Lecture 25. Bose-Einstein Condensation (Ch. 7 ) BEC and BEC of photons related phenomena (lasers) Townes Basov Prokhorov Nobel 1964 BEC in a weakly- BEC in a strongly- interacting system interacting system (atomic gases) (superfluid 4He) Nobel 2001 Einstein described the phenomenon of Landau Kapitsa condensation in an ideal gas of particles with Nobel 1962 Nobel 1978 nonzero mass in 1925. In the 1930’s Fritz London realized that superfluity 4He can be understood on terms of BEC. However, the analysis of superfluity 4He is complicated by the fact that the 4He atoms in liquid strongly interact with each other. 70 years after the Einstein prediction, the BEC in weakly interacting Bose systems has been experimentally demonstrated - by laser cooling of a system of weakly- interacting alkali atoms in a magnetic trap. Ideal Gas of Conserved Bosons Two types of bosons: (a) Composite particles which contain an even number of fermions. These number of these particles is conserved if the energy does not exceed the dissociation energy (~ MeV in the case of the nucleus). (b) Particles associated with a field, of which the most important example is the photon. These particles are not conserved: if the total energy of the field changes, particles appear and disappear. The chemical potential of such particles is zero in equilibrium, regardless of density. The occupancy cannot be negative for any ε, thus, for Critical density for bosons: bosons, μ≤0 (ε varies from 0 to ∞). Also, as T→0, μ → 0 2/3 2/3 ∞ g()ε ⎡ 2s+ 1⎛ 2 m⎞ ⎤s2+ 1⎛ 2 mk⎞ T ∞ x 2/1 n = dε=⎢ g() ε = ⎜ ⎟ ε 2/1 ⎥ = ⎜ B ⎟ dx ∫ expβ ε− μ −1 4π2 2 4π 2 2 ∫ exp x−βμ −1 0 []() ⎣⎢ ⎝ h ⎠ ⎦⎥ ⎝ h ⎠ 0 [] ∞ x 2/1 Since μ≤0, the maximum possible dx ≈ 3.1 π value of n is obtained when μ = 0, ∫exp()x − 1 μ 0 T 2/3 s2+ 1⎛ 2 mk⎞ T B 3/2 ncr = 3.1 π 2 ⎜ 2 ⎟ = 6.2 nQ 2 4π ⎝ h ⎠ h ⎡ n ⎤ TC ≅ ⎢ ⎥ 2π mk 2 . 6s + 2 1 where nQ is the quantum concentration, B ⎣ ()⎦ which varies as T 3/2 Approaching the Critical Density... What happens if we reduce the temperature at a given density n until ncr drops below n? ∞ x dx n= A ×() k T2/3 2/3 n B ∫ exp x−βμ −1 s2+ 1⎛ 2 mkB ⎞ T 0 [] ncr = 3.1 π ⎜ ⎟ 4π 2 ⎝ 2 ⎠ h μ (n,T) TC T T 3/2 2 ⎡ n ⎤ h (apart from a numerical factor of order unity, TC is equal TC = 2π ⎢ ⎥ mk2B .⎣ 6()s 2+ ⎦ 1 to the Fermi temperature of a Fermi gas with density n) As β becomes larger (T smaller), lμl must decrease (μ - increase) to maintain fixed density. However, since μ cannot be positive for a BE system, at some T=TC the decrease of lμl cannot compensate the rapid decrease of the T-dependent pre-factor. Paradox: we wanted to maintain n= N/V fixed, and now we see that we are not allowed to do this at sufficiently low temperatures. Of course there is no physical reason why we cannot continue lowering the temperature at fixed density (or increasing the density at fixed temperature). Particles in the Ground State are Missing... Resolving the paradox: The problem is caused by the behavior of the 3D density of states and our use of the continuum approximation. Because g(ε)=0 at ε=0, our calculations of n ignored all the particles in the ground (ε=0) state. At low energies, we have to take into account the discreteness of the quantum states. g(ε) n(ε) ) ε ( n × × = ) gε()∝ ε ε ( g ε ε ε For a Fermi system, this neglect would be of no consequence – there are only 2 fermions at zero energy. For a Bose system, the number of particles in the ground state at sufficiently low T becomes huge: 1 1 k T f ()0 = ≈ =B >>1 exp−()βμ −1 −βμ μ Because μ→0 at a non-zero T, the ground state can accommodate any number of particles! It was a bad idea to ignore the ground level. 2 2 The ground state of an ideal Bose gas: h 2 2 2 h =ε 1() + 1 + 1 = 3 1 8mL2 8mL2 In this lowest state, all the wavefunctions look like that: (we’ve chosen the energy of this state as our zero energy). Bose-Einstein Condensation We can discuss the ideal Bose gas in the same terms of a phase transition. That is, at a critical value of temperature, TC, μ(n,T) reaches the limit of μ = 0 and stops increasing. Beyond this point, the relation 2/3 2⎛ π 2 m ⎞ ∞ εd ε n = ⎜ 2 ⎟ ∫ π ⎝ h ⎠ 0 expβ[]() ε− μ −1 is no longer able to keep track of all the particles – we miss the particles in the ground state. Below TC, bosons begin to condense into the ground state. The abrupt accumulation of bosons in the ground state is called Bose-Einstein condensation. μ (n,T) 2 3/2 2 h ⎡ n ⎤ 0 . 53⎛ h ⎞ 3/2 T = 2π TC ()= s 0 = ⎜ ⎟ ()n TC T C ⎢ ⎥ k ⎜ 2π m ⎟ mk2B .⎣ 6()s 2+ ⎦ 1 B ⎝ ⎠ 2/3 2/3 The eq. n(T) with μ = 0 still works at ⎛ 2πmk⎞ T ⎛ T ⎞ n = 6.2 B = n⎜ ⎟ TT< T<TC for calculating the number of ε >0 ⎜ 2 ⎟ ⎜ ⎟ []C particles not in the ground state: ⎝ h ⎠ ⎝ TC ⎠ 2/3 ) The density of ε ⎡ ⎤ ( ⎛ ⎞ T < T particles in the T n C n=0 n −ε n =⎢1 n ⎜ − ⎟ ⎥ TT[]< C × ⎜ ⎟ ) ground state: ⎢ ⎝ TC ⎠ ⎥ ε ⎣ ⎦ ( g n nε>0 ε a tremendous number of particles all sitting n0 in the very lowest available energy state TC T BE Condensation vs. Gas-to-Liquid Condensation Classical physics analogy: let’s fill a container with a non-ideal gas and start lowering the temperature. The gas density remains constant until the condensation of gas (vapor) occurs. Since the density of liquid is much higher than that of gas, the gas density decreases with T. Despite this similarity, the Bose-Einstein condensation is an entirely different phenomenon. One essential difference is that in the gas-to-liquid transition, which is due to interparticle attraction, the liquid and gas phases occupy different regions of space. The BE condensation is driven by exchange interactions. Each particle in the BE condensate has a wave function that fills the entire volume of the container. The BE condensation is the condensation in the momentum space. The phase separation occurs in the k-space, not in the coordinate space. The condensed bosons have essentially zero momentum (i.e., their wave vector k is as small as the size of the container permits). A common misconception about BE condensation is that it requires “brute force” cooling, when kBT becomes much smaller than the differences of energies of the quantum states of the system - that would be a trivial effect, though the temperatures would have been inaccessibly low for any macroscopic system. The point is that condensation can happen at much higher temperatures, when kBT is still large compared to the inter- level Δε. Some Numbers... Interestingly, the expression for the critical temperature that we got works even for liquid helium, despite the fact that this is a strongly-interacting liquid, not a dilute gas. For 4He: 145 kg/m3 n = 3= . 6 ⋅ 4 10mol/m23 = . 2 ⋅ 28 10atoms/m3 4 g/mol 2 3/2 −34 2 0 . 53⎛ h ⎞ ⎛ N ⎞ 0 . 53 6 . 6⋅ 10 3/2 T = ⎜ ⎟ = ( 2 .) 2 10⋅28 ≈ 3 . 1 K C ⎜ ⎟ ⎜ ⎟ −23 −27 () kB ⎝ 2π m ⎠ ⎝ V ⎠1 . 382⋅ 10 4×π 1 × . 7 ⋅ 10 - only ~40% higher than the actual superfluid transition temperature (2.17 K). To ensure weakness of interactions, the experimenters should work with dilute atomic gases. In this case, the critical temperature is much lower. One of the record-high values of TC – for atomic hydrogen (BEC was achieved in 1998 at MIT). The density was 20 3 1.8·10 atoms/m : which, according to our equation for TC, corresponds to TC = 51 μK, in a nice agreement with the exp. value TC = 50 μK. 2 3/2 −34 2 0 . 53⎛ h ⎞ ⎛ N ⎞ 0 . 53 6 . 6⋅ 10 3/2 T = ⎜ ⎟ = ( 1 .) 8⋅ 1020 ≈ 5 . ⋅ − 15 K 10 C ⎜ ⎟ ⎜ ⎟ −23 −27 () kB ⎝ 2π m ⎠ ⎝ V ⎠1 . 382⋅ 10 1×π 1 × . 7 ⋅ 10 The calculated TC = 51 μK is in a nice agreement with the exp. value TC = 50 μK. At this density, the distance between the atoms is ~ 104 times greater than the Bohr radius, and the interatomic interactions are extremely weak. The condensation is driven by statistics rather than by interactions! Realization of BEC in a Dilute 87Rb Vapor In principle, the lighter the bosons, the greater TCBE condensation of For example, the . excitons (light-induced electron-hole pairs) in semiconductors has been observed before the BE condensation in dilute gases (electron is a ferion, mbut an electron-hole pair has an integer spin). First observation of the BEC with weakly-interacting gases was observed with relatively heavy atoms of 87rubidium-87 atoms were confined within a 10,000 Rb. “box” with dimensions ~ 10 μm (he density ~ 10t19 m-3). The spacing between the energy levels: 2 −34 2 3h 3 6 .( 6⋅ 10) −32 −10 Δε~ ε1 =2 = 1 . 1≈2 10 ⋅ J ⇒ 8 ⋅ 10 K 8mL878 1× . 7 ⋅− 1027 () × −5 10 The transition was observed at ~ 0.1 μK.