Bose–Einstein condensation of photons in an erbium–ytterbium co-doped fiber cavity

Bose–Einstein condensation (BEC) is a special many-boson phenomenon that was observed in atomic particles at ultra-low temperatures. Later, BEC was also shown for non-atomic bosons, such as photons. Those experiments were usually done in micron-size cavities, where the power (particle number) was varied, and not the temperature, until condensation was reached. Here we demonstrate BEC of photons in a few-meters-long one-dimensional (1D) erbium–ytterbium co-doped fiber cavity at, below and above room temperature, between 100 K and 415 K. The experiments were done at about the 1550 nm wavelength regime having a few to tens of μW intra-cavity light power (107−108 photons). By varying the power and also the temperature, we found linear dependence of the condensation on power for various temperatures and of the critical power (for condensation) on temperature. These findings agree, functionally and quantitatively, with the theoretical BEC prediction without any adjustable parameter.


Supplementary Note 1: The rate equations model with numerical calculations
To understand the thermalization and the condensation processes in EDF and EYDF cavities, with and without cutoff filters, and to distinguish between these cases, we give a simple numerical model that describes the signal and pump propagation along the fibre (z-axis). Most features can be obtained by the following simple distributed rate equations (and some important aspects can be obtained even by a single lumped rate equation that we describe below) 1 Supplementary equation (1) (3) and (4) describe the population dynamics of the ground and excited levels 12 , NN . We use the two-level approximation that assumes very fast transition from the pump excited levels (Yb) to the excited levels 2 (Er). We note that each of these levels is broad and comprises a band of sublevels. The parameters description and their values are listed in Supplementary Table 1.

Parameter Symbol Value Unit
Position along the fibre z 0-10 m Frequency

Supplementary Note 2: Core vs. cladding pumped cavities, with and without a cutoff filter
The pump/signal-core-mode overlap parameter p  controls the pump behavior: For core pumping (in the EDF case) p ~1, and for the cladding pumping EYDF case p  is small (slightly above zero). The filter is modeled by the transfer function () F  ) that multiplies the signal at z=9 m (1 m before the output coupler). We assume for simplicity an ideal filter: Supplementary Fig. 1 shows the results obtained with the model for cladding and core pumped cavities with a cutoff filter. Fig.1a shows for the cladding pumping case (EYDF), thermalization spectra and then a continuous transition to condensation as the pumping power is increased. This picture fits the experimental results in the present paper. For the core pumping case, Fig. 1b shows development to thermalization spectra and a jump and abrupt transition to condensation (and hysteresis) when the pumping is increased 1,2 .
To better understand the differences between the two cases we show in Supplementary  Fig. 1c,d the spectra at z=1 m, close to the pump origin, and in Supplementary Fig. 1e,f the population ratio 2 d / N N as a function of z. For the cladding pumping case the spectra at z=1 m are similar to the output spectra, and the population ratio remains almost constant, independent of z. (In the experiment we keep the pumping and the population ratio even more uniform by pumping from two sides). This means that every point in the cavity sees the same conditions, and even the lumped system rate equation that is 1 : However, for the core pumped case, below threshold, the pump is mostly absorbed after a short distance from the input and therefore the excited levels population is high only near the pump input. The signal in this region shows typical spectra of erbium emission, but as it propagates in the un-pumped region it thermalizes, mainly by absorption. Above threshold, there is enough signal to maintain the excited state population at the required threshold value which is equal to the cladding pumping case. This state is therefore similar to the condensate state of the cladding pumping case.
As shown in Supplementary Fig. 1, our model also explains the tail in the spectrum below the cutoff frequency ( 1568 ) nm   . Although the filter is ideal, there is a buildup of spontaneous emission at the 1 m long fibre between the filter and the output coupler. This spontaneous emission is not affected by the cavity conditions since it takes less than one roundtrip to build it up. We believe that the spontaneous emission in one roundtrip which is unaffected by the cavity mirrors or the density of states is also the reason for the spectrum tail in the microcavity experiment 3 . Supplementary Fig. 2 shows the results obtained with the model for cladding and core pumped cavities without filters. Supplementary Fig. 2a,b shows the output power spectra for the two cases. For the cladding pumping case (Fig. 2a), the spectra show thermalization and then oscillation, (probably lasing) in multiple modes between (1600-1620) nm. The transition between thermalization to lasing is continuous and resembles the condensation transition, but since the cutoff mode is not well defined, and the emission and absorption cross sections at the region (1600-1620) nm don't agree with the McCumber (or Kennard-Stepanov) relation 1,4,5,6,7,8 there is an undefined state of oscillation. The results shown in Supplementary Fig. 2b for the core pumping case can be compared to the experimental results we reported in Supplementary Ref. 1 that show a broad thermalization region and a jump and abrupt transition to spectra with a broad thermalization region and oscillation at the band edge, above 1600 nm, when the pumping was increased. The spectra for the cladding pumping case can be compared to the experimental results in Supplementary Fig. 3 shown here. Without a filter there is no BEC power accumulation near the cutoff wavelength at 1568 nm and the thermalization region is extended to wavelengths above 1600 nm up to the point where the emission-absorption region ends. The experimental results show oscillation at ~1610 nm on top of the thermalized spectra in a large but defined band.

Cladding pumped cavity with filter
Core pumped cavity with filter Supplementary Figure 1: Results obtained with the model for cladding and core pumped cavities with a cutoff filter. The output power spectra for (a) -cladding pumped cavity, and for (b) -core pumped cavity. The spectra inside the cavity at z=1 m for (c) -cladding pumped cavity, and for (d) -core pumped cavity. The population ratio of the excited level N2/Nd as a function of z for (e) -cladding pumped cavity, and for (f) -core pumped cavity.

Cladding pumped cavity without filter
Core pumped cavity without filter Supplementary Figure 2: Results obtained with the model for cladding and core pumped cavities without cutoff filters. The output power spectra for (a) -cladding pumped cavity, and for (b) -core pumped cavity. The spectra inside the cavity at position z=1 m for (c)cladding pumped cavity, and for (d) -core pumped cavity. The population ratio of the excited level N2/Nd as a function of z for (e) -cladding pumped cavity, and for (f) -core pumped cavity. Figure 3: Experimental spectra measurements at 296 K of a cladding pumped cavity without a filter. The dashed line shows the theoretical slope of the BE distribution for this temperature.

Supplementary Note 3: Additional data on the emission and absorption cross sections and the amplified spontaneous emission spectra
We give here additional data supporting the claims for thermalization and BEC. If the cavity has a low loss-rate that supports many absorption-emission cycles the signal spectra will not be dominated by the spectral shape of the emission and absorption cross sections. Instead, it depends only on their ratio, which is temperature dependent. To show that the spectra we measured in the cavity are indeed very different from the usual gain spectra, we experimentally measured the amplified spontaneous emission (ASE) spectra. Supplementary Fig. 4 shows experimental ASE in a 1 m long open double-clad EYDF that we used in the present paper for different pump power levels at three temperature values 100 K, 296 K and 413 K that cover the range that we used. They show a small effect of the temperature in the ranges we used. We emphasize that the thermalizaion drastically transforms the whole spectrum to a very broad BE distribution (where the initial spectrum is less important to this change), and the generation of BEC near the cutoff when the power is high enough, as we see in this paper and Supplementary Ref. 1 (on thermalization). Since there is a difference between spontaneous emission that is usually taken for short distances and ASE (that is here taken for 1 m propagation in the fibre) we used for our modeling the spectra given in Supplementary Fig. 5. Figure 4: Experimental spectra of ASE in an open 1 m EYDF. They are given for three temperatures and different pumping power levels (that are not specified since only a small part of it is absorbed in the fibre in the cladding pumping case.) The temperatures, 100 K, 296 K and 413 K, cover the range that we used (room, lowest and highest values). They show that there is a relatively small effect of the temperature on the ASE spectra in the temperature range that we used. We emphasize that upon a long enough propagation the thermalizaion drastically transforms the whole spectrum to a BE distribution and eventually to BEC.