1. INTRODUCTION

Precision phase measurements are invaluable for a wide range of applications, including gravitational wave detection and biological sensing [1–3]. The sensitivity of phase-measuring devices, such as interferometers, is characterized by the uncertainty of a single phase measurement $\mathrm{\Delta }\varphi$. When using coherent optical fields in a Mach–Zehnder interferometer, the phase sensitivity is bounded by the standard quantum limit (SQL) $\mathrm{\Delta }\varphi =1/\sqrt{N}$, where $N$ is the mean photon number in a single phase measurement.

The phase sensitivity of an interferometer can be improved by using quantum resources [3–5]. By injecting squeezed states [6–8] or other nonclassical states [3,9–11] into a conventional Mach–Zehnder interferometer, for example, one can achieve phase sensitivities that surpass the SQL.

An alternative to injecting quantum states into a Mach–Zehnder interferometer is to replace the passive beam splitters in such a device with active nonlinear optical elements [12]. A particular configuration that has drawn considerable interest is the SU(1,1) interferometer [12], which is expected to achieve phase sensitivities beyond the SQL even in the presence of loss [13–16]. The SU(1,1) interferometer takes two input modes and, using parametric down-conversion or four-wave mixing (4WM), produces a two-mode squeezed state that is used for phase sensing, as shown in Fig. 1. We consider the seeded version of the SU(1,1) interferometer, where the phase-sensing two-mode state consists of two bright beams. These beams are then recombined in another nonlinear process, and any changes in the phase sum of the two modes relative to the pump phase can be inferred by detecting the light at the outputs, either by using homodyne detection or direct intensity detection.

 

Fig. 1. Overview of the SU(1,1)-type phase-measurement device: a coherent state and a vacuum state are mixed with a pump beam in a nonlinear optical medium (NLO) to produce a two-mode quantum state. A phase shift $\varphi$ is applied to the seeded arm. The detection scheme is referenced to the input phase ${\varphi }_0$. (a) The full SU(1,1) interferometer recombines the two-mode squeezed state with a pump beam, phase-shifted by $\pi /2$, in a second nonlinear process, and then detects the output modes using (i) two homodyne detectors (HDs), (ii) one HD on the unseeded mode, (iii) direct intensity detection of only the unseeded mode, or (iv) direct detection in both modes. (b) The truncated SU(1,1) interferometer consists of the first nonlinear interaction, followed by two homodyne detectors (scheme v) that directly collect the two-mode quantum state.

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We present a variation on the SU(1,1) interferometer that replaces the detection schemes shown in Fig. 1(a), including the second nonlinear interaction, with the simpler arrangement shown in Fig. 1(b). This scheme eliminates losses associated with the second nonlinear process, such as imperfect mode matching of the beams and absorption in the medium. We show theoretically that the phase-sensing ability of this seeded “truncated SU(1,1) interferometer” is the same as that of the full SU(1,1) interferometer shown in Fig. 1(a) (i) and surpasses the sensitivity of the direct-detection schemes shown in Fig. 1(a) (iii, iv). We build a version of the truncated SU(1,1) interferometer using 4WM and show how the potential phase sensitivity of this device beats the SQL over a range of operating points.

2. THEORETICAL ANALYSIS OF THE PHASE SENSITIVITY

To analyze the phase sensitivity of the full and truncated SU(1,1) interferometers, it is useful to consider the phase-sensing potential of the quantum state separately from the phase sensitivity of the entire device, which depends on the chosen detection scheme [17–20]. The phase-sensing potential of the quantum state is identical for both the full and truncated SU(1,1) interferometers, and is related to the quantum Fisher information ${\mathcal{F}}_Q$ according to $\mathrm{\Delta }\varphi =1/\sqrt{{\mathcal{F}}_Q}$, where ${\mathcal{F}}_Q$ is a function of both the two-mode quantum state at the output of the nonlinear medium in Fig. 1 and the particular mode(s) in which the phase object(s) are placed [21]. The phase sensitivity of the measurement is related to the classical Fisher information ${\mathcal{F}}_C\le {\mathcal{F}}_Q$, where $\mathrm{\Delta }\varphi \ge 1/\sqrt{{\mathcal{F}}_C}$ describes the phase sensitivity of the entire device [21]. To optimize the phase-sensing ability of a device, one should choose a measurement scheme such that ${\mathcal{F}}_C$ is as close as possible to ${\mathcal{F}}_Q$ [20].

Given a measured observable $X$, the phase sensitivity of a measurement device $\mathrm{\Delta }\varphi$ can be evaluated from the signal-to-noise ratio (SNR),

For the present purposes, we make the simplifying assumption that the two modes have identical losses (see Ref. [23] for a more complete model) and that the seed photon number ${|\alpha |}^2\gg 1$. The sensitivity depends on the operating point set by the local oscillator phases. Specializing to the case where ${\varphi }_c=\pi /2$ (the phase quadrature), but allowing ${\varphi }_p$ to vary, one can use the formalism in Ref. [13] and show from Eq. (1) with $\mathrm{SNR}=1$ that the variance of the phase estimation for the truncated SU(1,1) interferometer is

Figure 2 shows the variance of the phase estimation ${\mathrm{\Delta }}^2\varphi$ in the case of no loss as a function of the gain of the nonlinear process that produces the two-mode squeezed state. The variance shown in Fig. 2 is defined for the best operating point, e.g., curve (v) is for ${\varphi }_p=\pi /2$ in Eq. (3). We also show the variance in phase estimation for the full SU(1,1) interferometer when detecting just the conjugate (unseeded) mode using homodyne detection (scheme ii) or direct intensity detection (scheme iii), as well as direct detection in both modes (scheme iv). (See Supplement 1 for more detailed calculations of the sensitivity for these alternative detection schemes.) Balanced homodyne detection of the joint quadratures substantially improves the phase sensitivity in the seeded truncated and full SU(1,1) interferometers.

 

Fig. 2. Theoretical prediction for the variance of the phase estimation ${\mathrm{\Delta }}^2\varphi$ times ${|\alpha |}^2$, in the case of no loss, as a function of gain, calculated from Eq. (1) with $\mathrm{SNR}=1$ [24]. The black, solid curve (i, v) defines the optimal phase sensitivities for the full and truncated SU(1,1) interferometers using both HDs (schemes i and v). This curve is also equivalent to ${|\alpha |}^2/{\mathcal{F}}_C$ for each detection scheme. The red (short-dashed) and black (long-dashed) curves are the phase sensitivities for the full SU(1,1) interferometer using just a conjugate detector (HD or intensity, schemes ii and iii), and for both intensity detectors (scheme iv), respectively. The gray circles are ${|\alpha |}^2/{\mathcal{F}}_Q$, and represent the fundamental phase sensitivity of the two-mode squeezed state.

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In addition, Fig. 2 shows the fundamental bound on the phase sensitivity of the two-mode quantum state used here, as calculated from the quantum Fisher information ${\mathcal{F}}_Q$. For the seeded two-mode squeezed state and phase object placement considered here (see Fig. 1) [20],

We note that if the phase object were placed in the lower-power, vacuum-seeded arm, ${\mathcal{F}}_Q$ is smaller [20]. The result in Eq. (4) also requires that the measurement has an external phase reference, as shown in Fig. 1, so that the measured sensitivity is independent of the phase of the input beam [21]. One can see in Fig. 2 that ${\mathrm{\Delta }}^2{\varphi }_{\mathrm{tSUI}}$ asymptotically approaches the $1/{\mathcal{F}}_Q$ bound, and that even at low gain ($G\gtrsim 2$), ${\mathrm{\Delta }}^2{\varphi }_{\mathrm{tSUI}}\approx 1/{\mathcal{F}}_Q$. The phase-sensing ability of the truncated SU(1,1) interferometer is thus not only equivalent to that of the full SU(1,1) interferometer, but is an optimal measurement choice for sufficiently high gain. For very low gains ($G\lesssim 2$), one can still reach the bound set by the quantum Fisher information using the truncated SU(1,1) interferometer by varying the classical gain of one of the homodyne detectors [24].

In a practical device the losses play an important role and cause the minimum detectable phase shift to increase. In the full SU(1,1) interferometer the internal losses between the nonlinear interactions have a more detrimental effect on the sensitivity than external losses after the second nonlinear interaction [14]. In the truncated SU(1,1) interferometer, all losses are effectively internal losses. While the truncated SU(1,1) interferometer avoids additional internal loss arising from coupling the beams into a second nonlinear device, the full SU(1,1) interferometer has an advantage of being less sensitive to detector losses (see Supplement 1).

In the particular case of Gaussian quantum states and homodyne measurements, the classical Fisher information ${\mathcal{F}}_C$ is related to the measurement observable $X$ by [20]

The first term is the inverse of the sensitivity limit determined from the SNR in Eq. (1). If the second term is nonzero, one can achieve a better phase sensitivity by using the change in the distribution as a function of phase [25]. In our scheme, the most sensitive operating point corresponds to the minimum of the joint phase quadrature, at which point ${\partial }_{\varphi }\mathrm{\Delta }X=0$.

3. EXPERIMENTAL DEMONSTRATION OF THE TRUNCATED SU(1,1) INTERFEROMETER

We construct the truncated SU(1,1) interferometer using 4WM in a ${}^{85}\mathrm{Rb}$ vapor cell, as depicted in Figs. 3(a) and 3(b) [23]. The nonlinear interaction among the applied beams and the atoms results in an amplified probe and the creation of a correlated conjugate field, where the probe and conjugate fields constitute a two-mode squeezed state. After the cell, the probe and conjugate fields are sent to separate balanced homodyne detectors, as shown in Fig. 3(c).

 

Fig. 3. (a) The experimental setup consists of a strong pump beam (power $\approx 400\mathrm{mW}$, $1/e^2$ beam waist 0.7 mm) and orthogonally polarized weak probe beam (power $\approx 100\mathrm{nW}$, $1/e^2$ beam waist 0.4 mm) propagating at a small angle ($\approx 4\mathrm{mrad}$) through a warm rubidium vapor (temperature 105 °C). After the cell, the pump beam is blocked using a Glan–Taylor polarizer, and the probe and conjugate beams propagate to separate balanced HDs. The local oscillators for each HD are generated using a similar 4WM setup, resulting in local oscillator powers of $\approx 1\mathrm{mW}$. The sum of the AC photocurrents is sent to the spectrum analyzer. (b) We pump the D1 $5S_{1/2}\to 5P_{1/2}$ transition in ${}^{85}\mathrm{Rb}$. Here, $\mathrm{\Delta }\approx 700\mathrm{MHz}$, and ${\nu }_{HF}=3.03\mathrm{GHz}$ is the hyperfine splitting between the $F=2$ and $F=3$ ground state sublevels. The 4WM generates a conjugate field (blue) that is 6.06 GHz higher in frequency than the applied probe field (red). (c) The probe HD setup consists of a local oscillator (LO) propagating through an electro-optic modulator (EOM) and reflecting off a piezo-electric-transducer-mounted mirror (PZT). The LO mixes with the probe beam on a 50/50 beam splitter, and each output is sent to a subtracting detector. The conjugate HD setup does not contain an EOM.

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To detect a specific quadrature, we use piezo-mounted mirrors in the path of each local oscillator to control the phases of each homodyne setup, ${\varphi }_p$ and ${\varphi }_c$. The AC components ($\approx 1\mathrm{MHz}$) of the two homodyne detectors are summed and sent to a spectrum analyzer. The DC components are used to lock the phase of each homodyne detector.

By locking the homodyne detectors to the DC component of the signal, we effectively lock the homodyne detectors to the input phase of the seed beam, indicated by the phase reference line in Fig. 1. This allows us to assess the potential phase-sensing ability of the device without having to build a fully phase-stable interferometer. Likewise, by measuring noise at a convenient high frequency $\mathrm{\Omega }=1\mathrm{MHz}$, we avoid technical noise sources. In the absence of technical noise, the noise floor at 1 MHz would extend to DC. The optimal phase sensing is achieved when both homodyne detectors are locked to the phase quadrature of their input beam. This occurs where ${\varphi }_p$ and ${\varphi }_c$ are locked so that the individual DC homodyne signals are zero and the slopes are the same (see Supplement 1).

An electro-optic phase modulator (EOM) imposes a weak sinusoidal phase shift $\varphi (t)=\sqrt{2}\delta \varphi \cos (2\pi \mathrm{\Omega }t)$ on the probe local oscillator, where $\delta \varphi$ is the root mean square amplitude of the phase shift. The signal resulting from this modulation appears on the spectrum analyzer as a peak, shown in Fig. 4(a), with an SNR of ${\mathrm{SNR}}_{\mathrm{tSUI}}$. The signal is generated in the probe detection arm and is determined by $\delta \varphi$ and the probe and local oscillator powers. The noise level is determined by both detection arms, i.e., by the level of squeezing generated by the 4WM process. The reduction in the noise level due to squeezing is the key to achieving quantum-enhanced precision in our system. Both signal and noise vary with the operating point as set by the local oscillator phases. Absent low-frequency technical noise, the modulation frequency could be lowered while keeping a constant SNR. Thus we use this AC signal to infer a sensitivity to DC phase shifts for comparison to the theory given above.

 

Fig. 4. (a) An example SNR measurement (resolution bandwidth 30 kHz) using two-mode squeezed beams (blue, solid) and coherent beams (red, dashed), with identical optical power in the phase-sensing (probe) beam path. (b) The SNRI as a function of ${\varphi }_p$. The inset shows the region where we observe an improvement beyond the SQL. The red curve is from Eq. (3) with best-fit parameters $\eta =0.65$ and intrinsic gain $G=3.3$. The black, dashed curve shows the potential improvement (${\mathrm{SNRI}}_{{\mathcal{F}}_C}$) that could be obtained by additional signal processing as derived from the classical Fisher information in Eq. (5) using those fit parameters. The uncertainties are statistical, one standard deviation for 10 measurements.

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For comparison to this quantum-enhanced measurement procedure, we follow a similar procedure using coherent fields. We turn off the 4WM process by blocking the pump and increase the intensity of the seed beam to create a coherent state probe beam with the same power as the one generated by the 4WM process, resulting in the dashed curve in Fig. 4(a) with an SNR of ${\mathrm{SNR}}_{\mathrm{coh}}$. In this case, ${\mathrm{SNR}}_{\mathrm{tSUI}}-{\mathrm{SNR}}_{\mathrm{coh}}\approx 4\mathrm{dB}$.

A previous demonstration of a full SU(1,1) interferometer [15] uses the detection scheme defined by curve (ii, iii) in Fig. 2, which is not the optimal choice of measurement even in the case of no loss. We note that the conventions used in Ref. [15] imply a 3 dB improvement in SNR even at $G=1$, where there are no quantum enhancements.

To characterize the operation of the truncated SU(1,1) interferometer, we compare the ${\mathrm{SNR}}_{\mathrm{tSUI}}$ for a particular choice of local oscillator phases to the optimal ${\mathrm{SNR}}_{\mathrm{coh}}$ (i.e., the SNR when the local oscillator phases are set to give the largest ${\mathrm{SNR}}_{\mathrm{coh}}$). The phase sensitivities are then related to the SNR improvement ($\mathrm{SNRI}={\mathrm{SNR}}_{\mathrm{tSUI}}-{\mathrm{SNR}}_{\mathrm{coh}}$) by

To measure the SNRI as a function of operating point, we lock ${\varphi }_c$ to detect the phase quadrature of the conjugate beam and scan ${\varphi }_p$. The resulting variation in SNRI is shown in Fig. 4(b). The peaks at ${\varphi }_p=\pm \pi /2$ correspond to the operating points with the maximum signal. At ${\varphi }_p=\pi /2$, the joint quadrature noise is at its minimum (maximum squeezing). At ${\varphi }_p=-\pi /2$, the joint quadrature noise is at its maximum (maximum antisqueezing). The solid curve is a theoretical fit of Eq. (3) to the data with best-fit parameters $\eta =0.65$ and intrinsic gain $G=3.3$. The experimentally measured gain is 2.7, which includes some loss in the vapor cell. For simplicity, we take both arms to have identical loss parameters $\eta$.

The classical Fisher information for this detection scheme in the presence of loss is shown by the dashed line in Fig. 4(b), which is calculated from Eq. (5) using the fit parameters defined above and ${\mathrm{SNRI}}_{{\mathcal{F}}_C}=-10{\log }_{10}[(1/{\mathcal{F}}_C)/{\mathrm{\Delta }}^2{\varphi }_{\mathrm{coh}}]$. The close overlap of this curve with that derived from ${\mathrm{\Delta }}^2{\varphi }_{\mathrm{tSUI}}$ implies that the second term in Eq. (5) is negligible compared to the first.

The standard quantum limit, $\mathrm{\Delta }{\varphi }_{\mathrm{SQL}}$, is usually taken to mean the best sensitivity of a Mach–Zehnder interferometer with intensity detection at both output ports. We compare the truncated SU(1,1) interferometer to a truncated version of a Mach–Zehnder interferometer in which the second beam splitter is replaced with homodyne detectors. The measured signal $X$ is taken to be the difference in the quadrature signals from the two homodyne detectors. At its optimum operating point, this configuration has the same sensitivity as the standard Mach–Zehnder. The SQL depends on the mean number of photons used in the measurement. In the seeded SU(1,1) interferometer, the two beams have different photon numbers, $N_p$ and $N_c$ (with $N_c<N_p$). Taking the number of photons going through the phase object as the essential resource, we select as the SQL the sensitivity of an ideal Mach–Zehnder with $N_p=\eta G{|\alpha |}^2$ photons detected in the phase-sensing arm, i.e., ${\mathrm{\Delta }}^2{\varphi }_{\mathrm{SQL}}=1/(2N_p)$.

To compare our results to the SQL, we need to compare the measured ${\mathrm{SNR}}_{\mathrm{coh}}={(\delta \varphi )}^2{N}_p$ to that expected for the truncated Mach–Zehnder. We determine $N_p$ in terms of the probe power and estimated losses, and we find that the measured ${\mathrm{SNR}}_{\mathrm{coh}}$ agrees with the expected value (see Supplement 1). Thus we can conclude that the SNRI measured here shows that the truncated SU(1,1) interferometer achieves a 4 dB improvement in ${\mathrm{\Delta }}^2\varphi$ over the SQL, including losses. Further, if we could eliminate all detection losses for the coherent state measurement (see Supplement 1), then ${\mathrm{\Delta }}^2{\varphi }_{\mathrm{tSUI}}$, which includes losses, surpasses that by 3 dB. Finally, in the ideal limit of no losses, where the phase sensitivity corresponds to curve (v) in Fig. 2, the SNRI for a gain of 3.3 would be approximately 10.5 dB. The measured value of 4 dB is lower than the ideal value due to losses in the system.

4. CONCLUSIONS

We have constructed a novel variation on the SU(1,1) interferometer that removes one nonlinear interaction, and we have measured the SNR of this device as a function of operating point. This arrangement provides a simpler means to achieve quantum-enhanced phase sensitivities, and even with $\approx 35\%$ loss we have demonstrated up to 4 dB improvement over the comparable classical device.