Continuous-wave squeezed vacuum states of light via self-phase modulation

Continuous-wave (cw) squeezed vacuum states of light have applications in sensing, metrology and secure communication. In recent decades their efficient generation has been based on parametric down-conversion, which requires pumping by externally generated pump light of twice the optical frequency. Currently, there is immense effort in miniaturizing squeezed-light sources for chip-integration. Designs that require just a single input wavelength are favored since they offer an easier realization. Here we report on the first direct observation of cw squeezed vacuum states generated by self-phase modulation. The wavelengths of input light and of balanced homodyne detection are identical, and 1550 nm in our case. At sideband frequencies around 1.075 GHz, a nonclassical noise reduction of (2.4 +/- 0.1) dB is observed. The setup uses a second-order nonlinear crystal, but no externally generated light of twice the frequency. Our experiment is not miniaturized, but might open a route towards simplified chip-integrated realizations.

Introduction -Squeezed vacuum states of continuouswave (cw) quasi-monochromatic light have been improving the sensitivity of the gravitational-wave detector GEO 600 during its observational runs since 2010 [1,2] and was tested for integration in LIGO [3]. More recently, proof-of-principle experiments used the unique feature of these states for one-sided device independent quantum key distribution (QKD) [4], oblivious transfer (OT) [5], and the calibration of photo-diode quantum efficiencies without knowledge of the incident light power [6]. Noteworthy, the latter three experiments are impossible by scaling the power of light beams being in coherent states.
State-of-the-art devices for the generation of squeezed states of light are based on cavity-enhanced (type I) degenerate parametric down-conversion in a nonlinear crystal such as MgO-doped LiNbO 3 or quasi-phase-matched periodically poled KTP [7][8][9]. Here, the nonlinear process is of second order and requires pump light at twice the optical frequency, which needs to be spatially overlapped with the cavity mode of fundamental frequency. The fine-tuning of phase-matching between the two wavelengths is realised via the temperature of the crystal. The largest squeeze factors observed so far are 15 dB at 1064 nm and 13 dB at 1550 nm below vacuum noise [6,10]. More than 10 dB of two-mode-squeezing, i.e. the continuous-variable entanglement between two beams of light was realised in [11]. The limitations to the squeeze factors were in all cases set by photon loss. The states were produced in about 10 mm long crystals surrounded by 4 cm long standing-wave cavities, and coupled to freely propagating beams. The complete optical setups had footprints of one to two square meters to allow for individually optimized mode-matchings and electro-optical phase controls [12]. Commercial applications of squeezed light, however, demand more compact devices. Microoptomechanical devices and integrated optics could offer ways for miniaturization [13][14][15][16][17].
Other approaches for the generation of squeezed light are non-degenerate four-wave mixing [18,19] and selfphase modulation (fully degenerate four-wave mixing). Self-phase modulation (SPM) has the convenient feature that just a single optical frequency needs to be supplied for the generation and observation/exploitation of quantum noise squeezing. Fig. 1 (a) illustrates the selftransformation from a coherent state to a squeezed state via SPM when propagating through a medium having an intensity-dependent refractive index. The latter corresponds to the optical Kerr effect and was theoretically analysed in [20][21][22]. SPM is based on a third-order nonlinear coupling of the light with itself, which requires high intensities. Ref. [23] reported 5 dB of squeezing due to SPM on light pulses after propagation through a 50 m glass fibre loop. Ref. [24] observed 6.8 dB on light pulses after propagation through a 13 m fibre. The generation of squeezed states in the cw regime is more challenging. To the best of our knowledge there is just a single such experiment based on all-optical SPM [25]. About 1.5 dB of squeezing on a cw beam of 0.45 mW power was generated. Squeezing, however, was not observed with respect to the shot noise level of the detector's local oscillator beam but only with respect to a slightly higher shot noise level that was calculated and included the contribution from the signal beam power. SPM due to radiation pressure on a micro-mechanical device was recently used to produce cw squeezed light. About 0.2 dB and 1.7 dB were achieved, respectively [13,14] These optomechanical experiments required operation in vacuum and cryogenic cooling. The direct observation of cw squeezed vacuum states of light generated by all-optical SPM has not been achieved so far.
Here, we report the direct observation of squeezed vacuum states of light at the telecommunication wavelength of 1550 nm produced by all-optical self-phase modulation. A 70 mW beam that initially was in a coherent Figure 1: (a) Phase-space description of the transformation from a coherent state to a squeezed state through self-phase modulation.X andŶ are the amplitude and phase quadrature amplitude operators defined with respect to a local oscillator field. The shaded areas correspond to quantum uncertainties. The phase shift depends on the light's intensity, which is proportional to the photon number operatorn =â †â . The latter are the creation and annihilation operators. (b) Illustration of how the intensity dependent phase shift is realized in our setup. During transmission through a PPKTP crystal some fraction of the light, here represented by the reflectivity R1 of a fictitious mirror, is frequency converted and reconverted. In this case, the optical propagation phases ϕ1 and ϕ0 differ. The converted fraction R1 depends onn.
state transformed itself to a (2.4 ± 0.1) dB squeezed state by transmission through a 10 mm long periodically poled KTP (PPKTP) crystal inside a traveling-wave cavity. The temperature of the crystal was precisely adjusted (around the design temperature of about 60 • C) to a minimum of the sinc-squared function to prevent secondharmonic generation. A second cavity subsequently filtered the undepleted 70 mW light. The nonclassical noise suppression of the squeezed vacuum states was observed with conventional balanced homodyne detection.
Experimental implementation - Fig. 1 (b) illustrates the fundamental principle of how we realised self-phase modulation. First of all, the PPKTP-crystal temperature is set to a conversion minimum next to the global conversion maximum. At such a temperature (almost) no up-converted light leaves the crystal. Light that is up-converted from 1550 nm to 775 nm during propagation through the first half of the crystal is fully converted back to 1550 nm during propagation through the second half of the crystal. The conversion efficiency at the crystal's centre depends on the intensity of the 1550 nm field, which is a variable of non-zero quantum uncertainty. The unmatched propagation phases together with the intensitydependent conversion efficiency generate an intensity dependent phase shift of the 1550 nm light beam, which results in self-phase modulation. This special type of a third-order nonlinearity was called cascaded second-order nonlinearity [26,27] and was previously used to reduce classical fluctuations of laser light [28,29]. The work reported in [25] also used a cascaded second-order non-  In the conversion maximum the resonance profile was symmetric. At other temperatures, the phases of 1550 nm and 775 nm light were less-well matched and self-phase modulation resulted in asymmetric peaks, with maximal asymmetry in the conversion minima. As predicted by theory, the asymmetries were found to be more pronounced for higher light intensities and were independent of scanning speeds.
convex mirrors M 1 and M 2 was −500 mm. Together with the concave mirrors M 3 and M 4 , which had radii of curvatures of 100 mm, a waist of 30 µm was formed in the centre of the PPKTP crystal. The produced squeezed sidebands and the bright carrier field left the bow-tie resonator in reflection and were spatially separated by an output mode-cleaner (OMC). The FSR of the OMC was precisely twice that of the squeezing resonator and was fine adjusted manually with a high-voltage driven piezo-mounted mirror. The filtering by the OMC had the purpose to produce a signal field without carrier to allow for (conventional) BHD. Our filtering scheme prevented the detection of frequencies within the linewidth of the squeezing resonator and at frequencies of even number multiples of the squeezing resonator's FSR. Squeezed vacuum states at frequencies of odd multiples of the squeezing resonator's FSR propagated to the BHD where they were superimposed with a local oscillator at a 50/50 beam splitter. The BHD was home-built [30] and had a bandwidth of 1.5 GHz allowing for the detection of squeezed states at about 358 MHz and 1074 MHz. Measurement results -Self-phase modulation with lowest loss is achieved at crystal temperatures at which the SHG efficiency is minimal. To identify these, we varied the temperature of the crystal between 20 • C and 88 • C while having an input light power of 8.8 mW at 1550 nm, see Fig. 3. For all measurement points, the length of the squeezing resonator was stabilized on reso- nance with the Pound-Drever-Hall (PDH) method. Light converted to 775 nm was coupled out via mirror M 3 , passed through a dichroic beam splitter to filter 1550 nm light, and was detected with a power meter. The global SHG conversion maximum was found at 40.5 • C. The first and the second high-temperature-sided conversion minima were observed at 61.2 • C and 81.8 • C, respectively. At conversion minima linear scans of the cavity length revealed significantly deformed resonance profiles, as shown in Fig. 3 (right inset). At 40.5 • C the resonance profile was not deformed (left inset). For these measurements we recorded the out-coupled power at 1550 nm behind mirror M 2 . The shape of the resonance profiles did not depend on the scanning speed (which was of the order of microseconds per full width at half maximum). This result showed that absorbed light in combination with the crystal's thermal expansion coefficient did not have an influence on the resonance profiles. For squeezing generation, the temperature of the PPKTP crystal was adjusted to 61.2 • C, and 70 mW of 1550 nm light was mode-matched to the squeezing resonator, whose length was stabilized close to resonance. The bright light field and the squeezed sidebands left the bow-tie resonator at mirror M 1 and were separated by the OMC. The remaining squeezed vacuum field was overlapped with the local oscillator beam on the 50/50 beam splitter were a fringe visibility of 97 % was achieved. Fig. 4 represents the highest squeeze factor we reproducibly observed. Trace (a) is the measured vacuum noise level of our local oscillator power of 4 mW, which was recorded when the signal path of the BHD was blocked. With the signal path open the variance of the squeezed field was evaluated with a zerospan measurement at a sideband frequency of 1074 MHz, which corresponded to the third FSR of the bow-tie resonator. During the measurement, the relative phase between the signal field and the local oscillator was periodically changed with a piezo-actuated mirror that was located in the local oscillator path. As depicted by trace (b) in Fig. 4, a nonclassical noise reduction of (2.4 ± 0.1) dB below the vacuum noise level and an anti-squeezing value of (7.5±0.1) dB was detected. If optical loss was the only decoherence process, these values would request a total detection efficiency of just 47 % [9]. Our independent measurements of the squeezing resonator's escape efficiency ((84 ± 2) %), the OMC transmission ((89 ± 1) %), remaining SHG ((98 ± 1) %), and the detection efficiency of BHD ((90±4) %) suggested, however, a total detection efficiency of about (66 ± 5) %. At the sideband frequency of the first FSR, we observed (in the same conversion minimum) (2.0±0.1) dB of squeezing together with (9.5 ± 0.1) dB of anti-squeezing, see Fig. 5. For this measurement we increased the input power to 85 mW. We could not observe any higher squeeze factor at 358 MHz. The difference of the two squeeze factors is not easy to explain since the optical loss and phase noise [31] were identical in these measurements. Other parameters such as the performance of the balanced homodyne detector at these frequencies were also identical. A potential explanation for the higher squeeze factor at higher frequency is thermally-driven internal phase noise. Thermal energy results in microscopic vibration inside the (nonlinear) medium that translates into broadband random phase modulations of the carrier light, which usually falls off with frequency [32,33]. In one of the earliest fibre squeezing experiments cooling the fibre down to 4 K reduced noise and allowed the observation of squeezed states [19]. We had observed such noise in a previous similar setup [29] and were able to describe it as broadband phase noise with a 1/f slope of its noise power [22]. The limitation of the squeeze factors in this work, however, cannot be described by the same slope and we could not clearly confirm the presence of internal phase noise. Fig. 5 shows the highest achievable squeeze factors at 358 MHz for different temperatures of the PPKTP crystal. The highest squeeze and anti-squeeze factors are in the first conversion minimum. This observation clearly supports our claim that the dominant nonlinear process for squeezed state generation at this temperature was self-phase modulation. At other temperatures, where a significant amount of light power at 775 nm was coupled out, squeezed states were partly produced by non-linear depletion of the 1550 nm light. In the conversion maximum the latter effect was the only one. Non-linear depletion was previously used for the generation of squeezed states as reported in [34,35]. anti-squeeze factors observed at the sideband frequency of 358 MHz. The highest squeezed factor was achieved in the first conversion minimum and is attributed to self-phase modulation. Also at the conversion maximum as well as at other operational points of the experiment squeezed states were produced. This is due to another process, namely the nonlinear depletion of the fundamental wave, also called SHG squeezing, and was previously demonstrated in Refs. [34,35].
Summary and conclusion -We report the first generation of a continuous-wave squeezed vacuum field via self-phase modulation (SPM). The carrier light of 70 mW at 1550 nm was subsequently subtracted by a filter cavity. We directly observed a nonclassical noise-reduction of up to 2.4 dB. Since the squeezed vacuum was not accompanied by any carrier light, the shot-noise reference corresponded to the power of the balanced homodyne detector's local oscillator alone. Varying the phase of the local oscillator enabled quantum tomography on the squeezed states. The value of 2.4 dB was observed at a sideband frequency of 1074 MHz, which corresponded to three-times the free spectral range of the squeezing resonator. Our setup also allowed the squeezed light measurement at about 358 MHz, where we found a slightly smaller value of 2.0 dB. The SPM was realized in a second-order nonlinear crystal whose temperature was set to minimal second-harmonic generation. Limitations to the observable squeeze factor were optical loss and phase noise of the carrier light. The latter potentially arose due to imperfect stabilization of the length of the squeezing resonator, which is a problem that should get strongly reduced in miniaturized integrated realizations. SPM, however, is well-known to be susceptible to internal phase noise, i.e. thermally-driven inelastic scattering of the carrier light inside the medium [19,24,32,33,36]. The design of miniaturized squeezed light sources using SPM need to take this issue into account.