Multimode Single-Pass Spatio-temporal Squeezing

We present a single-pass source of broadband multimode squeezed light with potential application in quantum information and quantum metrology. The source is based on a type I parametric down-conversion (PDC) process inside a bulk nonlinear crystal in a non-collinear configuration. The generated squeezed light exhibits a spatiotemporal multimode behavior that is probed using a homodyne measurement with a local oscillator shaped both spatially and temporally. Finally we follow a covariance matrix based approach to reveal the distribution of the squeezing among several independent temporal and spatial modes. This unambiguously validates the multimode feature of our source. © 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
Squeezed states, in which squeezing is distributed to a set of independent optical modes, constitute an important quantum resource in the field of continuous-variable quantum information technology [1], as for example, one-way quantum computation [2], and quantum communication [3]. Besides, multimode squeezed light is a promising tool in metrological applications, in particular for multi-parameter estimation with quantum enhanced sensitivity [4,5]. Examples include quantum imaging via spatially multimode squeezing [6,7], and quantum improved synchronization of distant clocks exploiting temporal/spectral multimode squeezed light [8]. The aforementioned wide range of potential applications are closely linked to the ever-increasing ability to generate, control, and detect multimode quantum light, thanks to the development of optical technologies such as spatial light modulators, optical frequency combs, multi-pixel detectors, to name a few.
Squeezed light is commonly obtained from parametric down conversion (PDC) in a secondorder nonlinear crystal placed inside an optical cavity, a so-called optical parametric oscillator (OPO). The optical cavity enhances the nonlinear interaction as well as it confines the squeezed light to a single spatial mode. Multimode squeezing has been produced by exploiting different degrees of freedom of light such as temporal/spectral [9], spatial [10], and orbital angular momentum [11]. However, the OPO resonator limits the squeezing bandwidth to the resonator bandwidth. A promising alternative to produce broadband multimode squeezing is a singlepass PDC source, pumped with a pulsed laser which features an optical frequency comb in the frequency domain [12]. The single-pass design with a pulsed pump ensures that squeezing sustains at every pulse of the PDC output [13,14]. Nonlinear waveguide based single-pass PDC sources are interesting to obtain high level of squeezing due to tight confinement of light [15]. However, waveguide-based designs traditionally suffer from high loss that deteriorates the purity of the generated squeezed state [16]. Furthermore, most of the existing studies on single-pass PDC sources, to the best of our knowledge, only followed a single-mode approach to characterize the generated squeezed light.
In this study, we report multimode analysis of a single-pass source of broadband squeezed light based on a type I PDC process inside a bulk nonlinear crystal, in a non-collinear configuration. This particular design of our source leads to the fact that the generated squeezed light exhibits multimode features in both spatial and temporal/spectral degrees of freedom. To characterize the spatiotemporal multimode feature, we implement a homodyne detection in which the local oscillator (LO) is shaped both spatially and temporally. Finally, we construct covariance matrices from the results of the homodyne measurement and perform eigenvalue decomposition of the measured covariance matrices, providing a definite proof for multimode squeezing.

Theoretical model
A theoretical analysis of a single-pass pulsed optical parametric amplifier in the temporal domain is reported in [12,17], where the temporal modes in a waveguide are considered. In our case, we employ a bulk crystal where the spatial modes are not fixed by the waveguide geometry and, therefore, spatiotemporal coupling effects become important [18]. It is convenient to consider the fields in the Heisenberg picture and in the slowly varying envelope approximation: where the indices s and p indicate, respectively, the down converted field (oscillating at a central angular frequency of ω s ) and the pump of the process (with a central angular frequency of ω p = 2ω s ). The creation operator can be written in the slowly varying envelope approximation: a s, p ì q, z, Ω =Â s, p ì q, z, Ω e ik z ( ì q,Ω) withÂ being the slowly varying envelope operator and k z ( ì q, Ω) = k(ω s + Ω) − ì q 2 being the projection of the wave vector along the propagation axis z. The coordinates of interest are Ω, the detuning from the central frequency ω s = ω p /2, and ì q, the transverse wave-vector that corresponds to the position vector ì x in the Fourier plane of a lens. Usually being an intense coherent beam, and in the small pump depletion limit, the pump can be treated classically and the operator can be replaced by its mean value,Â p ( ì q, z, Ω) → A p ( ì q, z, Ω). The gain of the process is proportional to the non-linear susceptibility χ (2) , to the pump peak amplitude α p , and to the length of the crystal ℓ c , g ∝ χ (2) α p ℓ c . As shown for example in [19], in the low gain regime we can write the amplifier input-output relation: where the kernel K is given by: Here, A p is the Fourier transform of the spatiotemporal profile of the pump and ∆ is the wavevector mismatch along the pump propagation direction z: In our case of negative uniaxial crystal, the signal experiences the ordinary index n o while the pump encounters the extraordinary index n ext , which depends on its orientation with respect to the optical axis of the nonlinear crystal: where θ 0 is the angle between the optical axis and the pump propagation axis in the perfect phase matching condition for the desired non-collinear angle.
Applying an eigenvalue decomposition to the matrix obtained by discretizing the kernel in equation (3), the description of the problem simplifies [20], and if the kernel is complex a Takagi decomposition can be used [21]. Considering the initial fieldÂ in s as a linear combination of the eigenmodes, S k ( ì q, Ω), of the integral operator related to the kernel: we simplify the equation of the amplifier, Eq. (2), as follows: where Λ k is the corresponding eigenvalue. The quadratures of the eigenmodes are then independently squeezed with a squeezing parameter proportional to gΛ k : A detailed simulation of the spatio-temporal modes of a single-pass squeezer is shown in [22]. In our experiment as detailed in the following sections, we attempt to probe the spatiotemporal eigenmodes, S k ( ì q, Ω), as well as the corresponding squeezing parameter 2gΛ k of our single-pass source by employing a homodyne detection with a spatial and temporal shaping of the local oscillator.

Experimental set-up
A scheme of the experimental set-up is shown in figure 1. A titanium:sapphire (Ti:Sa) femtosecond oscillator, delivering 22 fs pulses at a repetition rate of 156 MHz, constitutes the main frequency comb with a spectral FWHM of 43 nm centred at 795 nm. At the laser output, the pulses are separated by a beam splitter and 40% of the laser power is reserved for the LO while the rest is sent to the squeezer. A 1 mm long bismuth-triborate (BiBO) crystal generates the pump for the PDC process, centered at a wavelength of 397.5 nm with a spectral FWHM of 1.82 nm, by frequency doubling the Ti:Sa pulses. The pump power is subsequently increased by a factor of seven using a synchronous cavity so that 120 mW (0.77 nJ energy per pulse) are oscillating inside the linear cavity when it is locked with a Pound-Drever-Hall mechanism in transmission. The synchronous cavity also allows to clean the transverse profile of the pump beam.
The PDC crystal is placed at the waist (49 µm) of the cavity. We choose a 2 mm long beta barium borate (BBO) as the nonlinear crystal which is set to optimize the type I PDC in the non-collinear configuration. The angle between the pump and the down converted beam is fixed at 1.8°such that the down converted photons can be collected without passing through the cavity end-mirror to avoid losses. At the PDC output we get two beams which are then combined on a 50-50 beam splitter [23]. Finally, we implement a homodyne detection scheme in one of the beam splitter outputs to measure the quadratures. By shaping the temporal and spatial profiles For the temporal shaping of the LO we use a pulse shaper based on a 792 × 600 spatial light modulator (SLM) in a 4 f line. A periodic saw-tooth grating is applied for each wavelength of the 43 nm FWHM spectrum. The groove depth and position allow to control respectively the amplitude and phase for each spectral component [24]. For the spatial shaping, we cut the beam at half power with a razor blade into left-cut (L) and right-cut modes (R), which are by construction orthogonal modes. As shown in figure 1, the distances from the homodyne beamspliter to the cut plane in the LO path and the Fourier plane of the squeezed beam are equal. In this way, the LO cut plane corresponds to the Fourier plane of the squeezed light beam.

Measurements
Our first goal is to measure the amount of squeezing in different temporal/spectral as well as spatial modes. It is known that the temporal/spectral squeezed modes of the pulsed PDC process closely resemble Hermite-Gaussian (HG) modes if the spatiotemporal coupling is neglected [12,17]. Therefore, we implement different HG pulse shapes on the LO, thanks to the pulseshaper in the LO path. We observe squeezing for the first four HG modes, where the first one, HG 0 , has a spectral FWHM of 15 nm. Figures 2a and 2b show the spectral profile of the LO for the 0th-(HG 0 ) and 1st-order HG mode (HG 1 ), respectively. The traces in figures 2c and 2d report the noise variance of the homodyne signal, respectively, for HG 0 and HG 1 mode as a function of time while scanning the LO phase. The squeezing spectra are recorded with a spectrum analyser at a sideband frequency of 10 MHz with a 100 kHz resolution bandwidth and 30 Hz video bandwidth, meaning that the HG 0 case, −0.25(4) dB for HG 1 , and it reduces to lower values for the higher order modes as expected from theory [25], so that we find −0.19(5) dB for HG 2 and −0.19(6) dB HG 3 . The reason behind this low squeezing, compared to the existing studies of single-pass sources, is due to the limited pump pulse energy as well as not so tight confinement of the pump beam.
In order to analyse multimode squeezing in the spatial domain, the LO beam is cut into two orthogonal modes, L and R. The homodyne traces for the whole beam and the half-cut beams are shown in figure 3 with the corresponding spatial shape of the beam at the homodyne beam splitter in the insets. In these measurements, the HG 0 temporal mode is chosen for the LO. We find that both the half-cut modes are squeezed, −0.18(4) dB for mode L and −0.11(4) dB for mode R, though less compared to the whole beam, which is at −0.26(4) dB.

Covariance matrix analysis
Finally, we follow a covariance matrix approach to disclose the underlying independent squeezed modes, to unequivocally validate the multimode nature of the generated squeezed light [26,27]. The covariance matrix V is made of four blocks: where V X ij = X iXj , V P ij = P iPj and V X P ij = X iPj + P jXi 2 . HereX i andP i indicates the quadratures of the squeezed state related to the i-th mode of the LO basis. With this notation the diagonal elements of the covariance matrix report the variance of the X quadrature, X 2 i or the P quadrature P 2 i which are both equal to unity if the i-th mode is in a vacuum state. The off-diagonal elements represent the quantum correlations between the X or P quadratures related to different mode of the LO basis. A Bloch-Messiah decomposition of the covariance matrix gives a basis of eigenmodes whose quadratures are not correlated. If, in this basis, there exist more than one eigenvalue, which are different from 1 (the variance of the vacuum state), then the generated state of light can be genuinely labelled as a multimode [27].
Our measurement set up does not allow us to measure the correlations between the X and P quadratures of the modes i.e. the V X P block. Correlations could be observed between the X and P quadratures if the pump was complex at the center of the crystal, however, in our case, the chirp acquired by the pump can be neglected and thus we can neglect this block of the covariance matrix. We therefore consider only the V X (X block) and V P (P block) and diagonalize them separately.
The quadrature variances of each squeezed mode are extracted from the corresponding homodyne trace. We relate the averaged maxima and minima of the homodyne trace to P 2 i and X 2 i , respectively. To find the off diagonal elements of each block, we consider the following relation: where X i +X j 2 is extracted from the homodyne measurement with a local oscillator in (HG i + HG j ) mode. In this way, all the elements of the X and the P block of the covariance matrix are measured.

Temporal domain
We first report the results of the covariance matrix analysis in the temporal/spectral domain. In figure 4 the X and the P block in the basis of the first four HG modes are shown on the left. For clarity, we remove the identity (the vacuum contribution) from the X and P blocks. These two matrices are independently diagonalized and their eigenvalues are shown on the right of figure  4.
As shown in figure 4, there are more than one eigenvalues that have an absolute value higher than 1 for the P block and lower than 1 for the X block, validating the multimode feature of the generated squeezed light. As can be seen from the eigenvalues, their absolute value is higher for the P block eigenvalues (anti-squeezing) than for the X block eigenvalues (squeezing). This means that even in this diagonal basis the state is not pure. A reduced purity can be caused by losses in the experimental measurement or by an imperfect spatial or temporal overlap. Since in a single-pass squeezing source the losses can be neglected, then this reduced purity is probably due to a imperfect spatial overlap in this case.
The large errors on the eigenvectors, unfortunately, do not allow us to infer the exact shape of the eigenmodes related to the eigenvalues. This is probably because the HG basis is close to the eigenmode basis, as can be seen from the X and P blocks in figure 4. As a result, the relative errors on the off-diagonal elements of the blocks become quite high, thus increasing the errors on the eigenvectors. Nevertheless, as can be seen from the right part of figure 4, the error bars on the eigenvalues are low enough to claim that the system is truly multimode in temporal domain.

Spatial domain
As in the temporal case, we analyse the spatial properties of our system through the covariance matrix with the basis of L and R modes. It is interesting to point out that we can record a covariance matrix in this spatial mode basis for each of the temporal mode that we used in the time domain analysis. We have then four different two-dimensional spatial covariance matrices in the L and R mode basis related to each temporal/spectral mode. In figure 5, we present the X and P blocks of the spatial covariance matrices related to each temporal mode HG i . Right to the X and P blocks, the eigenvalues and eigenvectors for each matrix are reported.
For each of the temporal mode, we find that both the spatial modes are squeezed and, interestingly, the distribution of squeezing changes with the temporal modes. This is linked to the spatiotemporal coupling effect as explained in the theory section. To the best of our knowledge this is the first experimental measurement of the quadrature correlations in the spatial domain related to different time-frequency modes. The error bars on the eigenvectors are too big to infer the exact shape of the squeezed modes. Nonetheless, for each temporal mode, we can see that the second eigenvector (the one related to the lowest eigenvalue) has always a π phase shift between the left and the right part of the beam, while the first one does not. This indicates that the first squeezed mode in the spatial domain is a T E M 00 while the second is a flip-mode [10], and interestingly this feature is valid for every temporal/spectral mode. These measurements demonstrate that our source is multimode also in the spatial domain. However, it does not necessarily mean that the system has only two spatial squeezed modes. In fact, the number of detected eigenmodes is limited by the capabilities of our spatial mode shaping. In order to completely unveil the spatial distribution of squeezing, for example, a supplementary spatial-light modulator could be included in the LO path.

Conclusion and perspective
In this study, we present a single-pass source of multimode squeezed light based on a noncollinear type I PDC process, pumped with an optical frequency comb. Though single-pass squeezing was already observed long ago, its multimodal nature was never experimentally demonstrated in the continuous variable domain. In this paper, we showed that we can measure the amount of squeezing in different temporal modes as well as in different spatial modes. Furthermore, we adopt a covariance matrix based approach to find the principal squeezed modes both in temporal and spatial domain, that serves up a clear signature of the multimodal behavior of our source. An interesting perspective for this source concerns the generation of a dual-rail cluster state, which has been shown to be a good candidate for quantum information [28]. As clear from figure 1, the non-collinear configuration of our source allows, in principle, to put an inter-pulse delay between the signal and the idler pulses before recombining them on the beam splitter. The state generated in this way would be a large-scale dual-rail cluster state as reported in ref. [29]. Another exciting perspective of the present source with a seed beam is that it would be possible to generate bright beams, when seeding the PDC, with entangled temporal properties as demonstrated in the spatial domain [30].