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Abstract
We experimentally demonstrate programmable multimode phase-sensitive amplification multiplexed in the frequency domain for flexible control of parallelly generated squeezed states. We utilize the unique phase-matching condition of a type-II periodically poled potassium titanyl phosphate (PPKTP) crystal and pulse shaping technique to fully control the frequency-domain parallel generation of squeezed states in the optical telecommunication band. We experimentally verify that the independent programmability of phase-sensitive optical parametric amplification (OPA) for the modes corresponding to different frequency bands can be achieved by shaping the pump laser pulse from optical parametric gain measurements using a coherent probe light generated by a degenerate synchronously pumped optical parametric oscillator.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Squeezed light has been utilized for various experiments, such as continuous-variable entanglement generation [1], gravitational wave detection [2,3], and quantum imaging [4]. Especially when thinking about its application to practical universal quantum computation [5,6], the ideal squeezed state needs to be scalable, pure, and high-level. To the best of our knowledge, an up to -15.7 dB squeezed state has been experimentally demonstrated to date [7].
In general, a high squeezing level is beneficial for various applications. However, the squeezing level must be controlled in several experiments, for example, for adjusting the amplitude of coherent superposition in the experiment of Schrödinger's cat state generation [8]. Recently, quantum emulation based on Gaussian boson sampling was proposed by Huh, et al. [9]. According to their report, the molecular vibronic transition spectra based on the Franck-Condon principle can be simulated by the Gaussian boson sampler, which consists of squeezed state generation, a linear optical circuit, and photon counters. In this scheme, the squeezing levels of parallelly generated modes need to be independently controlled for a specific target molecule.
So far, numerous methods for large-scale squeezed state generation have been proposed and demonstrated in the time domain [10], the frequency domain [11,12], and the spatial domain [13]. In this paper, we focus on a method to independently control the squeezing level and phase of parallelly generated squeezed states in different frequency bands for specific applications such as programmable Gaussian boson sampling [14]. We utilize the frequency degree of freedom of light to parallel squeezed state generation, similar to the wavelength-division multiplexing (WDM) technique, which has been widely used in optical telecommunication. WDM enables expansion of the available mode number without expanding the spatial mode, as shown in Fig. 1. At the same time, we achieve control of squeezed states by applying a pulse shaping technique to the femtosecond pump laser pulse to control the spontaneous parametric down-conversion (SPDC) process.
Fig. 1. (a) Conventional multimode squeezed state generation by spatially split pump laser pulses. (b) Wavelength-multiplexed squeezed state generation.
For a detailed explanation of our method, let us discuss the Hamiltonian of the SPDC process [15]. When considering a mode-locked pulse laser as a pump light source, the SPDC process should be described in a multimode picture as follows:
It is known that spectral purity ${ p}$ is given by ${ p} = {1}/{ K}$. Roslund et al. reported multimode squeezed state generation in the frequency domain by using a type-I bismuth borate (BiBO) crystal whose Schmidt number is relatively large [11]. They generated large-scale squeezed states in parallel and measured several of them by selective homodyne detection.
The Schmidt number roughly corresponds to the number of multimode squeezed states. Therefore, a large Schmidt number ${ K}$ is required to generate a highly multimode squeezed state.
However, it is difficult to independently control all squeezed states generated by a broadband pump pulse because each spectral component of the pump pulse contributes to generating several squeezed states complicatedly, as shown in Fig. 2(a). Recently, Arzani et al. applied an optimization algorithm to obtain an optimum pump laser pulse spectrum which can optimize individual squeezed modes generated by spontaneous parametric down-conversion of a frequency comb [15]. However, to fully control the independent squeezed states, the pump spectra and the squeezed states need to be in one-to-one correspondence, as shown in Fig. 2(b). This one-to-one correspondence can be achieved with a type-II periodically polled KTiOPO4 (PPKTP) crystal pumped by a near-infrared (NIR) laser pulse [16]. Signal and idler pulses are known to exhibit symmetric group velocity against that of the pump pulse in the type-II PPKTP crystal. As a result, narrow-band idler and signal pulses are generated due to the short interaction length. In previous research, Jin et al. reported an experimental demonstration of highly pure single-photon generation using a type-II PPKTP crystal [17]. They experimentally obtained joint spectral intensity and calculated its Schmidt number by assuming that pump pulses had a flat spectral phase. The results show that a highly pure single-photon pair can be generated in the wide telecommunication band ranging from 1565 to 1615 nm with a Schmidt number of ∼1.05. Very recently, Rodriguez et al. explored the type -II SPDC generates several 2-mode squeezed states [18].
Fig. 2. Conceptual scheme of photon pair generation by SPDC pumped by a broadband laser pulse (a) with broadband spectral correlation, which is achieved for example with a type-0 PPKTP, and (b) without broadband spectral correlation, which can be achieved for example with a type-II PPKTP.
We apply this unique property of type-II PPKTP crystals to squeezed state generation. Photon pair generation via the SPDC process is equivalent to continuous-variable entanglement generation. Thus, a squeezed state can be extracted by linearly interfering with the entangled modes. By spectrally shaping the pump pulse into a pulse with 2 isolated spectral peaks and launching the shaped pulse into a PPKTP crystal, we parallelly generate a frequency-domain squeezed states in a programmable manner.
We show detailed calculations and experiments in the following sections. First, we numerically calculate the characteristics of spectral purity by calculating the JSA. Next, we describe the experiments in detail. To obtain a probe laser pulse for detecting squeezed states, we construct a stable degenerate synchronously-pumped optical parametric oscillator (SPOPO) using a feedback cavity stabilization system. We verify spectrally multiplexed 2 squeezed states generation and their independent control by classical phase-sensitive parametric gain measurements with the shaped probe laser pulse. Finally, we confirm the programmability of 2 squeezed states.
2. Numerical simulation
We calculated JSAs and Schmidt numbers ${K}$ under the following conditions: a 7.2-mm-long type-II PPKTP crystal, a 785 nm pump laser pulse with a Gaussian temporal waveform, and a fundamental spatial mode for all. The pump pulse is launched into the y-axis of the crystal, and then yields signal and idler photons in the y- and z-axis. We employed the Sellmeier equations reported in Refs. [16] and [19]. We used the nonlinear coefficient d24=3.6 pm/V for the SPDC of the type-II PPKTP crystal.
We determined the optimal bandwidth of the pump pulse by calculating the Schmidt number${\boldsymbol \; K}$. The calculation results are shown in Fig. 3. The full width at half maximum (FWHM) of the optimal pump pulse bandwidth was 1.47 nm, and the corresponding Schmidt number ${K}$ and purity ${p}$ were 1.24 and 0.8065, respectively.
Fig. 3. (a) Spectrum width of the pump pulse (FWHM) vs. Schmidt number (red curve) and purity (blue curve). (b) Enlarged view of (a) around the optimal point.
Figure 4 depicts the PEA, PMA, JSA, and Schmidt coefficient under the optimal condition.
Fig. 4. (a) Pump envelope amplitude, (b) Phase matching amplitude, (c) Joint spectrum amplitude, and (d) Schmidt number of 7.2 mm type-II PPKTP pumped by 785 nm mode-locked pulse laser with spectral width of 1.47 nm (FWHM).
Next, we consider the case where the pump pulse is shaped into a double Gaussian spectrum. Figure 5(a) shows the calculated PEA when the double Gaussian spectrum was used. The 2 center wavelengths were set to 783 and 787 nm. Figure 5(b) shows the PMA, which is the same as Fig. 4(b). Figures 5(c) and 5(d) are the JSA and the Schmidt coefficient, respectively. From the Schmidt coefficients, we obtained Schmidt number ${ K}$ = 2.35 and purity ${ p}$ = 0.43. The Schmidt number was close to 2. This implies that 2 spectrally multiplexed squeezed modes were generated. We will show the spectral shape of the 2 dominant Schmidt modes later in this paper when describing probe laser pulses shaped for OPA gain measurements.
Fig. 5. (a) Pump envelope amplitude, (b) Phase matching amplitude, (c) Joint spectrum amplitude, and (d) Schmidt number obtained when 7.2 mm type-II PPKTP was pumped by the double spectral peak pulse.
A squeezed state is generated as a result of 50:50 linear interference between the signal and the idler photons. If a linear optical beam splitter is used for the interference, the signal and the idler need to be spectrally overlapped. Since the PMA was slightly curved due to the high-order dispersion, the available mode number will be limited in this scheme.
Furthermore, the preciseness of this method is deteriorated by the side lobe of the sinc function seen in the PMA because it causes some correlation between parallelly generated different modes. This can be improved with the method proposed by Graffitti et al. in which they engineered the poling period of a PPKTP crystal to eliminate the side lobe [20].
3. Single-mode phase sensitive parametric amplification
Figure 6 shows the experimental setup for spectrally multiplexed squeezed modes generation and detection. We utilized a 7.2-mm-long type-II PPKTP crystal for generating squeezed states in SPDC. The poling period of this crystal was 46.1 μm. The crystal was used under a Peltier-module-based temperature control system and was kept at 32.0°C.
Fig. 6. The experimental setup for squeezing with a 7.2 mm type-II PPKTP crystal. HWP: Half-wave plate; DM: Dichroic mirror; AL: Achromatic lens; LPF: Long-pass filter; PBS: Polarizing beam splitter; PD: Photodetector; upper right corner: birefringence compensation by 2 KTP crystals.
In general, balanced homodyne detection, which is a phase-sensitive detection in which signal light and local oscillator light are mixed by an optical beam splitter and measured by balanced photodiodes, is used to evaluate squeezed state. The quadrature amplitude is vulnerable to light loss, so the quantum efficiency of the photodiode should be near 100%. In addition, the local oscillator light must exactly match the signal light in temporal, spectral, and spatial modes. On the other hand, since the amplified signal by OPA is resistant to light loss, we estimated the squeeze level by classical parametric amplification gain measurements [21].
Under the assumption of single-mode squeezing, squeezing gain ${{ G}_{{ sq}}}{ \; }$ and anti-squeezing gain ${{ G}_{{ asq}}}$ are given by
The OPA was pumped by the NIR pump pulse, which is split from the Ti:sapphire laser pulse, with a spectral width of 10 nm (4.8 THz). We prepared the frequency-halved probe pulse, which is coherent to the pump pulse (details are provided in the Appendix). There could be another possibility to conduct this experiment with a 1550 nm mode-lock laser and its second harmonic pulses. However, with the exception of Cr-doped YAG mode-lock lasers, where high-performance laser crystals are difficult to obtain, mode-lock fiber lasers show considerable noise due to the beat of ASE output. Therefore, fiber lasers cannot be used for experiments of quantum optics. SPOPO is almost essential to generate a 1550-nm probe pulse.
The probe pulse was filtered by a 6-nm bandpass filter at a center wavelength of 1570 nm. This bandwidth (0.73 THz) was close to that of the single-mode squeezed state calculated in Section 2. Therefore, this filtered probe pulse could extract a phase-sensitive OPA gain corresponding to single-mode squeezed state from a broadband multimode squeezed state generated by the broadband pump pulse.
We combined the probe and the pump beams with a dichroic mirror (DM) and focused them to spot diameters of 15 and 21 μm with an achromatic lens (f = 50 mm), respectively. We obtained a squeezed state by interference between the signal and idler entangled pair. Therefore, we launched the probe pulse at 45° polarization to collect the OPA gain as a result of polarization interference between the horizontally polarized signal photons and the vertically polarized idler photons. Moreover, we inserted two 3.6 mm KTP crystals before and after the PPKTP crystal to compensate for the birefringence of the PPKTP crystal. The squeezed state formed at 45° polarization at the center of the crystal. We compensated for the second half of the birefringence of the crystal with a 3.6 mm KTP crystal after the PPKTP crystal. We precisely adjusted the birefringence compensation by slightly rotating the KTP crystals. The visibility of the polarization interference was 85%. Finally, we obtained the linearly polarized OPA gain by extracting 45° polarization with a half-wave plate (HWP) and polarizing beam splitter (PBS).
We measured the pump power dependence of the parametric gain in the range of 10 to 440 mW. We obtained the maximum amplification and de-amplification levels of -0.66 dB and 2.04 dB when pumping at the maximum power. One example of phase scan data of phase-sensitive amplification is shown in Fig. 7(a). Figure 7(b) shows the power dependence of the amplification gains.
Fig. 7. (a) Classical parametric gain versus relative phase at a pump power of 400 mW. During the measurements, we manually added the phase difference between the pump and the signal pulses. In this figure, one cycle corresponds to the phase difference of π. (b) Plots of classical parametric gain versus pump power.
By substituting the amplification and deamplification gain at ${{ P}_{max}}$=440 mW into Eqs. (5) and (6), we obtained the mode-matching efficiency ${ \eta }\, = { \; }\,$0.18 and squeezing parameter ${ r\;= }\,$ 0.72. We calculated squeezing and anti-squeezing levels at each pump power by using the parameters at the maximum pump power [the solid lines in Fig. 7(b)].
If ${ \eta }$=1, the maximum squeezing level would be improved by -6.3 dB. Poor mode-matching efficiency significantly deteriorated the squeezing levels in our experiment. The mode-matching efficiency can be factored into the products of spatial, polarization, and temporal mode matching. The contribution of the latter 2 factors is estimated to be not so significant in our measurements. Therefore, we speculate that the main reason is the spatial mode mismatching, which can be improved by the adjustment of the beam size and numerical aperture [22,23].
4. Independently programmable multimode squeezers
4.1 Spectral shaping with a 4-f pulse shaper
We demonstrate independent control of parallelly generated squeezed states which correspond to the different frequency bands. For the spectral control of phase and amplitude, we applied 4-f pulse shaping to pump pulses. Then, we also shaped the probe pulse to evaluate the squeezed state. We spectrally shaped them into double-peak Gaussian spectra to demonstrate a squeezer with 2 frequency modes. Figure 8 shows the schematic view of this experimental setup.
Fig. 8. (a) The experimental setup for multimode squeezing and detection. LCOS-SLM: Liquid crystal on silicon-spatial light modulator; HWP: Half-wave plate; DM: Dichroic mirror; PBS: Polarizing beam splitter; PD: Photodetector. In this schematic view, spectrally shaped pump and signal pulses were illustrated in red and yellow colors, respectively. Those 2 peaks correspond to the spectral peak, not temporal peak, because we manipulated only the spectral amplitude. (b) The relative spectral phase is always 0 between the 2 spectra. Therefore, the temporal shape is a single pulse with amplitude modulation at a beat frequency between 2 spectral components. This pulse shape corresponds to an in-phase double spectral peak pulse.
We utilized a 4f-type pulse shaper using a 2D liquid crystal on a silicon-spatial light modulator (LCOS-SLM, Santec Corp.) placed at the Fourier plane. The spectral amplitude and phase of femtosecond pulses were shaped based on the method proposed by Frumker and Silberberg, [24]. The reflected beam from the LCOS-SLM was spectrally recombined by the cylindrical lens and the grating. The diameter of the probe pulse was 4.72 mm. We employed a blazed reflection grating with an angular dispersion of 0.74 nm/rad and blazed wavelength of 1550 nm. The first-order diffraction efficiency was ∼60%. We formed a spectral Fourier plane with a cylindrical lens (f = 200 mm). The spot size of each wavelength component can be calculated from its diameter and focal length; it was 84.7 μm. Since this diameter was sufficiently larger than the pixel size of the LCOS-SLM (10 μm), the spectral resolution of the pulse shaper was mainly determined by the diameter of each spectral component. We determined that the spectral resolution of the pulse shaper was ∼0.3 nm (1/e2).
In Fig. 9(a), we showed the spectra of the first 2 Schmidt modes with a Schmid number of 0.68: which were analyzed in the Schmidt mode decomposition shown in Fig. 5 for type-II PPKTP SPDC when the pump pulse is shaped into a double Gaussian spectrum. These 2 Schmidt modes exhibit 2 spectral peaks corresponding to the double Gaussian spectrum of the pump pulse. However, when we linearly combine these 2 Schmidt modes [Mode A and B in Fig. 9(a)] in the frequency domain with the same function as a beam splitter, we can generate 2 Gaussian spectra [Mode 1 and 2 in Fig. 9(b)] with a different center frequency. There is no significant spectral overlap between these 2 basis-transformed spectral eigenmodes. Therefore, in the experiment we shaped the probe pulse into a double-peak Gaussian spectrum, which corresponds to a superposition of the 2 eigenmodes. The center wavelengths of these Gaussian peaks were located at 1566 and 1574 nm, and these spectral widths were 2.8 and 2.2 nm (FWHM), respectively. The spectrum of the shaped probe pulse is shown in Fig. 10(a). During the experiment, the shaped probe laser power was ∼10 mW.
Fig. 9. (a) Spectra of the first 2 Schmidt modes (Mode A and B) with a Schmid number of 0.68, which were analyzed in the Schmidt mode decomposition shown in Fig. 5. (b) Type-II PPKTP SPDC when the pump pulse is shaped into a double Gaussian spectrum. (b) Mode transform by linear combination of Mode A and B in the frequency domain with a beam splitter. Two Gaussian spectra with a different center frequency are generated.
Fig. 10. The spectrum of (a) the shaped probe pulse and (b) the shaped pump pulse. Since we manipulated only the spectral amplitude, the relative spectral phase is always zero. The temporal shape is a single pulse with amplitude modulation at a beat frequency between 2 spectral components.
We shaped the pump pulse with another pulse shaper as well. The diameter of the pump pulse was 3.3 mm. We employed a transmission blazed grating with an angular dispersion of 0.47 nm/rad and blaze wavelength of 800 nm. The first-order diffraction efficiency was >90%. We spectrally collimated the angularly dispersed beam with a cylindrical lens (f = 200 mm). The spot size of each wavelength component was 60.6 μm. Since this diameter was also larger than the pixel size, the spectral resolution of the pulse shaper was mainly determined by the diameter. We calculated that the spectral resolution of the pulse shaper was ∼0.14 nm (1/e2).
The pump pulse was shaped into a double-peak Gaussian spectrum so that each peak corresponded to the spectral peaks of the shaped probe pulse in SPDC. The center wavelengths of these Gaussian peaks were located at 783 nm and 787 nm, and both spectral widths were 1.2 nm (FWHM). The spectrum of the shaped pump pulse is shown in Fig. 10(b). We refer to the long- and short-wavelength components as “Mode 1” and “Mode 2” in the following description.
4.2 Verification of independent control of spectrally multiplexed 2-mode OPA
As shown in Fig. 8, we launched the spectrally shaped pump and probe pulses into the type-II PPKTP crystal. By measuring the classical parametric amplification gains, we confirmed the independent controllability of these OPAs. Each pump power was 30 mW.
We simultaneously detected the amplified probe power at “Mode 1” and “Mode 2” by dividing them with a diffractive grating. We shaped the pump pulse into the following five spectral patterns: (a) in-phase between “Mode 1” and “Mode 2,” (b) opposite-phase between “Mode 1” and “Mode 2,” (c) only “Mode 1,” (d) only “Mode 2,” and (e) nothing.
The relation between the relative phase and the parametric gain of 2 base-transformed orthogonal modes is shown in Fig. 11. In the case of Fig. 11(a), we obtained in-phase parametric amplification and the parametric gain was ∼0.2 dB. This gain almost agrees with that shown in Fig. 7(b). The parametric gains of “Mode 1” and “Mode 2” are slightly different because of the difference in the mode-matching efficiency ${\boldsymbol \eta }$. Next, we set the relative phase of the pump pulses between “Mode 1” and “Mode 2” to ${\boldsymbol \pi }$ [Fig. 10(b)]. In this case, we confirmed that “Mode 1” was deamplified when “Mode 2” was amplified. This revealed that the phase of squeezing operators can be independently controlled by pulse shaping of the pump pulse. Figures 11(c) and 11(d) show the result when one of the 2 pump pulses was eliminated. In this experiment, we obtained the phase-sensitive amplification only in the noneliminated mode. This reveals that even if the original Schmidt modes [Mode A and B in Fig. 9(a)] contain 2 spectral peaks corresponding to the double spectral peaks of the pump pulse, there is no significant correlation between these base-transformed “Mode 1” and “Mode 2”. Nothing happened without the pump pulses [Fig. 11(e)].
Fig. 11. Classical parametric gain in cases of (a) a pump pulse with in-phase double spectra, where the both modes are synchronously amplified or deamplified, and (b) a pump pulse with opposite-phase double spectra, where “Mode 1” is deamplified when “Mode 2” is amplified. When the pump pulse contains only a spectrum corresponding to “Mode 1” (c) or “Mode 2” (e), no OPA gain appears in the orthogonal spectrum. (e) No pump pulse is launched. During the measurements, we manually added the phase difference between the pump and signal pulses. In these figures, one cycle corresponds to the phase difference of π.
From these experiments, we concluded that we successfully demonstrated independent parallel control of 2 frequency-multiplexed squeezers by using a pulse shaping technique and the type-II PPKTP crystal. In our experiment, the squeezing level was limited due to the mode-matching effect and the limited pump power. If the mode-matching efficiency approaches 1, higher squeezing levels can be obtained. Since the saturation of the squeezing level was not observed, higher pump power enables access to the higher squeezing levels. A waveguide-based squeezer [18] or an externally enhanced pump laser cavity will be helpful.
The available mode number in this scheme is limited by the high-order dispersion in a PPKTP crystal, which degrades the property of JSA. We will be able to reduce the effect of the dispersion by dividing the PPKTP crystal into multiple thin crystals and inserting dispersion compensation between them. We believe that further improvement of our method will enable large-scale high-quality multimode squeezing operation in parallel in the frequency domain.
5. Conclusion
We experimentally demonstrated parallel generation and independent control of frequency-domain squeezed states using the type-II PPKTP crystal, which exhibits unique phase match properties and achieves a one-to-one correspondence between the pump spectra and squeezed states. The spectrally shaped broadband pump laser pulse successfully controlled the squeezed states multiplexed in the frequency domain.
Appendix: SPOPO
Here, we describe the synchronously-pumped optical parametric oscillator (SPOPO) that we made as a probe light source in detail. We employed a mode-locked Ti:sapphire laser (Mai Tai HP, Spectra-Physics Inc.) as a pump light source. Its pulse duration, center wavelength, and repetition rate were 100-fs [full width at half maximum (FWHM)], 785 nm, and 80 MHz, respectively. Since we generate a squeezed state in the half frequency band of the Ti:sapphire laser, a coherent 1570 nm probe laser pulse is needed to evaluate the squeezed state. We divided the 785 nm pulse with a polarization beam splitter (PBS) into 2 beams for probe pulse generation and for squeezed state generation. The pump pulse for the probe generation was dispersion compensated with a prism pair to eliminate the relatively large amount of second-order dispersion caused in the electro-optic modulator (EOM).
Fig. 12. Experimental setup for the SPOPO. HWP: Half-wave plate; PBS: Polarizing beam splitter; EOM: Electro-optic modulator; BPF: Bandpass filter; PZT: Piezoelectric transducer; OC: Output coupler; LPF: Longpass filter; PD: Photodetector.
The carrier-envelope phase (CEP) of the generated probe pulse needs to be locked to that of the pump pulse and the squeezed state. Therefore, we generated the probe pulse by utilizing a degenerate SPOPO that automatically locks the CEP of the output beam to that of the pump pulse [25]. Figure 12 shows the schematic view of the experimental setup for the coherent probe light generation.
SPOPO was composed of an optical cavity with a nonlinear crystal and an external feedback circuit for stabilization of the cavity length. As a nonlinear gain medium, we used a 3-mm-long type-0 periodically poled 5 mol.% MgO-doped congruent LiNbO3 (PPLN) crystal (HC Photonics Corp.) whose poling period was 19.47 μm. We constructed a ring cavity with 2 concave mirrors (RC = 200 mm), three high reflector (HR) mirrors, and an output coupler whose transmittance was 3%. We carefully designed the cavity mechanics so that the angle of incidence to the concave mirrors was reduced as much as possible for the reduction of astigmatism. The angle of incidence to the concave mirror was 2.5° in both the experiment and the calculation. We analyzed the beam propagation inside the cavity by using an ABCD matrix. Figure 13 shows the calculation results. The distance between the center of the PPLN crystal and the concave mirror was set to 101.4 mm, and we then obtained a spot radius of 28.6 μm (sagittal) × 31.3 μm (tangential). The Rayleigh length of the cavity mode was 4.87 mm and longer than the crystal length. We adjusted the spot size of the pump pulse to obtain high mode-matching efficiency. We focused the 785 nm pump pulse with a lens whose focal length was f = 125 mm and the spot radius was then to 26.2 μm. We confirmed a non-degenerate operation when high-order spatial mode oscillation occurred in the SPOPO. Therefore, we focused the pump beam tighter than the cavity’s fundamental mode so that the cavity’s high-order mode could be suppressed.
The 2 concave mirrors in the SPOPO were anti-reflection (AR) coated on both sides for the pump pulses and HR coated on the concave side for the signal pulse at 1570 nm. The three HR mirrors and the output coupler were also HR coated for the signal pulse so that we could align the cavity with the partially reflected pump pulse. We roughly adjusted the cavity length based on the repetition rate of the pump pulse train, then precisely adjusted it by checking interference on a complementary metal oxide semiconductor (CMOS) camera between the partially reflected pulse from the input concave mirror and the pulse that traveled around the cavity. We aligned the cavity mirrors so that the interference pattern became concentric. After we got the concentric interference pattern and slightly adjusted the cavity length again, we successfully obtained the laser oscillation. During the oscillation, we could see the green light that was generated via sum-frequency generation between the signal and pump pulse. As we mentioned above, the 1570 nm probe light generated from the SPOPO needed to be coherent with the 785 nm pump pulse. In the nondegenerate SPOPO, the CEP is not locked to that of the pump pulse and is allowed to flexibly change depending on the cavity length. On the other hand, the CEPs of the idler and signal beams in the SPOPO cavity must be the same under the degenerate operation because they interfere with each other inside the cavity. If the CEPs of the signal and idler beams are different, they destructively interfere and impede stable laser oscillation. Therefore, the cavity length, which determines the CEPs of the idler and signal beams in the SPOPO cavity, must be controlled precisely for stable laser oscillation. We controlled the cavity by using the Pound-Drever-Hall (PDH) locking method in addition to measures for the reduction of the external noise. As shown in Figs. 14(a) and 14(b), we assembled the SPOPO on the optical breadboard placed on seismic isolation rubbers. Additionally, we covered the entire experimental setup with aluminum plates and acrylic plates to isolate the setup from the window and acoustic waves. Since our Ti:sapphire laser (Mai Tai HP) had an excess phase noise when pump optimization feedback was working inside the laser, we turned off the pump optimization loop.
Fig. 14. (a) SPOPO on a seismic isolated board. (b) Seismic isolation rubbers on the back of the board. (c) A piezoelectric stack on a copper pillar.
We applied the 20-MHz phase modulation to the pump pulse train with a resonant EOM (EO-PM-R-20-C1, Thorlabs, Inc.) for the PDH locking. The part of the output from the SPOPO was reflected by a polarizing beam splitter (PBS) and spectrally filtered by an optical bandpass filter (center wavelength: 1570 nm; FWHM 3 nm; NIR01-1570/3-25, Semrock Inc.). The filtered signal was detected by an InGaAs-amplified photodetector (gain 104, cutoff 150 MHz, PDA05CF2, Thorlabs, Inc.). We obtained the time differential value of the cavity length fluctuation by mixing the intensity signal from the photodetector and the 20-MHz modulation signal with a mixer (ZP-3+, Mini-Circuits). Here, the optimal phase shift was applied to the 20-MHz modulation signal with a phase shifter (PS-909, Synergy Microwave Corp.) so that the error signal was maximized. After 100-times amplification with a handmade inverting amplifier, the mixed signal was sent to an analog proportional integral derivative (PID) controller (SIM960, Stanford Research Systems). The setpoint of the PID controller was 0 V and then the obtained feedback signal was sent to a piezoelectric stack (AE0203D08F, Thorlabs, Inc.) on which the cavity mirror was mounted. The piezoelectric stack was mounted on a 1-inch copper pole [Fig. 14(c)]. The cavity mirror mounted on the piezoelectric stack was 1/4 inch to reduce its weight. The difference in weight between the copper pole and the piezoelectric stack was important for the improvement of the frequency response property of the feedback system. We used vacuum epoxy (Torr Seal, Varian Inc.) as an adhesive between these components because of its relatively high rigidity.
Our whole feedback system enabled the stable degenerate operation of the SPOPO laser. The oscillation threshold was ∼350 mW. The stability of the output power, the error signal from the PID controller, and the spectrum of the output pulse are shown in Fig. 15. We confirmed stable oscillation over 30 min and obtained 44 nm spectral width (FWHM) under the degenerate oscillation. We also verified that the insertion of fused silica wedge plates of adequate thickness into the SPOPO cause spectral broadening up to a width of 60 nm (FWHM) because of the dispersion compensation. However, since we did not require a broad spectrum due to the relatively narrow bandwidth of the squeezed state, the plates were not inserted during the squeezed state detection experiment.
Funding
Japan Society for the Promotion of Science (JP11K11111); Core Research for Evolutional Science and Technology (JPMJCR1772).
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 may be obtained from the authors upon reasonable request.
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