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Abstract
We report on the waveguide-based generation of pulsed squeezed light at 795 nm, suitable for quantum enhanced measurements with rubidium atoms. Pulsed ultraviolet second harmonic light with a power of more than 400 mW is produced using a periodically poled LiNbO3 (PPLN) waveguide and is injected into another PPLN waveguide to generate quadrature squeezing. We find that the phase of the second harmonic pulse is shifted within a pulse, and we attribute the shift to heating due to blue-light induced infrared absorption (BLIIRA) from a comparison between the experiment and a numerical simulation. A squeezing level of −1.5(1) dB is observed in homodyne detection when we apply a linear phase shift to the local oscillator. The experiment and simulation imply that the squeezing level can be further improved by reducing BLIIRA.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Squeezed light lies at the heart of today’s quantum optics and is finding wide application [1]. It has been used for improving the sensitivity of various measurements including atomic magnetometry [2–4], rotation sensing [5], and gravitational wave detection [6]. Pulsed squeezed light combined with atoms may open intriguing possibilities in quantum memory [7] and interferometric sensing [8]. Stroboscopic quantum nondemolition measurements, which can generate spin squeezing even in a large atomic ensemble [9,10], may enable Heisenberg-limited sensitivity by using squeezed probe pulses [11,12].
In such atomic experiments the wavelength of the squeezed light should be near atomic resonance, and the D1 line (795 nm) of rubidium (Rb) is often used [2–4,13–16]. Optical parametric amplification (OPA) is usually used for generating continuous-wave (cw) squeezed light near the D1 line [2,4,14–16]. An alternative for generating 795 nm squeezed light is an atomic squeezer [3,17–21]. Although the atomic squeezer can produce cw or pulsed squeezed light [18], short pulse squeezing is limited due to the evolution time requirement for reaching a steady state of each atomic level [21]. An optical waveguide is an appropriate choice for pulsed squeezed light generation [13,22] as it can generate high-quality squeezed light thanks to the large nonlinearity resulting from strong confinement of the light. A waveguide squeezer is also attractive for practical applications due to its potential compactness and stability [23]. However, there have been few reports of the generation of squeezed light for atomic experiments using a waveguide. In Ref. [13], squeezing 0.9 dB of cw light at 795 nm was achieved using two ridge MgO-doped periodically poled LiNbO$_3$ (PPLN) waveguides for second harmonic generation (SHG) and OPA.
An obstacle to producing squeezed light for atomic experiments using waveguides is generating a strong second-harmonic pump beam, typically in the ultraviolet (UV) region. UV light power generated by SHG in a waveguide is typically several tens of mW [13,16,24], while using a bulk periodically poled KTiOPO$_4$ inside a cavity to generate a few hundreds of mW UV light has been reported [2,4,25–31]. The poor efficiency of UV generation with a waveguide has been attributed to temperature nonuniformity in the waveguide due to large absorption of UV light [24]. However, a detailed evaluation of the thermal inhibition of UV generation in an actual waveguide is currently lacking and a mechanism for inhibiting strong UV light generation with a waveguide is yet to be clarified [32]. It also remains unclear to what extent we can improve 795 nm squeezing using strong UV pump light. It should be noted that improvement in SHG efficiency into (near) UV light is by itself important aside from motivation to generate squeezed light for atomic experiments, because powerful UV light source is widely required.
We develop a waveguide-based squeezer in the context of quantum-enhanced atomic magnetometry. The compactness and stability is promising for future applications, for example, portable magnetometry. The cavity approach has realized squeezing level of 5.6 dB at 795 nm [29], but constructing a good and compact cavity is demanding. 1.6 dB of squeezing at 795 nm using a monolithic cavity has been reported [33]. While the atomic squeezer is useful for some atomic experiments and is possibly compact, the atomic squeezer is not suitable for producing largely detuned squeezed light (typically of GHz detuning) required for the magnetometry based on Faraday rotation. The squeezed light frequency from a waveguide can be adjusted by changing the temperature. We note that the optimal detuning for magnetometry becomes larger when the squeezed light is used for the probe [12]. A holy grail in quantum magnetometry is the sensitivity enhancement by the combination of squeezed probe light and spin squeezing, which should lead to an excellent sensitivity. Pulsed light with a width much shorter than the reciprocal of the Larmor frequency, typically several tens or hundreds of kHz, enables spin squeezing by stroboscopic measurement [9,10]. Parametric amplification in a waveguide can generate such pulsed squeezed light without light chopping, inducing additional optical losses and preventing good squeezing. Therefore, it is important to overcome drawbacks of a waveguide squeezer for quantum sensing.
In this paper, we report the generation of pulsed squeezed light at 795 nm using ZnO-doped PPLN waveguides. Pulsed second harmonic light of 1 $\mu$s with a peak power of more than 400 mW is generated from a waveguide at a duty cycle of 0.01 and is injected into another waveguide to generate squeezed 795 nm pulsed light. We observe a quadrature squeezing of $-1.5(1)$ dB by compensating a phase shift in the second harmonic pulse. The phase shift is attributed to a temperature rise during pulse propagation due to blue-light induced infrared absorption (BLIIRA) rather than UV absorption, based on a comparison between a numerical simulation and the experimental results. The simulation also indicates that BLIIRA dominantly deteriorates the efficiency in SHG and squeezing.
The paper is organized as follows. We describe the experimental setup in Sec. 2. In Sec. 3, we present the experimental results. The numerical simulations of SHG including BLIIRA are described in Sec. 4. Section 5 discusses the achievable squeezing level using waveguides. We conclude the paper in Sec. 6.
2. Experimental setup
Figure 1 shows the experimental setup. CW light at 795 nm with a maximum power of 3 W is generated from a diode laser master oscillator power amplifier system (DLC TA pro, Toptica Photonics) with a wavelength set close to the D1 line resonance of Rb. The light power is modulated using an acousto-optic modulator (AOM) to produce a series of pulses. The AOM is driven by an amplified rf wave from a function generator. Each pulse length is 1 $\mu$s. The pulse repetition rate is normally set to 10 kHz.
Fig. 1. Experimental setup for generation of pulsed squeezed vacuum at 795 nm. FG: function generator; AOM: acousto-optic modulator; HWP: half-wave plate; PBS: polarizing beam splitter; EOM: electro-optic modulator.
The 795 nm pulses are injected into a ZnO-doped PPLN optical waveguide (PPLN1), manufactured by NTT Electronics Corp., for SHG. The core size of the waveguide is 12 $\mu$m $\times$ 13 $\mu$m and the length is 30 mm. The waveguide is designed to have a poling period of 7.54 $\mu$m to fulfill the third-order quasi-phase matching condition at 795 nm near room temperature. It is placed on a copper mount, the temperature of which is controlled using a Peltier device. A fraction of the 795 nm pulse passes through a fiber-coupled electro-optic modulator (EOM) (NIR-MPX800-LN-0.1, iXBlue) and is used as a probe for classical parametric amplification or a local oscillator for homodyne detection of squeezed light. The EOM was used for compensation of the phase shift in the pulse, as explained later.
The second harmonic light from the waveguide pumps another waveguide (PPLN2) for OPA. The length of the OPA waveguide is 22 mm and the other features are the same as those of the SHG waveguide. The waveguide temperature is controlled in a similar manner as for the SHG waveguide. Signal light at 795 nm output from the waveguide is detected using a balanced homodyne measurement. The signal and a local oscillator with orthogonal polarization are combined at a polarizing beam splitter (PBS) and the difference of the $\pm$45$^{\circ }$ linear-polarization components of the total field is measured using a custom-built balanced pulse detector. The detector is composed of high quantum efficiency Si photodiodes (S3883, Hamamatsu), a charge sensitive preamplifier (A250, Amptek), and three pulse shaping amplifiers (A275, Amptek) [34,35].
3. Results
Figure 2(a) shows the second-harmonic (SH) peak power, $P_{2\omega }$, from the SHG waveguide as a function of the peak power of the fundamental wave (FW) coupled to the waveguide, $P_{\omega }$. The data at each coupled power plotted in Fig. 2(a) were taken with the mount temperature set at a value to maximize the SHG power and the waveguide coupling was optimized for each temperature. We obtained second-harmonic light of a peak power of 424 mW at the maximum pump power of 1.28 W. We also plot the conversion efficiency, defined by $P_{2\omega }/P_{\omega }$, in Fig. 2(a). The conversion efficiency reached 0.3 when $P_{\omega }\approx 0.5$ W. A maximum conversion efficiency of 0.34 was obtained at $P_{\omega }=$ 0.65 W and the conversion efficiency slightly decreased for higher $P_{\omega }$.
Fig. 2. Properties of the SHG using the 30 mm waveguide. (a) Output power and conversion efficiency. (b) Mount temperature dependence of the conversion efficiency. (c) Power dependence of the optimal temperature. The dashed curve is an empirical quadratic fit to the data.
Typical temperature dependence of the conversion efficiency is shown in Fig. 2(b), where we plot the conversion efficiency as a function of the waveguide mount temperature. The optimal mount temperature and acceptance bandwidth (FWHM) for $P_{\rm {\omega }}$ = 495 mW were 29.6 $^{\circ }\rm {C}$ and 0.40 $^{\circ }\rm {C}$, respectively. When $P_{\rm {\omega }}$ was increased to the maximum, the optimal mount temperature shifted to 29.4 $^{\circ }\rm {C}$. This shift suggests a temperature rise in the waveguide. The optimal temperature as a function of $P_{\omega }$ is plotted in Fig. 2(c). We note that the phase matching temperatures in Figs. 2(b) and 2(c) differ due to a change of laser frequency of approximately 2 GHz. We also observed a narrowing of the acceptance width to 0.24 $^{\circ }\rm {C}$ and an asymmetric temperature dependence at the maximum pump power. Such narrowing of the acceptance width and asymmetry in the SHG using a waveguide have been reported previously [16,24,36–38].
We measured the classical parametric gain in the OPA waveguide. In this experiment, we used the second harmonic light as a pump beam and a probe beam at 795 nm, which was picked off by a polarizing beam splitter after the AOM for pulse generation. The pump and probe beams were both vertically polarized in the OPA waveguide. The relative phase between the pump and probe beams could be changed using a piezo actuator attached to a mirror mount. The output pulse was measured with a high-speed photodiode (not shown in Fig. 1). The output of the photodiode was recorded using a data acquisition card. We measured the photodiode voltage in the middle of each pulse, which gave the parametric amplification gains at various relative phases. As the optical path lengths slowly changed, the relative phase took a value of $+\pi /2$ or $-\pi /2$ at some time and the gain became maximum or minimum after a sufficient number of measurements. This also happened when a phase shift occurred during a pulse, which we show below. We obtained the amplification and deamplification gains ($G_+$ and $G_-$) from the maximum and minimum output powers (795 nm) in a series of data, respectively. $G_{\pm }$ is plotted in Fig. 3(a) for several pump powers, $P_{2\omega }$. The pump power was varied by changing the fundamental power for SHG through the diffraction efficiency of the AOM for pulse generation. The pump power was controlled in the same manner in the experiments detailed below, unless otherwise noted. The injected peak probe beam power, $P_{\rm probe}$, into the OPA waveguide was fixed at 2.6 mW. The gains monotonically grew with increasing $P_{2\omega }$, as shown in Fig. 3(a). The maximum and minimum gains of $+2.2$ dB and $-1.8$ dB were obtained at $P_{2\omega }$ = 408 mW.
Fig. 3. Properties of OPA and BLIIRA in the 22 mm waveguide. (a) Pump power dependence of the classical parametric gains. The data were taken with a pulse repetition rate of 1 kHz. The red curve is a fitting curve using Eq. (1). (b) BLIIRA factor as a function of $P_{\rm 2\omega }$. The inset shows the BLIIRA coefficient $\alpha _{\rm BL}$ versus $P_{\rm 2\omega }$. The solid black line is a linear fit.
We note that BLIIRA is not negligible in this gain measurement. To deduce BLIIRA, we measured $\eta _{\rm BL}=\it P'_{\rm probe}/ P_{\rm probe}$, where $P'_{\rm probe}$ and $P_{\rm probe}$ are the probe power transmitted through the OPA waveguide with and without the second harmonic light, respectively. We refer to $\eta _{\rm BL}$ as the BLIIRA factor hereafter. In measuring $P'_{\rm probe}$, we injected horizontally polarized second harmonic light into the waveguide, which did not cause parametric amplification. We also confirmed that the BLIIRA is independent of the polarization of the second harmonic light at a weak pump power, where no significant parametric amplification occurs. The measured BLIIRA factor, $\eta _{\rm BL}$, is plotted in Fig. 3(b). Figure 3(b) also shows the BLIIRA coefficient $\alpha _{\rm BL}$, defined by $\eta _{\rm BL} = e^{-\alpha _{\rm BL}L}$, where $L = 22$ mm is the waveguide length. We observed a kink in $\eta _{\rm BL}$ (and $\alpha _{\rm BL}$) around $P_{2\omega } = 5$ mW. Above $P_{2\omega } = 10$ mW, $\alpha _{\rm BL}$ is well fitted by a linear function as $\alpha _{\rm BL} = \gamma _1 \it P_{2\omega } + \alpha _{\rm BL0}$, where $\gamma _1$ is 19.4(4) ${\rm m}^{-1}W^{-1}$ and $\alpha _{\rm BL0}$ is 0.50(7) ${\rm m}^{-1}$. We note that the UV intensity in the waveguide was estimated to be up to several GW/m$^2$ owing to strong confinement of light. The BLIIRA of a LiNbO$_3$ crystal with 795 nm light has not been studied for such intense second harmonic light as far as we know. The measured gains, $G_{\pm }$, were fitted by
where $\eta$ is the effective mode-matching efficiency between the probe and pump beams and $r$ is the effective gain coefficient. The fitting gives $r$ = 0.49(5) $\rm {W}^{-1/2}$.We measured the waveform in classical parametric amplification with an oscilloscope. When a strong fundamental beam was injected into the SHG waveguide and the pump beam power into the OPA waveguide was strong, the signal waveform skewed as shown in the leftmost panel of Fig. 4(a). The skew can be ascribed to the change in the relative phase between the pump beam and the probe beam as they propagate in the waveguide (note that the pulse length is much longer that the waveguide length). Different waveforms were observed for different shots because the initial relative phase varied slowly for each shot due to the scan of the local oscillator phase with a period of 5 s. A waveform deformation occurred when we decreased the pump power into the OPA waveguide while keeping the fundamental beam power into the SHG waveguide high, as shown in the middle panel of Fig. 4(a). The pump beam power was reduced by an HWP and a PBS (not shown in Fig. 1), additionally inserted after the SHG waveguide, in taking these data. However, when we decreased the fundamental beam power into the SHG waveguide, the gain was almost constant over a pulse, as shown in the rightmost panel of Fig. 4(a), indicating that no significant phase change occurred in this case. These results suggest that the phase shift is related to the fundamental beam power and should occur in the SHG waveguide.
Fig. 4. Temporal variation in the OPA signal. (a) Waveforms of the signal beams exhibiting classical parametric amplification with injected fundamental and pump peak powers of ($1.9$ W, $0.4$ W), ($1.9$ W, $0.1$ W), ($0.5$ W, $0.1$ W) for the leftmost, middle, and rightmost panels, respectively. The red and black data were taken for different shots. The blue curve was taken without the pump beam and is shown for reference. (b) Phase shift rate as a function of $P_{2\omega }$. The inset shows the measured phases for three different pulses at $P_{2\omega }= 389$ mW.
We estimated the change of the relative phase between the fundamental beam and the pump beam, $\phi (t)$, where $t$ is the time in a pulse, using the following equation:
where $V(t)$ is the voltage from a photodetector measuring the signal light power and $V_{\textrm {max}}$ and $V_{\textrm {min}}$ are the maximum and minimum voltages of 20 pulses taken with time intervals of several hundred milliseconds with fixed $P_{2\omega }$. While scanning the phase, it is expected that the maximum and minimum amplification should occur during measurement of the 20 pulses. The variation in $P_{2\omega }$ among pulses was less than 1 %. The estimated phase, $\phi (t)$, was almost linear with respect to the time for every shot. We plot $\phi (t)$ measured in three experimental runs with $P_{2\omega }= 389$ mW in the inset of Fig. 4(b). We fitted a linear function to the observed $\phi (t)$ to obtain the phase shift rate. The phase shift rate for $P_{2\omega }= 389$ mW was 4.1(3) rad/$\mu$s. We show the dependence of the phase shift rate on $P_{2\omega }$ in Fig. 4(b).A squeezed vacuum was generated by injecting only the pump beam into the OPA waveguide. We cut the collinear fundamental wave with an interference filter and dichroic mirror, shown in Fig. 1. The squeezing level, $S_-$, and antisqueezing level, $S_+$, were measured using homodyne detection. A local oscillator in the homodyne detection generated 3.8 $\times 10^{8}$ electrons per pulse at the pulsed balanced detector. The electric noise of the detector was more than 20 dB below the optical shot noise for the local oscillator strength, and is subtracted in the results shown below. The output of the pulsed balanced detector was recorded with a data acquisition card. We calculated the variance of the detector output over 2000 successive pulses to evaluate the noise level. As we measure the detector output for every pulse with an interval of 100 $\mu$s, the detector output acquired during 200 ms is used to calculate each variance value plotted in the time trace, Figs. 5(b) and 5(c). We scanned the local oscillator phase during the measurement to observe the squeezing and antisqueezing. Figure 5 shows the noise levels. The noise level in Fig. 5 is normalized by that for the local oscillator (without the squeezed vacuum). We define $S_{\pm }$ as the averaged values of several normalized local maximum (minimum) noise levels.
Fig. 5. Measured noise levels without phase compensation. (a) Squeezing and antisqueezing levels as a function of the pump power (black open diamonds). The red squares are the estimated values from the data with phase compensation (see text). (b)(c) Time traces of noise levels for (b) $P_{2\omega } =$ 246 mW and (c) 407 mW. The red filled circles and black asterisks represent data with and without the squeezed vacuum, respectively.
$S_{\pm }$ for different $P_{2\omega }$ are plotted in Fig. 5(a). We obtain the best squeezing of $S_{-} = -1.30(9)$ dB at a pump power of 246 mW. $S_{+}$ at the same pump power is $+1.65(3)$ dB. The squeezing level deteriorates when $P_{2\omega }$ is further increased, in contrast to the behavior of the classical parametric gains. We show time traces of the normalized noise levels for the optimal pump power ($P_{2\omega }$ = 246 mW) and $P_{2\omega }$ = 407 mW in Figs. 5(b) and 5(c), respectively. For $P_{2\omega } =$ 407 mW, the noise did not become significantly smaller than the shot noise level for any local oscillator phase. The degradation of the squeezing level for high $P_{2\omega }$ was also due to the change in the relative phase between the squeezed vacuum and the local oscillator during the pulse, caused by the phase shift in the pump pulse. Since the detector for the homodyne measurement integrates the photocurrent over a pulse, a large change of the phase in a pulse results in an averaged output of the amplified and deamplified noises, degrading the observed (anti-)squeezing level.
We improved the observed squeezing levels by compensating the relative phase shift between the squeezed vacuum and the local oscillator. We applied a linear phase shift to the local oscillator by the fiber-coupled EOM. The measured squeezing and antisqueezing levels with phase compensation are plotted in Fig. 6. In taking these data, $P_{2\omega }$ was changed using an additional HWP and PBS (not shown in Fig. 1) after the SHG waveguide and $P_{\omega }$ was fixed to maintain the phase shift at the SHG waveguide. The phase gradient in the compensation pulse was adjusted to minimize the change in the classical parametric amplification gain in each pulse. The squeezing level monotonically increased as $P_{2\omega }$ was increased, while the squeezing level does not improve for $P_{2\omega } > 250$ mW with no phase compensation. We fitted the experimental results by Eq. (1), reading the efficiency, $\eta$, as $\eta =\eta _{\rm WG} \it \eta _{\rm detect}$, where $\eta _{\rm WG}$ and $\eta _{\rm detect}$ represent the transmittance in the waveguide and the total detection efficiency, respectively. Here we approximate $\eta _{\rm WG}$ by $e^{-\alpha _{\rm BL} L/2}$, assuming that a generated squeezed vacuum experiences a loss on average of over a half length of the waveguide. We obtained $\eta _{\rm detect} =$ 0.60(7) and $r=$ 0.52(6) by the fitting. We estimated $\eta _{\rm detect}$ from independent experiments as $\eta _{\rm detect}= \eta _{\rm trans} \eta _{\rm QE} \eta _{\rm match} = 0.49$, where $\eta _{\rm trans}=0.95$ is the transmittance of the optical system after the waveguide, $\eta _{\rm QE}=0.82$ is the measured detector quantum efficiency, and $\eta _{\rm match}$ is the mode-matching efficiency between the squeezed light and the local oscillator. We estimated $\eta _{\rm match}$ by the square of the visibility between the probe beam and the local oscillator as $(0.79)^2=0.62$. The current value of $\eta _{\rm match}$ will be greatly improved if the spatial mode matching between the squeezed vacuum and the local oscillator is optimized. The discrepancy in $\eta$ may be partly due to the power dependence of $r$. Although we assumed that $r$ in Eq. (1) is constant to estimate $\eta$, $r$ can change due to a slight mode-mismatch or insufficient phase compensation introduced by a pump power change. We optimized the beam alignment and phase compensation at a high pump power in taking the data in Fig. 6(a). The squeezing and antisqueezing levels reached $-1.5(1)$ dB and $+2.1(1)$ dB, respectively, when $P_{2\omega } = 400$ mW with the HWP and PBS for power control removed. Figure 6(b) shows the time trace for the best squeezing. The best squeezing and antisqueezing were not obtained with the HWP and PBS due to the extra power loss and possible change in spatial mode of the pump beam. We note that the waveguide is not single-mode and the observed (anti-)squeezing level depends on the pump beam mode.
Fig. 6. Noise levels with phase compensation. (a) Squeezing and antisqueezing levels with phase compensation as a function of the pump power. (b) Time traces of the squeezing level for $P_{2\omega }$ = 400 mW. The red and black points represent data with and without the squeezed vacuum, respectively.
Using the applied compensation phase shift rate and the observed (anti-)squeezing level, we can calculate the squeezing level if phase compensation was not applied in this experiment. The calculated levels (red plot in Fig. 5(a)) are in reasonable agreement with the experimental data without phase compensation; the squeezing level becomes maximum around $P_{2\omega }$ = 250 mW and decreases when $P_{2\omega }$ is further increased. This supports the claim that the phase shift decreased the squeezing level. The calculated values are slightly lower than the experimental values for any $P_{2\omega }$. This discrepancy may be due to insufficient spatial mode-matching caused by the HWP and PBS in taking the data in Fig. 6(a), as mentioned above.
4. Numerical simulation
We performed a numerical simulation of the SHG process. We found that the thermal effect in the SHG waveguide should be included in a model to explain the experimental observations, even though the average beam power injected into the SHG waveguide was less than 30 mW due to the small duty cycle of 0.01. The change in temperature of the waveguide, $T$, during a pulse can be described by
where $\rho$, $C$, and $K$ are the density, specific heat, and heat conductance, respectively. For LN, $\rho =4600$ $\mathrm {kg}/\mathrm {m}^3$, $C=650$ $\mathrm {J}/(\mathrm {kg}\, \mathrm {K})$, and $K = 4.6$ $\mathrm {W}/(\mathrm {m}\, \mathrm {K})$ [39]. The heat per volume, $q$, is given by where $\alpha _1$ and $\alpha _2$ are the absorption coefficients of IR and UV light, respectively, and $\beta _{2}$ is the two-photon absorption coefficient of the UV light. $I_1$ and $I_2$ are the fundamental and second harmonic light intensities in the waveguide, respectively. As $\alpha _1$ is small and difficult to determine from our experiments, we used the value for MgO-doped LN, $\alpha _1=0.3$ $\mathrm {m}^{-1}$ [40]. We estimated $\alpha _2$ from the transmittances of UV light through 22 mm and 30 mm waveguides, 0.57 and 0.55, respectively. Assuming the coupling efficiencies are the same for both waveguides, $\alpha _2$ was estimated to be 4.5 $\mathrm {m}^{-1}$. This value is consistent with the literature value of 5 $\mathrm {m}^{-1}$ for MgO-doped LN [40]. We used $\alpha _{\textrm {BL}} = \gamma I_2 + \alpha _{\textrm {BL}0}$, based on our BLIIRA measurement. We calculated $\gamma$ to be $3.0\times 10^{-9}$ W/m using a calculated electric field mode profile in the waveguide and assuming the coupling efficiency of the UV light into the waveguide to be 50 %. We estimated that the two-photon absorption of the UV light was negligible and set $\beta _{2}=0$, considering the known value of $\beta _{2}$ for 532 nm [41] and the generated UV power.In evaluating the temperature change, we assumed that the waveguide temperature at the beginning of the pulse was homogeneous. This assumption is valid if the temperature inhomogeneity in the waveguide caused by a pulse sufficiently diminishes between pulses. We approximated the temperature at a time $\tau$ after starting the pulse as
where $T(0)$ is the initial waveguide temperature. We adopted a one-dimensional model and neglected thermal diffusion along the waveguide [42]. With these crude approximations, the temperature rise in a pulse of 1 $\mu$s was estimated to be on the order of $0.1$ K mainly due to the BLIIRA in our experiment. This temperature rise caused a phase mismatch in the SHG, degrading the conversion efficiency. It also induced a phase shift, which we attribute to the observed phase shift in a pulse, as explained later.For a more quantitative evaluation, we numerically calculated the field evolution in the SHG process adopting the above heating model. Here, we write the electric field as $E_j=A_je^{\mathrm {i}(\omega _it-k_jz+\phi _j)}$, where $A_j$ describes the field amplitude, $\omega _j$ is the angular frequency, $k_j$ is the propagation constant in the waveguide, and $\phi _j$ is the phase deviation from the trivial phase of $\omega _j t-k_j z$. The indices $j=1$ and $2$ denote the fundamental and second-harmonic fields, respectively. When the change in $k_j$ along the waveguide axis is sufficiently small and the field inhomogeneity across a cross-section is neglected, the field evolution can be described by the one-dimensional coupled-mode equations [37,42]
We related the field amplitudes to beam powers by
The simulated conversion rates are shown in Fig. 7. In the simulation, the initial temperature, $T(0)$, was adjusted to maximize the output second harmonic power (at $z=L$) in the middle of the pulse ($\tau =0.5~\mu$s), to reproduce the experimental procedure. This simulation clearly indicates that BLIIRA degrades the conversion efficiency for high $P_{\omega }$, while the degradation due to the UV linear absorption is not so significant. That is, the inefficient conversion in our experiment should mainly be due to BLIIRA, rather than to the UV (linear) absorption. The simulation with BLIIRA and the associated heating reproduces the slight decrease of the conversion rate at high $P_{\omega }$ in the experiment. The decreased efficiency in the SHG due to the BLIIRA is related to the inhomogeneous temperature in the waveguide, as shown in the inset of Fig. 7. We cannot satisfy the QPM condition at all places in the waveguide due to the temperature inhomogeneity. The large heating in the middle of the waveguide in the high power case can be understood as follows: the second harmonic light and the BLIIRA-induced heating initially grow but the fundamental power decreases due to the BLIIRA and the heating then decreases.
Fig. 7. Numerically simulated conversion efficiencies. The circles represent simulation results with no absorptions (green), with linear absorptions only (purple), with all absorptions but without heating due to BLIIRA (yellow), and with all absorptions and heating (blue) from top to bottom. The red diamonds represent the experimental data. Inset: Simulated temperature distribution in the waveguide during a pulse at $P_{\omega }=0.45$ W (left, top) and (b) 1.65 W (right, bottom). The curves represent the distributions at $\tau =$1 $\mu$s (green), $\tau =0.75$ $\mu$s (purple), $\tau =0.5$ $\mu$s (yellow), $\tau =0.25$ $\mu$s (red) and $\tau =0$ $\mu$s (blue).
The remaining discrepancy between the experiment and the simulation (including all absorptions and heating) may be due to an error in the estimation of $I_j$ accounting for the heating, as mentioned above. The temperature distribution at the start of each pulse, which we neglect in the model, may also lead to worse conversion efficiencies because of a larger temperature inhomogeneity. In fact, we obtained a slightly better conversion efficiency for a smaller duty cycle, implying that the temperature inhomogeneity accumulates over pulses. Another possible cause of discrepancy is the fact that the intensity distribution was neglected across a cross-section. The thermal inhomogeneity across the cross-section makes quasi-phase matching more difficult. The thermal lens effect might further degrade the matching efficiency. A 3D electric field simulation coupled with heat propagation may be used for a more satisfactory simulation; however, it is out of the scope of this work.
Finally, we examined the phase change in the second harmonic light during a pulse. The phase at the output end of the waveguide is the sum of $\phi _2$ and the phase delay caused by the refractive index change due to the heating. The simulated phases at several $\tau$ in a pulse are shown in Figs. 8(a) and 8(b). We fitted the sum of the thermally induced phase shift and $\phi _2$ by a linear function to obtain the phase change rate with respect to time. The change rate from the simulation reproduces the experimental data well, as shown in Fig. 8(c). We note that the phase change rate in the experiment is larger than in the simulation. The results for the conversion efficiency and the phase change rate imply that the simulation may slightly underestimate the heating. As our simulation reasonably reproduces the observed conversion rate and phase shift, we believe the model well represents the SHG process in the actual waveguide and conclude that the observed phase shift in each pulse is primarily due to the heating by BLIIRA.
Fig. 8. Phase shift of the second harmonic light during a pulse. Phase at the end of the waveguide as a function of $\tau$ for (a) $P_{\omega }=0.45$ W and (b) 1.65 W. The blue squares and yellow diamonds represent $\phi _2$ and the thermally induced phase (aside from $\phi _2$), respectively. The red circles denote the total phase. The red solid line is a linear fit to the total phase. (c) Simulated (blue circles) and measured phase change rate (red diamonds).
5. Discussion
One obstacle to obtaining a high squeezing level using waveguides has been preparing a strong pump light. We generated second-harmonic light at 397 nm with a peak power more than 400 mW using a waveguide. The peak power was very high compared with previous results: Continuous generation of UV light with a waveguide generated a few tens of mW [13,24]. The high peak power is due to the small duty cycle of 0.01, reducing the light-induced heat. We showed that the nonlinear coefficient of the ZnO-doped PPLN waveguide at the UV region is comparable to that for a longer wavelength [37,45], while it effectively appears to be small for a continuous operation [24].
Figure 9(a) shows the estimated squeezing levels for several BLIIRA coefficients. We used the experimentally obtained squeezing parameter ($r$) and we assumed $\eta _{\rm detect}=1$ in all estimations. The simulation implies we can observe a squeezing level better than 2 dB using the currently available pump beam of approximately 400 mW if we improve the detection efficiency. A better squeezing level will be obtained with a stronger pump beam. The squeezing level, however, saturates around $P_{\rm 2\omega }=1$ W due to the BLIIRA factor if the BLIIRA coefficient is the same as the measured coefficient. The squeezing level at a very high pump power ($>1.2$ W) is rather degraded. This calculation indicates that the squeezing in the waveguide is largely degraded by BLIIRA, as argued for LiTaO$_3$ waveguides for 840 nm pulsed squeezed light generation [46]. The situation improves if the BLIIRA is reduced. If the BLIIRA is reduced by a factor of 3, the expected squeezing level reaches 3 dB at around 0.6 W of $P_{2\omega }$, which we can obtain with the fundamental beam power of the current experiment. Such a reduction of the BLIIRA can be achieved by raising the crystal temperature [47,48].
Fig. 9. Estimated squeezing levels versus pump power. (a)The solid lines represent the estimated squeezing level with BLIIRA coefficients of 100 % (red), 33 % (yellow), and 0 % (purple) of the measured $\eta _{\rm BL}$. The dotted lines represent 95 % confidence intervals. In the estimation, we set $\eta _{\rm detect}=1$. The black diamonds plot the experimental results shown in Fig. 6(a). The vertical dashed line (i) represents a pump power of $P_{2\omega }=424$ mW, which is typical in the current experiment. The dashed lines (ii) and (iii) represent the simulated pump power for $P_{\omega } = 1.28$ W with 33 % and 0 % BLIIRA, respectively. (b) Estimation for the waveguides with the first-order matching. We set $\eta _{\rm detect}=0.85$ for this estimation. The shaded areas represent 95 % confidence intervals.
The situation also greatly improves for a waveguide fulfilling the first-order quasi-phase matching condition, which has a three times better ratio of the squeezing parameter over the loss. We estimated squeezed levels for waveguides of the first-order matching in Fig. 9(b). We presented squeezing levels for waveguides of 11 mm and 22 mm lengths with the same BLIIRA coefficient as that in the experiment and half of that. We set the detection efficiency as $\eta _{\rm detect}=0.85$ in these estimations. The simulation indicates that a squeezing level greater than 5 dB might be obtained for the 22mm waveguide with reduced BLIIRA. The expected squeezing level is comparable to the best one at 795 nm so far [28], obtained with the cavity approach.
6. Conclusion
In conclusion, we have reported the generation of pulsed squeezed light at 795 nm with the best observed squeezed level of $-1.5(1)$ dB using two ZnO-doped PPLN waveguides. The improvement in the squeezed level is due to the strong pump ultraviolet light and compensation of the phase shift during a pulse. The phase shift has been identified as occurring in the SHG waveguide and is ascribed to the heating of the SHG waveguide due to the BLIIRA through a numerical simulation. The phase compensation technique should be effective for improving the observed squeezing level in other experiments using an absorbing crystal for pulsed squeezed light generation. The numerical simulation also shows that the inefficiency in the SHG is likely caused by the BLIIRA and the resulting waveguide heating rather than by the UV absorption. We have also identified the BLIIRA as a dominant limitation in obtaining a better squeezing level. Our results show the possibility of generating high-quality squeezed light at 795 nm using waveguides.
Funding
MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) (JPMXS0118070326); Japan Society for the Promotion of Science (19K03703).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but can be obtained from the authors on request.
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