Robust, long-lived optical quantum memories are important components of many quantum information and communication protocols. We demonstrate coherent generation, storage, and retrieval of excitations on a long-lived spin transition via spontaneous Raman scattering in a rare-earth ion-doped crystal. We further study the time dynamics of the optical correlations in this system. This is the first demonstration of its kind in a solid and an enabling step toward realizing a solid-state quantum repeater.
© 2013 OSA
Robust quantum memories based on long-lived internal atomic states in optically dense ensembles are important for various quantum information applications, particularly for long-distance quantum communication . We demonstrate storage and retrieval of collective excitations on a long-lived spin transition in a rare-earth ion-doped crystal (REIC), which acts as a “frozen” atomic ensemble. This is an enabling step toward a long-lived solid-state quantum memory that can store single collective excitations. In addition, we observe qualitatively different time dynamics than have been seen previous studies in atomic gas systems that have shorter relevant time scales [2, 3].
Solid-state materials have numerous advantages over atomic gases for practical quantum memories. They are more compatible with existing solid-state hardware and offer the possibility of integrated, on-chip processing. They do not exhibit spatial diffusion, which limits potential coherence times in atomic gases. And while atomic diffusion can be mitigated using optical lattices or microtrap arrays, leading recently to storage times longer than 100 ms, the experimental complexity involved in cooling and trapping atoms or ions is typically much greater than for solid-state systems . REIC systems are particularly promising for quantum memory applications and have been studied for nearly two decades in quantum optical information schemes [5, 6]. In particular, they have all the benefits of solid-state materials and additionally exhibit long hyperfine coherence times, demonstrated up to seconds and theoretically potentially much longer [7–11].
To date, various quantum-memory configurations have been demonstrated in REIC, including a gradient echo memory, electromagnetically induced transparency, and atomic frequency comb schemes [5, 7, 12, 13]. Recently, experiments in REIC have achieved entanglement between a single collective atomic excitation and a photon in both bulk crystal and waveguide and between two crystals [14–16]. The echo and atomic frequency comb schemes require precise optical pulses for coherent population transfer to utilize the long storage times of the spin transitions in REIC. We implement a protocol in which collective excitations are stored directly on the long-lived spin transition via spontaneous Raman scattering without precise optical π-pulses and has built in entanglement purification when used in a quantum repeater scheme .
In this work, collective excitations on a long-lived spin transition are stored and retrieved via Raman scattering similar to the method of ref. . An ensemble of atoms with a Λ-type energy level configuration is prepared with all atoms in one of the metastable ground states, labeled |g〉. An optical write field couples state |g〉 to an excited state and spontaneously scatters atoms to the initially unoccupied ground state |s〉. Each scattering event that transfers an atom from |g〉 to |s〉 is accompanied by emission of a single photon, shifted in frequency by the splitting between the ground states, called the heralding photon. All atoms in the ensemble are equally likely to be the source of a detected signal photon, so the scattering event is a collective excitation of the ensemble stored on the |g〉 → |s〉 transition. Collective excitations can be converted back into optical fields by applying a read field that scatters the population in state |s〉 back to state |g〉. For each collective excitation read out in this way, a single retrieved photon, shifted in frequency from the read field by the |g〉 → |s〉 splitting, is emitted in a spatial mode defined by the phase-matching constraint kwrite + kread = kheralding + kretrieved.
The particular implementation in Pr3+:Y2SiO5 is shown in Fig. 1. We label the optical fields correlated with generation and retrieval of collective excitations as heralding and retrieved, respectively. This protocol has been demonstrated in many materials and can be used in implementations of single photon sources and quantum repeaters. The storage time of the collective excitation is an important figure for both these applications. This time sets a limit on the on-demand nature of a single photon source and must be longer than the time needed to reliably generate entanglement for a practical quantum repeater. The storage time can be as long as the coherence time of the |g〉 → |s〉 transition, up to 30 seconds in REIC [8, 9].
2. Experimental setup and spectral hole-burning
A challenge of working with REIC is the inhomogeneous broadening of the optical transition due to crystal field variations at different dopant sites [18, 19]. This inhomogeneous broadening is ≈1 GHz in our sample and, similar to other REICs, much larger than the |g〉 → |s〉 splitting of the Λ-system (Fig. 1). To implement our write-read protocol, we use spectral hole-burning techniques to prepare a spectrally narrow ensemble of ions in a particular ground state and pump all ions in other resonant frequency classes to non-interacting states [6, 19, 20]. The final spectral profile consists of two 6 MHz-wide transmissive spectral trenches, one covering the write and retrieved transitions and the other the read and heralding transitions, with the spectrally narrow absorbing feature in the write trench. We use a hole-burning sequence modified from ref.  to generate absorbing features ≈100 kHz wide. The third, auxiliary ground state is used to shelve unwanted population, allowing for the wide spectral trenches around the four optical transitions of interest.
We use a 2 cm-long 0.005 % doped Pr3+:Y2SiO5 crystal held at ≈2 K to limit phonon excitations and apply a small static magnetic field to zero the earth’s field and reduce the inhomogeneous broadening of the spin transition. We address the 3H4 →1D2 transition at 606 nm with a frequency-stabilized dye laser. This transition has three doubly-degenerate (in zero magnetic field) ground and excited hyperfine states . The upper two hyperfine ground states and the lower two hyperfine excited states comprise our double-Λ system (Fig. 1). The write and read fields overlap in the crystal at a small angle (θ ≈ 0.5 °) and we collect the co-propagating heralding and retrieved fields with a single-mode fiber directly between the write and read fields (Fig. 2). This geometry leads to a small phase mismatch Δk = kheralding + kretrieved − (kwrite + kread) ≈ (8π/λ)sin2(θ/4) ≈ 2 cm−1. The advantages of this geometry, however are that we detect in only a single spatial mode and the small angle keeps the bright write and read fields from saturating the detector.
The write field is ≈ 1 mW/mm2 and red-detuned 600 kHz from the |g〉 → |ew〉 transition and the read field is ≈ 3 mW/mm2 and resonant with the |s〉 → |er〉 transition. We implement the spectral hole-burning procedure described above before each write-read trial and repeat the full sequence 37,500 times. A balanced heterodyne detection system is used to obtain the intensities of the heralding and retrieved fields as functions of time (Fig. 2). This detection scheme provides the frequency discrimination necessary to detect the weak heralding and retrieved fields in the presence of scattered light from the bright write and read pulses.
As a first step toward generating, storing, and retrieving single collective excitations, we operate in a high gain regime where each write pulse causes many collective excitations in the detected mode. We detect an average of 105 heralding photons per write pulse. We operate in this high-gain regime to obtain sufficient signal for the silicon PIN photodiodes in our heterodyne detection configuration.
The write and read pulses are 10 μs long transform limited pulses separated by 35 μs. We find the intensities of the heralding and retrieved fields as functions of time relative to the excitation pulses, Ih(th) and Ir(tr). Figure 3(a) shows the mean number of photons in the heralding and retrieved fields (solid lines) and the timing of the write and read pulses. The heralding field is delayed with respect to the write field due to the strong absorption of the nearly resonant write field. The retrieved field is magnified 100x and the write and read intensities are not to scale with respect to the heralding and retrieved fields. We discuss the shapes of these emission profiles below. The retrieval efficiency we observe ((<1 %) is much lower than has been previously demonstrated in atomic gas systems (>50 %) or in other quantum memory protocols in REIC (69 %) [12, 21], but similar to other spin-state storage protocols [22, 23]. One likely factor contributing to this inefficiency is the inherent phase-mismatch in our geometry discussed above, which we plan to address in future work.
3. Optical correlations
We characterize the write-read scheme with the normalized second-order intensity correlation functions of the heralding and retrieved fields. The correlation of fields j and k is defined as24, 25]. indicates correlation beyond random coincidences. We calculate the auto- and cross-correlations of the fields as functions of the detection times of the fields.
Collective excitations in a single mode, similar to a single mode of spontaneous emission from a medium, exhibit thermal number statistics. The number of collective excitations fluctuates shot to shot according to the thermal number distribution P(n) = 〈n〉n/(1+〈n〉)(n+1) where 〈n〉 is the mean number of collective excitations. The heralding and retrieved fields should thus exhibit a value of 2 for the second order intensity auto-correlation function . We find the auto-correlation functions of the heralding and retrieved fields to be near 2 at all detection times with and , suggesting that the fields are largely free of leakage from the classical excitation fields, which exhibit an auto-correlation function equal to 1.
We model the cross-correlation between the heralding and retrieved fields assuming some uncorrelated (noise) emission in each channel. Collective excitations follow the thermal number distribution noted above, and we note that the intensity and photon number are related by a multiplicative factor that cancels in the normalized expression of eq. 1. We define the ratio of the average intensity of uncorrelated emission to the average intensity of correlated emission for each field (μh and μr). From this, it is straightforward to use eq. 1 to obtain and expression for the cross correlation in terms of the mean number of collective excitations and the noise parameters. We find that
We note that for the case of no noise in either field, . Thus, in the high gain regime with large 〈n〉, the cross-correlation has an upper bound near 2 and is reduced in the presence of noise in either field.
4. Experimental results
The measured cross-correlation between the detected heralding and retrieved fields is shown in Fig. 3(b) as a function of the detection times of both fields, th and tr. The highest value of the cross-correlation is near the theoretical maximum for the high gain regime, 2.0±0.1 (statistical uncertainty). The correlation between the time separated heralding and retrieved fields is a signature of generation, storage, and retrieval of collective excitations. We note that the cross-correlation remains largely constant over the main part of the retrieved emission, seen as a broad maximum along the vertical axis. However the correlation peaks early during the heralding emission, and drops for later heralding detection times.
We check potential sources of correlations other than the generation and retrieval of collective excitations. We operate in a strongly phase mismatched configuration by applying co-propagating write and read pulses while collecting the heralding and retrieved light in a single spatial mode at a small angle (the only difference from Fig. 2 is that the read field propagates in the write mode). The heralding and retrieved fields in the detected mode should not be correlated with each other. The heralding field should be correlated with retrieved light in an undetected mode on the other side of the write/read mode at the same small angle. Unlike the small phase mismatch in the primary geometry, in which there is no better phase-matched mode for the retrieved field to propagate in, this phase mismatch should eliminate correlations rather than just reducing retrieval efficiency. This configuration yields values of at all times, well below the values for the (nearly) phase-matched case. The lack of correlations in this configuration implies that the correlations in the phase-matched case indicate coherent generation, storage, and retrieval of collective excitations.
5. Temporal dynamics
The overall temporal behavior in our system is qualitatively different than what has been seen in cold atomic gas systems . Atomic gases exhibit decoherence-dominated temporal dynamics in which the storage time (tr − th) is the primary factor. Due to decoherence over even short storage times, atomic gases show decreased correlations as storage time increases. Our system exhibits cross-correlation that is largely constant during the retrieved field, suggesting that there is negligible decoherence of the collective excitation during storage. This is consistent with previously reported coherence times for the spin transition Pr3+:Y2SiO5, which are longer than the storage time demonstrated here . However, the reduction of the cross-correlation for later detection times of the heralding field suggests an additional noise process with a temporal profile delayed with respect to the generation of collective excitations that causes emission in the heralding channel.
We use equation 2 and our measured values of to estimate the ratios of noise to correlated emission in the fields (μh and μr) as functions of th and tr. We assume some fluctuation of the mean collective excitation rate, 〈n〉, due to drift of experimental parameters such as laser intensity and pointing and find ranges of μh and μr that are consistent with the data. Our results for the fraction of emission that is correlated with the generation and retrieval of collective excitations are plotted as the shaded bands in 3a along with the measured emission in each field. While the retrieved field is composed primarily of correlated emission, we find that the heralding field contains significant noise and that the correlated and noise photons in the heralding field exhibit distinctly different temporal dynamics. Namely, we find that the correlated field in the heralding channel peaks earlier than the overall emission, and that the tail of the emission is largely noise. This temporal separation allows for gating of the heralding field to improve the fidelity of the heralding process.
In conclusion, we have demonstrated generation, storage, and retrieval of collective atomic excitations on the ground hyperfine transition in a rare-earth ion-doped crystal, indicated by correlations between the optical fields heralding the generation and retrieval of collective excitations. We develop a theoretical model to describe the temporal dynamics of our observed correlations and show that we are currently limited by neither decoherence nor noise. This is an enabling step toward building a quantum repeater with long-lived, solid-state quantum memory nodes.
S.E.B. and M.J.S. would like to acknowledge the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (Project No. CE110001029)
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