Over the past 30 years, squeezed states of light have been exploited as an important quantum resource in different fields, such as continuous-variable quantum teleportation ^[1,2] and quantum computing ^[3], quantum error correction coding ^[4,5], quantum state engineering ^[6–8], quantum imaging ^[9,10], and precision measurement and sensing ^[11–17]. Because the noise of certain quadrature components of squeezed light is lower than that of coherent light, squeezed light is capable of surpassing the shot-noise limit (SNL) for the high precision measurement of a physical parameter. Obviously, squeezed light sources are necessary to optimize and upgrade the performance of an optical sensing system. The main methods used to generate squeezed states are the second and third nonlinear processes, e.g., parametric downconversion (PDC) ^[18] and four-wave mixing (FWM) ^[19]. Squeezed states of light have been obtained with different experimental systems, such as atomic ensembles ^[20–22], fibers ^[23–25], mechanical resonators ^[26,27], micro-cavities ^[28–30], and optical parametric oscillator (OPO) cavities ^[31–33]. In this review, we focus on the single-mode squeezed light generated in a cavity, as it is a mature method to achieve high-quality squeezed states of light. To date, the highest squeezing factor of 15 dB has been obtained by employing a sub-threshold OPO cavity ^[33].

Single-mode squeezed light has proven very useful in many practical systems of precision measurement, such as phase estimation ^[16] and tracking ^[12], optical magnetometers ^[11], biological measurement ^[13], small displacement measurement ^[34], and gravitational wave (GW) detection ^[14,35], with the purpose being to improve the signal-to-noise ratio (SNR). For instance, with squeezing, the error of a tracked phase was well below the coherent-state limit ^[12]. Squeezing was applied in a magnetometer with the sensitivity enhanced by 3.2 dB ^[11] and biological measurements for which quantum noise was surpassed by 42% ^[13]. Moreover, quantum-squeezing-enhanced fiber Sagnac ^[36] and Mach–Zehnder interferometers (MZIs) ^[37] were demonstrated in 2010 and 2017, respectively. Goda et al. improved the displacement sensitivity of a prototype GW detector by 44% with a squeezed vacuum ^[38]. Subsequently, the GEO 600 was the first GW detector employing squeezing leading to a broadband noise reduction of up to 3.5 dB ^[35]. Thereafter, noise of the LIGO interferometer was beyond the SNL of 2.15 dB with squeezed-light injection, and the frequency range was extended down to 150 Hz ^[14].

It should be noted that other quantum light sources can also be utilized in quantum sensing. For example, quantum correlation ^[39–41] and quantum entanglement created by multimode squeezed states ^[42–54] have potential applications in quantum computing ^[3], quantum networks ^[53], and squeezing-enhanced imaging ^[55]. The phase sensitivity of the interferometer can be theoretically improved to the Heisenberg limit of 1/N ^[56–62] by using the NOON state ^[60] or SU(1,1) interferometry ^[56,62]; here N is the average photon number for detection. One can see the references for details.

In this paper, we review the recent progress and applications of single-mode squeezed states of light for optical sensing and tracking. In Section 2, we present the basic principle and the primary measurement schemes such as the interferometer, homodyne detection, and heterodyne detection with squeezed-light injection. The applications of these schemes such as phase tracking, position and rotation measurement, magnetic field measurement, GW detection, and fiber sensing are introduced in Section 3. The future prospects for squeezing are presented in Section 4.

A nearly monochromatic electromagnetic field can be regarded as a function of a pair of non-commuting quadrature components in phase space, which are normally called amplitude quadrature X^ and conjugate phase quadrature Y^. According to Heisenberg’s uncertainty principle, non-commuting observables cannot be precisely measured simultaneously, and their uncertainty relation is Δ2X^Δ2Y^≥1.(1)For a coherent state, the fluctuations of different quadrature components are identical in phase space and equal to zero-point fluctuations (also called shot noise), i.e., Δ2X^=Δ2Y^=1. Figure 1(a) shows a representation of a coherent state using the phasor diagram. When measuring weak signals such as GWs with a coherent state, the experimental data generally exhibit these fluctuations. To improve the sensitivity of the measurement while obeying Heisenberg’s uncertainty principle, squeezed light is introduced in which the fluctuations of one quadrature component are squeezed at the expense of an expansion for the other. In this case, the fluctuations are given by Δ2X^=e−2r<1 and Δ2Y^=e2r>1, where r is called the squeezing factor ^[63]. During a measurement process, one can use the squeezed quadrature component to enhance the precision. In Fig. 1, we present several typical coherent states and squeezed states.

To obtain squeezed light, the main methods employ phase-dependent nonlinear processes, such as FWM and PDC. In 1985, Slusher et al. first observed squeezing using FWM ^[19]; then Wu et al. obtained 3-dB squeezed light via a PDC process ^[18]. Thereafter, researchers have generated squeezed states using versatile methods with significant technical improvements ^[33,64–67]. Because the performance of a quantum-sensing system strongly depends on the squeezed light, researchers have made great efforts in the improvement of the squeezing factor. To date, the greatest squeezing is 15 dB below the SNL, which is obtained through an optical parametric amplification (OPA) process in a periodically poled KTiOPO4 (PPKTP) crystal ^[33].

OPA happens in a nonlinear crystal with a second-order susceptibility χ(2). A pump field and a signal field are injected into the crystal. In the OPA process, the pump field is converted to the signal and idler fields. Because of energy conservation, the sum of the frequencies of the signal and idler fields equals the frequency of the pump field. Similarly, the momentum conservation requires phase matching of the interacting wavevectors. In addition, the temperature of the nonlinear crystal must be carefully controlled for proper phase matching.

Taking the vacuum state as an example, the manner in which the nonlinear process generates squeezed light is depicted in Fig. 2 ^[68]. Consider a short segment of a χ(2) nonlinear crystal interacting with a vacuum state Evac,fin and coherent state E2fin, where E denotes the amplitude of the electric field, and f and 2f denote the optical frequencies of the vacuum state and the coherent state, respectively. The red and orange regions in Fig. 2 represent the fluctuations of the vacuum state before and after the OPA process, which describes the squeezing effect. The dielectric polarization P(E) inside the crystal generates a vacuum-squeezed state via OPA. Because of phase mismatch, the unwanted components such as E4fout are greatly suppressed, which ensures the purity of the squeezed light. The graph of P(E) illustrates the phase dependence of the noise of the vacuum field on the pump field.

The standard method to detect squeezing is balanced homodyne detection (BHD). A strong local oscillator (LO) and a squeezed light with the same optical frequency interfere on a 50/50 beam splitter (BS) and then are coupled to a pair of balanced detectors [Fig. 3(a)]. After isolating the low-frequency signal and reserving the first-order terms, the fluctuation of the BHD output is given by Δi−≅α(δX^cosθ+δY^sinθ),(2)where α is the amplitude of the LO, δX^ and δY^ are the fluctuations in the amplitude and phase quadrature components, respectively, and θ is the relative phase between the LO and the squeezed light. The output contains the fluctuations of the squeezed light multiplying by the amplitude of the LO. By modifying the relative phase θ, one is able to measure the fluctuations of any quadrature component δX^θ=δX^sinθ+δY^cosθ and then reconstruct the state ^[69].

Most optical precision measurements are based on interferometry, homodyne detection, and heterodyne detection. The optical phase difference of the two arms provides a measure of the signal sought. However, vacuum noise introduced through the unused port limits the measurement precision ^[70]. Replacing vacuum noise by injecting squeezed light into the unused port can significantly suppress the noise and improve the SNR. Many experiments have demonstrated squeezing-enhanced precision measurement. In this section, we briefly review two typical configurations, i.e., the squeezing-enhanced MZI ^[71] and the polarization interferometer ^[72].

An MZI [Fig. 3(b)] takes in an incident monochromatic coherent light beam and a vacuum-squeezed beam, mixes them at BS1(50/50), and splits them into two arms. The two beams are brought together again to interfere at BS2(50/50) and finally are detected by two photodetectors. The frequencies of the two incident light beams are same. The relative phase ϕ of the two arms is kept equal to π/2 for the optimal SNR. Without injection of a squeezed light, the minimal measurable phase difference is Δϕmin=1N,(3)where N is the average photon number of the coherent beam of light. If a vacuum-squeezed component is used in detection, the sensitivity Δϕ improves by a factor e−r for a lossless process. In practice, the enhancement is degraded by the system loss. The above analysis is also valid for other types of interferometers, such as the Sagnac interferometer and the Michelson interferometer ^[73].

The first squeezing-enhanced experiment based on MZI was performed by Xiao et al. ^[71]. The squeezed vacuum was produced from an OPO cavity and injected through the unused port of the MZI. A phase dither was placed in one arm. As shown in Fig. 4, a 3-dB improvement of the SNR was observed using squeezed light.

A squeezing-enhanced polarization interferometer [Fig. 5(a)] ^[72] was proved to be very sensitive in the detection of polarizations and magnetic fields ^[11]. A KTP crystal was placed in a cavity to generate squeezed light, which was subsequently coupled with a coherent light using a polarization beam splitter (P1) and then passed through a polarization rotator (PR). A half-wave plate and the second polarization beam splitter (P2) served as a BS for BHD where the coherent light acted as an LO. When the rotator was removed and the squeezed light was blocked, the fluctuation in the BHD output corresponded to an SNL. The polarization rotation signal was detected by BHD. Traces (a), (b), and (c) in Fig. 5(b) were obtained with squeezed light, the vacuum state, and anti-squeezed light input at the unused port, respectively. With squeezing injection, the SNR was improved by 1.8 dB relative to the vacuum.

Besides, heterodyne detection is capable of measuring the difference in frequency of two fields to avoid low-frequency noise. The technique was applied to Doppler anemometry to obtain the velocity of gas flows and improved the SNR by 1 dB using amplitude-squeezed light ^[74].

The measurement of the point spot of a laser beam is used in many areas such as atomic force microscopy ^[75], ultraweak absorption measurements ^[76], and single-molecule tracking in biology ^[13,77]. The position of the spot can be defined as the mean position of all photons in the beam, the measurement precision of which is constrained by quantum fluctuation. The transverse displacement resolution of a TEM00 laser beam is given by ^[78] DQNL=w02Ntot,(4)where w0 is the waist of the beam and Ntot is the total photocurrent; QNL is an abbreviation for the quantum noise limit.

Any displacement modulation of the TEM00 mode is transferred to the amplitude of higher-order modes, whereas if the modulation is very small, the TEM10 mode mainly contributes to the displacement signal. Therefore, the displacement measurement is based on extracting the TEM10 mode component from the displaced TEM00 mode. There are two conventional ways to detect the displacement, i.e., split detection and TEM10 homodyne detection. The advances in both schemes are reviewed below.

One way to measure the displacement of a laser beam based on split photodetectors uses the two-pixel detector. In 2000, Fabre et al. proposed that the minimum measurable displacement DQNL in a two-pixel detector by a TEM00 Gaussian beam is defined by ^[55] DQNL=π8w0Ntot,(5)which is degraded in comparison to Eq. (4). This reduction arises from the mismatch between the noise mode of the split detection (i.e., the “flipped” mode) and the TEM10 mode introduced by displacement. To further enhance the sensitivity in split detection, one needs to fill the input beam with a squeezed noise mode, specifically, a squeezed “flipped” mode.

In 2002, Treps et al. used a spatial squeezed state for the displacement measurement and achieved a sub-QNL sensitivity ^[10]. They combined a 3.5-dB squeezed-vacuum TEM00 mode generated by an OPA cavity and its coherent “flipped” mode. Introducing a small transverse displacement modulation by two electro-optic modulators (EOMs) [Fig. 6(a)], they reduced the noise floor to a level of about 2.4 dB below the QNL. In 2003, they extended their work to measure two-dimensional sub-QNL displacements using a three-mode nonclassical state, which was generated by coupling three orthogonal transverse modes, two in a squeezed-vacuum state and one in a bright coherent field [Fig. 6(b)] ^[34]. In this experiment, they simultaneously achieved a 3.05±0.1dB horizontal noise reduction and a 2.0±0.1dB vertical noise reduction, with the smallest displacement detectable improved from 2.3 Å (1 Å = 0.1 nm) to 1.6 Å.

Another detection scheme, TEM10 homodyne detection, can achieve QNL for very small displacements ^[78]. This scheme uses a homodyne detection setup involving a TEM10 mode of the LO (Fig. 7). The noise mode of this detection is the TEM10 mode, which perfectly matches the information to be extracted, thereby accounting for its 100% efficiency detection in theory.

In this scheme, filling the input beam with a squeezed-vacuum TEM10 mode enables measurements beyond the QNL, which outperforms the split detection for equal values of squeezing. In 2006, Delaubert et al. ^[79] achieved a 2-dB enhancement in small displacement measurements using the TEM10 detection scheme. In 2014, Sun et al. ^[80] obtained 2.2±0.2dB spatial squeezing and reduced the minimum measurable displacement from 1.17 Å to 0.99 Å.

Measurements of rotation signals are similar to those of displacements and are also constrained by quantum mechanics; the rotation angle and angular momentum obey Heisenberg’s uncertainty principle ^[81,82]. It is conventional to use a beam carrying orbital angular momentum such as the Laguerre–Gaussian modes ^[83] in spatial rotation measurements based on its vortex wavefront ^[84–86]. However, such measurement schemes are restricted to a resolution limit depending on the photon number and the orbital angular momentum of the beam ^[82]. Using nonclassical states such as the squeezed states of Laguerre–Gaussian modes can help beat such limits and reduce the noise below the SNL ^[87]. Liu et al. defined an n-order orbital angular position (OAP)-squeezed state, which is the combination of a bright LG0,nsin mode coherent state and a squeezed-vacuum LG0,−nsin state ^[87]. Using a first-order OAP-squeezed state in metrology, they achieved a 3-dB enhancement in rotation measurement over the QNL, with the sensitivity being improved from 24.9nrad/Hz to 17.7nrad/Hz.

Optical phase estimation (OPE), targeting quantum-limited precisions with limited photon numbers, is one of the key techniques in optical communications and precise metrology ^[88,89]. Adaptive homodyne measurement of an optical phase is an efficient and sophisticated method to approach an ideal canonical phase measurement ^[90,91]. If the adaptive control works properly, the sensitivity of the phase estimation is solely limited by the intrinsic phase uncertainty of the probe beam. Squeezed states ^[92,93] provide a further quantum enhancement in the phase estimations by suppressing quantum noise of the probe beams ^[94]. From theory, Wiseman et al. have demonstrated that an adaptive homodyne measurement with coherent states can estimate a fixed or time-varying optical phase with quantum-limited precision, and, moreover, it can be surpassed with the squeezed states ^[91,94,95]. In this section, we review how OPE provides an advance in sensitivity with the use of squeezed states.

The first experiment of a single-shot adaptive phase measurement was conducted by Armen et al. in 2002 ^[96]. They performed an adaptive homodyne measurement on weak coherent states with a fixed phase, and proved that the adaptive homodyne measurement can beat the heterodyne measurement in terms of phase estimation variances. Their system approached the intrinsic quantum uncertainty and paved the way for squeezing-enhanced phase estimation experiments.

In 2015, Berni et al. demonstrated quantum-enhanced ab initio OPE with squeezed-vacuum states and a real-time Bayesian adaptive estimation algorithm ^[16]. They generated 5.69±0.07dB squeezed-vacuum states using a cavity-enhanced degenerate PDC. The input, a squeezed-vacuum state, underwent a fixed phase shift [Fig. 8(a)], which was then measured by homodyne detection. The LO of the homodyne detection was feedback-controlled based on Bayesian inference. The quantum Fisher information for a pure 6-dB squeezed-vacuum state [Fig. 8(b)] was larger than that for a coherent state within a specific range of phase shift, which indicates that for an optimal measurement this system has one certain phase operating point. They performed a pre-estimation using the initial measurement results to make a rough estimate of the phase with Bayesian inference. This information was used to shift the LO to the nearest optimal phase operating point, and then they measured the homodyne outcomes with a larger number of samples to obtain the final estimate. The estimation variances versus the different input phases and the number of homodyne samples (Fig. 9) show that adaptive-feedback control with squeezing enhancement outperforms the quantum Cramer–Rao bound (QCRB) for coherent states and approaches the optimal CRB for squeezed states.

Like single-shot phase estimations, time-varying phase estimations have also been investigated. In 2010, Wheatley et al. demonstrated a continuous adaptive phase measurement of a stochastically varying optical phase of coherent states, which provided better precision than the standard (non-adaptive) quantum limit ^[97]. They used real-time filtering and post-processed smoothing with an improvement of 2.24±0.14 in terms of mean square error (MSE) compared with the standard quantum limit.

In 2012, squeezing-enhanced time-varying OPEs were realized by Yonezawa et al. ^[12]. They used a phase-squeezed state to track a time-varying optical phase with a range of approximately ±1 radian. The phase-squeezed states were prepared by an OPO (Fig. 10) and modulated with a stochastically varying signal. The modulated phase-squeezed beams were detected by adaptive homodyne detection with an overall efficiency of 85%. A Kalman filter was used to design the phase-locked loop to track the stochastically varying optical phase. The phase tracking results (Fig. 11) show that there exists an optimum squeezing level in terms of MSE of the estimate. This is because increasing anti-squeezing affects the precision of the estimate. With the optimal squeezing level, the MSEs [red crosses in Fig. 11(b)] were below the coherent-state limit [trace (i)]; quantum-enhanced phase tracking was realized with large phase changes.

Optical phase tracking with squeezing enhancement has a variety of applications, one of which is optomechanical motion sensing. Iwasawa et al. have worked on measuring the motion of a mirror under an external stochastic force using optical phase tracking techniques (Fig. 12) ^[98]. They estimated the position, momentum, and force of the mirror with optical probe beams in coherent and phase-squeezed states. The estimations with coherent states [Fig. 12(b), green circles in right panel] were nearly reaching the coherent QCRB [trace (ii)]. The estimations with squeezed states (red diamonds) showed a clear enhancement and beat the coherent QCRB, although the estimation precisions did not reach the squeezed-state QCRB [trace (iv)] because of the impurity of the squeezed state. These results demonstrate the potential of the squeezing-enhanced phase estimation method for future quantum metrological applications.

Ultrasensitive magnetometers have found versatile applications, from geomagnetic field detection ^[99] to GW detection ^[100], nuclear-magnetic-resonance signal detection ^[101] to biomagnetism ^[102]. In 1992, Kupriyanov et al. proposed using squeezed light to detect magnetic fields ^[103]. Subsequently, many methods incorporating squeezed light have been reported to enhance the SNR, including quantum non-demolition detection, the quantum Kalman algorithm, and spin-squeezed atom ensembles, with the potential to reduce noise to the Heisenberg limit ^[104–108]. This section mainly covers sub-SNL magnetometers using polarization-squeezed light.

The nonlinear magneto-optical rotation (NMOR)-based magnetometer ^[109] is a common type of atomic optical magnetometer. It consists of a polarization interferometer where the polarization rotator is replaced by a well-prepared atomic ensemble [Fig. 5(a)], for example, a spin-oriented atom ensemble (Fig. 13) ^{[108,110,111]}. In this system, the polarization of the probe beam rotates under a weak external magnetic field because of the interaction between the probe beam and the atoms. According to Heisenberg’s uncertainty principle, this magnetometer is ultimately confined by quantum noise including atomic projection noise and optical shot noise. A polarization-squeezed state of light can improve the performance of NMOR magnetometer.

Generally, the noise var(S2out) detected by an NMOR-based magnetometer is ^[11,112] var(S2out)=var(S2in)+G2Ntot24var(Fz),(6)where Fz denotes the collective atomic spins in the z direction, S represents the Stokes operators, G is the interaction strength, and Ntot is the total number of photons. The optical shot noise and the atomic projection noise are described by var(S2in) and var(Fz), respectively. In regard to the optical noise, the fluctuation of a bright coherent beam is depicted as a sphere of uncertainty on the surface of the Poincare sphere, whereas, for the S2in parameter, squeezed polarized light is represented by an ellipsoid ^[113,114] (Fig. 14). In this sense, the optical noise can be reduced by injecting a polarization-squeezed beam; therefore, a sub-QNL magnetometer can be built. The polarization-squeezed probe beam of such a magnetometer can be produced by mixing a squeezed vacuum with a coherent beam in a polarization beam splitter ^[11].

In 2010, Wolfgramm et al. used 3.6-dB polarization-squeezed light by mixing a vertically polarized vacuum-squeezed beam and a horizontally polarized LO to probe a hot unpolarized ensemble of rubidium atoms. They improved the sensitivity by 3.2 dB beyond the SNL ^[11]. In 2012, Horrom et al. enhanced the sensitivity to 1pT/Hz in a sub-kilohertz frequency band using 2-dB low-frequency squeezed light generated by an atomic squeezer ^[115]. In 2014, Otterstrom et al. combined the squeezer and the magnetometry together, producing an in situ NMOR-based magnetometer ^[15]. In this experiment, they generated two-mode relative-intensity squeezed states via FWM obtaining an enhancement in sensitivity from 33.2pT/Hz to 19.3pT/Hz. In 2016, Lucivero et al. achieved 1.5-dB noise suppression in a high atomic density (n≈1.3×1013cm−3) system ^[116].

GWs propagating at the speed of light are generated by accelerated mass distributions, such as coalescing binary black holes and neutron stars ^[117]. Several initial detectors, such as LIGO, Virgo, GEO 600, and TAMA 300, made joint observations from 2002 to 2011 and have evolved into a global network ^[38]. In 2015, Advanced LIGO first observed a GW from a binary black-hole merger ^[118], the event symbolizing a new era in GW astronomy. In 2017, a GW from a binary black-hole coalescence was first detected by a three-detector observation, i.e., Advanced LIGO and Advanced Virgo ^[119]. In this section, we review the principle of Advanced LIGO and the squeezing-enhanced interferometer of LIGO.

Advanced LIGO consists of two similar kilometer-scale Michelson interferometers, located in Hanford and Livingston, U.S. [Fig. 15(a)]. When a GW passes through the two interferometers, the two arms of each interferometer are strained weakly, leading to a change in length ^[73]. This small change in length is converted to a weak phase signal between the two arms of each interferometer. The interferometers operate close to a dark fringe, which is the most sensitive point for GW detection. The light sources are 1064 nm wavelength Nd:YAG lasers, stabilized in intensity, frequency, and beam geometry. To detect the weak GW signal, enlarging the light power and suppressing noise are necessary. In the interferometer, arm cavities and power recycling cavity are used to increase the light power ^[120], and a partially transmitting signal-recycling mirror at the output is used to optimize signal extraction ^[121,122]. Moreover, the BS and arm cavities are all suspended as pendula on a seismic isolation platform. The optical path must be in vacuum to prevent phase fluctuations caused by Rayleigh scattering ^[14]. Figure 15(b) shows the sensitivity versus frequency of Advanced LIGO.

Although a GW has been detected, higher sensitivity and a broader frequency range are needed for future GW astronomy. Injecting nonclassical light is one powerful method for improvements, and squeezed light is the optimal choice ^[123]. A squeezing of 10 dB amounts to a 10-fold increase in light power, without introducing thermal deformation of the sensitivity from a high light power ^[124]. After proof-of-principle experiments ^[38,71,72], squeezed light was demonstrated in the GEO 600 ^[35] and LIGO ^[14] with sensitivity improvements.

The LIGO device with squeezing injection (Fig. 16) has a light source (H1 laser), Nd:YAG laser stabilized in frequency and intensity, emitting a 15 W light beam to a Michelson interferometer. The light in the arm cavities is power-enhanced to about 40 kW by the cavities. The interferometer operates close to a dark fringe, outputting light of about 30 mW power to the antisymmetric port. After going through the output Faraday isolator, a squeezed-vacuum state enters the interferometer from the antisymmetric port replacing the vacuum state. The vacuum-squeezed state is produced using PDC in a PPKTP crystal placed in an OPO cavity. To control the squeezing angle, a frequency-shifted laser beam is also sent through the interferometer, together with the squeezed-vacuum state. Before final detection, an output mode cleaner is used to filter out the frequency-shifted laser beam, where the squeezed vacuum passes. For enhancing sensitivity and mitigating noise from the squeezed source, mode match between the squeezing and coherent states is needed in addition to critically controlling the squeezing angle and reducing the back-scattered light from the OPO cavity and the total loss in detection ^[14].

By injecting a ∼10-dB squeezed-vacuum state, the sensitivity improvement is up to 2.15 dB in the shot-noise-limited frequency band down to 150 Hz (Fig. 17). In addition, frequency-dependent squeezing ^[125,126] and two-mode squeezing were proposed ^[127] in optimizing GW detection in practice.

Fiber interferometers are widely used in high-sensitivity strain sensors ^[128], fiber-optic gyroscopes ^[129], and quantum state detections ^[130], because fiber systems have unparalleled advantages such as immunity to electromagnetic disturbances, flexible multiplexing, and long-distance sensing ^[131,132]. We review next those experiments using fiber interferometer with squeezed-light injection.

As stated in the previous section, when N classical probes are used, the sensitivity of the optical measurement device is fundamentally limited by the shot noise. This precision can be surpassed using squeezed states. In a fiber system, the most challenging part for squeezing applications is the optical loss introduced by the fiber interferometer from decoherence effects and degradation of the squeezing factor ^[36]. Moreover, although squeezed states at high frequencies (MHz) are easy to achieve, they are not compatible with fiber sensing because the parameters to be measured for fiber position and strain sensing systems typically vary in the kilohertz frequency band ^[37]. To make full use of the squeezed state in fiber interferometers, many techniques have been developed, and a series of experiments has demonstrated the feasibility of quantum-enhanced fiber sensing ^{[36,37,133,134]}.

In 2010, Mehmet et al. ^[36] first applied squeezed states in fiber-based Sagnac interferometers and realized sensitivities beyond SNL. A 10 m long fiber Sagnac interferometer was constructed and operated with a continuous-wave laser operating at 1550 nm wavelength, for which low-loss fibers are available. To compensate for the strong divergence and to minimize coupling loss, aspheric lenses and antireflecting coatings were used, and the modes of the two beams were carefully matched. The single-path transmission was 98% for these fiber-based Sagnac systems. The squeezed vacuum was injected in the unused port of the interferometer, and a noise reduction of 4.5 dB was achieved. When a 6 MHz phase modulation signal was added (Fig. 18), the results performed by BHD showed that the SNR with squeezed-light injection [Fig. 18, trace (a)] outperforms the SNR with vacuum state injection [Fig. 18, trace (b)].

In addition to high-frequency quantum-enhanced measurements in Sagnac interferometers being realized in fiber systems, a squeezing-enhanced fiber MZI for low-frequency phase measurement has also been demonstrated. In 2017, Liu et al. ^[37] reported a fiber MZI for phase measurements in the kilohertz frequency band beyond the SNL with a high-frequency (MHz range) squeezed state. The local light imported into one port of the interferometer was amplitude-modulated (AM) at the megahertz frequency to avoid low-frequency noise. Another input beam was the squeezed-vacuum field, which was generated by a NOPA and coupled to a polarization-maintaining fiber for quantum detection. The interference signal was then measured by BHD, and a 2-dB phase noise suppression below the SNL was achieved at a frequency of tens of kilohertz. Figure 19(a) shows the phase noise reduction compared with shot noise at several frequencies; Figs. 19(b)–19(d) present experimental results obtained at 30 kHz, 80 kHz, and 150 kHz, respectively. With the injection of an AM coherent field and a squeezed state, the precision of the low-frequency phase measurement was enhanced and the noise reduction was independent of the modulation depth.

Apart from various fiber interferometers for phase measurements, adaptive homodyne systems have also been developed to estimate the phase signal in fiber systems. Not confined to measuring single frequency signals, in 2018, Zhang et al. designed a quantum-limited fiber homodyne system to detect low-frequency random phase signals beyond the π range ^[135]. They used a double sideband mode as a signal beam on which a random phase signal was imposed using a piezoelectric transducer. The signal was then detected with a homodyne detection system combined with two feedback loops, one working below 100 Hz to prevent the system from picking up ambient noise and the other serving as a Kalman filter to realize an optimal estimation. The optical parts were packed to minimize ambient noise, and all the optical components were polarization-maintaining fiber devices for a higher interferometer visibility in the fiber system. From phase tracking results (Fig. 20), they demonstrated quantum-limited fiber phase tracking over a large angular range at a photon flux of ∼106. One can further improve such a system with a squeezed state of light.

These results in fiber interferometers open a door to real-world quantum communications and quantum sensing based on continuous-variable quantum optics ^[136,137].

Due to the rapid developments over the past 30 years, quantum squeezing has become much closer to practical applications than ever before. Stable squeezed-light sources above 10 dB are readily achievable in many quantum optic laboratories, and comparable squeezed-light sources are even commercially available now. The designs and configurations for different quantum-sensing systems are diversified according to the specific parameters to be measured. The techniques of signal processing and tracking have been greatly optimized to fully utilize the unique advantages of the available quantum squeezing. It should be noted that squeezed light at the current stage is still to be developed for further applications. For example, a robust squeezed source is required for certain applications under environmental disturbance. In addition, on-chip generation of high-quality squeezing is necessary for scalable quantum sensing in an integrated platform ^[26–30]. In this review, we have focused on several fields that are close to the practical utilizations of quantum-squeezed states of light. One can surely expect that many significant and practical applications are yet to appear. From this point of view, this review just serves as a starting point for stimulating future research on the sub-shot-noise measurement using quantum-squeezed states of light. One should not be surprised to see new applications of quantum-squeezed states in various frontier fields such as laser frequency stabilization ^[138], optic gyroscopes ^[139], frequency combs ^[140], and single-particle detection ^[141].