Efficient cavity-assisted storage of photonic qubits in a solid-state quantum memory

We report on the high-efficiency storage and retrieval of weak coherent optical pulses and photonic qubits in a cavity-enhanced solid-state quantum memory. By using an atomic frequency comb (AFC) memory in a $Pr^{3+}:Y_2 SO_5$ crystal embedded in a low-finesse impedance-matched cavity, we stored weak coherent pulses at the single photon level with up to 62% efficiency for a pre-determined storage time of 2 $\mu$s. We also confirmed that the impedance-matched cavity enhances the efficiency for longer storage times up to 70 $\mu$s. Taking advantage of the temporal multimodality of the AFC scheme, we then store weak coherent time-bin qubits with (51+-2)% efficiency and a measurement-device limited fidelity over (94.8+-1.4)% for the retrieved qubits. These results represent the most efficient storage in a single photon level AFC memory and the most efficient qubit storage in a solid-state quantum memory up-to-date.

Photonic quantum memories are devices that can store quantum information carried by photons and retrieve it at a later time [1].They can be used as synchronization devices for probabilistic quantum processes and, as such, find applications in quantum light engineering, photonic quantum computation and quantum networking.In particular, they are crucial elements for the implementation of a large scale quantum internet [2] by enabling longdistance distribution of entangled states using quantum repeaters [3,4].
Rare-earth-ion doped crystals are interesting for quantum memory applications because they enable ensemblebased storage without the need for complex laser cooling and trapping schemes.They provide a large number of atoms with optical and spin transitions that are naturally trapped in a solid-state matrix.The ions exhibit large coherence times at cryogenic temperature, both for the optical and spin transitions, which have led to demonstrations extremely long storage times for classical light [28,29].In addition they also feature an inhomogeneous broadening over several GHz which can be tailored using spectral hole burning techniques and can be used as a resource for temporal and spectral multiplexing [30].In particular, the atomic frequency comb (AFC) protocol [31], based on a comb-shaped structure in the inhomogeneous broadening, exhibits intrinsic temporal multi-modality [32,33].It has also proven to support frequency [34,35] and spatial [36,37] multiplexing.
The AFC protocol, first demonstrated with weak coherent qubits in 2008, plays an important role in rareearth based quantum memories, as it is the only protocol so far that has allowed the storage of externally generated single and entangled photons in such devices.This enabled the demonstration of quantum repeater building blocks, e.g.quantum entanglement between quantum memories and telecom photons [38][39][40], light-matter quantum teleportation [41,42] or heralded entanglement between spatially separated quantum memories [32,43].Moreover, the AFC intrinsic temporal multiplexing capability allows a significant speed-up in entanglement distribution rate [32,33,44] in quantum networks as well as a natural platform for the storage of time-bin encoded states.
However, reaching high efficiencies for AFC memories requires a large optical depth which is challenging to achieve with mm or cm sized doped crystals.The highest efficiency demonstrated so far in free-space crystals is around 30 % for single photon qubits [45], 38 % for weak coherent states [46] and 40 % for classical light [30].Moreover, the AFC efficiency, is typically limited by re-absorption to 54 % [31] for retrieval in the forward direction.A possibility to surpass this limit is provided by cavity-assisted AFC, where the crystal is embedded in a low finesse cavity in the impedance-matched regime [47].In the last decades, several experiments employing this technique have been carried out [48][49][50], all achieving remarkable efficiencies with respect to the single-pass AFC efficiencies observed in the respective materials.In this paper, we report on the highest AFC efficiency achieved so far with up to 62 % for weak coherent pulses and 51 % for time-bin qubits.
For an AFC memory in a low-finesse cavity, the impedance matching condition is given by [47]: Here, R in is the reflectivity of the coupling mirror, R out the reflectivity of the second mirror, d is the effective optical depth of ions averaged over the comb peaks.In the case of an AFC with square peaks, this last quantity can be evaluated as d = d FAFC [31], where d is the optical depth of the crystal and F AF C is the finesse of the comb, namely the ratio between the distance among the peaks and their width.If Eq. 1 is fulfilled, for perfect mode matching and if there is no intracavity loss, the input light is coupled to the cavity and fully absorbed in the crystal.In that case, the efficiency of the AFC is given by [47]: where η deph depends on F AF C .η cav can in principle reach 100 % even for excited state storage, if F AF C ≫ 1, R out = 1 , d ≪ 1 and for negligible intra-cavity loss.
In the presence of an intra-cavity round-trip loss ϵ, R out can be replaced by R out − ϵ in Eq. 2 to model a realistic experiment [49].For a given intra-cavity loss, there is an optimum average optical density d for which the efficiency is maximal.In this experiment, we use two Pr 3+ :Y 2 SiO 5 crystals, one functioning as a memory crystal and the other one as time-bin qubit analyzer using an AFC-based interferometer.
The 3 H 4 − 1 D 2 -transition used is at 605.977 nm, in the visible regime (orange light).This transition is shown in the bottom part of Fig. 1.Both crystals are cooled down to 3.2 K in a closed-loop Montana cryostat.Fig. 1 shows a schematic of the optical setup used.The laser to prepare the spectral structure and input pulses, is a frequency-doubled external cavity diode laser at 606 nm.The fundamental wavelength at 1212 nm is locked to an ultra-stable cavity to achieve a linewidth below 3 kHz.Acousto-optic modulators (AOMs) are used in doublepass to create the pulses used to prepare the AFCs, to lock the cavity and to be stored in the memory.
By means of a PBS, two different beams of light, the signal beam and the locking beam, are overlapped and coupled into the cavity.The locking beam is shifted by 640 MHz and has orthogonal polarization with respect to the signal beam, in order to minimize interactions with the ions.A half-wave plate (HWP) is then used to orient the signal beam polarization along the interacting axis of the crystal.Then, a BS splits the light in two paths, one going to a reference detector, while the other one goes to the cavity.The highly asymetric cavity is formed by two mirrors with reflectivities R in =0.4 and R out =0.97, leading to a finesse F cav = 6.5.The cavity is built around the vacuum chamber and the Pr 3+ :Y 2 SiO 5 -crystal.We pick up half of the reflected light by means of the BS, and we couple it to a single-mode fiber.Depending on whether we store qubits or not, this light is then out-coupled in free-space and sent to the analyzing crystal or directly detected in a single-photon avalanche detector (SPAD).After crossing the analyzing crystal, the light is coupled into a single-mode fiber and sent to the SPAD.The locking light, transmitted through the cavity is detected using a photo diode, and read by an Arduino microcontroller.This microcontroller feeds back on the voltage applied to the piezo that controls the cavity length in order to keep the cavity on resonance.
The preparation beam used to create AFCs in the crystal is sent on another path, crossing the signal beam inside the crystal with an angle of 7 • .We prepare AFCs using spectral hole burning techniques described in [51].The AFC is prepared with a target average OD of d = 1  2 ln( Rout Rin ) = 0.46 which ensures impedancematching according to Eq. 1.
The preparation beam is not coupled to the cavity mode.Analogously, a strong preparation beam addresses the filter crystal, in order to prepare an AFC where the transmitted pulses have the same amplitude as the stored and retrieved pulses to form an unbalanced Mach-Zehnder interferometer for qubit analysis.
A new AFC is prepared at every cycle of the cryostat, i.e. every second.
Each experimental cycle is divided into two phases: (1) a preparation and locking phase lasting around 160 ms, during which the cavity is stabilized and the comb structure is prepared.During this phase, all the shutters on the detector path are closed to avoid strong light to reach the APDs.(2) A measurement phase, lasting up to 80 ms, when pulses to be stored are sent to the quantum memory.In the second phase, the locking light shutter and preparation shutter close, and only the single photon light is entering the cavity path.To emulate single photons, we use weak coherent input states containing on average µ in < 1 photons per pulse, in front of the cavity.We send 1000 pulses separated by a time ranging from 13 µs to 80 µs, depending on the AFC storage time τ AF C .Given the unbalanced reflectivity of the cavity mirrors, the AFC echoes are mostly leaving the cavity through the input mirror.We detect those echoes in the same path as the input pulses.
To account for instabilities in laser power or alignment, every other cryostat cycle we prepare a high OD absorption feature instead of an AFC inside the crystal.This effectively blocks the cavity and, as a consequence, we only detect a signal corresponding to 40 % of the input light, that gets reflected from the first mirror.
The storage efficiency is estimated by comparing these reflected pulses (normalized to 100 % to have an estimation of the input pulse) with the AFC echoes retrieved from the quantum memory.To ensure that the AFC effi- ciency is not overestimated, the coupling to the SM-fiber is optimized for input mode, i.e. the spatial mode that is reflected from the first mirror when the cavity is blocked.We assume that the spatial mode originating from the cavity is not coupled with a higher efficiency to the fiber than the reflected mode that we use for comparison.We detect the reflected pulses and the AFC echoes with a single photon avalanche detector (PerkinElmer SPCM-AQR-16-FC, 25 Hz dark counts).Note that we do not need to account for losses along the path, since the echo and the reflected input are transmitted through the same path.The efficiency can be understood as a device efficiency where the cavity around the vacuum chamber and Pr 3+ :Y 2 SiO 5 -crystal constitute the quantum memory.
We first demonstrate the storage and retrieval of weak coherent pulses with µ in =0.33 for a pre-determined storage time τ AF C = 2 µs.Fig 1 c shows an example of the associated atomic frequency comb, with a measured finesse of 5.8 and an average OD of d =0.4.Fig. 2 shows a time histogram with the input pulse and the AFC echo.The storage and retrieval efficiency is η=(62±2) %, which is to our knowledge the highest AFC efficiency reported up-to-date.The reflected part of the pulse from the cavity is as low as 6 %, demonstrating a high degree of both mode and impedance matching.The average efficiency measured with ten independent measurements taken over a few days is (55 ± 2) %.For these results, we are sending 1000 pulses per second to the cavity, using only the first 12.2 ms of the measurement phase.The measurement times are ranging between 2 min and 10 min, which ensures a good statistics during all the trials.Taking into account the measured intra-cavity loss of 3 %, the finesse of the AFC and the current d = 0.40, the maximum efficiency is estimated to be 67 %.The recorded input pulse (cyan pulse) is the reflected 40 % of the original input pulse, so it is normalized to 100 % (dark blue pulse).The ratio between the echo and the original pulse yields (62 ± 2) % storage efficiency.The reflection from the cavity (light blue pulse) was 6 %, indicating that we achieved a good mode-match and a good impedance-match.The detection window for input and echo is 2 µs.
We then measured how the efficiency depends on the storage time.For each storage time, we optimized the AFC finesse and height to achieve approximate impedance-matching.
Fig. 3 shows the recorded efficiencies as a function of storage time.In order to demonstrate the enhancement, we record both the cavity efficiencies and the single pass efficiency with the same AFC, using coherent pulses with µ in =0.22.Note that the single pass efficiency could still be improved if the OD and finesse of the AFC were optimized for single pass storage.
The cavity allows us to observe AFC echoes for storage times up to τ AF C = 70 µs, where the efficiency was still η = 2 %.By fitting our data with the function η = η 0 exp − 4τ AF C T eff 2 [51], where η 0 is the extrapolated efficiency at zero storage time and τ is the storage time, we can extract T eff 2 =(89±4) µs, compatible with previous values reported in free space [32].Fit to Efficiency -T2: 89±4 s Efficiency without cavity Efficiency with FIG.3: Efficiency versus storage time for 9 different storage times.The blue dots represent the efficiencies with weak coherent pulses, the green ones are the efficiencies in single-pass (without cavity) recorded at the classical level.The orange curve is an exponential fit to the blue points yielding the effective T eff 2 of our atomic frequency comb.The combs for the first 2 efficiencies are shaped through the hole-burning technique, while the remaining 7 combs are prepared through the coherent method [51].
Another important parameter is the bandwidth of the memory.As observed in previous experiments with impedance-matched cavities [48,52,53], the abrupt change in OD between the inhomogeneously broadened line and the spectral window that is created prior to the AFC preparation leads to strong dispersion and to a dramatic change in the group velocity of light.As such, the free spectral range of the cavity increases therefore, the cavity linewidth decreases.A measurement of the cavity line width at the center of the spectral window gives 1.16 MHz.To assess how this reduction in line width affects the efficiency of spectrally broader pulses, we performed a storage experiment where we change the bandwidth of the input pulses and measure the storage efficiency.The setup for the experiment is the same as the one presented above, with 2 µs storage time; in this case, we changed the temporal duration of the Gaussian pulses from 120 ns up to 1 µs, corresponding to pulse bandwidths ranging from 3.7 MHz to 412 kHz.Since the line width of the laser is around 3 kHz and therefore much smaller than these values, we can assume that these pulses are Fourier-limited.
Fig. 4 shows the results of this experiment.As expected, the efficiency starts dropping as soon as the pulse spectral width exceeded the cavity line width.Despite the efficiency decrease, we can still store a 3.7 MHz broad pulse with 31.7 % efficiency.Moreover, this measurement tells us that we would be able to store 1.8 MHz pulses with 48 % efficiency.This bandwidth corresponds to the FIG.4: Storage efficiency versus pulse bandwidth for Gaussian pulses and 2 µs storage time.The input photon number per pulse is 0.32 photons per pulse for the longest pulses and has been decreased for the shorter pulses as the pulse peak power was kept constant.As it is expected, the broader the pulse becomes, the more the storage efficiency decreases, due to the filtering effect of the cavity.We attribute the initial bump to the fact that the separation between spectral lines of the AFC teeth starts being comparable in width with the pulse bandwidth, leading to a decreased number of peaks absorbing the pulse.one of single photons generated in our cavity enhanced spontaneous parametric down-conversion source [54].We attribute the slightly lower efficiency for very small bandwidths to the fact that in this regime, the separation of the AFC teeth is comparable to the spectral width of the pulses.Therefore, the pulses do not intercept all the AFC teeth, but only a few, and any inhomogeneity in the teeth preparation leads to a decrease in efficiency.
Finally, we show that our fixed-storage-time quantum memory can efficiently and faithfully store photonic quantum bits.To this aim, we generate, store, retrieve and analyze weak coherent time-bin qubits of the form |ψ⟩ = 1/ √ 2(|e⟩ + e iδ |l⟩), where |e⟩ (|l⟩) is the early (late) component.
The qubit is 1 µs-long, with each Gaussian pulse being 510 ns long (FWHM), and contains on average µ in =0.25 photons per qubit.
The memory storage time was 2 µs.The qubit echo is emitted in the reflected path, coupled to a SM fiber and routed towards the analyzing crystal.
Inside this crystal, we prepared another AFC that we used as an unbalanced Mach-Zehnder inteferometer: we tune the OD of the inteferometric comb in such a way that the probabilities of transmitting the input without absorption (short path) or absorbing and re-emitting them as echoes (long path) [55], are equal.The long path delay, i.e. the AFC storage time, of this interferometer corresponds to the time between the two input time-bins (1 µs): in this way, the process corresponding to the early time bin taking the long path in the analyzer and the late time-bin taking the short path in the analyzer will be superposed.This enables interference between the two processes and from the visibility of the interference fringe we can assess the fidelity of the equatorial qubit storage.The phase ∆Φ of the can be scanned by shifting the AFC spectral position with respect to the central frequency of the qubit [31,55].A schematic of this qubit storage and interference is included in Fig. 1.
In a first step, we prepare a transparent spectral window inside the filter crystal, to assess the storage efficiency of our qubit memory.We recorded two sets of data, exhibiting 51 % efficiency.We attribute the lower efficiency compared to the previous single pulse data to the larger bandwidth of the qubit pulses.Nevertheless, this efficiency is to our knowledge the highest demonstrated for qubit storage in a solid-state quantum memory.
We then prepared the AFC inside the analyzing crystal and recorded the detection probability as a function of the analyzer phase ∆Φ for various input qubit states (|±⟩ = 1/ √ 2(|e⟩ ± |l⟩) and |L/R⟩ = 1/ √ 2(|e⟩ ± i |l⟩).The interference fringes are shown in Fig. 5 and the fitted visibilities are reported in Table I.The final average value of V coh = (89.9± 3.9) %, calculated by averaging the measurements corresponding to the same basis with equal weights.From this visbility, we deduce an average fidelity for the superposition states of =(95 ± 2) % [56].To complete the assessment of the storage fidelity, we also measured the states on the poles of the Bloch sphere, |e⟩ and |l⟩.We find a fidelity of F pole = (94.6 ± 0.3) %.Finally, the fidelity averaged over the Bloch sphere is F total = 2 3 F coh +  without the quantum memory leads to a qubit fidelity of F interf = (94.2± 1.0) %.This shows that the storage fidelity that we measured was limited by the creation and analysis system.Therefore we can estimate the fidelity of the memory alone as F memory = F total /F interf = 100 % +0 % −2 % .The measured fidelity is above the threshold of 2/3 for measure-and-prepare strategies for single photon input states.However, since we use weak coherent states, the threshold fidelity is increased when we include the Poissonian statistics of the input state [17,57].For a photon number around 0.25 and a memory efficiency of 51 % (for qubits), we find a threshold of 75 %.Our measured fidelity is much higher than this bound, showing that our device performs genuine quantum storage.
To summarize, we have demonstrated a cavityenhanced atomic frequency comb memory at the single photon level.Using an impedance-matched cavity configuration, we showed the storage of weak coherent pulse with efficiencies up to 62 % for a fixed storage time of 2 µs, which represents the most efficient AFC memory at the single photon level to date.In addition we demonstrated the storage of photonic time-bin qubits with an efficiency of 51 %, and a fidelity of 95 % for the retrieved, only limited by the qubit preparation and analysis system.This represents the most efficient storage of photonic qubits in solid-state quantum memories up-to-date.
The efficiency enhancement due to the cavity was observed for AFC storage times up to 70 µs.
The maximum efficiency achievable in our current setup was limited by intra-cavity loss, by the limited finesse of the AFC, and by the non-unity reflectivity of the second mirror.Intra-cavity loss could be reduced by installing the cavity inside the vacuum chamber of the cryostat, or by coating the crystal facets, to avoid losses at the window surfaces.If loss can be reduced, the effective d could also be reduced which would allow us to reach even higher efficiencies.For a loss of 1 % and comb finesse of 10, the maximum efficiency would reach 91 %.Another limitation of the current configuration is the limited bandwidth of the memory due to the slow light effect.This effect could be alleviated by reducing the absorption in the AFC, e.g. by detuning the light away from the center of the inhomogeneous absorption line or by using a shorter crystal.In this work, light was stored for pre-determined AFC storage times in the excited state of the Pr ions.While this could be already useful for some quantum repeater protocols [34], on-demand read-out could also be implemented by sending strong control pulses to store the light in the spin states of Pr ions [58].
Finally, a Faraday rotator could be used in combination with a PBS to replace the beam-splitter and double the detected signal.These improvements would open the door to storing single photons (e.g.heralded single photons from cavity-enhanced spontaneous parametric down-conversion source [54]) with high efficiencies in solid-state quantum memories, which would represent a critical resource for the realization of functional quantum repeaters.

FIG. 1 :
FIG. 1: (a): Sketch of our experimental setup for qubit storage.The two pulses constituting the time-bin qubit are depicted as magenta and blue.The slightly fainter pulses are the echoes, preceded by residual reflections from the cavity due to imperfect mode and impedance matching.(b): Level scheme of the 3 H 4 − 1 D 2 transition of Pr 3+ :Y 2 SiO 5 shows 3 hyperfine levels in both the ground and excited state.The optical transition, where the AFC is prepared, is ±1/2g -±3/2e.(c): Plot of a single pass absorption measurement of the AFC.The finesse of the AFC is 5.8.

FIG. 2 :
FIG.2: Input pulse and its echo (red pulse) after 2 µs.The recorded input pulse (cyan pulse) is the reflected 40 % of the original input pulse, so it is normalized to 100 % (dark blue pulse).The ratio between the echo and the original pulse yields (62 ± 2) % storage efficiency.The reflection from the cavity (light blue pulse) was 6 %, indicating that we achieved a good mode-match and a good impedance-match.The detection window for input and echo is 2 µs.

FIG. 5 :
FIG. 5: Interference fringes for different phases of the qubit.The plots show the detection probability per trial (p D ) versus the interferometer phase change, expressed in units of π.All these measurements were recorded with 2000 pulses per second.The detection window was 1 µs.

TABLE I :
Fringe visibility for different values of the phase between qubit components.