Near-video frame rate quantum sensing using Hong-Ou-Mandel interferometry

Quantum sensing and metrology is a rapidly growing field due to its outstanding features in measuring physical parameters with great precision and accuracy, outperforming the pre-existing technologies based on the principles of classical physics [1]. In recent years, quantum sensing has enabled the measurement of several key physical quantities, including the measurement of the electrical field [2], magnetic field [3], vacuum [4], temperature [5], and pressure [6] with unprecedented precision and accuracy. Among various quantum sensors as presented in Ref. [7], Hong-Ou-Mandel (HOM) interferometry, since its discovery [8], has found a large number of applications within quantum optics for a variety of advantages, including easy development and implementation, sensitivity only to photon group delay and not to phase shifts [9].
HOM interference, purely a non-classical phenomenon, is observed for two photons that are identical in all degrees of freedom, including spin, frequency, and spatial mode, when simultaneously entering on a lossless, 50:50 beam splitter through different input ports bunch together into one of the output ports. The coincidence probability between the two output ports shows a characteristic low or zero coincidence, known as the HOM interference dip, directly related to the level of indistinguishability or the degree of purity of the photons [10]. As a result, the change in coincidence counts can be treated as a pointer to estimate the time delay between photon properties, forming the basis of the HOM interferometer-based quantum sensors to sense any physical process influencing the photon delay. As such, efforts have been made to use the HOM-based quantum sensors to characterize the single photon sources [11], precise measurement of time delays between two paths [8, 12, 13], characterization of ultrafast processes [14], measurement of frequency shifts [15, 16] and the spatial shift [17]. On the other hand, the use of Fisher Information (FI), a measure of the mutual information between the interfering photons, has enabled the maximum precision to access a lower limit decided by the Cramér-Rao bound [18, 19]. In fact, the use of peak FI of the HOM interferometer, the point of maximum sensitivity, has enabled five orders of magnitude enhancement in the resolution of time delay measurements [12, 13, 20]. A typical FI-based measurement approach involves tuning the HOM interferometer for the peak FI point and performing iterative measurement to estimate any optical delay between the two photons. However, the number of iterations/measurements and hence the total time needed to reach the targeted precision decreases with the increase of FI value. On the other hand, the FI value can be increased by optimizing the parameters affecting the visibility and the width of the HOM interference dip. Despite the experimental demonstration of nanometer path length (few-attosecond timing) precision [12, 21] and a recent theoretical study on the tailoring of the spectral properties [22, 23] of photon pairs to achieve precision beyond the value reported in the laboratory condition, the width of the HOM interference dip governed by the bandwidth of the pair photons remained the limiting factor for optimal measurement precision [12, 13, 24, 25] in HOM interferometer based dynamic (real-time) or fast sensing applications. As such, ultra-broadband photon sources have long been hailed as a vital prerequisite for ultra-precise HOM interferometry. On the other hand, the dynamic sensing process requires the single-photon source to have high brightness. On the other hand, a new strategy has been reported very recently where entangled photons have been used to produce the HOM beating effect at very different frequencies (visible and infrared) [21]. As the frequency of the beating is proportional to the frequency detuning of the entangled photons [21, 26] the HOM beating, in conjunction with FI analysis, enabled the measurement of low-frequency (0.5 - 1 Hz) vibration with a precision of $\sim$2.3 nm using low (18000) number of photons pairs [21]
Nonlinear spontaneous parametric down-conversion (SPDC) [27, 28] process-based sources, where the annihilation of a high energy pump photon produces two daughter photons simultaneously, have evolved as a workhorse for quantum optics experiments, including HOM based sensors. However, the development of high brightness single-photon source requires a nonlinear crystal with high figure-of-merit (FOM) [29], long crystal length, and high intensity of the pump beam. As such, periodically poled nonlinear crystals can be used due to their high intrinsic FOM. However, the increase in the crystal length reduces the spectral bandwidth of the single-photon source. On the other hand, the increase of input pump intensity and spectral bandwidth of the generated photons using ultrafast laser [30] has its own limitations in terms of expensive and bulky system architecture for any practical on-field uses. Therefore, it is imperative to explore a continuous-wave (CW) pump-based bright single-photon source with high brightness and spectral bandwidth for ultra-precise HOM interferometry.
Recently, we reported a CW pumped high brightness, degenerate single-photon source at 810 nm using periodically-poled potassium titanyl phosphate (PPKTP) crystal in non-collinear, type-0, phase-matched geometry [31, 32]. Using the same phase-matching geometry, here, we report on the experimental demonstration showing the dependence of spectral bandwidth of degenerate SPDC photons and subsequent change in the HOM interference dip width on the length of the PPKTP crystal. Pumping the PPKTP crystal of length 1 mm, using a single-frequency, CW diode laser at 405.4 nm we observed the spectral width of the SPDC photons to be as high as 163.42 $\pm$ 1.68 nm producing a HOM interference dip having a full-width-at-half-maximum (FWHM) as small as 4.01 $\pm$ 0.04 $\upmu$m. Using such small HOM interference dip width, we have dynamically measured the threshold vibration amplitude corresponding to the optical delay (twice the vibration amplitude of the piezo mirror) introduced between the photons as small as 205$\pm$0.75 nm at a frequency as high as 8 Hz. On the other hand, the reduction of crystal length from 30 mm to 1 mm shows a 17 times enhancement in the magnitude of the peak FI value and achieves a targeted precision (say $\sim$5 nm or $\sim$16.7 attosecond optical delay) for a number of iterations/measurements (or the total experiment time) as low as 3300 (19 minutes).

The spectral brightness of SPDC photons generated from a nonlinear crystal of length, $L$ can be expressed as follows,

$$I_{SPDC}\sim\mbox{sinc}^{2}\left(\Delta kL/2\right)$$

(1)

where $\Delta k$ is the momentum mismatch among the interacting photons (pump and down-converted pair-photons) given as follows

$$\Delta k=k_{p}-k_{s}-k_{i}-\frac{2\pi m}{\Lambda}$$

(2)

Here, $k_{p,s,i}$ = $\frac{2\pi n_{p,s,i}}{\lambda_{p,s,i}}$ is the wavevector and $n_{p,s,i}$ is the refractive index of the nonlinear crystal for the pump, $\lambda_{p}$, signal, $\lambda_{s}$, and idler, $\lambda_{i}$, wavelengths. $\Lambda$ is the grating period of the periodically-poled nonlinear crystal to satisfy the quasi-phase-matching (QPM) condition. The odd integer number, $m$, is the QPM order. For maximum efficiency, we consider $m$ = 1. The momentum mismatch between degenerate photons around the central angular frequency, $\omega_{p}/2$, can be represented as [33, 34]

$$\Delta k=\Delta k|_{\omega=\omega_{p}/2}+M|_{\omega=\omega_{p}/2}{\Delta\omega}+\frac{1}{2}K|_{\omega=\omega_{p}/2}{\Delta\omega}^{2}+....$$

(3)

Assuming the QPM is achieved at degenerate SPDC photons with angular frequencies of pump, signal, and idler as $\omega_{p}$, $\omega_{s}$, and $\omega_{i}$, respectively, for the grating period, $\Lambda$, the first term of the right-hand of Eq. (3) can be made zero. In this situation, owing to the energy conservation, $\omega_{p}$ = $\omega_{s}$ + $\omega_{i}$, the spectral bandwidth of the down-converted photons for the fixed pump frequency defined as $\Delta\omega$ = $\Delta\omega_{s}$ = -$\Delta\omega_{i}$ =$\omega-\omega_{p}/2$, is decided by the second term, M = $[\partial k_{s}/\partial\omega-\partial k_{i}/\partial\omega]_{\omega=\omega_{p/2}}$, known as group-velocity mismatch. However, for degenerate $(\omega_{s}=\omega_{i}=\omega_{p}/2)$, type-0 SPDC process, the group-velocity mismatch term vanishes or has negligible value. Under this condition, the spectral acceptance bandwidth of the SPDC photons is essentially determined by the much smaller third term of Eq. (3), commonly known as group-velocity dispersion with a mathematical form $K=\partial^{2}k/\partial\omega^{2}|_{\omega=\omega_{p/2}}$. Under this condition, the spectral bandwidth of the SPDC photons can be written as,

$$\Delta\omega\propto\frac{1}{\sqrt{KL}}$$

(4)

Thus, one can utilize the length of the nonlinear crystal as the control parameter in the SPDC process to generate paired photons of broad spectral bandwidth near the degeneracy. On the other hand, it is well known that the optical delay range or the width, $\sigma$, of the HOM interference is proportional to the coherence length of the photon wave packets [9, 35] or the ensemble dephasing time of SPDC photons [36] bunching on the beam splitter. Although one can find the width, $\sigma$, of the HOM interference dip from the Fourier transform of the spectral density function of the paired photons [37, 38], we used the coherence length, $L_{c}$, of the photon wave packets and Eq. (4) to find the dependence of HOM interference dip width on the crystal length as,

$$\sigma\hskip 5.0pt\sim\hskip 5.0ptL_{c}\hskip 5.0pt\propto\hskip 5.0pt2\pi c\sqrt{KL}$$

(5)

where c is the speed of light in vacuum. It is evident from Eq. (5) that the width, $\sigma$, of the HOM interference dip can be controlled by simply varying the crystal length even in presence of the single-frequency, CW diode laser as the pump.

The schematic of the experimental configuration is shown in Fig. 1. The details of the experiment can be found in the Methods section. A single-frequency, CW diode laser providing 20 mW of output power at 405.4 nm is used to create downconverted photons in the periodically poled KTiOPO₄ (PPKTP) crystals of 1 $\times$ 2 mm² aperture but of five different interaction lengths, $L$ = 1, 2, 10, 20, and 30 mm. All the crystals have a single grating period of $\Lambda$ = 3.425 $\upmu$m corresponding to the degenerate, type-0 ($e$→$e+e$) phase-matched parametric down-conversion (PDC) of 405 nm at 810 nm.

The noncollinear down-converted photon pairs having annular ring spatial distribution with signal and idler photons on diametrically opposite points are guided in two different paths and finally made to interact on the two separate input ports of a balanced beam splitter (BS). The combination of the piezo-electric actuator (PEA) (NF15AP25), placed on a motorized linear translation stage (MTS25-Z8), is used to provide both fine and coarse optical delay. The photons from the output ports of the BS, after being extracted by long pass filters (F1, F2), are coupled into the single mode fibers, SMF2 and SMF3, through the fiber couplers, C3, and C4, respectively. The collected photons are detected using the single-photon counting modules, SPCM1-2 (AQRH-14-FC, Excelitas), and counted using the time-to-digital converter (TDC). All the data is recorded at a typical pump power of 0.25 mW with a coincidence window of 1.6 nanoseconds.

The ultimate limit on the precision in measurement is decided by the Cramér-Rao bound [18, 19] defined as,

$$Var(\tilde{x})\hskip 5.0pt\geq\hskip 5.0pt\frac{1}{NF(x)}$$

(6)

where $\tilde{x}$ represents an unbiased estimator to estimate the physical parameter and var($\tilde{x}$) is the variance in the measurement. $F(x)$ represents the Fisher information (FI), a metric describing the amount of information available from a probability distribution of an unknown parameter, for estimating the information of the parameter $x$ in a single measurement, and N is the total number of measurements performed in the experiment. In the case of HOM interference, the mathematical form of FI is primarily determined by the temporal mode profile of the signal and idler photons [12]. As the down-converted photons generated through the type-0 SPDC process have a Gaussian spectral profile, the FI can be modeled as,

$$F=\frac{4s^{2}\alpha^{2}(\gamma-1)^{2}(1+\gamma)}{(e^{s^{2}}-\alpha)(\alpha-\alpha\gamma+e^{s^{2}}(1+3\gamma))\sigma^{2}}$$

(7)

here, $\alpha$, $\gamma$, and $\sigma$ represent the HOM interference visibility, the rate of loss in photon detection, and the FWHM width of the HOM curve, respectively. $s=\frac{x}{\sigma}$ is the parameter controlling the distinguishability of the photon arriving at the beam splitter. As evident from Eq. (7), keeping all parameters constant, one can increase the magnitude of FI, $F$ by employing the HOM interference of smaller dip width ($\sigma$). Again, it is evident from Eq. 5 that for a given laser parameter, the dip width of the HOM interference varies with the square root of the crystal length. Since the crystal length, a physical parameter controllable in the experiment, using Eq. (5) in Eq. (7), we can derive a general expression for FI in terms of crystal length, L, as,

$$F=\frac{4x^{2}\alpha^{2}(\gamma-1)^{2}(1+\gamma)}{A^{4}(e^{\frac{x^{2}}{A^{2}L}}-\alpha)(\alpha-\alpha\gamma+e^{\frac{x^{2}}{A^{2}L}}(1+3\gamma))L^{2}}$$

(8)

Here, A is a constant determined by the crystal properties. It is evident from Eq. (6) that for a fixed number of measurements/iterations, $N$, the precision can be enhanced, or for a fixed precision, the number of measurements/iterations, $N$, can be reduced with the increase of the magnitude of $F$ while saturating the Cramér-Rao bound. Therefore, Eq. (8) sets the foundation for our further study of precision augmented quantum sensing.

In conclusion, we have experimentally demonstrated easy control in the spectral bandwidth of the pair-photons through proper selection of the length of the non-linear crystal. Using a 1 mm long PPKTP crystal, we have generated paired photons with spectral width as high as 163.42$\pm$1.68 nm even in the presence of a single-frequency, CW, diode laser as the pump. The use of photon pairs with such a high spectral bandwidth in a HOM interferometer results in a narrow-width HOM interference dip enabling sensing of static displacement as low as 60 nm and threshold vibration amplitude as low as $\sim$205 nm with a resolution of $\sim$80 nm at a frequency measurement up to 8 Hz. Further, we experimentally observed the dependence of FI on the spectral bandwidth of the pair photons and hence the length of the nonlinear crystal. We observed a 17 times enhancement in the FI value while reducing the crystal length from 30 mm to 1 mm. Such an increase in the peak FI value (35.06 $\pm$ 1.65) x $10\textsuperscript{-4}$ $\upmu$m^-2, which is nearly 24 times higher than the previous study [12], saturates the Cramér-Rao bound to achieve any arbitrary precision (say $\sim$5 nm) in a lower number of iterations ($\sim$ 3300), $\sim$11 times lower than the previous reports. Multiplying the number of coincidence counts to the number of iterations, one can find the total number of photons used in the experiment to achieve such high precision. In our case, this provides 2 $\times$ $10^{6}$ number of photon pairs to achieve the desired precision. This is somewhat higher than the 2 $\times$ $10^{4}$ photons used e.g. in the experiments by Kwait et.al.[21]. However, our total measurement time is still lower thanks to the high photon flux rates, therefore allowing us to achieve overall faster sampling rates and sensitivity to higher vibration frequencies. The accessibility of high precision in lower iterations or time establishes the potential of HOM-based sensors for real-time, precision-augmented, in-field quantum sensing applications. Unlike the use of bulky and expensive ultrafast pump lasers of broad spectral bandwidth to enhance the spectral bandwidth of the down-converted photons, the generation of broadband photons using a single-frequency diode laser in small crystal length is beneficial for any practical applications.

The schematic of the experimental configuration is shown in Fig. 1. A single-frequency, CW diode laser providing 20 mW of output power at 405.4 nm is used as the pump laser in the experiment. The spatial mode filtering unit comprised of a pair of fiber couplers (C1 and C2) and a single-mode fiber (SMF1) is used to transform the laser output in a Gaussian (TEM₀₀ mode) beam profile. The laser power in the experiment is controlled using a combination of $\lambda$/2 plate (HWP1) and a polarizing beam splitter cube (PBS1). The second $\lambda$/2 plate (HWP2) is used to adjust the direction of linear polarization of the pump laser with respect to the orientation of the crystal poling direction to ensure optimum phase matching conditions.The pump laser is focused at the center of the nonlinear crystal (NC) using the plano-convex lens, L1, of focal length, $f1$ = 75 mm to a beam waist radius of $w_{o}$ = 58 $\upmu$m. Periodically poled KTiOPO₄ (PPKTP) crystals of 1 $\times$ 2 mm² aperture but of five different interaction lengths, $L$ = 1, 2, 10, 20, and 30 mm, are used as the nonlinear crystal (NC) in the experiment. The crystals have a single grating period of $\Lambda$ = 3.425 $\upmu$m corresponding to the degenerate, type-0 ($e$→$e+e$) phase-matched parametric down-conversion (PDC) of 405 nm at 810 nm. To ensure quasi-phase-matching, the PPKTP crystals are housed in an oven whose temperature can be varied from room temperature to 200^oC with temperature stability of $\pm$0.1^oC. The down-converted photon pairs, after separation from the pump photons using a notch filter (NF), are collimated using the plano-convex lens, L2 of focal length, $f2$ = 100 mm. Due to the non-collinear phase-matching, the correlated pair (commonly identified as “signal” and “idler”) of down-converted photons have annular ring spatial distribution with signal and idler photons on diametrically opposite points. Using a prism mirror (PM), the signal and idler photons are separated and guided in two different paths and finally made to interact on the two separate input ports of a balanced beam splitter (BS). Since we need to vary temporal delay between the photons at the BS, and the translation of the optical component disturbs the alignment, we have devised the alignment preserving delay path consisting of a mirror, M1, $\lambda/2$ plate (HWP3), PBS2, $\lambda/4$ plate (QWP), mirror, M2 on the delay stage, and finally, the beam splitter, BS. The $\lambda/2$ plate rotates the photon’s polarization state from vertical to horizontal to ensure complete transmission through PBS2. The $\lambda/4$ having the fast axis at +45^o to its polarization axis transforms the photon polarization (horizontal) into circular (left circular). However, the handedness of the photon polarization gets reversed (right circular) on reflection from the mirror, M2, at normal incidence. On the return pass through the $\lambda/4$ plate, the right circular polarized photons convert into vertical polarized photons and are subsequently reflected from the PBS2 to one of the input ports of the BS. Due to the normal incident of the photons on the moving mirror, M2, the varying delay of the photons does not require tweaking of the experiment. The combination of the piezo-electric actuator (PEA) (NF15AP25), placed on a motorized linear translation stage (MTS25-Z8), is used to provide both fine and coarse optical delay. The range (resolution) of the PEA and motorized linear stage as specified by the product catalog is 25 $\upmu$m (0.75 nm), and 25 mm (29 nm), respectively. Throughout the report, the optical delay of the photon is twice the displacement of the mirror, M2, if otherwise mentioned. On the other hand, the photons of the other arm are guided using mirrors, M3 and M4, to the second input port of BS such that the optical path length of both arms are the same. The photons from the output ports of the BS, after being extracted by long pass filters (F1, F2), are coupled into the single mode fibers, SMF2 and SMF3, through the fiber couplers, C3, and C4, respectively. The collected photons are detected using the single-photon counting modules, SPCM1-2 (AQRH-14-FC, Excelitas), and counted using the time-to-digital converter (TDC). All optical components used in the experiment are selected for optimum performance at both the pump and SPDC wavelengths. All the data is recorded at a typical pump power of 0.25 mW with a coincidence window of 1.6 nanoseconds.