Analysis of a photon number resolving detector based on fluorescence readout of an ion Coulomb crystal quantum memory inside an optical cavity

The goal of the initialization is preparation of a cold Coulomb crystal with the ions in the state |g〉 = |D_5/2,m_J = + 5/2〉. The preparation consists of several steps similar to the preparation described in [19]. A magnetic field of a few Gauss along the cavity axis defines the quantization axis, and laser cooling is achieved by applying two counter-propagating light beams along the cavity axis. The light is resonant with the ${\rm S}_{1/2}{\leftrightarrow } {\rm P}_{1/2}$ transition at 397 nm, and the beams are left- and right-hand circularly polarized, respectively (see figure 3). Atoms that fall into the D_3/2 state are repumped by a laser at 866 nm applied from the side with its polarization orthogonal to the cavity axis, equivalent to left- and right-hand circularly polarized with respect to the quantization axis. Once the ions are sufficiently cold, the cooling laser is turned off. Two additional lasers pump the ions to the m_J = + 5/2 Zeeman sub-level of the D_5/2 state. The first of these lasers drives the σ⁺-transitions from S_1/2 to P_3/2. The second laser is resonant with the ${\rm D}_{5/2} {\leftrightarrow } {\rm P}_{3/2}$ transition, and its propagation direction and polarization are chosen such that σ⁺ and π transitions are addressed simultaneously. Atoms that spontaneously decay from P_3/2 to D_3/2 are reintroduced into the optical pumping process by the laser cooling repumper. The typical duration of the cooling and pumping procedure is 25 μs [22].

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Efficient optical pumping is very important for high-fidelity measurements of the photon number in the probe field. Ions that are not in the state |g〉 after the initialization may offset the fluorescence in the final step of the detector protocol, leading to an overestimation of the photon number. Herskind et al [19] state an optical pumping efficiency to the D_3/2 level with m_J = + 3/2 of 97%. Optical pumping into the m_J = + 5/2 sub-state of the D_5/2 state is expected to have a similar efficiency. The efficiency is limited by imperfect polarization of the pumping light, leading to a distribution of the remaining ions over the other D_5/2 sub-states. The spontaneous decay rate into D_3/2 of only 2π × 0.18 MHz is very weak compared to the pumping and repumping fields, so only a tiny fraction of the ions will end up in S_1/2 or D_3/2. The number of these ions can be estimated by monitoring the ultraviolet fluorescence during the optical pumping, and the result subtracted from the photon number measurement at the end of the protocol. Alternatively, one can extend the dead time of the detector by a variable amount and let optical pumping proceed until the moment when the monitored fluorescence ceases. This signals that the relevant energy levels are empty, and the detector is ready to receive the probe pulse. We note that the larger the total number of ions, the more difficult it is to transfer all ions to D_5/2, making a moderate number of ions the preferred choice.

In the second step of the detector protocol, the photons in the probe pulse are converted into collective excitations in the metastable state |m〉. The procedure is the same as the absorption of photons into a quantum memory for light, based on an ensemble placed inside an optical cavity [23]. We consider a storage scheme based on EIT, where the probe pulse is resonant with the σ⁻ transition $\vert {\rm D}_{5/2}, m_J{=}+5/2\rangle {\leftrightarrow } \vert {\rm P}_{3/2}, m_J{=}+3/2\rangle$. At the same time a strong coupling field is acting on the transition $\vert {\rm P}_{3/2}, m_J{=}+3/2\rangle {\leftrightarrow } \vert {\rm D}_{3/2}, m_J{=}+1/2\rangle$ (see also figure 2). The strength of the coupling field determines the width of the EIT window and the group velocity of the probe pulse. One can coherently convert the quantum state of the probe pulse into a collective excitation by reducing the group velocity to zero, that is, by adiabatically turning off the coupling field.

The most essential parameter of such memories is the cooperativity C = g²N/κγ, where g is the coupling rate between a single ion and a cavity photon, N is the effective⁵ number of ions interacting with the mode of the cavity [19], κ is the cavity decay rate and γ is the rate of decoherence on the transition $\vert g\rangle {\leftrightarrow } \vert e\rangle$ in the protocol. The cooperativity determines the maximally obtainable photon conversion efficiency η = C/(1 + C) [24]. In principle, probe pulses of any temporal shape can be stored with optimal efficiency by adapting the shape of the coupling field. However, the adiabatic conversion of photons into collective excitations requires that the temporal length T of the photonic probe pulse is much larger than 1/Cγ [24, 25]. Optimal storage and retrieval with an efficiency of η² ≃ 45% has been demonstrated using EIT in rubidium vapour [26] without cavity. Using a low-finesse cavity, a retrieval efficiency of 73% was recently obtained with cold atoms [27] using a Raman scheme that has the same adiabaticity conditions as EIT.

The efficiency of the light storage determines the overall detection efficiency, as long as saturation effects are avoided by ensuring that the number of photons in the probe pulse is much smaller than the number of ions. The experimental parameters obtained in [19] on the transition $\vert {\rm D}_{3/2}, m_J{=}+3/2\rangle {\leftrightarrow } \vert {\rm P}_{1/2}, m_J{=}+1/2\rangle$ are (g = 2π × 0.53 MHz, N ≃ 1500, κ = 2π × 2.15 MHz, γ = 2π × 11.9 MHz), giving a maximum cooperativity⁶ of C ≃ 16. For the transition $\vert {\rm D}_{5/2}, m_J{=}+5/2\rangle {\leftrightarrow } \vert {\rm P}_{3/2}, m_J{=}+3/2\rangle$ used in our protocol, the theoretical values for g and γ are slightly higher. Hence, a cooperativity of C ≃ 15 should be straightforward to obtain, which gives a detection efficiency above 93%.

The duration of the storage process is essentially given by the duration of the probe pulse. We find that Cγ ≈ 10⁹ s⁻¹, allowing, in principle, for the optimal storage of Fourier-limited pulses of duration down to about T ≃ 50/Cγ = 50 ns [24], but even a more conservative value of 1 μs is negligible compared to the duration of the entire protocol.

To avoid unwanted Doppler effects related to rf-induced micromotion (see references in the caption of figure 2), the coupling field has to co-propagate with the probe field inside the cavity, as indicated in figure 2. In this case the geometrical constraints on the spatial profile of the coupling field reduce the optimal storage efficiency by an amount that depends on the radial extension of the Coulomb crystal. However, this reduction can be rendered negligible by carefully choosing the size of the crystal, adjusting the strength of the coupling field or constraining the radius by the addition of a second calcium isotope [25].

In the last step of the protocol, the number of ions transferred to the D_3/2 state is measured by fluorescence collection. Fluorescence collection is routinely applied in trapped-ion-based quantum computing, and a single-ion state-discrimination with an error probability below 10⁻⁴ has been reported [28]. The fluorescence is induced by the same lasers that were used during the cooling stage (figure 3(a)). Ions in one of the D_3/2 will emit fluorescence on the ${\rm S}_{1/2}{\leftrightarrow } {\rm P}_{1/2}$ transition. The amount of fluorescence is proportional to the number of ions undergoing the optical cycling. For unity absorption efficiency and the ideal initialization of the detector, this number is equal to the number of photons originally present in the probe pulse. Since the detection laser is 12 nm detuned from the ${\rm D}_{5/2} {\leftrightarrow } {\rm P}_{3/2}$ transition, the ions that remained in D_5/2 are not affected at all. However, because the lifetime of the D-states is finite (τ_D = 1.15 s), ions in D_5/2 can still enter the fluorescence cycle by spontaneously decaying into S_1/2.

We now analyse the process of fluorescence collection in a more quantitative way. The Poissonian statistics of the detected fluorescence photons is taken into account, and the fidelity of the photon number estimation discussed. Let us start by considering η = 1 and N_in photons in the probe pulse. Assuming that all the involved optical transitions are saturated, every ion spends about 1/4 of its time in P_1/2, from where it spontaneously decays into S_1/2 at a rate of γ_PS = 2π × 20.7 MHz. Letting Θ denote the amount of solid angle covered by the collection system and η_D the overall detection efficiency at the relevant wavelength of 397 nm, the photon detection rate per ion is given by R = γ_PSΘη_D/16π. A typical detector has η_D = 0.4, and a lens with 4 cm diameter at a working distance of 7 cm can cover about Θ/4π = 2% of the full solid angle and still image the whole crystal, albeit with some distortion. These parameters give R = 260 kHz. The total number of photons collected after a time t will follow a Poissonian distribution with the mean

$$\begin{equation} \mu_{{\rm in}}(t) = N_{{\rm in}}\,R\,t. \end{equation} \tag{ 1 }$$

At any intermediate time t', the mean number of ions having decayed from D_5/2 is N(1 − e^−t'/τ_D), neglecting the time passed since the initial state preparation. After a time t, the mean number of collected fluorescence photons from these ions is

$$\begin{equation} \mu_{{\rm decay}}(t) = \int_0^t N(1-{\rm e}^{-t'\tau_{\rm D}}) R\, {\rm d}t' = N\, R\, \tau_{\rm D} \left[({\rm e}^{-t/\tau_{\rm D}} - 1) + t/\tau_{\rm D} \right]. \end{equation} \tag{ 2 }$$

The probability of detecting N_fl fluorescence photons after a time t is then given by

$$\begin{equation} \fl p_{N_{{\rm fl}}}(t) = \sum_{n=0}^{N_{{\rm fl}}} {\rm Po}[n; \mu_{{\rm in}}(t)] {\rm Po}[N_{{\rm fl}}-n; \mu_{{\rm decay}}(t)] = {\rm Po}[N_{{\rm fl}}; \mu_{{\rm in}}(t) + \mu_{{\rm decay}}(t)], \end{equation} \tag{ 3 }$$

where Po[n;μ] denotes the Poisson distribution with the mean μ. So, the spontaneously decaying ions shift the amount of fluorescence to slightly higher values without changing the shape of the distribution. This is illustrated in figure 4(a) for N = 1500 ions.

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The time required for fluorescence collection depends on the amount of confidence in the estimated input photon number that one wants to obtain. We consider a simple strategy, where the input is estimated as N_in photons if the amount of fluorescence is larger than a threshold that corresponds to the point where the Poisson distributions for N_in and N_in − 1 input photons cross, see figure 4(a). As a measure of the fidelity, we will consider the probability that N_in input photons lead to an estimate that is different from N_in. This error probability is plotted as a function of the collection time in figure 4(b) for N_in = 1,3 or 10 photons. The larger the N_in, the more the fluorescence needed to reduce the error below a certain level. For the parameters considered here and an error probability below 10%, one can distinguish up to three photons after t ≃ 130 μs and ten photons after t ≃ 430 μs.

In the case where η < 1, the number of ions transferred to |m〉 follows a binomial distribution, that is, not all photons are necessarily converted into collective excitations. In fact, the probability that all input photons are converted is η^N_in, which sets a lower bound on the error probability of p^min_err = 1 − η^N_in. We note that this lower bound is valid for any photon number resolving detector with non-unity efficiency. For our parameters, the bound is obtained after t ≃ 180 μs for N_in = 1 and t ≃ 250 μs for N_in = 3.

After the fluorescence detection, the ions will have to undergo a brief period of laser cooling before reinitialization back into D_5/2. Based on previous experiments [19, 20], this procedure is expected to take less than 100 μs.

To calculate the repetition rate, we add up the durations of the individual steps of the protocol, neglecting the short duration of the light storage. Using the numbers stated in the previous sections, we obtain T_total ≃ (25 + 200 + 100) μs, giving a repetition rate of about 3 kHz.