9c from the 1,064-nm laser by two acousto-optic modulators.<\/p>\n

In this experiment, we choose \u03b4\u03c9c\/2\u03c0\u2009=\u20093\u2009MHz, well inside the NOPO cavity bandwidth but also far from the resonance of the atomic spin oscillator. An electronic reference phase for the control beam \u03d5c is provided by driving the acousto-optic modulators by two outputs of an ultralow-phase-noise direct digital synthesizer.<\/p>\n

The control beam experiences the parametric interaction and is amplified while maintaining its phase, resulting in the output field \\({\\alpha }_{{\\rm{c}}}^{{\\rm{s}}}\\). The interaction of control and pump provides the simultaneous generation of another coherent field, \\({\\alpha }_{{\\rm{c}}}^{{\\rm{i}}}\\), centred at frequency \u03c9i\u2009\u2212\u2009\u03b4\u03c9c. In this case, the field phase is the combination of the pump and signal control beam phases, \\({\\phi }_{{\\rm{c}}}^{{\\rm{i}}}={\\theta }_{{\\rm{p}}}-{\\phi }_{{\\rm{c}}}\\). The classical beams generated by this process propagate together with their respective entangled fields to the homodyne detections. We demodulate the photocurrents using the electronic reference. This provides an error signal proportional to the phase difference between the entangled fields and the local oscillators, measured by the two homodyne detectors. The photocurrent from the homodyne detectors Is(t) contains the information from the signal-arm quadrature qs(\u03b8s) and the beat note of the local oscillator with the control field shifted from the relevant signal band by \u03b4\u03c9c<\/p>\n

$${I}_{{\\rm{s}}}(t)\\propto {q}_{{\\rm{s}}}({\\theta }_{{\\rm{s}}})+{\\alpha }_{{\\rm{c}}}^{{\\rm{s}}}\\cos (\\delta {\\omega }_{{\\rm{c}}}t-{\\phi }_{{\\rm{c}}}),$$<\/p>\n

\n (5)\n <\/p>\n

with the second term allowing us to select and set the homodyne LO phase \u03b8s. In turn, the outcome of the measurement of the idler field by means of the homodyne detection can be presented as<\/p>\n

$${I}_{{\\rm{i}}}(t)\\propto {Q}_{{\\rm{i}}}({\\theta }_{{\\rm{i}}})+{\\alpha }_{{\\rm{c}}}^{{\\rm{i}}}\\cos (\\delta {\\omega }_{{\\rm{c}}}t-{\\phi }_{{\\rm{c}}}^{{\\rm{i}}}-{\\phi }_{{\\rm{i}}}),$$<\/p>\n

\n (6)\n <\/p>\n

in which the first term is the idler quantum field contribution and the second term is the beat note between the second control field that has passed through the NOPO channel and the respective local oscillator. The demodulated signal at \u03b4\u03c9c\/2\u03c0 provides a tool to control the homodyne phase \u03b8i\u2009=\u2009\u03d5i\u2009+\u2009\u03b4\u03b8i, whereas the phase offset \u03b4\u03b8i can be tuned separately using the quarter-wave plate (see the set-up in Extended Data Fig. 1<\/a> and Fig. 1<\/a>). An exact definition of Qi(\u03b8i) is given in the next section.<\/p>\n

The frequency offset of \u03b4\u03c9c\/2\u03c0\u2009\u2243\u20093\u2009MHz guarantees that locking of those phases is not affected by the atomic spin oscillator. The procedure described above provides a set of well-defined phases that we use as references for the scan of \u03b8s involved in the demonstration of frequency-dependent conditional squeezing.<\/p>\n

Idler field and light\u2013atoms interaction<\/p>\n

The 1.6-GHz detuning of the 852-nm NOPO idler field from atomic resonance is achieved by changing the frequency of the 1,064-nm laser, which alters the NOPO cavity length and its resonance condition. The 852-nm laser lock follows the change, enabling fine-tuning of the idler field with precision close to the NOPO cavity bandwidth.<\/p>\n

To enable interaction between the idler field and the atomic spin ensemble, we overlap the idler output of the NOPO with an orthogonally polarized probe beam. This is realized by combining the linearly polarized idler and probe beams on a polarizing beam splitter, which transforms the quadrature fluctuations of the idler field into Stokes operator fluctuations55<\/a>,56<\/a>. For a linearly polarized, strong coherent probe beam with a relative phase \u03d5i between the idler and the probe, the quadratures map onto the Stokes operators as \\({q}_{{\\rm{i}}}({\\phi }_{{\\rm{i}}})=\\sqrt{2}{{\\mathcal{S}}}_{2}^{{\\rm{in}}}\/| {\\alpha }_{{\\rm{pr}}}| \\) and \\({q}_{{\\rm{i}}}({\\phi }_{{\\rm{i}}}+\\pi \/2)=\\sqrt{2}{{\\mathcal{S}}}_{3}^{{\\rm{in}}}\/| {\\alpha }_{{\\rm{pr}}}| \\), in which \u03b1pr is the amplitude of the probe field, \\(\\{{{\\mathcal{S}}}_{0},{{\\mathcal{S}}}_{1},{{\\mathcal{S}}}_{2},{{\\mathcal{S}}}_{3}\\}\\) are quantum Stokes operators obeying \\([{{\\mathcal{S}}}_{2},{{\\mathcal{S}}}_{3}]=i{{\\mathcal{S}}}_{1}\\) (and cyclical permutations thereof) and \\({q}_{{\\rm{i}}}({\\phi }_{{\\rm{i}}})=\\cos ({\\phi }_{{\\rm{i}}}){p}_{{\\rm{i}}}+\\sin ({\\phi }_{{\\rm{i}}}){x}_{{\\rm{i}}}\\) is the quadrature of the optical field entering the atomic ensemble.<\/p>\n

The idler field encoded into the polarization state is processed by the atomic ensemble. The output Stokes parameters are then given by \\({{\\mathcal{S}}}_{2}^{{\\rm{out}}}={{\\mathcal{S}}}_{2}^{{\\rm{in}}}+{{\\mathcal{K}}}_{a}{{\\mathcal{S}}}_{3}^{{\\rm{in}}}+{{\\mathcal{N}}}_{{\\rm{a}}}\\) and \\({{\\mathcal{S}}}_{3}^{{\\rm{out}}}={{\\mathcal{S}}}_{3}^{{\\rm{in}}}\\), in which \\({{\\mathcal{N}}}_{{\\rm{a}}}={N}_{{\\rm{a}}}| {\\alpha }_{{\\rm{pr}}}| \/\\sqrt{2}\\); \\({{\\mathcal{K}}}_{{\\rm{a}}}\\) and Na are defined in the main text.<\/p>\n

The light emerging from the spin ensemble is measured by the polarization homodyne detection. The diagonal linear and circular polarization operators \\({{\\mathcal{S}}}_{2}\\) and \\({{\\mathcal{S}}}_{3}\\) are measured by passing light through a half-wave plate and an extra quarter-wave plate, respectively, followed by the polarizing beam splitter, as shown in Extended Data Fig. 1<\/a>. The resulting measured Stokes quadrature operator is \\({{\\mathcal{R}}}_{{\\rm{i}}}(\\delta {\\theta }_{{\\rm{i}}})={{\\mathcal{S}}}_{2}^{{\\rm{out}}}\\cos (\\delta {\\theta }_{{\\rm{i}}})+{{\\mathcal{S}}}_{3}^{{\\rm{out}}}\\sin (\\delta {\\theta }_{{\\rm{i}}})={Q}_{{\\rm{i}}}(\\delta {\\theta }_{{\\rm{i}}})| {\\alpha }_{{\\rm{pr}}}| \/\\sqrt{2}\\), in which the phase \u03b4\u03b8i is set by the orientation of the quarter-wave plate.<\/p>\n

The phase \u03d5i is set by a coherent control loop that monitors the phase offset between the probe beam and the control field at the idler homodyne detector. The control field follows the same propagation path as the idler beam but is unaffected by the atomic oscillator owing to a frequency offset, \u03b4\u03c9c. The phase offset \u03d5i measured by the coherent control loop combines the encoding phase and \u03b4\u03b8i.<\/p>\n

Throughout those measurements, we set \u03b8i\u2009=\u20090, so that the quadrature of the idler detected outside the bandwidth of the spin dynamics bandwidth (\u03a9\u2009\u226b\u2009|\u03a9a|) is pi. In the setting not engaging virtual rigidity, that is, in the absence of a quarter-wave plate (\u03b4\u03b8i\u2009=\u20090), the coherent control loop sets the encoding phase \u03d5i\u2009=\u20090, corresponding to \\({{\\mathcal{S}}}_{2}^{{\\rm{in}}}={p}_{{\\rm{i}}}| {\\alpha }_{{\\rm{pr}}}| \/\\sqrt{2}\\) and \\({{\\mathcal{S}}}_{3}^{{\\rm{in}}}={x}_{{\\rm{i}}}| {\\alpha }_{{\\rm{pr}}}| \/\\sqrt{2}\\). With such settings, the final idler photocurrent within the relevant range of frequencies \u03a9 is proportional to \\({P}_{{\\rm{i}}}={p}_{{\\rm{i}}}+{{\\mathcal{K}}}_{{\\rm{a}}}(\\varOmega ){x}_{{\\rm{i}}}+{N}_{{\\rm{a}}}(\\varOmega )\\). For general \u03b4\u03b8i, we have Qi(\u03b4\u03b8i)\u2009=\u2009Xisin(\u03b4\u03b8i)\u2009+\u2009Picos(\u03b4\u03b8i).<\/p>\n

Optimal quantum noise reduction by Wiener filtering in the idler channel<\/p>\n

The squeezing we demonstrate for the signal arm is conditioned on the idler-arm measurement<\/p>\n

$${Q}_{{\\rm{s}}| {\\rm{i}}}(\\varOmega ,{\\theta }_{{\\rm{s}}})={q}_{{\\rm{s}}}(\\varOmega ,{\\theta }_{{\\rm{s}}})+g(\\varOmega ,{\\theta }_{{\\rm{s}}},\\delta {\\theta }_{{\\rm{i}}}){Q}_{{\\rm{i}}}(\\varOmega ,\\delta {\\theta }_{{\\rm{i}}}).$$<\/p>\n

\n (7)\n <\/p>\n

To achieve the maximal frequency-dependent squeezing of the signal field based on the idler measurement record, a Wiener filter g(\u03a9,\u2009\u03b8s,\u2009\u03b4\u03b8i) is designed for each detection phase \u03b8s, which provides an optimal estimate of the correlated quantum noise in the idler channel. By subtracting the filtered idler quantum noise from the signal noise, the optimal reduction of the signal quantum noise limited by the remaining uncorrelated noise is achieved11<\/a>,31<\/a>,57<\/a>.<\/p>\n

Our protocol is compatible with both causal and non-causal filtering g(\u03a9,\u2009\u03b8s,\u2009\u03b4\u03b8i) of the idler measurement record. Causal filtering, which uses only past idler measurements to reduce the signal-arm noise at a given time, is required for using the sensor for, for example, real-time signal tracking or adaptive sensing. On the other hand, the non-causal filtering uses the full idler measurement record and is relevant for sensing scenarios in which the signal is extracted by post-processing of the full measurement record.<\/p>\n

In this work, we focus on non-causal conditioning, thereby emphasizing the maximally achievable squeezing. In this case, the optimal gain is given by the idler spectral density \\({S}_{{Q}_{{\\rm{i}}}}\\) and the cross-spectral density \\({S}_{{q}_{{\\rm{s}}},{Q}_{{\\rm{i}}}}\\) of the two entangled channels as<\/p>\n

$$g(\\varOmega ,{\\theta }_{{\\rm{s}}},\\delta {\\theta }_{{\\rm{i}}})=-\\frac{{S}_{{q}_{{\\rm{s}}},{Q}_{{\\rm{i}}}}(\\varOmega ,{\\theta }_{{\\rm{s}}},\\delta {\\theta }_{{\\rm{i}}})}{{S}_{{Q}_{{\\rm{i}}}}(\\varOmega ,\\delta {\\theta }_{{\\rm{i}}})},$$<\/p>\n

\n (8)\n <\/p>\n

which leads to the power spectral density of the optimized signal<\/p>\n

$${S}_{{Q}_{{\\rm{s}}| {\\rm{i}}}}(\\varOmega ,{\\theta }_{{\\rm{s}}})={S}_{{q}_{{\\rm{s}}}}(\\varOmega ,{\\theta }_{{\\rm{s}}})-\\frac{| {S}_{{q}_{{\\rm{s}}},{Q}_{{\\rm{i}}}}(\\varOmega ,{\\theta }_{{\\rm{s}}},\\delta {\\theta }_{{\\rm{i}}}){| }^{2}}{{S}_{{Q}_{{\\rm{i}}}}(\\varOmega ,\\delta {\\theta }_{{\\rm{i}}})}.$$<\/p>\n

\n (9)\n <\/p>\n

The optimal filter (equation (8<\/a>)) automatically takes into account the signal homodyne detection phase \u03b8s, deleterious noise introduced in the idler path and phase shifts as a result of the atomic dissipation through decoherence, as represented by the complexity of the response function \\({{\\mathcal{K}}}_{{\\rm{a}}}(\\varOmega )\\) (see equation (2<\/a>)). Furthermore, the Wiener filter can potentially compensate for imperfect matching between the idler and the quantum sensor response introduced in the signal arm in a particular sensing application. The analysis of the experimental data using this approach is detailed in Supplementary Information Section\u00a0II<\/a>.<\/p>\n

Theory of frequency-dependent conditional squeezing<\/p>\n

Figures 2<\/a> and 3<\/a> present comparisons of the measured squeezing with the theoretical model, which we describe here. The model consists of equations of motion of the spin oscillator and input\u2013output relations describing its interaction with light. It accounts for contributions of thermal and broadband noise to the response of the spin system, as well as for the effects of imperfect readout efficiency (\u03b7i,out\u2009\u03b7i,in\u2009\u03b7s\u2009Qs|i, which, in turn, is minimized using Wiener filter theory as outlined in the preceding subsection. Detailed calculations are provided in Supplementary Information Section\u00a0IC<\/a>.<\/p>\n

Here we present a general formula for an arbitrary phase offset \u03b4\u03b8i imposed by the quarter-wave plate and giving rise to virtual rigidity. The case of no virtual rigidity can be obtained by setting \u03b4\u03b8i\u2009=\u20090. The spectrum of the optimally conditioned signal, normalized to the signal shot noise SSN level, in the signal-arm detection quadrature \u03b8s is given by<\/p>\n

$$\\begin{array}{l}\\frac{{S}_{{Q}_{{\\rm{s}}| {\\rm{i}}}}}{{S}_{{\\rm{SN}}}}\\,=\\,1-{\\eta }_{{\\rm{s}}}+\\frac{{\\eta }_{{\\rm{s}}}}{\\cosh (2r)}\\times \\\\ \\,\\,\\,\\,\\left[{\\cosh }^{2}(2r)-\\frac{{\\sinh }^{2}(2r){| \\cos ({\\theta }_{{\\rm{s}}})-\\sin ({\\theta }_{{\\rm{s}}}){{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{eff}}}| }^{2}}{1+{| {{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{eff}}}| }^{2}+2\\frac{{\\varLambda }_{{\\rm{in}}}^{{\\rm{eff}}}+{\\varLambda }_{{\\rm{out}}}^{{\\rm{eff}}}+{S}_{{\\rm{th}}}^{{\\rm{eff}}}+{S}_{{\\rm{bb}}}^{{\\rm{eff}}}}{{\\eta }_{{\\rm{i}},{\\rm{in}}}\\cosh (2r)}}\\right],\\end{array}$$<\/p>\n

\n (10)\n <\/p>\n

in which dependencies on \u03a9 are omitted for brevity. The numerator of the second term in the brackets represents the correlation between the signal and idler, shaped by the backaction of the atomic spin oscillator. The denominator captures the spectrum of the idler signal, incorporating, as well as the backaction, the following four terms, responsible for the various deleterious effects mentioned above:<\/p>\n

$${\\varLambda }_{{\\rm{in}}}^{{\\rm{eff}}}(\\varOmega )=\\frac{1-{\\eta }_{{\\rm{i}},{\\rm{in}}}}{2}(1+| {{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{eff}}}(\\varOmega ){| }^{2})$$<\/p>\n

\n (11)\n <\/p>\n

$${\\varLambda }_{{\\rm{out}}}^{{\\rm{eff}}}(\\varOmega )=\\frac{1-{\\eta }_{{\\rm{i}},{\\rm{out}}}}{2{\\eta }_{{\\rm{i}},{\\rm{out}}}}\\frac{1}{| {g}_{{\\rm{VR}}}(\\varOmega ){| }^{2}}$$<\/p>\n

\n (12)\n <\/p>\n

$${S}_{{\\rm{th}}}^{{\\rm{eff}}}(\\varOmega )={\\left|{{\\mathcal{K}}}_{{\\rm{th}}}(\\varOmega )\\frac{\\cos (\\delta {\\theta }_{{\\rm{i}}})}{{g}_{{\\rm{VR}}}(\\varOmega )}\\right|}^{2}(1\/2+{n}_{{\\rm{th}}})$$<\/p>\n

\n (13)\n <\/p>\n

$${S}_{{\\rm{bb}}}^{{\\rm{eff}}}(\\varOmega )={\\left|{{\\mathcal{K}}}_{{\\rm{bb}}}(\\varOmega )\\frac{\\cos (\\delta {\\theta }_{{\\rm{i}}})}{{g}_{{\\rm{VR}}}(\\varOmega )}\\right|}^{2}(1\/2+{n}_{{\\rm{bb}}}).$$<\/p>\n

\n (14)\n <\/p>\n

In order of appearance, they represent the effects of suboptimal coupling of the idler field to the spin ensemble, \u03b7i,in, imperfect readout efficiency of the idler detector, \u03b7i,out, thermal occupation of the collective spin state, nth, governed by \\({{\\mathcal{K}}}_{{\\rm{th}}}(\\varOmega )=\\sqrt{2{\\gamma }_{{\\rm{a}}}{\\varGamma }_{{\\rm{a}}}}{\\varOmega }_{{\\rm{a}}}\/({\\varOmega }_{{\\rm{a}}}^{2}-{\\varOmega }^{2}-i{\\gamma }_{{\\rm{a}}}\\varOmega \\,+\\)\\({\\gamma }_{{\\rm{a}}}^{2}\/4)\\) and the broadband noise occupation number, nbb, governed by \\({{\\mathcal{K}}}_{{\\rm{bb}}}(\\varOmega )=\\sqrt{2{\\gamma }_{{\\rm{bb}}}{\\varGamma }_{{\\rm{bb}}}}{\\varOmega }_{{\\rm{a}}}\/({\\varOmega }_{{\\rm{a}}}^{2}-{\\varOmega }^{2}-i{\\gamma }_{{\\rm{bb}}}\\varOmega +{\\gamma }_{{\\rm{bb}}}^{2}\/4)\\). The effect of the virtual rigidity is captured by the gain<\/p>\n

$${g}_{{\\rm{VR}}}(\\varOmega )=1-{{\\mathcal{K}}}_{{\\rm{a}}}(\\varOmega ,{\\varOmega }_{{\\rm{a}}},{\\varGamma }_{{\\rm{a}}})\\frac{\\sin (2\\delta {\\theta }_{{\\rm{i}}})}{2}.$$<\/p>\n

\n (15)\n <\/p>\n

The effective backaction \\({{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{eff}}}(\\varOmega )\\equiv {{\\mathcal{K}}}_{{\\rm{a}}}(\\varOmega ,{\\varOmega }_{{\\rm{a}}}^{{\\rm{eff}}},{\\varGamma }_{{\\rm{a}}}^{{\\rm{eff}}})\\) is defined as the backaction coefficient used in equation (2<\/a>) but evaluated with the effective readout rate and the effective Larmor frequency<\/p>\n

$${\\varGamma }_{{\\rm{a}}}^{{\\rm{eff}}}={\\varGamma }_{{\\rm{a}}}\\frac{{\\cos }^{2}(\\delta {\\theta }_{{\\rm{i}}})}{\\sqrt{1-\\frac{{\\varGamma }_{{\\rm{a}}}}{2{\\varOmega }_{{\\rm{a}}}}\\sin (2\\delta {\\theta }_{{\\rm{i}}})}}$$<\/p>\n

\n (16)\n <\/p>\n

$${\\varOmega }_{{\\rm{a}}}^{{\\rm{eff}}}={\\varOmega }_{{\\rm{a}}}\\sqrt{1-\\frac{{\\varGamma }_{{\\rm{a}}}}{2{\\varOmega }_{{\\rm{a}}}}\\sin (2\\delta {\\theta }_{{\\rm{i}}})}\\,;$$<\/p>\n

\n (17)\n <\/p>\n

these expressions are valid insofar as the quantity appearing under the square roots is positive, as is the case for our system parameters.<\/p>\n

By minimizing equation (10<\/a>) as a function of \u03b8s for each Fourier component \u03a9 separately, we find the optimal angle \u03b8s\u2009=\u2009\u03a6VR(\u03a9) to be the solution to the set of equations<\/p>\n

$$\\tan (2{\\varPhi }_{{\\rm{VR}}}(\\varOmega ))=\\frac{-2{\\rm{Re}}[{({{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{eff}}}(\\varOmega ))}^{-1}]}{{| {({{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{eff}}}(\\varOmega ))}^{-1}| }^{2}-1}$$<\/p>\n

\n (18a)\n <\/p>\n

$${\\rm{s}}{\\rm{i}}{\\rm{g}}{\\rm{n}}[\\cos (2{\\varPhi }_{{\\rm{V}}{\\rm{R}}}(\\varOmega ))]={\\rm{s}}{\\rm{i}}{\\rm{g}}{\\rm{n}}(1-|{{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{e}}{\\rm{f}}{\\rm{f}}}(\\varOmega ){|}^{2}).$$<\/p>\n

\n (18b)\n <\/p>\n

In the limit \\({\\gamma }_{{\\rm{a}}}^{2}\\ll {({\\varOmega }_{{\\rm{a}}}^{{\\rm{eff}}})}^{2},{({\\varGamma }_{{\\rm{a}}}^{{\\rm{eff}}})}^{2}\\), we have \\({| {({{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{eff}}}(\\varOmega ))}^{-1}| }^{2}\\approx {{\\rm{Re}}}^{2}[{({{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{eff}}}(\\varOmega ))}^{-1}]\\) \\(\\approx \\,{[{({{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{eff}}}(\\varOmega ))}^{-1}]}_{{\\gamma }_{{\\rm{a}}}\\to 0}^{2}\\), in which equations 18a<\/a> and 18b<\/a> reduce to<\/p>\n

$${\\varPhi }_{{\\rm{VR}}}(\\varOmega )\\approx -\\,\\arctan ({{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{eff}}}(\\varOmega )){| }_{{\\gamma }_{{\\rm{a}}}\\to 0},$$<\/p>\n

\n (19)\n <\/p>\n

as presented in the main text. This expression, valid for general \u03b4\u03b8i, is used to generate the dashed curves presented as \u03a6j(\u03a9) in Figs. 2b<\/a> and 3b,d,f<\/a> for the various special cases labelled j\u2009\u2208\u2009{\u00b1,\u2009VR}. Evaluating equation (10<\/a>) at \u03b8s\u2009=\u2009\u03a6VR(\u03a9) yields the achieved degree of (frequency-dependent) squeezing; this amounts to replacing the factor<\/p>\n

$$\\begin{array}{c}{|\\cos ({\\theta }_{{\\rm{s}}})-\\sin ({\\theta }_{{\\rm{s}}}){{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{e}}{\\rm{f}}{\\rm{f}}}|}^{2}\\,\\to \\\\ \\,\\frac{1+|{{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{e}}{\\rm{f}}{\\rm{f}}}{|}^{2}}{2}\\left[1+\\sqrt{1-\\frac{4{{\\rm{I}}{\\rm{m}}}^{2}[{{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{e}}{\\rm{f}}{\\rm{f}}}]}{{(1+|{{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{e}}{\\rm{f}}{\\rm{f}}}{|}^{2})}^{2}}}\\right]\\,\\approx \\,1+|{{\\mathcal{K}}}_{{\\rm{a}}}^{{\\rm{e}}{\\rm{f}}{\\rm{f}}}{|}^{2},\\end{array}$$<\/p>\n

\n (20)\n <\/p>\n

in which the approximation is valid in the limit \\({\\gamma }_{{\\rm{a}}}^{2}\\ll {({\\varOmega }_{{\\rm{a}}}^{{\\rm{eff}}})}^{2},{({\\varGamma }_{{\\rm{a}}}^{{\\rm{eff}}})}^{2}\\).<\/p>\n

In the aforementioned limit, we may use equations (2<\/a>) and (19<\/a>) to derive an expression for the bandwidth \u03b4\u03a9SQL\u2009>\u20090 over which the rotation angle \u03a6(\u03a9) (\u2208[0,\u2009\u03c0] for specificity) changes by 45\u00b0 relative to its value at the effective spin oscillator resonance \\(\\varPhi ({\\varOmega }_{{\\rm{a}}}^{{\\rm{eff}}})=\\pi \/2\\), that is, it obeys \\(| \\varPhi ({\\varOmega }_{{\\rm{a}}}^{{\\rm{eff}}}+\\delta {\\varOmega }_{{\\rm{SQL}}})-\\varPhi ({\\varOmega }_{{\\rm{a}}}^{{\\rm{eff}}})| =\\pi \/4\\). The result is \\(\\delta {\\varOmega }_{{\\rm{SQL}}}=| {\\varOmega }_{{\\rm{a}}}^{{\\rm{eff}}}| \\)\\((\\sqrt{1+{\\varGamma }_{{\\rm{a}}}^{{\\rm{eff}}}\/| {\\varOmega }_{{\\rm{a}}}^{{\\rm{eff}}}| }-1)\\approx {\\varGamma }_{{\\rm{a}}}^{{\\rm{eff}}}\/2\\), in which the approximation holds under the extra assumption \\({\\varGamma }_{{\\rm{a}}}^{{\\rm{eff}}}\\ll | {\\varOmega }_{{\\rm{a}}}^{{\\rm{eff}}}| \\).<\/p>\n

As well as comparing our model with the squeezing achieved in the present experiment, we also used the model to predict the degree of broadband noise reduction obtained by reducing the two main imperfections of our system: broadband spin noise and thermal spin noise. The results are reported in Extended Data Fig. 3<\/a> and further details can be found in Supplementary Information Section\u00a0IIB<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"A detailed layout of the experimental set-up is shown in Extended Data Fig. 1. Below, we discuss the…\n","protected":false},"author":3,"featured_media":34291,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[25],"tags":[28376,10046,10047,492,8068,28377,5649,159,67,132,68],"class_list":{"0":"post-34290","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-physics","8":"tag-atomic-and-molecular-interactions-with-photons","9":"tag-humanities-and-social-sciences","10":"tag-multidisciplinary","11":"tag-physics","12":"tag-quantum-mechanics","13":"tag-quantum-metrology","14":"tag-quantum-optics","15":"tag-science","16":"tag-united-states","17":"tag-unitedstates","18":"tag-us"},"share_on_mastodon":{"url":"https:\/\/pubeurope.com\/@us\/114787210162804929","error":""},"_links":{"self":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/34290","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/comments?post=34290"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/34290\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media\/34291"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media?parent=34290"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/categories?post=34290"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/tags?post=34290"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}