Entanglement between more than two hundred macroscopic atomic ensembles in a solid

# Entanglement between more than two hundred macroscopic atomic ensembles in a solid

## Abstract

We create a multi-partite entangled state by storing a single photon in a crystal that contains many large atomic ensembles with distinct resonance frequencies. The photon is re-emitted at a well-defined time due to an interference effect analogous to multi-slit diffraction. We derive a lower bound for the number of entangled ensembles based on the contrast of the interference and the single-photon character of the input, and we experimentally demonstrate entanglement between over two hundred ensembles, each containing a billion atoms. In addition, we illustrate the fact that each individual ensemble contains further entanglement. Our results are the first demonstration of entanglement between many macroscopic systems in a solid and open the door to creating even more complex entangled states.

Beginning with Schrödinger (1), the question whether quantum superposition and entanglement can exist in macroscopic systems has been at the heart of foundational debates (2); (3); (4); (5) and inspired many experiments (6); (7); (8); (9); (10); (11); (12); (13); (14); (15); (16). One important type of entangled states corresponds to a single excitation that is delocalized over many systems. Such Dicke states (17) have been created for individual photons (18); (19) and cold atoms (20); (21); (22); (23). In solids, superradiance associated with a Dicke state was recently demonstrated for two superconducting qubits (24).

Here we create a Dicke state by storing a single photon in an atomic frequency comb (AFC) (25); (26); (27); (28). An AFC is a collection of atomic ensembles with different, equally spaced resonance frequencies. Such AFCs, which also serve as quantum memories for light, can be conveniently generated in rare-earth ion doped crystals through optical pumping. Each individual ensemble represents one “tooth” of the comb. Here we will focus on the entanglement that is created between these teeth by the absorption of a single photon. We can then treat each tooth as a single two-level system (a “qubit”) with collective states (where all atoms in the tooth are in the ground state) and (where a single atom in the tooth is excited).

In the ideal case, the absorption of a single photon by an AFC consisting of teeth creates the lowest-order Dicke state, widely known as the state,

 |W⟩=1√N(|100...0⟩+|010...0⟩...+|000...1⟩), (1)

where the first term corresponds to the case in which the first tooth has absorbed the photon etc. This is a multi-partite entangled state. However, the existence of this multi-partite entanglement under realistic experimental conditions, where neither the single-photon source nor the storage process are perfect, has not been demonstrated so far.

Our theoretical and experimental approach to demonstrating this multi-party entanglement is based on the re-emission of the single photon from the AFC, which is due to a collective interference effect. As time passes, the different terms in the above equation acquire different phases , where stands for the detuning of the tooth relative to the lowest-frequency () tooth, runs from to , and represents the angular frequency spacing between the teeth. The photon has a high probability of being re-emitted only at the “echo” times when all the phase factors are the same, that is, at times that are integer multiples of (25); (26). This echo emission is thus an interference effect in time that is very similar to spatial multi-slit diffraction (see Fig. 1). This analogy is at the heart of our approach, which we now describe in detail.

Results
Derivation of a lower bound for the entanglement depth.
Our method for demonstrating the entanglement between the teeth is inspired by the fact that, for multi-slit interference, the number of participating slits can be inferred from the sharpness of the interference pattern. We consider the ratio of the maximum photon re-emission probability in the first echo to the re-emission probability averaged over one period ,

 R=P(2πΔ)Δ2π∫3πΔπΔP(t)dt. (2)

We refer to as the “echo contrast”. The photon emission probability is proportional to (26); (29), where and is the dipole operator for tooth , and . Here we are using the Heisenberg picture, where observables rather than states are time-dependent. At the time of the first echo, , all teeth are in phase, giving . On the other hand, averaging the re-emission probability over a time interval centered at the echo time as in (2) leads to , which is nonzero only for . Then, the denominator of Eq. (2) results in the expression , i.e. the sum of the excitation probabilities for each tooth, yielding .

We now show that the echo contrast is closely related to the “entanglement depth” (30). A (generally mixed) quantum state of qubits has entanglement depth at least equal to if it cannot be decomposed into a convex sum of product states with all factors involving less than entangled qubits, i.e. at least one of the terms needs to be an -qubit entangled state.

First, we consider the case where exactly one photon is absorbed by the AFC. In this case . Let us suppose that is found experimentally to have a value of . This value can be achieved by the state , i.e. a Dicke state involving teeth and no excitation in the remaining teeth. Let us note that is permutation invariant (as is in the general case), so all permutations of the teeth are equivalent for our purpose. The above state has an entanglement depth of . All other states in the single-excitation subspace giving involve more than entangled teeth, i.e. they are of the form with more than non-zero coefficients that are not all equal and fulfilling . This is due to the fact that, in the single-excitation subspace of the -teeth Hilbert space, the Dicke state is the only eigenstate of with a non-zero eigenvalue (namely ), see Methods. Thus an experimental value implies an entanglement depth of at least in the case of perfect absorption of a single photon.

Next, we consider the more realistic case of absorption of a single photon with probability . In this case . Let us again suppose that the experiment gives . This can be achieved with a mixed state or coherent superposition state that has a probability weight for and a weight for . With the same line of arguments as for the case of the perfect absorption (), the entanglement depth is again at least .

The situation becomes more complex if there is a chance for the AFC to absorb more than one photon. In our experiment, the most significant higher-order probability is , the probability of absorbing two photons, which is very small but non-zero. We now have to consider states with higher-order components, for example, the fully separable state with . Since this state has the Dicke state of teeth, , as its component in the single-excitation subspace, it can give very large values of the echo contrast up to , which happens in the limit of small . On the other hand, in this limit, the two-excitation probability is . In our experiment, the values of and satisfy , see below. It is clear from this example that the values of and are important for deciding whether large values of imply large values of the entanglement depth. We have numerically determined a lower bound for the entanglement depth, , as a function of the echo contrast, , conditioned on the experimental values of and (see Methods). This bound can be very well approximated in our regime by the relation

 M>R−√2P2P1N. (3)

Note that the bound gives (which is consistent with the correct value ) for the fully separable state discussed above, for which , but also . Below we utilize the obtained numerical bound to find the minimum entanglement depth corresponding to the experimental value of the echo contrast.

Experimental Results. The principle of our experimental approach is shown in Fig. 1c, and the detailed experimental setup is shown in Fig.  2a. It consists of a source of heralded single-photons, an AFC and a time-resolved detector to register the retrieved photons. The heralded single-photon source is implemented by spontaneous parametric down conversion (SPDC), which probabilistically down-converts photons at  nm into pairs of photons with wavelengths centered around  nm and  nm. Provided a single pair was generated, detection of a  nm photon heralds the presence of a  nm photon. Measuring the cross-correlation function of the source, shown in Fig. 2b, allows us to determine the mean photon number per mode , where the mode is defined by the coherence time of the pump laser. We find (see Methods). This low mean photon number indicates that the probability of generating multi-pairs is very low (of the order per mode), thus confirming the nearly single-photon nature of our heralded source. This can also be seen from the heralded auto-correlation function in Fig. 2b. In combination with a precise estimation of the loss at various places of the setup and the absorption efficiency in the AFC (see Methods), this allows us to determine and using a detailed theoretical model of the setup (see Methods). We find and , where the uncertainties in and are dominated by the uncertainties in the optical depth of the AFC and in the mean photon number per mode, respectively.

The AFC for the 795 nm photons is implemented in a bulk Tm:LiNbO crystal. Optical pumping is used to spectrally shape the inhomogeneously broadened absorption line of the Tm atoms into a series of absorption peaks (teeth) spaced by angular frequency , see Fig. 2c. The number of teeth in the AFC is given by , where is the bandwidth of the AFC (fixed at  GHz); and can be changed by modifying . After a storage time of , the 795 nm photons are retrieved and detected by a single photon detector whose signal is sent to a time-to-digital converter (TDC). This generates a time histogram of the 795 nm photon detections such as the one in the inset of Fig. 3a.

The echo contrast can be directly extracted from the histogram by taking the ratio of the peak value of the echo to the average of the counts in a time interval centered on the echo and equal to the storage time, i.e. . Fig. 3a shows a plot of the ratio as a function of the number of teeth in the AFC. We see that the value of obtained from the raw data (solid blue circles) first increases with , reaches a maximum of for , and then saturates. The main factors that limit the value of are on the one hand the limited retrieval efficiency of the AFC and detector jitter, both of which limit the peak value reached by the echo, and on the other hand noise coming from multi-pair emissions of the source and detector dark counts, both of which increase the denominator of the echo contrast . After noise subtraction (red circles) and additional deconvolution of the finite detector response time (solid green squares), we find a maximum of and respectively, both for . The linearity of the echo contrast with , after compensating for the above effects (see Methods for details), is consistent with the expectation, since greater values of correspond to larger Dicke states.

In Fig. 3b, we plot our numerical bounds for the entanglement depth as a function of echo contrast for and taking into account the experimental values of and , see also Eq. (3). We include other values of for comparison. (Note that in all cases, so is kept essentially constant.) We conclude that the noise-and-jitter-corrected echo contrast of implies at least entangled teeth, where the uncertainty in the entanglement depth is dominated by the uncertainty in .

Each of the entangled teeth above in fact consists of many atoms (of the order , see Methods). As a consequence, each state in Eq. (1) is in fact a type of Dicke state itself. To illustrate this, we first create an AFC with (red trace in Fig. 4a, and extract a minimum entanglement depth of from the data. Using only the atoms that form one of the teeth in the first AFC, we then create a second – nested – AFC, again with . We experimentally determine the minimum entanglement depth for this secondary AFC to be . This procedure of sub-dividing an ensemble, while restricted by the laser linewidth in our experiment, is fundamentally only limited by the homogeneous linewidth of the atoms. In principle, it would be possible to demonstrate entanglement of a very large number of entangled subsystems (and even sub-subsystems etc.) in this way.

Discussion
The remaining mismatch between the obtained value of and the maximum possible value can be explained by other experimental limitations, such as the imperfect creation of the AFC and the Lorentzian spectral-density profile of the photons, which limit the contribution to the interference coming from the teeth that are further detuned. The state that is actually generated in the experiment is thus of the form (not all being equal) rather than a perfect state. However, this only means that our bound on the entanglement depth is conservative, because a state with unequal coefficients has to involve a greater number of entangled teeth in order to achieve the same value of . The most relevant type of decoherence in our experiment is irreversible dephasing due to the finite spectral width of each tooth. This effect accumulates over time and thus affects the observed value of . The entangled state at the time of absorption is therefore likely to have had a higher entanglement depth than what can be inferred from observing the echo, which is emitted some time later.

The multi-partite entanglement that we have demonstrated underpins the operation of AFCs for quantum storage. Looking forward, it has potential for use in quantum metrology (23), provided the storage efficiency can be increased substantially, which is possible e.g. using cavities (31). The AFC system allows one to address individual or sets of teeth in frequency space, which in principle makes it possible to characterize the created quantum state in much more detail. The large ratio of inhomogeneous to homogeneous linewidths in rare-earth doped crystals implies that one can create entanglement between larger numbers of teeth, possibly more than one hundred million. The length and the doping concentration of rare-earth materials can be easily increased, allowing one to additionally greatly increase the number of atoms in each tooth, and hence make each sub-system of the created Dicke state even more macroscopic. Storing entangled photons in AFCs created in separated crystals offers the possibility to study the nature of multi-partite entanglement at yet a higher nesting level. Furthermore our system allows storing more than one photon, which enables the creation of more complex entangled states such as higher-order Dicke states.

We note that entanglement between many individual atoms in a solid is demonstrated in a parallel submission by F. Fröwis et al., using a related, but complementary approach based on the directionality of the echo emission from an atomic frequency comb.

Methods
The Dicke state is the only single-excitation eigenstate of with non-zero eigenvalue.
Here we prove the statement that in the single-excitation subspace of the -teeth Hilbert space the Dicke state is the only state with a non-zero eigenvalue. For this purpose, it is useful to view each qubit as a spin-1/2 system, such that the dipole operators and are now viewed as spin raising and lowering operators, respectively. In this spin representation, the zero-excitation state has total spin and projection , and (which is also completely symmetric and thus belongs to the irreducible representation with maximum total spin) has total spin as well, but projection . In addition to , there are other basis states spanning the single-excitation subspace. They all have the same projection as the Dicke state (i.e. ), but total spin since they are not fully symmetric. They have the lowest possible projection value that is compatible with their total spin value, and for that reason, they are annihilated by the spin lowering operator . Hence, is the only eigenstate of in the single-excitation subspace that has a non-zero eigenvalue.

Deriving bounds on the entanglement depth for given , and . In order to derive lower bounds on the entanglement depth for given , and , it is convenient to - equivalently - derive upper bounds on for given , and . The fully separable example discussed in the text shows that can be much greater than the entanglement depth once is different from zero. For a given , we are allowed to use entangled states of (up to) teeth as the individual factors, where the factors in the fully separable example correspond to . Keeping in mind that in our experiment, it is clear that there should be a significant vacuum component in each factor. As for the non-vacuum part, the Dicke state is clearly a good choice because it is the single-excitation state that maximizes in each -teeth subspace. In fact, it is the optimum choice. Higher-order Dicke states have greater , but not , because they also have greater values for the total number of excitations . Moreover - in contrast to - they require a non-zero (alternatively stated, they make a non-zero contribution to ), whose value is one of our constraints. Higher-order non-Dicke states are clearly sub-optimal. We conclude that the optimal state should contain factors of the form . However, it is not immediately apparent how many such factors there should be. For example, consider the states and . The latter state has a component in the single-excitation subspace and will thus achieve a greater value of compared to . But it also makes a contribution to because of the cross-term that is proportional to , whereas makes no such contribution. This suggests that in general the optimal state may be a mixed state of the form

 ρ=∑iqi|ψi⟩⟨ψi|, (4)

with , , and

 |ψi⟩=(αi|0⟩⊗M+βi|W⟩M)⊗i⊗|0⟩⊗(N−i⋅M), (5)

where such that . Let us first consider the case where is divisible by , in which case one simply has . Note that it is optimal for the coefficients of the (non-vacuum) factors in each to be identical, in order to maximize the overlap with the state in the single-excitation subspace and thus maximize . The ratio and the probabilities and for the state are given by

 R =MP1+2P2k∑n=1n2qn(β2n)(1−β2n)n−1, (6) P1 =q1β21+k∑n=2nqn(β2n)(1−β2n)n−1, (7) P2 =k∑n=2n(n−1)2qn(β4n)(1−β2n)n−2. (8)

To derive the upper bounds on , we maximize Eq. (6) with the constraints of Eqs. (7) and (8) where the values of and are obtained from our experiment. We use a global numerical search algorithm that runs a local nonlinear programming solver which finds the maximum value of a constrained nonlinear multivariable function for multiple start points. It finally reports the global maximum value of for a given .

The numerical maximization shows that only the and components in (4) are non-zero in the optimal case, and the latter is close to one. This can be understood physically. The state has the largest possible number of non-separable states of size , giving a state in the single-excitation subspace, which maximizes . Thus, the weight of this state, , should be as large as possible, and indeed it comes out close to 1 in the maximization. However, the size of is constrained by the value of , which is very small in our case. As a consequence, this state only makes a small contribution to . The remaining contribution to is best provided by a state that does not increase , i.e. .

In general, is not always divisible by and one has such that (while ). In this case the same arguments apply with the small difference that . The optimum values for and are such that the state is obtained in the single-excitation subspace.

Figure 5 shows the maximum possible contrast, , as a function of entanglement depth, , constrained on our experimental values of and . In this figure, the curve is close to a straight line. This can be explained by noting that only and are non-zero (), and is very small, which makes the contribution of to negligible ( for any value of ) resulting in . Thus, one can obtain , which gives and thereby leads to (3).

Theoretical model for obtaining and from experimental data. The atomic state of the AFC system after absorption depends on the photon statistics of the source, on transmission loss, and on inefficiencies such as inefficient single-photon heralding and absorption in the AFC. We now describe a theoretical model that allows extracting and from experimental data.

In our experiment an intense laser beam passes through a spontaneous parametric down conversion crystal (SPDC) that converts photons into entangled pairs of photons traveling in different spatial modes and (see Fig. 6). A non-number-resolving photon detector , henceforth referred to as the heralding detector, is placed in mode , and the AFC system followed by another detector is located in mode . A detection by the detector heralds the presence of at least one photon in mode .

The combined state of the two down-converted photons in mode and can be written as

 |ψ⟩=∞∑n=0ξnn!(a†)n(b†)n|0⟩a|0⟩b, (9)

where

 |ξn|2=μn(μ+1)n+1 (10)

is the thermal photon number distribution. Here, () is the photon creation operator in mode (), is the probability of creating pairs of photons (one photon per pair in mode and one in ), and is the mean photon pair number per temporal mode (per coherence time of the pump laser). Let us note that our source is in fact slightly temporally multi mode, which implies that the distribution should be somewhere in-between thermal and Poissonian. However, here we conservatively assume thermal statistics, which leads to slightly larger values of and thus slightly lower values of the entanglement depth extracted from the measured data (Fig. 3).

Due to loss, a photon created in mode will reach the detector with probability of . It will be detected with a probability given by the detection efficiency . We model the combined (limited) channel transmission and detector efficiency using a beamsplitter with transmission probability of followed by a perfect detector. Thus, the the creation operator for mode transforms as

 a†→√ηaa†+√1−ηaa†l, (11)

where is the creation operator in the loss mode of this hypothetical beamsplitter. With this transformation we can write

 (a†)n=n∑k=0(nk)(√ηaa†)k(√1−ηaa†l)n−k. (12)

The state of the loss mode and the mode together, after a -photon detection in the detector , becomes

 |¯ψ⟩k=∞∑n=k√(nk)ξn√ηak√1−ηa(n−k)|n−k⟩al|n⟩b. (13)

Considering the fact that is a detector that does not resolve photon numbers, we sum over all possible values of in mode and trace out the loss channel from the state . This results in the state of the mode after heralding by as

 ρb =1N∞∑k=1∞∑n=k(nk)|ξn|2ηka(1−ηa)(n−k)|n⟩b⟨n| =∞∑n=1pn|n⟩b⟨n|, (14)

with

 N =∞∑k=1∞∑n=k(nk)|ξn|2ηka(1−ηa)(n−k), (15) pn =∑nk=1(nk)|ξn|2ηka(1−ηa)(n−k)N. (16)

In mode , each photon experiences loss (before entering the AFC) that is modeled by a beam-splitter with transmission probability . Each photon is then absorbed by the AFC with probability . Otherwise, it is transmitted through the AFC and detected by the detector , similarly modeled as a beamsplitter with transmission probability of followed by a perfect detector. Thus, the creation operator for mode transforms as

 b†→ √ηbηwb†+√ηb(1−ηw)ηtb†t +√ηb(1−ηw)(1−ηt)b†tl+√1−ηbb†l, (17)

where , , and are the creation operators in the transmission mode, loss mode before the AFC, and the loss mode after the AFC, respectively. Hence, the associated optical state becomes

 ρb=∑npn|Φn⟩⟨Φn|, (18)

where

 |Φn⟩= 1√n!(√ηbηwb†+√ηb(1−ηw)ηtb†t +√ηb(1−ηw)(1−ηt)b†tl+√1−ηbb†l)n|0⟩.

Conditioning on ”no detection” in the mode (since a detection would correspond to a photon that was not absorbed in the AFC) results in the density matrix

 ~ρb =1M∑n¯pn|~Φn⟩⟨~Φn|, (19)

where

 |~Φn⟩=1(ηbηw+Z2)n/2|¯Φn⟩,|¯Φn⟩=1√n!(√ηbηwb†+ZX†)n|0⟩,¯pn=pn(ηbηw+Z2)n,M=∑npn(ηbηw+Z2)n. (20)

We have used

 Z=√ηb(1−ηw)(1−ηt)+(1−ηb),x†=1Z(√ηb(1−ηw)(1−ηt)b†tl+√1−ηbb†l). (21)

We can rewrite the state in the number basis of the mode and the mode (defined in the previous equation) as

 ~ρb=1M{∞∑n=1pnn∑r,r′=1 √(nr)(nr′)√ηbηwr+r′Z2n−r−r′ |r,n−r⟩b,x⟨r′,n−r′|}. (22)

The probability that photons are present in the AFC is calculated as

 Pr=b⟨r|Trx~ρb|r⟩b,=1M∞∑n=rpn(nr)(ηbηw)rZ2n−2r (23)

Therefore, we can obtain the values of and by measuring , , , , and .

Experimental estimation of . The second order cross-correlation function gives information about the photon-number distribution of a two-mode field. Specifically, for the down-converted photons in mode (1532 nm) and mode (795 nm) at zero time delay (), it can be written as

 g(2)ab(0)=pab(0)papb, (24)

where () is the probability for a detection in mode () and the probability to detect a coincidence in a temporal window centered at . Experimentally we obtained as the ratio between the coincidence rate at (where photons in mode and exhibit maximum correlations) and the coincidence rate at a delay larger than the coherence length of the photons in mode and (where photon creation, and hence detection, in mode and is completely independent).

All experiments were performed at maximum pump power of , for which we measured . In the limit where , the cross correlation function can be written as (32). This allows us to estimate a value of . Note that the uncertainty in is the dominant contribution to the uncertainty of given in the manuscript. Our coincidence measurements were performed using home made logic electronics with a detection window of 5 ns. A characterization of as a function of the pump power can be seen in Fig. 2.

Experimental estimation of , , and . To estimate and , we change the setup represented in Fig. 6 and place the detector (with detection efficiency of ) before the AFC, i.e. the photons in mode only pass through the loss channel with probability before they reach . In this configuration, both and can be estimated from the coincidence detection rate and the single detection rates and . Since in our case, we can write

 Cab=μηa∗ηDaηb∗ηDb/τp,Sa=μηa∗ηDa/τp,Sb=μηb∗ηDb/τp, (25)

where is the coherence time of the pump laser. From the above relations, we find

 ηa=CabSb,ηb∗=CabSaηDb. (26)

We experimentally find and , for which we used . In the setup depicted in Fig. 6, we can write , where is the probability with which a photon that passes through the loss will enter the AFC. Note that the background absorption of the AFC , as shown in Fig. 7, can be treated as loss and is hence included in . Since varies for AFCs with different , and hence also varies. For instance, the AFC with and  GHz for which entanglement depth is calculated in Fig. 3b, results in . For the two AFCs shown in Fig. 4 with and bandwidths of  GHz and  GHz, we estimate and , respectively.

The probability with which a photon that is transmitted through the AFC is detected by the detector , can be written as . Here, is the probability that the photon passes through the loss channel after the AFC, reaches into the detector which has a detection efficiency of . We estimate .

Experimental estimation of AFC write efficiency . The write efficiency of the AFC is the probability for the AFC to absorb a photon. It can be calculated from the effective optical depth of the AFC as

 ηw=1−e−d1F, (27)

where is the peak absorption, is the finesse of the AFC and is the effective optical depth of the comb (see Fig. 7). For the different AFCs that we create, varies, and hence the write efficiency also varies. For instance, the write efficiencies for the broadband AFCs with number of teeth and are and , respectively. For the narrowband AFC with , which is shown in Fig. 4, we estimate .

The dominant contribution to the uncertainty in given in the manuscript comes from the uncertainty in the optical depth of around . The uncertainties associated with the measurements of the other efficiencies are negligible in comparison.

Heralded auto-correlation . The second-order zero-time autocorrelation function is a witness of non-classicality for single-mode fields. To measure , we add a balanced (50/50) beam-splitter (BS) before the detection of the photon in mode b (795 nm photon), and measure the probability to detect a coincidence between the two BS outputs (), as well as the single output probabilities ( and ), all conditioned on the detection of a photon in mode (1532 nm photon). We can now write the autocorrelation function as

 g(2)bb(0)=pb1b2(0)pb1pb2. (28)

For a perfect single-photon state , whereas for a coherent state . We find , see Fig. 2b, proving the nearly ideal nature of our heralded single-photon source. This value is consistent with the value of determined from measuring the cross-correlation function above (33).

Background noise estimation and subtraction. The collection of our experimental data suffers from imperfections that limit the the values of , but are unrelated to the detection of the heralded single photon. In our analysis we account for two such sources of imperfection, namely background noise and limited temporal resolution of the detector, where the latter is treated in the next section. The background noise causes a constant level of detection events, which we can assess by averaging over the detection events before a heralding photon detection as indicated by the red region in the insert of Fig. 3. The background noise is constant at about 0.9 counts per 80 ps data bin for a 15 minute measurement. Since this background noise is completely uncorrelated with the heralded photon we can subtract it in order to obtain a more accurate echo contrast. Clearly, the impact of such noise subtraction will increase when the echo signal is weakened, e.g. due to increased AFC storage times or reduction of the AFC bandwidth, as evidenced by the data shown in Figs. 3, 4 and 8.

We can determine the origin of the background noise from a series of independent measurements. Detector dark counts contribute about 12%, while noise due to leaked optical-pumping light and spontaneous emission from atoms excited during optical pumping amounts to approximately 11%. Finally, the photon pair-source may generate additional pairs other than that causing the heralding event, as discussed above in the context of the second order cross-correlation function. If the 795 nm photon member of one of these additional pairs reaches the detector after the AFC, it will appear as a background noise count. This is the main contribution to the background noise at around 77%. The relative contributions to the background noise depend slightly on the AFC efficiency and loss.

Deconvolution of the detector response. The temporal shape of the heralded photon is given by the Fourier transform of its spectral distribution. At the AFC input the spectrum is defined by the 6 GHz FP filter, whereas after re-emission from the AFC the photon spectrum will depend on the overall bandwidth as well as exact shape of the AFC. However, when the heralded photon is detected, the inherent detector jitter can smear out its temporal shape thus reducing the signal to noise ratio and thereby the echo contrast.

The joint detector response is measured by removing all spectral filtering elements and recording the distribution of correlated detection immediately after the, now extremely broad-band, pair-source. We find that the joint detector response is well approximated by a Gaussian function of full-width-at-half-maximum (FWHM)  ps. Fitting the echoes with a Gaussian function with amplitude and a FWHM of allows us to de-convolute the detector response and compute a corrected signal amplitude of . Note that for the broadband AFCs the output photon duration is on the order of  ps, which is significantly smeared by the detector jitter. As expected, the deconvolution of the detector response leads to a large increase of the echo contrast for the broadband AFCs and less of an increase for narrow-band AFCs, for which the temporal duration of the echoes exceeds the detector jitter — this interplay is evident in Figs. 4 and 8.

Number of atoms corresponding to a single AFC tooth. We calculate the number of atoms that correspond to a single AFC tooth using two complementary approaches. The first method utilizes the experimentally-determined absorption spectrum and the Tm-atom density of the Tm:LiNbO crystal, while the second method relies on single-ion spectroscopic properties.

For the first method, we note that the integrated absorption spectrum of an inhomogeneously-broadened transition of an atomic ensemble is , where is the absorption coefficient resulting from all transitions that feature a resonance frequency , and is the length of the medium. Similarly, the integrated absorption spectrum of a single AFC tooth is , where the integration is taken over the tooth. If a laser beam of cross-sectional area (defined by full-width at half-maximum of the intensity distribution of a Gaussian beam) is sent through a medium that features atom density , then the number of atoms within the beam is , and the number of atoms corresponding to a single tooth is .

For the second method we calculate the optical depth that corresponds to a single atom in a crystal using , where and are the spontaneous emission rate and homogeneous broadening of the transition, respectively, is the index of refraction, and (35). Since refers to an atom that features a linewidth of , we estimate the number of atoms that correspond to a single tooth using .

For the H H transition of Tm:LiNbO, the Tm-atom number density is cm, , kHz, kHz, and 497 cm (34), where . The laser beam that is used to determine the absorption spectra is collimated and has a cross-section of m, and the crystal length is mm. For our AFC with teeth, we measure MHz, giving and atoms per tooth. The difference between the two estimates may be due to the measurement uncertainty or to the fact that not all transitions that contribute to the absorption line have identical properties.

Echo contrast as a function of AFC bandwidth. In order to create a perfect state, the AFC structure should be uniformly illuminated by the incoming photon. This guarantees that each AFC tooth has the same probability to contribute in the photon absorption. In the experiment we use a GHz FWHM bandwidth photon with a Lorentzian profile given by the transmission profile of the used Fabry Perot filtering cavity (FP). The uniformity of the absorption over the different teeth should thus depend on the bandwidth of the AFC. In order to analyse this effect we measure, for a fixed storage time ( ns), the echo contrast as a function of the prepared AFC bandwidth (see Fig. 8). Here is expressed in terms of the percentage with respect the maximum attainable value given the known number of teeth created in each AFC. We observe that, as we narrow the AFC bandwidth, the relative value increases. This is consistent with the expectation that we are approaching the creation of an ideal state. Note that our derived bounds on the entanglement depth are still correct, albeit too conservative, for the case where the created state is not a perfect state, since a state with unequal coefficients has to involve more entangled teeth in order to generate the same value of .

Acknowledgments. We thank F. Fröwis and F. Bussières for very useful discussions and V. Kiselyov for technical support. This work was funded through NSERC and AITF. VBV and SWN acknowledge partial funding for detector development from the Defense Advanced Research Projects Agency (DARPA) Information in a Photon (InPho) program. Part of the detector research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. W.T. acknowledges funding as a Senior Fellow of the Canadian Institute for Advanced Research.

Author Contributions. N.S. and C.S. conceived the project with some input from W.T. The theoretical approach was developed by P.Z., S.G., K.H. and C.S., and the calculations performed by P.Z. and S.G. with guidance from C.S. The experiments were developed and performed by C.D., N.S., G.A., P.L., M.G., and D.O. with guidance from W.T. The detectors were designed and fabricated by V.V., F.M., M.S. and S.W.N. The paper was written by P.Z., C.D., P.L., G.A., S.G., M.G., D.O., N.S., W.T. and C.S.

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