Calorimetry of & symmetry breaking in a photon Bose-Einstein - PowerPoint PPT Presentation

Calorimetry of & symmetry breaking in a photon Bose-Einstein condensate Frank Vewinger Universität Bonn

What are we dealing with? System: 10 4 – 10 5 Photons „ ultracold “: 300K Box: Curved mirror cavity A few µm long 2

1) Photon BEC: HowTo 1) Photon BEC: HowTo 2) Thermodynamic Properties of Photons 2) Thermodynamic Properties of Photons 3) Fluctuations & Symmetry Breaking 3) Fluctuations & Symmetry Breaking Work done with Julian Schmitt Tobias Damm Jan Klaers (now@ETH Zürich) Martin Weitz 3

1) Photon BEC: HowTo 2) Thermodynamic Properties of Photons 3) Fluctuations & Symmetry Breaking 4

dye photon box The box: Dispersion 2 𝐹 = ℏ𝑑 2 ≈ ℏ𝑑 𝑙 𝑨 + 𝑙 𝑠 2 + 𝑙 𝑠 𝑙 𝑨 2 𝑜 𝑜 2 𝑙 𝑨 = 𝜌ℏ𝑑𝑟 𝜌ℏ𝑑𝑟 2 𝑠 2 + ℏ𝑑 𝐸 0 7 ᐧ λ 0 /2 2 + 2𝜌𝑟𝑜 𝑙 𝑠 𝑜 𝐸 0 𝑜 𝑆 𝐸 0 2 + 1 2 𝑛 0 Ω 2 𝑠 2 + 𝑙 𝑠 = 𝑛 0 𝑑 2 2𝑛 0 ⇒ Photons in the microcavity behave as - Massive particles - Two-dimensional - Harmonically trapped 5 Klaers, Vewinger & Weitz, Nature Physics 6 , 512 (2010)

dye photon box The box: Dispersion Dye 2 𝐹 = ℏ𝑑 2 ≈ ℏ𝑑 𝑙 𝑨 + 𝑙 𝑠 2 + 𝑙 𝑠 𝑙 𝑨 2 𝑜 𝑜 2 𝑙 𝑨 = 𝜌ℏ𝑑𝑟 𝜌ℏ𝑑𝑟 2 𝑠 2 + ℏ𝑑 𝐸 0 2 + 2𝜌𝑟𝑜 𝑙 𝑠 𝑜 𝐸 0 𝑜 𝑆 𝐸 0 2 + 1 2 𝑛 0 Ω 2 𝑠 2 + 𝑙 𝑠 = 𝑛 0 𝑑 2 2𝑛 0 ⇒ Photons in the microcavity behave as - Massive particles - Two-dimensional - Harmonically trapped ℏ𝜕 𝑎𝑄𝑀 ≫ 𝑙 𝐶 𝑈 6 Klaers, Vewinger & Weitz, Nature Physics 6 , 512 (2010)

dye photon box Dye 𝛽(𝜑) 𝑔(𝜑) Intensity [a.u.] TEM 6𝑛𝑜 TEM 7𝑛𝑜 TEM 8𝑛𝑜 600 400 500 ν [THz] Dye reservoir: - Thermalizes gas - Sets chemical potential 𝜈 𝛿 ℏ (𝜕 𝐷 −Δ) 𝑙 𝐶 𝑈 = 𝑥 ↓ 𝑁 ↑ ℏ𝜕 𝑎𝑄𝑀 ≫ 𝑙 𝐶 𝑈 𝑙 𝐶 𝑈 𝑓 𝑓 𝑥 ↑ 𝑁 ↓ 7 Klaers, Vewinger & Weitz, Nature Physics 6 , 512 (2010)

dye photon box Dye 𝛽(𝜑) 𝑔(𝜑) Intensity [a.u.] TEM 6𝑛𝑜 TEM 7𝑛𝑜 TEM 8𝑛𝑜 600 400 500 ν [THz] Dye reservoir: - Thermalizes gas - Sets chemical potential 𝜈 𝛿 ℏ (𝜕 𝐷 −Δ) 𝑙 𝐶 𝑈 = 𝑥 ↓ 𝑁 ↑ 𝑙 𝐶 𝑈 𝑓 𝑓 𝑥 ↑ 𝑁 ↓ 8 Klaers, Vewinger & Weitz, Nature Physics 6 , 512 (2010)

Scales Energy scales Dye   Trap frequency  150 µeV  Thermal energy k B T 25 meV  cutoff   Cavity cutoff 2 . 1 eV  Photon mass ≈ 10 −7 𝑛 𝑓  Critical particle number N  2  2 N   k T c   B   N c 80 . 000 @ 300 K     3  Critical phase space density n c  2 1 . 3 / µm 9 Klaers, Schmitt, Vewinger & Weitz, Nature 468 , 545 (2010) See also Marelic & Nyman, PRA 91 , 033826 (2015)

Scales Energy scales Dye   Trap frequency  150 µeV  Thermal energy k B T 25 meV  cutoff   Cavity cutoff 2 . 1 eV  Photon mass ≈ 10 −7 𝑛 𝑓  Critical particle number N  2  2 N   k T c   B   N c 80 . 000 @ 300 K     3  Critical phase space density n c  2 1 . 3 / µm 10 Klaers, Schmitt, Vewinger & Weitz, Nature 468 , 545 (2010) See also Marelic & Nyman, PRA 91 , 033826 (2015)

Scales Energy scales Dye   Trap frequency  150 µeV  Thermal energy k B T 25 meV  cutoff   Cavity cutoff 2 . 1 eV  Photon mass ≈ 10 −7 𝑛 𝑓  Critical particle number N  2  2 N   k T c   B   N c 80 . 000 @ 300 K     3  Critical phase space density n c  2 Brute force theory: 1 . 3 / µm Appl Phys B 105, 17 – 33 (2011) Microscopic models: de Leeuw, PRA 88, 033829 (2013). Kirton/Keeling, PRL 111,100404 (2013) 11 Kopylov et al., PRA 92, 063832 (2015) Klaers, Schmitt, Vewinger & Weitz, Nature 468 , 545 (2010) See also Marelic & Nyman, PRA 91 , 033826 (2015)

Experimental setup 12

Experimental setup 13

Experimental setup 14

Properties? 15

Properties? 16

1) Photon BEC: HowTo 2) Thermodynamic Properties of Photons 3) Fluctuations & Symmetry Breaking 17

Condensate Fraction 1000 1.0 2 𝑈 Condensate fraction n 0 /n 1 − 𝑈 𝐷 0.8 𝑙 B 𝑈 100 Signal (a.u.) 0.6 10 0.4 1 0.2 0 0.1 1.5 0.5 1.0 560 570 580 0 Wavelength λ (nm) Temperature T/T c Bose-Einstein distribution 𝑈 𝑈 𝑑 = 𝑂 c 𝑂 ≈ 80.000 18 Damm , Schmitt,… Nature Comm. 7,11340 (2016)

Internal Energy 5 4 c Energy (U- ℏ ω c ) / Nk B T classical limit 3 (Maxwell-Boltzmann) criticality 2 1 0 0 0.5 1.0 1.5 2.0 2.5 Temperature T/T 19 c

Internal Energy Atomic 5 Phys. Rev. Lett. 77, 4984-4987 (1996) systems 4 c Energy (U- ℏ ω c ) / Nk B T 3 2 Phys. Rev. A 90, 043640 (2014) 1 0 2.5 0 0.5 1.0 1.5 2.0 Temperature T/T 20 c

Specific Heat 5 cusp singularity at criticality 4 Specific heat C / Nk B 3 2 2 k B in 2D (equipartition theorem) 1 0 0 0.5 1.0 1.5 2.0 2.5 Temperature T/T 21 c Klünder & Pelster, EPJB 68, 457 (2009): C=4.38 k B T, TD limit

Specific Heat 5 Liquid 4 He Phys. Rev. B 68, 174518 (2003) 4 Specific heat C / Nk B 3 2 Science 335 , 563-567 (2012) Fermions ( 6 Li) 1 0 0 0.5 1.0 1.5 2.0 2.5 Temperature T/T 22 c Klünder & Pelster, EPJB 68, 457 (2009): C=4.38 k B T, TD limit

Entropy per Particle 5 𝑈 𝑈′ 𝑒𝑈 ′ 𝑇 𝑈 = 𝐷 𝑈′ 4 0 Entropy S / N 3 2 𝑇/𝑂 → 0 (third law of thermodynamics) 1 0 0 0.5 1.0 1.5 2.0 2.5 Temperature T/T 23 c

Entropy per Particle 5 𝑈 𝑈′ 𝑒𝑈 ′ 𝑇 𝑈 = 𝐷 𝑈′ 4 0 Entropy S / N 3 2 𝑇/𝑂 → 0 (third law of thermodynamics) 1 0 0 0.5 1.0 1.5 2.0 2.5 Temperature T/T 24 c

1) Photon BEC: HowTo 2) Thermodynamic Properties of Photons 3) Fluctuations & Symmetry Breaking 25

Coherence of a Bose-Einstein condensate P. W. Anderson (1986): "Do two superfluids which have never 'seen' one another possess a relative phase?" Spontaneous symmetry breaking F Damping of density fluctuations Phase selection: long-range order 2 T > T c g (2) ( 𝜐 ) P n 1 2 T < T c g (2) ( 𝜐 ) P n 1 Andrews et al., Science 275 (1997) n 𝜐 Öttl et al., PRL 95 (2005) 26 14

Coherence of a Bose-Einstein condensate P. W. Anderson (1986): "Do two superfluids which have never 'seen' one another possess a relative phase?" Spontaneous symmetry breaking Closed vs. open system F Isolation Reservoir ∆ n ,∆ E ≃ 0 ∆ n ,∆ E ≠ 0 Damping of density fluctuations Phase selection: long-range order 2 T > T c g (2) ( 𝜐 ) P n 1 2 T < T c g (2) ( 𝜐 ) P n 1 Andrews et al., Science 275 (1997) n 𝜐 Öttl et al., PRL 95 (2005) → BEC correlations in open environments? 27 14

Heat bath and particle reservoir for light Heat bath Molecules Photon M = 10 8 – 10 12 condensate , Δ𝐹, Δ𝑜 𝑜 𝑈, 𝜈 Grand canonical ensemble, Ω 𝑈, 𝑊, 𝜈 if 𝑁 ≫ 𝑜 Photon number distribution P n Master equation Statistics crossover 28 Klaers et al., PRL 108 (2012) Sob’yanin , PRE 85 (2012) 1 5

Limiting cases for BEC number statistics Canonical ensemble ( 𝑁 < 𝑜 2 ) Grand-canonical ensemble ( 𝑁 ≫ 𝑜 2 ) Poisson statistics ("quiet" BEC) Bose-Einstein statistics ("flickering" BEC) n =10 3 , M =10 7 n =10 3 , M =10 4 ɸ (t) ɸ (t) n(t) n(t) quasi-spin Time, Time, 2 2 𝑕 (2) (𝜐) Autocorrelation 𝑕 (2) (𝜐) 𝑕 2 0 = 2 𝜐 c(2) 𝑕 2 0 = 1 ≙ 1 1 Fluctuation level 𝜐 Delay, 𝜐 29 1 6

Experiment: intensity correlations of BEC Condensate fraction: Photon statistics Time evolution (PMT) 4% 1 16% 2 Probability, P n 3 28% 4 Reservoir (fixed size) 58% Time, t (ns) BEC photons Schmitt et al., Phys. Rev. Lett. 112 , 030401 (2014) see also: Physics 7 (2014) 30 1 7

Experiment: intensity correlations of BEC Condensate fraction: Photon statistics Time evolution (PMT) 4% 1 16% 2 Probability, P n 3 28% 4 Reservoir (fixed size) 58% Time, t (ns) BEC photons Crossover from Autocorrelation, g (2) ( 𝜐 ) Bose-Einstein 𝜐 c(2) ≃ 1.4 ns ( ) to Poisson statistics ( ) 0 2 4 6 8 Schmitt et al., Phys. Rev. Lett. 112 , 030401 (2014) Delay, 𝜐 (ns) see also: Physics 7 (2014) 31 1 7

Temporal phase evolution Response of condensate phase to statistical fluctuations? 32 1 8

Temporal phase evolution Response of condensate phase to statistical fluctuations? Canonical ensemble ( , second-order coherence) I ( t ) regular beating without phase jumps ⤷ Schmitt et al., Phys. Rev. Lett. 116 , 033604 (2016) 33 1 8

Temporal phase evolution Response of condensate phase to statistical fluctuations? Phase jumps Canonical ensemble ( , second-order coherence) Grand-canonical ensemble ( , intensity fluctuations) I ( t ) I ( t ) ( 𝚫 PJ-1 = 20ns) regular beating without phase jumps ⤷ Schmitt et al., Phys. Rev. Lett. 116 , 033604 (2016) 34 1 8