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Abstract
Correlated beams are important in classical and quantum communication as well as other technologies. However, classical amplifiers, which are essential for long transmission of correlated beams, degrade the correlation due to noise and due to the amplifier spectral response. We measure, with a novel high resolution single-shot measurement system, the impact of amplifiers on correlated beams. We develop a new method for analyzing the correlation between the signal and idler beams by choosing peaks in the pulses according to their power levels. We demonstrate how to tailor the correlation after the amplifier to obtain either higher or lower correlation. Our research may influence the future use of amplifiers in non-classical communication systems as well as the transmission of quantum information over long distances.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantum communication for prolonged and safe data transmission can revolutionize the telecommunication industry [1]. In some realizations, it is based on transmitting photon pairs and correlated photons over long fibers. Quantum communication was shown to increase the rate of information transfer [1] and improve the security [2]. Photon pairs are essential for quantum information [3,4], quantum key distribution (QKD) [1], and long range quantum telecommunication [5,6], especially for high bit rate QKD [7]. Other applications of correlated photons are improving measurement sensitivity [8] and reducing the noise in quantum networks [9]. Also, pure-state heralded photons, are possible building blocks of quantum computers and are usually obtained by filtering the correlated beams [10,11].
Still, long distance transmission of correlated photons is a challenge due to photon losses that lead to the fundamental rate-distance limitation, namely, the linear bound [12]. Classical amplifiers cannot beat the linear bound since they introduce noise and degrade the coherence [13,14]. There are different approaches to overcome this challenge, such as increasing the detection efficiency, by improving the detectors [15] or by matching the temporal modes [16]. Other approaches employ parametric amplifiers which preserve the quantum state of light [9,17], introduce quantum repeaters [1,2,18–20], or implement twin field QKD protocols [21–23]. Also, it is possible to skip the fibers altogether with free-space optical communication [24]. However, classical optical amplifiers are still more accessible, robust, simple to manufacture, and employ well established technology. We analyze the impact of classical amplifiers on correlated beams with a novel high spectral and temporal resolution, single-shot measurement system. The amplifier degrades the correlation between the beams because the amplification is a stochastic process [12], it adds noise [13,14], and it has a limited spectral response. We show that it is possible to improve the correlation by compensating for the amplifier spectral response by tailoring the power levels of the peaks of the photon pairs.
First, we investigate the influence of classical amplifiers on correlated beams in the spectral and temporal domains with high resolution and for each photon pair. Next, we developed a technique to tailor the spectral correlation of the photon pairs after the amplifiers. The classical optical amplifier serves as a filter on the correlated beams and therefore changes the correlation. With our technique we can obtain either stronger correlation for multi-photon protocols such as quantum entanglement [1,4] or lower correlation for generating pure-state single photons [10,11]. Our technique is based on sorting the peaks of the correlated beams according to their power, and retaining only specific peaks. We note that in our experiment we split the output wave with a beam-splitter which mixes it with a vacuum state and thereby introduces noise. However, in this work, we focus on the more significant noise generated by the amplifier [25–28] and demonstrate how to reduce it with a novel analysis scheme. We can also employ homodyne detection or other phase-sensitive detection for increasing the correlation of amplitude or phase quadratures.
2. Theoretical background
There are different methods for generating photon pairs, we focus on spontaneous four-wave mixing (SFWM) to generate correlated beams via highly nonlinear fiber (HNLF) [29]. Then, we amplify the correlated beams and study the effects of the optical amplifier on the correlation between the signal and the idler, both in time and frequency domains. For analysing the correlation of the beams, we evaluate the joint spectral function (JSF), which is a two-dimensional probability distribution of the generated frequencies [30]. The JSF allows us to study the states of the photon pairs [31,32], where a diagonal shape indicates high correlation between the signal and the idler, and Gaussian shape of the JSF indicates separable photon pairs with low correlation [10,11,33]. By Schmidt decomposition of the JSF, via the singular value decomposition (SVD), we can also find the spectral modes that compose the signal and the idler waves [30,34].
Our pump has a Gaussian frequency distribution:
where $\sigma$ is the bandwidth. The spectral correlation of the signal and idler results from energy conservation, $2\omega _p=\omega _i+\omega _s$ and defining $\nu _i \equiv \omega _p-\omega _i$, $\nu _s \equiv \omega _s-\omega _p$, leading to:By introducing narrow-band filters, we change the JSF into a function that can be separable with a lower $k$ parameter. Assuming that the narrow-band filter is:
where $\sigma _f$ is the bandwidth of the filter that is smaller compared to $\sigma$. Then, Eq. (3) becomes:
Fig. 1. Experimental setup for measuring the effect of amplifiers on correlated beams. We generate signal and idler waves via SFWM interaction in HNLF. We filter out the pump from the signal and the idler with the band-stop filter (BSF), split the beam with a broadband beam-splitter, and amplify them with Erbium-doped fiber amplifier (EDFA). Finally, we measure them in time with a time-lens system (TL), after filtering the signal with a band-pass filter (BPF), and in frequency with a time-stretch system (TS). The time-stretch system has a long dispersion compensating fiber (DCF) with dispersive parameter of 140 ps/nm. (a) The measured spectrum of the HNLF output, where the spectra of the pump, signal, and idler waves are around 1558 nm, 1578 nm, and 1538 nm, respectively. (b) The frequency response of the amplifier. (c) Representative measured spectrum of the signal wave with the time-stretch system. (d) Representative measured spectrum of the idler wave with the time-stretch system. (e) Representative measured temporal structure of the signal with the time-lens system. Insets (c), (d), and (e) show one event out of 80000 events that we measured.
3. Experimental system and results
A block diagram of our experimental setup is shown in Fig. 1. First, we generate the correlated beams via SFWM with ultra-short pump in HNLF. The laser pulse width is 90 fs, with an average intensity of 120 mW, and a repetition rate of 100 MHz. From this pulse, we filter a bandwidth of 3 nm for obtaining a 1 ps pulse as the pump. Next, we send it to the HNLF where two photons of pump turn into a pair of signal and idler photons, where their frequencies follow $2\omega _p=\omega _i+\omega _s$. The output spectrum from the HNLF is shown in Fig. 1(a), where the spectra of the pump, signal, and idler waves are around 1558 nm, 1578 nm, and 1538 nm, respectively. Next, we amplify the signal and idler waves and measure the amplification influence on their correlation. In order to measure the correlation between them, we measured the spectrum of each pair of photons by a time-stretch system (TS) [37] and the temporal structure of the signal with a time-lens system (TL) [38–41]. The time-stretch system maps the spectrum to time, with a spectral resolution of 0.17 nm. Representative measurements of the time-stretch output for the signal and the idler are shown in Fig. 1(c) and (d), respectively. The time-lens system magnifies the temporal structure of a signal by a nonlinear interaction with a chirped pump pulse [38–41]. The temporal resolution of our time-lens is 0.2 ps. Representative measurement of the signal in time is shown in Fig. 1(e).
We measure the spectra of the signal and the idler waves with the time-stretch system for over 80000 events, and show them in Fig. 2(a). Each event of the 80000 events is generated by a single round-trip of the laser. The idler is usually three times stronger than the signal since its wavelength matches the peak of the frequency response of the amplifier.
Fig. 2. (a) Measured spectra of 80000 events, showing both the idler (1538 nm) and the signal (1578 nm). (b) Peak-power cross-correlation. (c) Frequency cross-correlation. Both cross-correlations in (b) and (c) are indicating that the signal and the idler are generated by the same pump pulse.
In order to study the correlation between the photon pairs, we measure both the power correlation and the frequency correlation. For measuring the power correlation, we identify the peak power of the signal and the idler in each event. We denote the power of the signal peak for the $n$’th pulse as $P_n^{(s)}$, and the power of the idler peak as $P_n^{(i)}$. We calculate the correlation between the peak powers of the signal and the idler while shifting their round-trip one compared to the other. This cross-correlation, $f_P(\Delta n)=\Sigma _n P_{n+\Delta n}^{(s)} P_n^{(i)}$, is shown in Fig. 2(b), where $\Delta n$ is the round-trip shift between the pulse of the signal and the pulse of the idler. We can see that the correlation is high when the signal and the idler are taken from the same round-trip, namely $\Delta n=0$, indicating that they were generated by the same pump pulse. When shifting even a single round-trip left or right, the correlation disappears.
Due to energy conservation, the frequency gap between the pump and the signal or the idler waves remains constant, when the frequency of the signal increases, the frequency of the idler decreases, leading to a negative frequency correlation. We denote the frequency of the signal peak for the $n$’th pulse as $\nu _n^{(s)}$, and the frequency of the idler peak as $\nu _n^{(i)}$. Then, the cross-correlation is $f_\nu (\Delta n)=\Sigma _n \nu _{n+\Delta n}^{(s)} \nu _n^{(i)}$, shown in Fig. 2(c), where $\Delta n$ is the round-trip shift between the pulse of the signal and the pulse of the idler. We can see the strong negative frequency correlation between the signal and the idler, which occurs when the signal and the idler are generated by the same pump pulse. We note that although we got a negative frequency correlation, the value of the correlation is still 100 times lower than expected due to the noise of the amplifier.
Next, we measure the JSF of the signal and the idler waves, shown in Fig. 3(a). The JSF does not have a diagonal shape due to the low signal-to-noise ratio. The wavelength of the peak of the idler remains around 1535 nm regardless of the signal wavelength. This is a result of the amplifier frequency response, which has a peak around the same wavelength, and leads to the horizontal line in the JSF at 1535 nm. The signal amplification is much lower, and therefore the signal was less affected by the amplifier.
Fig. 3. (a) Evaluated JSF from the entire measured signal and idler waves for all 80000 events. (b) Evaluated JSF of the same events when eliminating the strongest peak of the idler. (c) Evaluated JSF of the same events when eliminating the two strongest peaks of the idler.
In order to compensate for this effect, we eliminate the strongest peak from the idler through post-processing calculations of the measured data, and calculate the JSF without it. That JSF, presented in Fig. 3(b), shows higher correlation, a slight increase from $k=3.001$ to $k=3.007$, and a $5 \%$ increase in the second eigenvalue compered to the first one. Then, we eliminate the second strongest peak of the idler as well and calculate the JSF, shown in Fig. 3(c). This leads to a further improvement of the correlation, with a higher $k=3.0429$, and a further increase of another $5 \%$ in the second eigenvalue compered to the first one.
We study all the different peaks in each of the 80000 events. We separate each event into its peaks, and then sort the peaks in the signal and the idler according to their intensity. We keep the 10 strongest peaks in the signal and idler. Then, we evaluate all the 100 possible JSFs, where for evaluating each JSF, we choose one peak from the signal and one peak from the idler out of the 10 strongest peaks. These 100 JSFs are shown in Fig. 4, organized in a 10X10 matrix, according to the chosen peaks of the signal and the idler waves. Peak number 1 in the signal or the idler is the strongest peak, and the JSF at the top left corner is the correlation between the strongest peak of the idler with the strongest peak of the signal. In the first row, that represents the correlation with the strongest peak of the idler, the wavelength of the idler remains at the same wavelength regardless of the signal wavelength, which means that there is no correlation between the signal and the strongest peak of the idler. In the second, third, and fourth rows, the diagonal shape is getting clearer, indicating a higher correlation. As we go down the rows, the peaks are getting weaker and closer to the noise. The Gaussian shape in those rows, indicates stochastic behavior of the noise. As we go to the right, the noise increases and the JSF stretches. As evident, the JSFs without the presence of the first peak of the idler, have higher correlation between the signal and the idler. Therefore, by choosing the peaks of the signal and the idler according to their power, we can tailor the desired correlation relation.
Fig. 4. Evaluated JSFs when choosing a single peak from the signal and a single peak from the idler out of the 10 strongest peaks. For example, the JSF at the top left corner is the JSF of the strongest peak of the idler with the strongest peak of the signal. This is different than the results in Fig. 3, where we only eliminate a single peak while here we eliminate all the peaks apart from one.
Finally, we verify that there are no temporal shifts in our spectral measurements, apart from the spectral dispersion. Since we measure the spectrum with a time-stretch system, which map the frequency to the temporal domain, these temporal shifts may lead to jitter in the spectral measurements. Therefore, we study the correlation between the temporal and the spectral domains, by evaluating the Wigner functions of the signal peaks [31–33,36,42]. Each Wigner function is a two-dimensional quasi-probability distribution of the time-frequency of each of the peaks from the signal beam. Thus, for evaluating each Wigner function, we choose a single peak from the temporal measurement of the signal wave and a single peak from the spectral measurement of the signal wave out of the 10 strongest peaks. In Fig. 5, we present the 10X10 Wigner functions of the peaks of the signal, sorted according to their peak intensity. The Wigner functions spread horizontally according to the chosen peak in the spectral domain and spread vertically according to the chosen peak in the temporal domain. For example, the Wigner function at the top left corner is the one of the strongest peak in time, and the strongest peak in frequency. The Wigner functions to its right correspond to weaker peaks in the spectral domain and the Wigner functions bellow correspond to weaker peaks in the temporal domain.
Fig. 5. Evaluated Wigner functions when choosing a single peak from the signal in time and a single peak from the signal in frequency out of the 10 strongest peaks. For example, the Wigner function at the top left corner is the one between the strongest temporal peak and the strongest spectral peak of the signal. This serves as a correlation test between the spectral peaks and the temporal peaks of the signal, and the results indicate that there is no correlation.
The results indicate that there is no correlation between frequency and time. The timing of the signal remains constant regardless of its wavelength, which leads to the horizontal line in the first row. As we go down the rows the peaks are getting weaker, the lines split and shift away from the timing of the first peak with an equal gap, because they are generated in an equal time-gap left or right to the stronger one. Weaker peaks appear further away from the main peak leading to the splitting into two parts. As we go to the right, the noise increases and the Wigner function stretches, since weaker peaks can be generated in wider bandwidth than stronger peaks. By these measurements, we justify our assumption that the temporal and spectral structures of the signal and the idler are generated simultaneously.
4. Conclusions
To conclude, we developed a novel single-shot measurement system with high resolution and measured the influence of amplification on correlated beams. Next, we developed a method to tailor the correlation relations of amplified signal and idler waves, generated by SFWM. The method is based on sorting the peaks of the signal and idler according to their power and choosing specific power levels. We studied all the possible JSFs for the first 10 strongest peaks both in the signal and the idler, and demonstrated either low or high correlation according to the peak power levels. Finally, with the help of the temporal measurement, we obtained the Wigner function of the 10 strongest peaks of the signal in time and frequency, indicating that the entire signal wave is generated at the same time. This is the first step in a research program aimed at analyzing spectral and temporal properties of non-classical light for communication and sensing applications with high resolution.
Funding
PAZY Foundation; Ministry of Science, Technology and Space (101821); Israel Science Foundation (205735, 2096/20).
Acknowledgments
This research was supported by the Israeli Science Foundation, grant number 205735, Israeli Ministry of Science and Technology, grant number 101821, Israel Innovation Authority (grant numbers 70002, 73795 and the XStable Eureka project), Pazy Foundation and Israeli Council for Higher Education.
Disclosures
The authors declare no conflicts of interest.
Data availability
All measured and calculated data will be provided upon request.
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