Quantum non-Gaussianity of frequency up-converted single photons

Nonclassical states of light are an important resource in today's quantum communication and metrology protocols. Quantum up-conversion of nonclassical states is a promising approach to overcome frequency differences between disparate subsystems within a quantum information network. Here, we present the generation of heralded narrowband single photons at 1550 nm via cavity enhanced spontaneous parametric downconversion (SPDC) and their subsequent up-conversion to 532 nm. Quantum non-Gaussianity (QNG), which is an important feature for applications in quantum information science, was experimentally certified for the first time in frequency up-converted states.


Introduction
Quantum non-Gaussian states of light cannot be expressed as mixtures of Gaussian states and form a subclass of nonclassical states. Important quantum information tasks cannot be performed with Gaussian states and Gaussian operations only, as for example stated in the no-go theorems for entanglement distillation [1,2,3], quantum error correction [4] or quantum computing [5]. Non-Gaussian states are therefore an important resource in modern quantum information processing. Filip and Mišta proposed a practical criterion to verify QNG characteristics of specific quantum states of light [6], which was used to analyse single photons from SPDC [7] and quantum dots [8]. The criterion was extended to show the QNG of noisy squeezed single photons [9] and to study its robustness [10]. An alternative approach to witness QNG is demonstrated in [11].
Quantum up-conversion is a versatile tool in today's quantum optics experiments. It has the potential to overcome frequency differences between disparate subsystems within a quantum information network. Today, the generation and fibre-based transmission of nonclassical states of light are most efficient at near infra-red wavelengths, for instance at the telecommunication wavelength of 1550 nm [12]. Future quantum memories based on trapped atoms [13], ions [14] or atomic ensembles [15] will potentially use shorter wavelengths, up to the visible spectrum. Furthermore, the up-conversion into the visible regime enables the efficient detection of infrared single photons with commercially available, low-noise and easy-to-handle silicon avalanche photodetectors [16].
To make quantum up-conversion a useful tool in quantum networks the photon statistics and in particular the nonclassical properties of the quantum states such as QNG have to be preserved. In their pioneering work, Huang and Kumar demonstrated quantum frequency conversion of a bright beam maintaining its nonclassical intensity correlations via second harmonic generation [17]. Recently, the up-conversion of squeezed vacuum states of light was shown [18]. Quantum up-conversion can also be realized with high efficiency at the single photon level via sum-frequency generation with a strongly attenuated coherent signal field [19,20,21]. In addition, nonclassicality of up-converted single photons from quantum dots has been demonstrated by measuring the second-order correlation function g (2) (0) [22]. However, quantum non-Gaussianity has not been certified for up-converted single photons yet. Here, we demonstrate for the first time QNG of frequency up-converted states.

Cavity enhanced SPDC
Spontaneous parametric down-conversion (SPDC) is a widely used technique to produce correlated photon pairs. It generally utilizes a non-linear crystal, which is pumped with a laser field. In our experiment, this crystal is embedded in an optical cavity. If such a device is operated above its oscillation threshold to produce bright fields in well defined spatial output modes, it is commonly known as an optical parametric oscillator (OPO). Far below the oscillation threshold, in a SPDC process photon pairs are generated to a very good approximation. The Heisenberg equations of motion for the cavity modes of interest are [23] dâ whereâ ± (â ±,in ) are the (input) mode operators for the upper (+) and lower (-) sideband, γ is the mean decay rate of the cavity, and the system Hamiltonian is given by The gain parameter ε = γ P/P th , with P th being the threshold pump power for parametric oscillation, is proportional to the mean pump field amplitude √ P. The set of differential equations can be solved with straightforward methods. The temporal correlation function of the two output modes is [23] Γ(τ) = â † +,out (t)â † −,out (t + τ)â −,out (t + τ)â +,out (t) where λ = γ − |ε|, µ = γ + |ε|. In the experiment the modes corresponded to propagating TEM 00 modes at 810 and 1550 nm. The modeâ −,out experiences additional filtering due to a slightly smaller SPDC cavity decay rate compared toâ +,out and transmission through the quantum up-conversion cavity. We model this with the transformation where κ is the bandwidth of the modelled extra filter. The temporal correlations then read where s(τ) = +1 for τ ≥ 0 and s(τ) = −1 for τ < 0. In Fig. 1 the effect of additional filtering of the signal mode is shown. The narrower the linewidth of the extra filter, the more the temporal correlations are smoothed out. no filter κ = 2.0γ κ = 1.0γ κ = 0.5γ

Quantum Up-Conversion
Quantum up-conversion increases the optical frequency of a mode without destroying its quantum coherences. We use sum-frequency generation, where a nonlinear crystal is pumped with a strong coherent pump field. A signal field interacts with the pump and is up-converted to a field with a frequency equal to the sum of the two initial frequencies. In 1990, Kumar showed that the quantum properties of the signal field can be fully transferred to the up-converted field [24]. The Hamiltonian for this process is given by [24] where ζ is the coupling constant, proportional to the mean pump amplitude and the secondorder susceptibility χ (2) . The evolution of the signal (fundamentalâ f and up-convertedâ u ) isâ After the time t c = π/2ζ , the input field is completely converted into the output state:â u (t c ) = a f (0).

Quantum Non-Gaussianity (QNG)
The property of quantum non-Gaussianity was originally proposed by Filip and Mišta [6]. Let G be the set of all Gaussian states and mixtures thereof. Any state ρ which is not in G is defined to be quantum non-Gaussian. The quantum non-Gaussian states can be conveniently identified using a simple and experimentally feasible criterion based on photon number probabilities. For a given (measured) vacuum probability p 0 a maximum single-photon probability p 1 achievable by a Gaussian state can be calculated. If the measured p 1 is higher, then ρ / ∈ G . A detailed description of how to apply this criterion can be found in [7]. It is also possible to define a witness of quantum non-Gaussianity, W = p 1 + ap 0 − W G (a), where a < 1 is a parameter specifying the witness and W G (a) represents the maximum of p 1 + ap 0 achievable by Gaussian states. Assuming Poissonian statistics of the measured coincidences, the statistical error of the witness can be determined and one can express the witness in numbers of standard deviations ∆W . A positive witness certifies QNG of the state, even though in general it can be a mixed state with a positive Wigner function. The experimental setup is depicted in Fig. 2. The main laser source was a Nd:YAG laser with 2 W output power, that was frequency doubled to 532 nm in a LiNbO 3 -based second harmonic generator (not shown in the figure). A major fraction of the 532 nm field pumped a monolithic non-degenerate, doubly resonant optical parametric oscillator (OPO) above threshold, providing up to 200 mW at 810 nm to pump the quantum up-converter (QUC). A smaller fraction of the 532 nm field was used as the pump field for another nonlinear cavity (SPDC), which was operated below threshold and generated twin photons at 810 nm and 1550 nm in a cavity enhanced SPDC process. The reflectivities of the mirrors of these two cavities were AR@532 nm, HR@810&1550 nm on the incoupling mirror and HR@532 nm, 96.5 %@810&1550 nm on the outcoupling mirror. The linewidths of the generated modes of the SPDC cavity at 810 nm and 1550 nm were 35 MHz and 19 MHz (FWHM), respectively, determined by scanning the cavity and comparing the Airy pattern with frequency markers introduced by a phase modulation [25].

Experimental Setup
The QUC cavity was modified from to what was reported in [26,18]. The new incoupling mirror had reflectivities of HR@532 nm, 97 %@810 nm and 91 %@1550 nm. The linewidth for the signal at 1550 nm was measured to be 68 MHz. With these modifications, the up-conversion efficiency could be increased to 90.2 ± 1.5 % for a pump power of 140 mW, determined with the methods described in [26]. Periodically poled potassium titanyl phosphate (PPKTP) crystals were embedded in all three cavities, actively stabilized to the phase matching temperatures.
The 810 nm heralding photons passed a Fabry Perot filter cavity to suppress uncorrelated modes. This quasi-monolithic cavity consisted of two mirrors separated by 2.5 mm, each with a reflectivity of 99 %@810 nm yielding a free spectral range of 60 GHz and a full width at half maximum of 190 MHz (Finesse ≈ 315). The transmitted photons were detected with a Si-APD (Perkin Elmer SPCM-AQRH-13).
The 532 nm photons were split on a balanced beamsplitter and detected with Si-APDs (Laser Components COUNT ® -250B). All APD signals were recorded with an oscilloscope (Agilent DSO7014A). In each measurement we took 2000 data streams with a duration of 2 ms each.

Results
The signal count rates were as high as 50 kHz, 120 kHz and 400 kHz, while increasing the gain parameter (i.e. the pump power) from ε = 0.10γ to ε = 0.16γ and ε = 0.28γ, respectively. The propagation efficiency (including 90.2 % up-conversion efficiency) was determined to be 68 % by operating SPDC above threshold and comparing the 1550 nm power with the up-converted field at 532 nm directly in front of the APDs. The APD quantum efficiency was 60-70 % (according to the manufacturer's datasheet). False triggering, due to insufficient filtering of the 810 nm mode, reduced the final detection efficiency to about 20 %. The counting statistics of APD-A and -B triggered on APD-T are shown for ε = 0.10γ in Fig. 3. The temporal profile of the up-converted photons clearly shows the smoothed out exponential decay predicted by Eq. (5). The theoretical curve was obtained by using a mean decay rate of γ = 31 MHz and an extra filtering effect of the signal mode with κ = 1.4γ. In this graph, the three-fold coincidence events do not show any significant contribution to the statistics.
When the pump power for SPDC is higher, the probability of generating more than one photon pair within the same coherence time increases. In Fig. 4 the counting statistics are shown for a measurement with ε = 0.28γ, i.e. 2.8 times the pump amplitude (≈ eight time the pump power) compared to the measurement shown in Fig. 3. The three-fold coincidences indicated by the grey circles now significantly contribute to the counting statistics.
The second order coherence function value, is calculated by g (2) where the probabilities p 0 and p 1 were estimated from the measured coincidences following the procedure described in [7]. Three-fold coincidences are estimated by 1 − p 0 − p 1 . The measured g (2) (0) values are plotted against the coincidence window ∆t in Fig. 5, for three different gain parameters ε. The coincidence window is the time interval, symmetric about the detection of a photon in APD-T, within which detections of photons in APD-A and APD-B are considered a coincidence detection. The g (2) (0) reaches values smaller than 0.05 for coincidence windows less than 34 ns, when ε = 0.10γ. For increasing coincidence windows the g (2) (0) value increases as more three-fold coincidence events are registered. If the gain parameter ε (i.e. the pump amplitude) is increased, the multiphoton probability (1 − p 0 − p 1 ) increases, as does g (2) (0). For very small coincidence windows (<5 ns) the absolute number of coincidences is small, causing the statistical fluctuations in Fig. 5. The sub-Poissonian statistics indicated by a g (2) (0) smaller than unity are a strong signature for the nonclassicality of the states. An even stronger feature of nonclassicality is a negative Wigner function, the hallmark of single-photon states [27]. However, due to mixture with vacuum, the negativity of such states decreases, or even vanishes, when losses exceed 50 %. Nevertheless, it is still possible to certify the quantum non-Gaussianity of the state, a useful feature for quantum information tasks. The measured probabilities p 0 and p 1 are therefore used to analyze whether the up-converted states ρ are quantum non-Gaussian or not. The result is shown in Fig. 6. In the plot the witness of QNG, W , in numbers of standard deviations, ∆W , is shown for measurements with three different gain parameters ε, dependent on the coincidence window size ∆t. For each ε and ∆t, the parameter a of the QNG witness W was optimized such as to maximize W /∆W . If the witness is positive, then ρ / ∈ G . For low gain ε = 0.10γ, the witness of QNG was positive for all coincidence windows smaller than 300 ns, reaching a maximum of 16 standard deviations at ∆t=34 ns. For higher gain, the multiphoton contribution increases, which lowers the witness. For ε = 0.16γ and ∆t ≥120 ns the witness is negative, indicating a state that possibly could be expressed as a convex mixture of Gaussian states. For even higher gain (ε = 0.28γ) the witness of QNG is never positive, since as the mode approaches thermal statistics [28], higher photon number contributions show up and background noise increases.

Discussion and Conclusion
We present the frequency up-conversion of heralded single photons from 1550 to 532 nm. For the first time, frequency up-converted states were shown to possess quantum non-Gaussianity and we achieved a significance of up to 16 standard deviations. The single photon contribution in the states with QNG certified with more than five standard deviations was up to 22.6 % (with ε = 0.16γ, ∆t = 80 ns). The g (2) (0) values were smaller than 0.05 (ε = 0.10γ, ∆t ≤ 34 ns). Our result shows that negative Wigner functions are also within reach in up-converted states when the optical losses were minimized. In particular, insufficient filtering of the trigger mode, limited quantum efficiencies of the APDs and propagation losses prevented us from fully benefiting from the high up-conversion efficiency, which was determined to be 90.2 ± 1.5 %. One or two additional filter cavities with different free spectral ranges would suppress uncorrelated modes more efficiently, but a more complex setup with technically elaborate cavity locking techniques would be required. Anti-reflection coated fibres would minimize the propagation losses, since the coupling efficiency could be increased and an optical isolator would no longer be required. Nonetheless, with such improvements we expect that our setup is capable of reaching a single photon contribution of 60 %, mainly set by the high up-conversion efficiency (≈ 90 %) and quantum efficiency of the APDs (70 %).