1. Introduction

Quantum memories, serving as nodes in quantum communication networks, play a pivotal role in mitigating the exponential decay of light in optical fibers [1]. Rare-earth-ion-doped crystals are commonly used as quantum storage materials due to their longer coherence time and wider inhomogeneous linewidth as compared to alkali metal vapours, making them more suitable for integration. Various rare-earth-ion-doped waveguide structures have been implemented for storing optical pulses [2–4]. Among them, Pr$^{3+}$ ion is a kind of rare earth ion extensively studied in quantum memories. Compared with Er$^{3+}$ ion, which has an odd number of electrons, Pr$^{3+}$ ion is insensitive to the electron spin magnetic moment, therefore, a long coherence time can be obtained even without the need for an extremely low temperature and a strong external magnetic field. Pr$^{3+}$ ion is also one of the few rare earth ions that have achieved the zero first-order Zeeman effect. When an external magnetic field in a specific direction is applied, Pr$^{3+}$ ions can exhibit magnetic insensitivity to external environmental magnetic fields, thereby showing an ultra-long coherence time [5]. In Pr:${\rm Y_2SiO_5}$ (Pr:YSO) crystal, the light pulse storage time as long as 1 minute was reported [6]. In addition, the wavelength of the stored photon (606 nm) is located within the visible spectral range, and research on quantum sources in visible spectral range is relatively mature [7,8]. Also, the dark count rate of a conventional single-photon detector is relatively low in the visible spectral range, and commercial EMCCD and iCCD can be used with high quantum efficiency, which is suitable for quantum image processing.

Currently, classical light pulses have been stored in Pr$^{3+}$ ions via electromagnetically induced transparency (EIT) and photon-echo-based protocol. The photon-echo-based protocol offers a diverse set of techniques including photon echo, controlled reversible inhomogeneous broadening (CRIB), gradient echo memory (GEM), and atomic frequency comb (AFC). However, limited by the storage mechanism, the efficiency based on the EIT effect in Pr:YSO crystals is usually lower than 1% [6,9,10]. With a complex optical cavity and waveform optimization technology, a storage efficiency of 76.3% has been reported in Pr:YSO crystals [10]. For the case of photon echo, the population inversion caused by $\pi$-pulse leads to spontaneous emission noise in the echo signal. The increased quantum noise renders traditional photon echo not suitable for single-photon storage. Various methods have been proposed to address this issue. For example, complex spectral structures can be prepared to avoid population inversion, including CRIB, GEM, AFC, and so on. Based on the orientation of the applied electric field or magnetic field and the transmission direction of the signal light, CRIB can be categorized into transverse CRIB and GEM. Hedges et al. [11] achieved a storage efficiency of 69% in Pr:YSO crystals using the GEM protocol. The AFC protocol has multi-mode storage capacity with relatively high storage efficiency. Amari et al. [12] achieved a storage efficiency of 35% in Pr:YSO crystals through the AFC protocol. Horvath et al. [13] achieved a storage efficiency of 36% by introducing the Stark effect into the AFC protocol. In addition, Sabooni et al. [14] raised the storage efficiency up to 56% using an optical standing wave cavity. Riedmatten et al. [15,16] realized single-photon storage in Pr:YSO crystals based on the AFC protocol.

One notes that all these methods require meticulous preparation processes of specific spectral structures. The revival of silenced echo (ROSE) scheme was proposed to address this challenge [17]. It utilizes two counter-propagating $\pi$-pulses with respect to the signal light to bring the population inversion back to the ground state. This approach effectively reduces the quantum noise in traditional optical echoes without the need for specific spectral tailoring. Also, the first echo, which is usually excited by the first $\pi$-pulse, does not emit actually due to spatial phase mismatch. Bonarotal et al. [18] achieved a background level of one photon in the echo mode, evenly shared between spontaneous emission and coherent noise. Based on the ROSE scheme, Ma et al. [19] proposed the noiseless photon echo (NLPE) protocol, which incorporates two additional energy levels to suppress the noise by frequency filtering, and achieved a fidelity of $0.952 \pm 0.018$ with a storage efficiency of $10.0 \pm 0.4$%.

As compared to all those protocols, the ROSE protocol is relatively simpler, easier to implement, and has higher storage efficiency while without the need for complex optimization. The ROSE protocol has been experimentally verified in Eu$^{3+}$ [20], Tm$^{3+}$ [18,21], and Er$^{3+}$ [21,22] doped crystals, but not for Pr$^{3+}$ ion. In this article, we realize the ROSE protocol in Pr:YSO crystals. Furthermore, we propose a method to optimize the crystal absorption by tuning the laser frequency in the inhomogeneous broadening, and therefore achieving light pulse storage with an echo retrieval efficiency of $24.4{\%}$ in Pr:YSO crystals. This method provides a simple way to dynamically optimize the light pulse storage efficiency and also suits for other atomic storage systems.

2. Experimental setup and principles

The experimental setup is illustrated schematically in Fig. 1(a). The laser employed in this study was a Matisse 2 DX dye laser from Spectra-Physics Inc. The wavelength of the laser beam was precisely tuned to 605.78 nm, corresponding to the atomic transition $^1$D$_2 \leftrightarrow ^3$H$_4$ of Pr$^{3+}$ ions in YSO crystals, and its linewidth was controlled to be less than 100 kHz through the Pound-Drever-Hall techniques. The laser beam was split into two beams, the signal beam and the control beam, via a beam splitter. Both beams were subject to intensity and frequency modulation through acoustic-optical modulators (AOM, MT110-B50A1-VIS, AA opto-electronic) respectively with a double-pass arrangement. The AOMs were digitally controlled by a FPGA card (PCI-7813R, NI). The signal beam was modulated into a Gaussian pulse with its central frequency corresponding to the $\pm \frac {3}{2}\left (e\right )\leftrightarrow \pm \frac {3}{2}\left (g\right )$ energy level transition frequency of the $^1$D$_2 \leftrightarrow ^3$H$_4$ transition of the Pr$^{3+}$ ion (shown in Fig. 1(b)) in Pr:YSO crystal. Subsequently, it was launched into a single-mode optical fiber via an optical coupler (PAF2-7A, Thorlabs Inc) and guided out through another optical coupler (PAF2P-18A, Thorlabs Inc.) for spatial mode optimization. The Gaussian signal pulse was then focused through a plano-convex lens with a focal length of 10 cm and launched into the Pr:YSO crystal. The optical power and the beam waist of the signal pulse were 78 $\mu$W and 56 $\mu$m, respectively. The control beam was modulated into a complex hyperbolic secant (CHS) light pulse by AOM through the LabVIEW program with an optical power of 9.1 mW. Similar to the signal pulse, the control CHS pulse was also coupled into a single-mode fiber and guided out for spatial mode optimization, and then was reflected by a polarization beam splitter and finally launched into the Pr:YSO crystal, enabling counter-propagation with respect to the signal pulse. The Rabi frequency of the control CHS pulses varied with time and was expressed as $\Omega (t)=\Omega _0~\textrm {sech}\left (\beta \left (t-t_{j}\right )\right )$, where $t_j=t_2$ or $t_3$ was the time sequence of the CHS pulses launched into the Pr:YSO crystal, $1/\beta$ and $\Omega _0$ were the pulse duration and the peak Rabi frequency of the CHS pulses, respectively. The frequency was swept around a central frequency $\omega _0$, also corresponding to the $\pm \frac {3}{2}\left (e\right )\leftrightarrow \pm \frac {3}{2}\left (g\right )$ energy level transition frequency of the $^1$D$_2 \leftrightarrow ^3$H$_4$ transition of the Pr$^{3+}$ ion in YSO crystal, and was expressed as $\omega (t)=\omega _0+\mu \beta ~\textrm {tanh}(\beta (t-t_{j}))$ with $t_j=t_2$ or $t_3$, respectively, where $\mu$ is a parameter related to the pulse width, $2\mu \beta$ is the sweep bandwidth of the CHS pulse. In the experiments, $\mu =1$ and $\beta =0.87$ MHz were employed, therefore, the pulse duration $\beta ^{-1}$ and the sweep bandwidth $2\mu \beta$ of the CHS pulses were 1.15 $\mu$s and 1.74 MHz, respectively. Note that we set the parameters of the two CHS pulses the same in the experiments. The CHS $\pi$-pulses are known to exhibit higher population inversion efficiency than other $\pi$ pulses such as square-wave $\pi$-pulses and are widely employed in photon echo experiments [23]. The mode overlap between the signal Gaussian pulse and the control CHS pulse within the crystal could be assessed by checking the reverse-coupling efficiency where the signal pulse was coupled into the single-mode optical fiber of the control CHS pulse. In our experiments, the reverse-coupling efficiency was measured to be 76%. To enhance the signal-to-noise ratio and minimize the back-reflection to the detector, the signal pulse was horizontally polarized and the CHS pulse was vertically polarized by using two half-wave plates (HWPs), as shown in Fig. 1(a).

 

Fig. 1. (a) Schematic diagram of the experimental setup. QWP: quarter-wave plate. HWP: half-wave plate. Lens: plano-convex lens with a 10-cm focal length. PBS: polarization beam splitter. BS: beam splitter. Coupler: optical fiber coupler. AOM: acousto-optic modulator. PD: photo-detector. The red arrows indicate the propagation directions of the signal pulse and the control CHS pulses, respectively. (b) Energy-level structure of Pr$^{3+}$ ions in Pr:YSO crystal.

Download Full Size | PDF

The Pr:YSO crystal (Physcience Opto-electronics Corp.), with a size of 4 mm $\times$ 5 mm $\times$ 10 mm and a Pr-doping concentration of 0.05%, was placed in a cryostat (Montana Inc.) maintained at a temperature of 3.4 K. The Pr:YSO crystal was of three mutually perpendicular optical axes referred as $\textbf {D}_1$, $\textbf {D}_2$ and b axis, respectively. The $\textbf {D}_1$ axis was along the 4-mm side, the $\textbf {D}_2$ axis was along the 5-mm side, and the b axis was along the 10-mm side, respectively. Both the signal pulse and the control CHS pulses propagated along the b axis. Due to the absence of an anti-reflection coating on the crystal surface, a slight inclination was introduced to eliminate photo-detectors from the stray/reflection lights. In the experiments, we utilized the fine energy levels of the $^1$D$_2 \leftrightarrow ^3$H$_4$ transition of Pr$^{3+}$ ions, which is associated with the $\pm \frac {3}{2}(g)$ ground state and the $\pm \frac {3}{2}(e)$ excited state, as shown in Fig. 1(b), because this energy level transition exhibited the highest absorption. To prepare such a two-level system for storage based on Pr$^{3+}$ ions in YSO crystal, they were initially populated in the $\pm \frac {3}{2}(g)$ ground state through spectral hole burning and anti-hole burning techniques. For details about the spectral hole burning and anti-hole burning techniques, please refer to our previous work in Ref. [25]. It is known that the inhomogeneous broadening of the $^1$D$_2 \leftrightarrow ^3$H$_4$ transition of Pr$^{3+}$ ions in YSO crystal is of the order of 10 GHz in full width at half maximum. The bandwidth of the spectral hole burned on the spectrum of the $^1$D$_2 \leftrightarrow ^3$H$_4$ transition of Pr$^{3+}$ ions in YSO crystal is of tens of MHz at a cryogenic temperature of 3.4 K. While the absorption spectral bandwidth of the transition $\pm \frac {3}{2}\left (e\right )\leftrightarrow \pm \frac {3}{2}\left (g\right )$ is only of hundreds of kHz, much narrower than the bandwidth of the burned spectral hole and the spectral separation between the fine energy levels of $^1$D$_2$ or $^3$H$_4$, as also shown in Fig. 1(b). Therefore, the transition $\pm \frac {3}{2}\left (e\right )\leftrightarrow \pm \frac {3}{2}\left (g\right )$ is spectrally well separated from other transitions such as $\pm \frac {1}{2}\left (e\right )\leftrightarrow \pm \frac {3}{2}\left (g\right )$ and $\pm \frac {5}{2}\left (e\right )\leftrightarrow \pm \frac {3}{2}\left (g\right )$ within the burned spectral hole [25], which justifies the appropriateness of the employed two-level system for storage in Pr:YSO crystal. In contrast to Er$^{3+}$ ions, the fine-structure interval of the $^1$D$_2 \leftrightarrow ^3$H$_4$ transition of Pr$^{3+}$ ions was of a few MHz, much smaller than the inhomogeneous broadening of Pr$^{3+}$ ions. As a result, the energy level preparation plays an important role in the storage. Without prior energy level preparation and when multiple storage processes are executed in rapid succession, the ground-state ions will be gradually depleted, leading to a gradual reduction in the absorption of the signal pulse, and consequently a reduction of the storage efficiency over time. Energy level preparation conducted in advance ensures population stabilization, thereby preserving the stability of storage efficiency.

In the light pulse storage and echo retrieval experiments, a signal $\pi /2$-pulse with a wave vector $\textbf {k}_1$ was launched into the crystal at time $t_1$. Then at time $t_2$, the first control CHS $\pi$-pulse with a wave vector $\textbf {k}_2$ was launched into the crystal in the opposite direction with respect to the signal pulse. However, due to the spatial phase mismatch, no echo was generated even at the time $t_1+2t_{12}$ (where $t_{12}=t_2-t_1$) though the coherence rephases. Nevertheless, due to the first control CHS pulse, the population on the ground state $\pm 3/2(g)$ was transferred to the excited state $\pm 3/2(e)$. At time $t_3$, a second CHS $\pi$-pulse with a wave vector $\textbf {k}_3$ was injected into the crystal in the same direction of the first CHS pulse, and the coherence rephases again at time $t_e=t_1+2t_{23}$ (where $t_{23}=t_3-t_2$). At this point, the spatial phase matching condition was $\textbf {k}_e=2\textbf {k}_3-2\textbf {k}_2+\textbf {k}_1$. Note that the two control CHS pulses were transmitted in the same direction and with the same wave vector, i.e., $\textbf {k}_2=\textbf {k}_3$. Therefore, at time $t_e$, an echo will be emitted from the ground state $\pm 3/2(g)$ with a wave vector $\textbf {k}_e=\textbf {k}_1$ which satisfied the phase matching condition. Moreover, such echo generated from the ground state is immune from the influence of spontaneous emission noise. In addition, the condition $\textbf {k}_e=\textbf {k}_1$ shows that the echo pulse and the signal pulse were propagated in the same direction, therefore, reducing the background noise originated from the control CHS pulse. The light pulse storage time for the entire ROSE process, denoted as $t_{\textrm{stor}}=t_e-t_1$, equals twice the temporal interval $t_{23}=t_3-t_2$. For more comprehensive information, please refer to the content in Appendix A.

3. Experimental results

Figure 2 depict the typical temporal sequence of the signal pulse, the control CHS pulses and the echo pulse respectively in the light pulse storage and echo retrieval process based on the ROSE protocol. Here, the laser frequency was tuned to be of 494736 GHz, i.e., at 605.78 nm. The pulse denoted as S in Fig. 2(a) represented a Gaussian signal pulse, which was launched at time $t_1=2.75~\mu \textrm {s}$. The first control CHS pulse, denoted as CHS$_1$ in Fig. 2(b) was launched at time $t_2=5.07~\mu \textrm {s}$. The second control CHS pulse CHS$_2$ was injected at time $t_3=10.44~\mu \textrm {s}$. Both control CHS pulses CHS$_1$ and CHS$_2$ were of identical parameters with $\mu =1$ and $\beta =0.87~\textrm {MHz}$. The pulse duration $1/\beta$ and the bandwidth $2\mu \beta$ of the control CHS pulses were 1.15 $\mu \textrm {s}$ and 1.74 MHz, respectively. Correspondingly, the echo R was generated at $t_e=13.48~\mu \textrm {s}$. One has to note that, during the AOM modulation of the signal pulse S, the steady-state condition was not satisfied on the rising and falling edges. As a result, there may be a slight discrepancy between the actual frequency and the set frequency of the signal pulse, leading to a slightly low absorption of the signal pulse during these transient phases. This can also be seen from Fig. 2(a) that the signal pulse S exhibits an intensity profile with a shallow valley in the center, indicating that the signal pulse was not in a perfect Gaussian pulse shape.

 

Fig. 2. (a) The temporal sequence of the signal pulse S and the echo pulse R. (b) The temporal sequence of the control CHS pulses.

Download Full Size | PDF

Theoretically, the echo retrieval efficiency $\eta$ decays exponentially and follows (see Appendix A)

 

Fig. 3. The dependence of the echo retrieval efficiency $\eta$ on the storage time $t_{\textrm{stor}}$. Here $t_{12}$ is the time interval between the signal pulse S and the first CHS pulse CHS$_1$. The solid squares are experimental results, and the red curves are the exponential fits.

Download Full Size | PDF

As shown by Eq. (1), the echo retrieval efficiency of the ROSE protocol depends on the optical depth of the crystal, therefore, it is possible to improve the echo retrieval efficiency by optimizing the optical depth of the crystal. One can easily get from Eq. (1) that the echo retrieval efficiency is optimized when $\alpha L=2$. In general, there are two primary methods to modulate the optical depth of a crystal. One approach involves altering the impurity doping concentration (here Pr-doping concentration) or the length of the crystal. However, when the doping concentration in crystal reaches a certain value, the interaction between impurity ions may only affect the inhomogeneous broadening but has little effect on the absorption of the crystal [24]. In addition, increasing the doping concentration will also reduce the decoherence time [24]. On the other hand, changing the crystal length is also a frequently employed method. Nevertheless, the length of the crystal is usually determined during the design and is not readily adjustable. As an alternative, one can build an optical cavity to enhance the interaction between the light field and the crystal. The second method to adjust the absorption of the crystal is by applying an external magnetic field [17]. However, it requires complex strong magnetic devices positioned around the cryostat.

Here, we proposed an alternative approach to modify the absorption of the crystal within the inhomogeneous broadening of the crystal. In the experiments, we modified the optical depth of the crystal by finely tuning the laser frequency within a narrow spectral range. Here, the laser frequency referred to the central frequency of the signal pulse and the control CHS pulse. The measured dependences of the crystal optical depth and the echo retrieval efficiency on the laser frequency detuning $\Delta \nu$ (with respect to a laser frequency $\nu _0=494740$ GHz) are shown in Fig. 4, where the storage time was set to be 15.6 $\mu$s and keeping other parameters the same. One sees that, within a 20-GHz frequency scanning range, the optical depth of the crystal gradually decreases with the increase of the laser frequency, as indicated by the red squares and red curve in Fig. 4. Correspondingly, the echo retrieval efficiency exhibits an initial increase up to a maximum and then followed by a decrease, as shown by the blue dots and blue curve in Fig. 4. The echo retrieval efficiency reaches the highest when the optical depth is approximately 2, in good agreement with the theoretical prediction [22]. The measured lowest optical depth in Fig. 4 was 0.05, at which the echo retrieval efficiency was about 3.2%. The above result shows that, by fine-tuning the laser frequency, the optical depth of the crystal can be dynamically modified, and consequently the echo retrieval efficiency can be optimized. In addition, the storage bandwidth is related to the frequency scanning range of the control CHS pulse. By increasing the frequency scanning range of the control CHS pulse, the storage bandwidth can be improved. However, it’s essential to adjust the Rabi frequency of the control CHS pulse accordingly to maintain the adiabatic condition.

 

Fig. 4. The dependence of the echo retrieval efficiency $\eta$ and the crystal optical depth $\alpha L$ on the frequency detuning $\Delta \nu$ with respect to the laser frequency $\nu _0= 494740$ GHz. The storage time was set to be 15.6 $\mu$s in the experiments. The blue dots are the experimental results of the echo retrieval efficiency, which corresponds to the left vertical axis. The red squares are the experimental results for the optical depth, which corresponds to the right vertical axis. The blue and red curves are a guide to the eye.

Download Full Size | PDF

4. Conclusion

In summary, we demonstrated the light pulse storage in Pr:YSO crystal based on the ROSE protocol, which offers the advantages of being immune from the spontaneous emission noise and easy separation of the echo pulse from the rephasing pulses. The echo retrieval efficiency of the light pulse was optimized by employing the CHS rephasing pulses and by optimizing the optical depth of the Pr:YSO crystal within the inhomogeneous broadening through finely tuning the frequency of the interacting light field. An echo retrieval efficiency as high as 24.4% and an optical coherence time of 34.6 ${\rm \mu} s$ were obtained at a cryogenic temperature of 3.4 K. In addition, both theoretical analysis and experimental results show the robustness of ROSE protocol, and the echo retrieval efficiency is predominantly determined by the time interval between the two rephasing CHS pulses. These results provide substantial promise for light pulse storage in Pr$^{3+}$-doped crystals, signifying its importance in quantum memories and related applications.