Polarization of a Bose-Einstein Condensate of Photons in a Dye-Filled Microcavity
PPolarization of a Bose-Einstein Condensateof Photons in a Dye-Filled Microcavity
S. Greveling, F. van der Laan, H. C. Jagers, and D. van Oosten ∗ Debye Institute for Nanomaterials Science & Center for Extreme Matter and Emergent Phenomena,Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands (Dated: July 13, 2018)We measure the polarization of a photon gas in a dye-filled microcavity. The polarization isobtained by a single-shot measurement of the Stokes parameters. We find that the polarization ofboth the thermal cloud and the Bose-Einstein condensate of photons (phBEC) does not differ fromshot to shot. In the case of the phBEC, we find that the polarization correlates with the polarizationof the pump pulse. The polarization of the thermal cloud is independent of parameters varied inthe experiment and is governed by a hidden anisotropy in the system.
Introduction — In many systems, ranging from con-densed matter physics and particle physics to cosmology,phase transitions and spontaneous symmetry breakingplay a crucial role. The simplest symmetry to be brokenis the U(1) symmetry; the symmetry of the overall com-plex phase of the wave function of a system. Phases thatshow the broken symmetry include superconductors [1],superfluids [2] and Bose-Einstein condensates (BEC) [3–5]. The associated phase transitions have been observedin many different systems under various conditions, yetthe actual breaking of the symmetry can only be observedindirectly, as the absolute phase in quantum mechanicsis not a measurable quantity.Richer physics can be observed when looking at thesymmetry breaking of more complex symmetries. Incondensed matter physics, beautiful examples have beenstudied using spin systems in the context of spinorBECs [6, 7], magnetism [8] and spintronics [9, 10], wheresymmetry breaking can lead to the formation of magneticdomains separated by domain walls. In these examples,the symmetry is broken because the system chooses aparticular direction for the spin degree of freedom.The realization of BEC of exciton-polaritons [11, 12]and of photons [13–15] opens up new possibilities. Afterall, both the phase and the direction of the electromag-netic field are observable quantities. In the case of thephBEC, an interesting property of the photon gas is itspolarization because the formation of a polarized con-densate from an unpolarized thermal cloud constitutes adirectly observable example of symmetry breaking. Thisraises the question: is the polarization symmetry spon-taneously broken and hence different for every single ph-BEC? This subject was recently discussed in theoreticalwork by Moodie et al. [16], who developed a model for thepolarization dynamics in a dye-filled microcavity whichtakes into account the polarization states of light, and theeffects of angular diffusion of the dye on the polarizationstate.In this letter, we experimentally study the polarizationof the photon gas by imaging the Stokes parameters on ∗ Corresponding author: [email protected] a single-shot basis. We find that both the thermal cloudand the condensate are polarized. We vary experimentalparameters to investigate the breaking of the symmetry.The polarization of the thermal cloud is independent ofthese parameters, and identical in every shot of the exper-iment. We conclude it is pinned by a hidden anisotropyin the system. However, the polarization of the conden-sate is fully determined by the pump polarization. Thesymmetry breaking is therefore induced.
Setup — The polarimetry setup used in this work isshown in Fig. 1. Light escaping through the back mirrorof the microcavity enters the polarimetry setup throughthe object plane, from where it is split and imaged ontofour areas of a single camera chip of a scientific com-plementary metal oxide semiconductor (sCMOS) camerawith a dynamic range of 16 bits . The image scale onthe camera is 702 nm / px. Special care is taken that thefour paths from the object plane to the sCMOS chip havethe same length, such that all four images are in focus.In each of the paths polarizing elements are placed suchthat they transmit light corresponding to I tot , I ◦ , I ◦ ,and I RHC , where I tot denotes the total intensity of thelight. The other subscripts denote the angle of linearpolarization of light, and right-handed circularly (RHC)polarized light.With these four intensities, the Stokes parameters areobtained [17] S = I tot ,S = I ◦ − I ◦ ,S = I ◦ − I − ◦ ,S = I LHC − I RHC , (1)which is one way to fully describe the polarization oflight.The total degree of polarization p is given by thesquared sum of S , S , and S : p = (cid:113) S + S + S (cid:30) S . (2) Andor, Zyla 5.5 sCMOS a r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec In order to work with the Stokes parameters, they arecombined into a vector known as the Stokes vector S : S = S S S S . (3)As one can observe from Fig. 1, the polarimetry setupconsists of many optical elements that could effect thepolarization of the experimental signal. Since we do notwant to measure each individual element separately, wecalibrate the setup independently. We define a cameravector C consisting of the signal in the four pixels corre-sponding to the same physical position. Using the Stokesvector we formally describe the effect of our polarimetrysetup as a 4 × M which transforms the Stokesvector into the camera vector, i.e. M · S = C , m . . . m ... . . . ... m . . . m · S S S S = C tot C ◦ C ◦ C RHC , (4)where C i denotes the components of the vector C .Using light sources with known Stokes vectors, we de-termine the matrix elements m ij for our polarimetrysetup. For more details on the calibration matrix, seethe supplementary information. After M is determined,we invert it to transform the camera vectors (for eachpixel) into a Stokes vector. We assume that each pixel ofthe chip is equally sensitive to polarization. FIG. 1. Schematic overview of the polarimetry setup. Thecolored paths indicate the different polarization states mea-sured on the Andor Zyla. From the intensities we determinethe two-dimensional (2D) Stokes parameters. A typical resultof the (2D) Stokes parameters, on a single-shot basis is shownin the inset. The contribution to the polarization for eachStokes parameter per pixel is indicated by the correspondingcolor bar.
Experiment — During the experiment the photon gasinside the cavity is imaged for each excitation pump pulseof 500 ns with a repetition rate of 8 Hz, using the po-larimetry setup. For each pump pulse we also take abackground measurement. We thus determine for eachpump pulse the full polarization of the photon gas insidethe cavity.The number of photons in the cavity is set by the powerof the pump pulse. To investigate whether the photondensity influences the degree of polarization, the pumppower is increased for each consecutive shot in the se-quence keeping the polarization of the pump pulse con-stant.The experimental sequence is performed for differentpump pulse polarizations and for a total of three differ-ent concentrations of Rhodamine 6G dissolved in ethy-lene glycol: 1 .
5, 10 .
5, and 14 . et al. [16], the rotational diffusion constant ofthe solvent can influence the polarization of the phBEC.We therefore also perform the experimental sequence forRhodamine 6G dissolved in methanol with a concen-tration of 1 . Results — Using the measured intensities and the in-verse of our matrices M , we determine the Stokes param-eters for every individual pixel. A typical result is shownin the inset of Fig. 1. . Here, the top left image showsthe total intensity, i.e. S , normalized to the maximumpixel count. In the false color image, the condensate andthermal cloud are clearly visible. The thermal cloud isidentified by the purple color, whereas the phBEC cor-responds to the bright yellow center of the image. Theother three images in the inset show the three 2D Stokesparameters S , S , and S , which are normalized to S .From S , S , and S one observes a clear difference − . − . . . . D e g r ee o f p o l a r i z a t i o n +1 . − . S . − . S . − . S p − − − N o r m a li ze d i n t e n s i t y S r [ µ m ] FIG. 2. Radial average of 2D Stokes parameters S (red), S (blue), and S (orange) and p (purple), averaged over 50identical phBECs. For each panel the degree of polarizationis plotted as a function of the distance from the center r .Additionally, each panel also contains the radial average of S (green), plotted on a logarithmic scale (right axis) as afunction of r . between the thermal cloud and the condensate, showingthat they have different polarizations. In the case of thephBEC, one observes that S and S are both close zero,but that S is close to 0 .
5, indicating that the phBECis mostly linearly polarized in the horizontal direction.For the thermal cloud, one observes that all three Stokesparameters fluctuate around zero in the periphery of theimages.Using 50 images containing a phBEC, we determine thecenter of the condensate with subpixel accuracy. Fromthe center we average the data radially outwards for eachStokes parameter. For photon gases created under iden-tical conditions, the 2D Stokes parameters do not dif-fer significantly from one another, which indicates thatthe polarization of the condensate and the thermal cloudare fixed in our system. We therefore average the radialStokes parameters over 50 BECs created under identicalconditions. An example is given in Fig. 2. In the firstthree panels the Stokes parameters S , S , and S areplotted as function of the distance r from the center ofthe trap. In the fourth panel of Fig. 2 the total degreeof polarization as given by Eq. 2 is plotted as a functionof r . The sharp peak close to the center is the phBEC,whereas the exponentially decaying signal for larger dis-tances corresponds to the thermal cloud. The root meansquare uncertainty for S , S , S , and p is indicated bytheir associated pastel color.From Fig. 2 one observes from the fourth panel that thephBEC is strongly polarized with a degree of polarizationat the trap center of p = 0 . ± .
23. For increasing r one observes that the degree of polarization decreases.From a range of r = 25 − µ m, the degree of polar-ization remains approximately constant with a value of p = 0 . ± .
12. As one can observe from the radial pro-file of S , r = 25 µ m corresponds to the spatial size of thephBEC; for larger distances only the thermal cloud re-mains. For r ≥ µ m, the Stokes parameters and theiruncertainty diverge due to near-zero experimental signalin the periphery. . . . . . . . . . N tot /N c . . . . . . D e g r ee o f p o l a r i z a t i o n EdgeCenter
FIG. 3. Degree of polarization for the center (orange) andedge(green) of the photon gas as a function of the condensatefraction. When N c is exceeded, a phBEC forms in the center. From the first three panels of Fig. 2 we observe thatthe main polarization contribution of the condensate islinear in the horizontal direction. The polarization of thethermal cloud does not have a large single polarizationcontribution, although S dominates.In Fig. 3 we plot the degree of polarization of the cen-ter and the edge of the photon gas as a function of thecopndensate fraction N tot /N c , where N tot denotes thetotal number of photons in the system and N c the crit-ical number of photons. For lower pump powers, i.e. N tot /N c <
1, only a thermal cloud is created. For N tot /N c ≥ N tot /N c <
1, the center measurement correspondsto a thermal cloud, but for N tot /N c ≥
1, it also con-tains a contribution from the condensate, which becomesmore important as N tot /N c becomes larger. The edgemeasurement denotes the degree of polarization of thethermal cloud, averaged over a range of r = 25 − µ m.For N tot < N c one observes from Fig. 3 that the degree E G : . m M E G : . m M E G : . m M M : . m M . . . . . . . D e g r ee o f P o l a r i z a t i o n CenterEdge
FIG. 4. Degree of polarization for the center (orange) andedge (green) of the photon gas as function of different dyesolutions and solvents. The average condensate fraction is1 . ± . − . − . . . . D e g r ee o f p o l a r i z a t i o n +1 . − . S . − . S . − . S p − − − N o r m a li ze d i n t e n s i t y S r [ µ m ] FIG. 5. Radial average of 2D Stokes parameters averagedover 50 identical phBECs created using a vertically polarizedpump pulse. The colors and labels are identical to those ofFig. 2. of polarization for the center and the edge of the thermalcloud are the same. For N tot > N c the degree of polar-ization in the center differs from that of the edge of thethermal cloud and the degree of polarization in the cen-ter increases for increasing N tot . This trend is consistentwith the assumption that the condensate is fully polar-ized in the horizontal direction, as discussed by Moodie etal. [16].In Fig. 4 we summarize the results of experiments withdifferent dye concentrations and solvents. Each bar inFig. 4 represents the averaged results of phBECs with acondensate fraction of 1 . ± .
07. As one can observefrom Fig. 4, the dye concentration does not significantlyinfluence the polarization results. The degree of polariza-tion and the main contribution remains the same for boththe condensate and the thermal cloud for every concen-tration. Changing the dye solvent from ethylene glycol tomethanol also does not significantly influence the results.All the results above are taken using the same polar-ization for the pump pulse: horizontal polarization. Theradially averaged Stokes parameters obtained using a ver-tically polarized pump pulse are shown in Fig. 5. Here,the radial profiles are averaged over 50 phBECs takenunder identical conditions, similar to to Fig. 2.From the figure one observes that the phBEC re-mains strongly polarized with a degree of polarizationof 0 . ± .
21. The result for S stands out as the polar-ization contribution described by this Stokes parameterchanged from the horizontal direction to the vertical di-rection. The polarization of the condensate follows thepolarization of the pump pulse. The main contributionsof S and S remain close to zero. The degree of polariza-tion for r = 25 − µ m remains approximately constantwith a value of p = 0 . ± .
12. For the thermal cloud,the contributions have not changed with respect to Fig. 2; the main contribution remains S . Conclusion — We investigate the symmetry break-ing properties of a phBEC in a dye-filled microcavity byimaging the polarization of the photon gas inside ourmicrocavity, on a single-shot basis. We show that thedegree of polarization is identical for every condensateand thermal cloud that we create under identical exper-imental conditions. For increasing condensate fractions,we show that the degree of polarization at the centerof the experimental signal increases. This is consistentwith the assumption that the phBEC is fully polarizedin the direction of the pump and thus yields a largercontribution for increasing condensate fractions, which isin agreement with the theoretical model by Moodie etal. [16]. The degree of polarization of the thermal cloudis not influenced by varying the condensate fraction, dyeconcentration, the dye solvent, and in particular is notsensitive to the pump polarization.We show that the dye concentration or the dye solventdoes not influence the results. Changing the pump polar-ization does not influence the polarization of the thermalcloud. The main contribution remains linear polarizationunder − ◦ , independent of the parameters we varied inthe experiment. The polarization therefore seems to begoverned by a hidden anisotropy. However, changing thepump polarization does change the polarization of the ph-BEC. The symmetry breaking is thus not spontaneous,but induced by the pump polarization. Acknowledgements — It is a pleasure to thank Arjonvan Lange, Javier Hernandez Rueda, Erik van der Wurff,Henk Stoof, Peter van der Straten, and Robert Nymanfor useful discussions. This work is part of the Nether-lands Organization for Scientific Research (NWO). [1] G. M. 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In the case that the optical elements used in our po-larimetry setup would be perfect, the calibration matrixwould be given by M theory = . . . . . . . (5)This is however not the case. Using a white light laserand an acousto optic tunable filter (AOTF) we determinethe calibration matrix for four different wavelengths; 570,580, 590, and 600 nm. Using achromatic λ/ λ/ M
570 nm = . − . − . − . .
39 0 .
38 0 . − . . − .
01 0 . − . . − .
01 0 .
33 0 . , (6) M
580 nm = . − . − . − . .
39 0 .
39 0 . − . .
46 0 .
00 0 . − . . − .
02 0 .
32 0 . , (7) M
590 nm = .
79 0 .
00 0 .
04 0 . .
40 0 .
40 0 . − . . − .
01 0 . − . . − .
02 0 .
36 0 . , (8)and M
600 nm = .
86 0 .
02 0 . − . .
44 0 .
44 0 .
01 0 . . − .
02 0 . − . . − .
02 0 .
38 0 . ..