1. Statistical methods for bosons Lecture 1. 19 th December 2012 by Bed ř ich Velický

2. 2 Short version of the lecture plan Lecture 1 Lecture 2 Introductory matter BEC in extended non-interacting systems, ODLRO Atomic clouds in the traps; Confined independent bosons, what is BEC? Atom-atom interactions, Fermi pseudopotential; Gross-Pitaevski equation for extended gas and a trap Infinite systems: Bogolyubov-de Gennes theory, BEC and symmetry breaking, coherent states Dec 19 Jan 9

3. The classes will turn around the Bose-Einstein condensation in cold atomic clouds comparatively novel area of research largely tractable using the mean field approximation to describe the interactions only the basic early work will be covered, the recent progress is beyond the scope

4. 4 Nobelists The Nobel Prize in Physics 2001 "for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates" Eric A. CornellWolfgang Ketterle Carl E. Wieman 1/3 of the prize USAFederal Republic of Germany USA University of Colorado, JILA Boulder, CO, USA Massachusetts Institute of Technology (MIT) Cambridge, MA, USA University of Colorado, JILA Boulder, CO, USA b. 1961b. 1957b. 1951

5. I. Introductory matter on bosons

6. 6 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable

7. 7 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Cannot be labelled or numbered

8. 8 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state Cannot be labelled or numbered

9. 9 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state

10. 10 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state

11. 11 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state fermionsbosons antisymmetric  symmetric 

12. 12 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state fermionsbosons antisymmetric  symmetric  half-integer spininteger spin comes from nowhere "empirical fact"

13. 13 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state fermionsbosons antisymmetric  symmetric  half-integer spininteger spin comes from nowhere "empirical fact" Finds justification in the relativistic quantum field theory

14. 14 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state fermionsbosons antisymmetric  symmetric  half-integer spininteger spin electronsphotons comes from nowhere "empirical fact" Finds justification in the relativistic quantum field theory

15. 15 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state fermionsbosons antisymmetric  symmetric  half-integer spininteger spin electronsphotons everybody knowsour present concern comes from nowhere "empirical fact" Finds justification in the relativistic quantum field theory

16. 16 Bosons and Fermions (capsule reminder) Independent particles (… non-interacting) basis of single-particle states (  complete set of quantum numbers)

17. 17 Bosons and Fermions (capsule reminder) Independent particles (… non-interacting) basis of single-particle states (  complete set of quantum numbers) FOCK SPACE Hilbert space of many particle states basis states … symmetrized products of single-particle states for bosons … antisymmetrized products of single-particle states for fermions specified by the set of occupation numbers 0, 1, 2, 3, … for bosons 0, 1 … for fermions

18. 18 Bosons and Fermions (capsule reminder) Independent particles (… non-interacting) basis of single-particle states (  complete set of quantum numbers) FOCK SPACE Hilbert space of many particle states basis states … symmetrized products of single-particle states for bosons … antisymmetrized products of single-particle states for fermions specified by the set of occupation numbers 0, 1, 2, 3, … for bosons 0, 1 … for fermions

19. 19 Bosons and Fermions (capsule reminder) Representation of occupation numbers (basically, second quantization) …. for fermions Pauli principle fermions keep apart – as sea-gulls

20. 20 Bosons and Fermions (capsule reminder) Representation of occupation numbers (basically, second quantization) …. for fermions Pauli principle fermions keep apart – as sea-gulls

21. 21 Bosons and Fermions (capsule reminder) Representation of occupation numbers (basically, second quantization) …. for bosons princip identity bosons prefer to keep close – like monkeys

22. 22 Bosons and Fermions (capsule reminder) Representation of occupation numbers (basically, second quantization) …. for bosons princip identity bosons prefer to keep close – like monkeys

23. Who are bosons ? elementary particles quasiparticles complex massive particles, like atoms … compound bosons

24. 24 Examples of bosons bosons elementary particles quasi particles photons phonons magnons simple particles N not conserved complex particles N conserved atomic nuclei excited atoms atoms

25. 25 Examples of bosons (extension of the table) bosons elementary particles quasi particles compound quasi particles atomic nuclei excited atoms ions molecules photons phonons magnons excitons Cooper pairs simple particles N not conserved complex particles N conserved atoms

26. 26 Question: How a complex particle, like an atom, can behave as a single whole, a compound boson ESSENTIAL CONDITIONS 1)All compound particles in the ensemble must be identical; the identity includes o detailed elementary particle composition o characteristics like mass, charge or spin 2)The total spin must have an integer value 3)The identity requirement extends also on the values of observables corresponding to internal degrees of freedom 4)which are not allowed to vary during the dynamical processes in question 5)The system of the compound bosons must be dilute enough to make the exchange effects between the component particles unimportant and absorbed in an effective weak short range interaction between the bosons as a whole

27. 27 Example: How a complex particle, like an atom, can behave as a single whole, a compound boson RUBIDIUM -- THE FIRST ATOMIC CLOUD TO UNDERGO BEC single element Z = 37 single isotope A = 87 single electron configuration

28. 28 Example: How a complex particle, like an atom, can behave as a single whole, a compound boson RUBIDIUM -- THE FIRST ATOMIC CLOUD TO UNDERGO BEC single element Z = 37 single isotope A = 87 single electron configuration 37 electrons 37 protons 50 neutrons total spin of the atom decides total electron spin total nuclear spin Two distinguishable species coexist; can be separated by joint effect of the hyperfine interaction and of the Zeeman splitting in a magnetic field

29. Atomic radius vs. interatomic distance in the cloud 29 http://intro.chem.okstate.edu/1314f00/lecture/chapter7/lec111300.html

30. II. Homogeneous gas of non-interacting bosons The basic system exhibiting the Bose-Einstein Condensation (BEC) original case studied by Einstein

31. 31 Cell size (per k vector) Cell size (per p vector) In the (r, p)-phase space Plane wave in classical terms and its quantum transcription Discretization ("quantization") of wave vectors in the cavity volume periodic boundary conditions Plane waves in a cavity

32. 32 IDOS Integrated Density Of States: How many states have energy less than  Invert the dispersion law Find the volume of the d-sphere in the p-space Divide by the volume of the cell DOS Density Of States: How many states are around  per unit energy per unit volume Density of states

33. 33 Ideal quantum gases at a finite temperature: a reminder mean occupation number of a one- particle state with energy 

34. 34 Ideal quantum gases at a finite temperature mean occupation number of a one- particle state with energy 

35. 35 Ideal quantum gases at a finite temperature fermionsbosons NN FD BE mean occupation number of a one- particle state with energy 

36. 36 Ideal quantum gases at a finite temperature fermionsbosons NN FD BE mean occupation number of a one- particle state with energy 

37. 37 Ideal quantum gases at a finite temperature fermionsbosons NN FD BE mean occupation number of a one- particle state with energy 

38. 38 Ideal quantum gases at a finite temperature fermionsbosons NN FD BE mean occupation number of a one- particle state with energy  Aufbau principle

39. 39 Ideal quantum gases at a finite temperature fermionsbosons NN freezing out FD BE mean occupation number of a one- particle state with energy  Aufbau principle

40. 40 Ideal quantum gases at a finite temperature fermionsbosons NN freezing out ? FD BE mean occupation number of a one- particle state with energy  Aufbau principle

41. 41 Ideal quantum gases at a finite temperature fermionsbosons NN freezing out BEC? FD BE mean occupation number of a one- particle state with energy  Aufbau principle

42. Bose-Einstein condensation: elementary approach

43. 43 Einstein's manuscript with the derivation of BEC

44. 44 A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) atoms: mass m, dispersion law system as a whole: volume V, particle number N, density n=N/V, temperature T.

45. 45 A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) atoms: mass m, dispersion law system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem:

46. 46 A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) atoms: mass m, dispersion law system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem: Always  < 0. At high temperatures, in the thermodynamic limit, the continuum approximation can be used:

47. 47 A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) atoms: mass m, dispersion law system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem: Always  < 0. At high temperatures, in the thermodynamic limit, the continuum approximation can be used: It holds For each temperature, we get a critical number of atoms the gas can accommodate. Where will go the rest?

48. 48 A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) atoms: mass m, dispersion law system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem: Always  < 0. At high temperatures, in the thermodynamic limit, the continuum approximation can be used: It holds For each temperature, we get a critical number of atoms the gas can accommodate. Where will go the rest? This will be shown in a while

49. 49 A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) atoms: mass m, dispersion law system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem: Always  < 0. At high temperatures, in the thermodynamic limit, the continuum approximation can be used: It holds For each temperature, we get a critical number of atoms the gas can accommodate. Where will go the rest? To the condensate This will be shown in a while

50. 50 Gas particle concentration The integral is doable: Riemann function use the general formula

51. 51 Gas particle concentration The integral is doable: Riemann function

52. Bose-Einstein condensation: critical temperature

53. 53 Gas particle concentration The integral is doable: Riemann function CRITICAL TEMPERATURE the lowest temperature at which all atoms are still accomodated in the gas:

54. 54 Critical temperature The integral is doable: Riemann function CRITICAL TEMPERATURE the lowest temperature at which all atoms are still accomodated in the gas: atomic mass

55. 55 Critical temperature A few estimates: systemMnTCTC He liquid4 1  10 28 3.54 K Na trap23 1  10 20 2.86  K Rb trap87 1  10 19 95 nK CRITICAL TEMPERATURE the lowest temperature at which all atoms are still accomodated in the gas:

56. 56 Digression: simple interpretation of T C Rearranging the formula for critical temperature we get mean interatomic distance thermal de Broglie wavelength The quantum breakdown sets on when the wave clouds of the atoms start overlapping

57. 57 de Broglie wave length for atoms and molekules Thermal energies small … NR formulae valid: At thermal equilibrium Two useful equations thermal wave length

58. 58 Ketterle explains BEC to the King of Sweden

59. Bose-Einstein condensation: condensate

60. 60 Condensate concentration GAS condensate fractionfraction

61. 61 Condensate concentration GAS condensate fractionfraction

62. 62 Condensate concentration GAS condensate fractionfraction

63. 63 Where are the condensate atoms? ANSWER: On the lowest one-particle energy level For understanding, return to the discrete levels. There is a sequence of energies For very low temperatures, all atoms are on the lowest level, so that

64. 64 Where are the condensate atoms? Continuation ANSWER: On the lowest one-particle energy level For temperatures below all condensate atoms are on the lowest level, so that question … what happens with the occupancy of the next level now? Estimate:

65. 65 Where are the condensate atoms? Summary ANSWER: On the lowest one-particle energy level The final balance equation for is LESSON: be slow with making the thermodynamic limit (or any other limits)

66. III. Physical properties and discussion of BEC Off-Diagonal Long Range Order Thermodynamics of BEC

67. 67 Closer look at BEC Thermodynamically, this is a real phase transition, although unsual Pure quantum effect There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions) BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space. This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave BEC is a macroscopic quantum phenomenon in two respects:  it leads to a correlation between a macroscopic fraction of atoms  the resulting coherence pervades the whole macroscopic sample

68. 68 Closer look at BEC Thermodynamically, this is a real phase transition, although unsual Pure quantum effect There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions) BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space. This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave BEC is a macroscopic quantum phenomenon in two respects:  it leads to a correlation between a macroscopic fraction of atoms  the resulting coherence pervades the whole macroscopic sample

69. 69 Closer look at BEC Thermodynamically, this is a real phase transition, although unsual Pure quantum effect There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions) BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space. This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave BEC is a macroscopic quantum phenomenon in two respects:  it leads to a correlation between a macroscopic fraction of atoms  the resulting coherence pervades the whole macroscopic sample

70. 70 Closer look at BEC Thermodynamically, this is a real phase transition, although unsual Pure quantum effect There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions) BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space. This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave BEC is a macroscopic quantum phenomenon in two respects:  it leads to a correlation between a macroscopic fraction of atoms  the resulting coherence pervades the whole macroscopic sample

71. 71 Closer look at BEC Thermodynamically, this is a real phase transition, although unsual Pure quantum effect There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions) BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space. This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave BEC is a macroscopic quantum phenomenon in two respects:  it leads to a correlation between a macroscopic fraction of atoms  the resulting coherence pervades the whole macroscopic sample

72. 72 Closer look at BEC Thermodynamically, this is a real phase transition, although unsual Pure quantum effect There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions) BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space. This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave BEC is a macroscopic quantum phenomenon in two respects:  it leads to a correlation between a macroscopic fraction of atoms  the resulting coherence pervades the whole macroscopic sample

73. Off-Diagonal Long Range Order Beyond the thermodynamic view: Coherence of the condensate in real space Analysis on the one-particle level

74. 74 Coherence in BEC: ODLRO Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix. Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state: Off-Diagonal Long Range Order

75. 75 Coherence in BEC: ODLRO Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix. Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state:

76. 76 Coherence in BEC: ODLRO Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix. Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state:

77. 77 Coherence in BEC: ODLRO Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix. Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state:

78. 78 Coherence in BEC: ODLRO Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix. Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state:

79. 79 OPDM for homogeneous systems In coordinate representation depends only on the relative position (transl. invariance) Fourier transform of the occupation numbers isotropic … provided thermodynamic limit is allowed in systems without condensate, the momentum distribution is smooth and the density matrix has a finite range. CONDENSATE lowest orbital with

80. 80 OPDM for homogeneous systems: ODLRO CONDENSATE lowest orbital with

81. 81 OPDM for homogeneous systems: ODLRO CONDENSATE lowest orbital with DIAGONAL ELEMENT r = r' for k 0 = 0

82. 82 OPDM for homogeneous systems: ODLRO CONDENSATE lowest orbital with DIAGONAL ELEMENT r = r' DISTANT OFF-DIAGONAL ELEMENT | r - r' |  Off-Diagonal Long Range Order ODLRO

83. 83 From OPDM towards the macroscopic wave function CONDENSATE lowest orbital with MACROSCOPIC WAVE FUNCTION expresses ODLRO in the density matrix measures the condensate density appears like a pure state in the density matrix, but macroscopic expresses the notion that the condensate atoms are in the same state is the order parameter for the BEC transition

84. Capsule on thermodynamics

85. 85 Homogeneous one component phase: boundary conditions (environment) and state variables

86. 86 Homogeneous one component phase: boundary conditions (environment) and state variables The important four

87. 87 Homogeneous one component phase: boundary conditions (environment) and state variables The important four The one we use presently

88. 88 Digression: which environment to choose? THE ENVIRONMENT IN THE THEORY SHOULD CORRESPOND TO THE EXPERIMENTAL CONDITIONS … a truism difficult to satisfy  For large systems, this is not so sensitive for two reasons System serves as a thermal bath or particle reservoir all by itself Relative fluctuations (distinguishing mark) are negligible  Adiabatic system Real system Isothermal system SB heat exchange – the slowest medium fast the fastest process  Atoms in a trap: ideal model … isolated. In fact: unceasing energy exchange (laser cooling). A small number of atoms may be kept (one to, say, 40). With 10 7, they form a bath already. Besides, they are cooled by evaporation and they form an open (albeit non-equilibrium) system.  Sometime, N =const. crucial (persistent currents in non-SC mesoscopic rings) S B temperature lag interface layer

89. 89 Thermodynamic potentials and all that Basic thermodynamic identity (for equilibria) For an isolated system, For an isothermic system, the independent variables are T, V, N. The change of variables is achieved by means of the Legendre transformation. Define Free Energy

90. 90 Thermodynamic potentials and all that Basic thermodynamic identity (for equilibria) For an isolated system, For an isothermic system, the independent variables are T, V, N. The change of variables is achieved by means of the Legendre transformation. Define Free Energy New variables: perform the substitution everywhere; this shows in the Maxwell identities (partial derivatives)

91. 91 Thermodynamic potentials and all that Basic thermodynamic identity (for equilibria) For an isolated system, For an isothermic system, the independent variables are T, V, N. The change of variables is achieved by means of the Legendre transformation. Define Free Energy Legendre transformation: subtract the relevant product of conjugate (dual) variables New variables: perform the substitution everywhere; this shows in the Maxwell identities (partial derivatives)

92. 92 Thermodynamic potentials and all that Basic thermodynamic identity (for equilibria) For an isolated system, For an isothermic system, the independent variables are T, S, V. The change of variables is achieved by means of the Legendre transformation. Define Free Energy

93. 93 A table isolated systemmicrocanonical ensemble internal energy isothermic systemcanonical ensemble free energy isothermic-isobaric systemisothermic-isobaric ensemble free enthalpy isothermic open systemgrand canonical ensemble grand potential How comes ? is additive, is the only additive independent variable. Thus, Similar consideration holds for

94. Grand canonical thermodynamic functions of an ideal Bose gas General procedure for independent particles Elementary treatment of thermodynamic functions of an ideal gas Specific heat Equation of the state

95. 95 Quantum statistics with grand canonical ensemble Grand canonical ensemble admits both energy and particle number exchange between the system and its environment. The statistical operator (many body density matrix) acts in the Fock space External variables are. They are specified by the conditions Grand canonical statistical operator has the Gibbs' form

96. 96 Grand canonical statistical sum for ideal Bose gas Recall

97. 97 Grand canonical statistical sum for ideal Bose gas Recall

98. 98 Grand canonical statistical sum for ideal Bose gas Recall

99. 99 Grand canonical statistical sum for ideal Bose gas Recall

100. 100 Grand canonical statistical sum for ideal Bose gas Recall

101. 101 Grand canonical statistical sum for ideal Bose gas Recall valid for - extended "ïnfinite" gas - parabolic traps just the same

102. Using the statistical sum 102

103. 103 Ideal Boson systems at a finite temperature fermionsbosons NN ? FD BE Equation for the chemical potential closes the equilibrium problem:

104. 104 Thermodynamic functions for an extended Bose gas For Born-Karman periodic boundary conditions, the lowest level is Its contribution has to be singled out, like before:    **

105. 105 Thermodynamic functions for an extended Bose gas For Born-Karman periodic boundary conditions, the lowest level is Its contribution has to be singled out, like before:    HOW TO PROCEED Start from . This we did already. Below T C Thus, the singular term in  is negligible for, there is none in 

106. 106 Thermodynamic functions for an extended Bose gas The following equation of state results Integrating by parts, Eqs.  and  are found as nearly identical This is an extension of the result known for classical gases Series expansion ** **  **

107. 107 Thermodynamic functions for an extended Bose gas Series expansion above the critical temperature: Results simplify in the condensation region: All temperature dependences are explicit

108. 108 Specific heat of an ideal Bose gas A weak singularity … what decides is the coexistence of two phases

109. 109 Specific heat: comparison liquid 4 He vs. ideal Bose gas Dulong-Petit (classical) limit  - singularity weak singularity quadratic dispersion law linear dispersion law

110. 110 Isotherms in the P-V plane For a fixed temperature, the specific volume can be arbitrarily small. By contrast, the pressure is volume independent …typical for condensation ISOTHERM

111. 111 Compare with condensation of a real gas CO 2 Basic similarity: increasing pressure with compression critical line beyond is a plateau Differences: at high pressures at high compressions

112. 112 Fig. 151 Experimental isotherms of carbon dioxide (CO 2 }. Compare with condensation of a real gas CO 2 Basic similarity: increasing pressure with compression critical line beyond is a plateau Differences: at high pressures at high compressions  no critical point Conclusion: BEC in a gas is a phase transition of the first order

113. IV. Non-interacting bosons in a trap

114. 114 Useful digression: energy units

115. 115 Trap potential (physics involved skipped) Typical profile coordinate/ microns  ? evaporation cooling This is just one direction Presently, the traps are mostly 3D The trap is clearly from the real world, the atomic cloud is visible almost by a naked eye

116. 116 Trap potential Parabolic approximation in general, an anisotropic harmonic oscillator usually with axial symmetry 1D 2D 3D

117. 117 Ground state orbital and the trap potential level number 200 nK 400 nK characteristic energy characteristic length

118. 118 Ground state orbital and the trap potential level number 200 nK 400 nK characteristic energy characteristic length

119. 119 Ground state orbital and the trap potential level number 200 nK 400 nK characteristic energy characteristic length

120. 120 Filling the trap with particles: IDOS, DOS 1D 2D "thermodynamic limit" only approximate … finite systems better for small meaning wide trap potentials For the finite trap, unlike in the extended gas, is not divided by volume !!

121. 121 Filling the trap with particles 3D Estimate for the transition temperature particle number comparable with the number of states in the thermal shell For 10 6 particles, characteristic energy

122. 122 Filling the trap with particles 3D Estimate for the transition temperature particle number comparable with the number of states in the thermal shell For 10 6 particles, characteristic energy

123. 123 The general expressions are the same like for the homogeneous gas. Working with discrete levels, we have and this can be used for numerics without exceptions. In the approximate thermodynamic limit, the old equation holds, only the volume V does not enter as a factor: In 3D, Exact expressions for critical temperature etc.

124. 124 How sharp is the transition These are experimental data fitted by the formula The rounding is apparent, but not really an essential feature

125. 125 Seeing the condensate – reminder Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix. Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state:

126. 126 OPDM in the Trap Use the eigenstates of the 3D oscillator Use the BE occupation numbers Single out the ground state zero point oscillations absorbed in the chemical potential

127. 127 OPDM in the Trap Use the eigenstates of the 3D oscillator Use the BE occupation numbers Single out the ground state zero point oscillations absorbed in the chemical potential Coherent component, be it condensate or not. At, it contains ALL atoms in the cloud Incoherent thermal component, coexisting with the condensate. At, it freezes out and contains NO atoms

128. 128 OPDM in the Trap, Particle Density in Space The spatial distribution of atoms in the trap is inhomogeneous. Proceed by definition: as we would write down naively at once Split into the two parts, the coherent and the incoherent phase

129. 129 OPDM in the Trap, Particle Density in Space Split into the two parts, the coherent and the incoherent phase

130. 130 OPDM in the Trap, Particle Density in Space Split into the two parts, the coherent and the incoherent phase The characteristic lengths directly observable

131. 131 Particle Density in Space: Boltzmann Limit We approximate the thermal distribution by its classical limit. Boltzmann distribution in an external field:

132. 132 Particle Density in Space: Boltzmann Limit We approximate the thermal distribution by its classical limit. Boltzmann distribution in an external field: For comparison:

133. 133 Particle Density in Space: Boltzmann Limit We approximate the thermal distribution by its classical limit. Boltzmann distribution in an external field: Two directly observable characteristic lengths For comparison:

134. 134 Particle Density in Space: Boltzmann Limit We approximate the thermal distribution by its classical limit. Boltzmann distribution in an external field: Two directly observable characteristic lengths For comparison: anisotropy given by analogous definitions of the two lengths for each direction

135. 135 Real space Image of an Atomic Cloud the cloud is macroscopic basically, we see the thermal distribution a cigar shape: prolate rotational ellipsoid diffuse contours: Maxwell – Boltzmann distribution in a parabolic potential

136. 136 Particle Velocity (Momentum) Distribution The procedure is similar, do it quickly:

137. 137 Thermal Particle Velocity (Momentum) Distribution Again, we approximate the thermal distribution by its classical limit. Boltzmann distribution in an external field: Two directly observable characteristic lengths Remarkable: Thermal and condensate lengths in the same ratio for positions and momenta

138. 138 Three crossed laser beams: 3D laser cooling the probe laser beam excites fluorescence. the velocity distribution inferred from the cloud size and shape 20 000 photons needed to stop from the room temperature braking force propor- tional to velocity: viscous medium, "molasses" For an intense laser a matter of milliseconds temperature measure- ment: turn off the lasers. Atoms slowly sink in the field of gravity simultaneously, they spread in a ballistic fashion

139. 139 Three crossed laser beams: 3D laser cooling the probe laser beam excites fluorescence. the velocity distribution inferred from the cloud size and shape 20 000 photons needed to stop from the room temperature braking force propor- tional to velocity: viscous medium, "molasses" For an intense laser a matter of milliseconds Temperature measurement: turn off the lasers. Atoms slowly sink in the field of gravity simultaneously, they spread in a ballistic fashion

140. 140 Three crossed laser beams: 3D laser cooling the probe laser beam excites fluorescence. the velocity distribution inferred from the cloud size and shape 20 000 photons needed to stop from the room temperature braking force propor- tional to velocity: viscous medium, "molasses" For an intense laser a matter of milliseconds Temperature measurement: turn off the lasers. Atoms slowly sink in the field of gravity

141. 141 Three crossed laser beams: 3D laser cooling the probe laser beam excites fluorescence. the velocity distribution inferred from the cloud size and shape 20 000 photons needed to stop from the room temperature braking force propor- tional to velocity: viscous medium, "molasses" For an intense laser a matter of milliseconds Temperature measurement: turn off the lasers. Atoms slowly sink in the field of gravity simultaneously, they spread in a ballistic fashion

142. 142 Three crossed laser beams: 3D laser cooling the probe laser beam excites fluorescence. the velocity distribution inferred from the cloud size and shape 20 000 photons needed to stop from the room temperature braking force propor- tional to velocity: viscous medium, "molasses" For an intense laser a matter of milliseconds Temperature measurement: turn off the lasers. Atoms slowly sink in the field of gravity simultaneously, they spread in a ballistic fashion

143. 143 BEC observed by TOF in the velocity distribution

144. 144 BEC observed by TOF in the velocity distribution Qualitative features:  all Gaussians  wide vs.narrow  isotropic vs. anisotropic

145. BUT The non-interacting model is at most qualitative Interactions need to be accounted for

146. 146 Importance of the interaction – synopsis Without interaction, the condensate would occupy the ground state of the oscillator (dashed - - - - -) In fact, there is a significant broadening of the condensate of 80 000 sodium atoms in the experiment by Hau et al. (1998), The reason … the interactions experiment perfectly reproduced by the solution of the Gross – Pitaevski equation

147. Next time !!!

148. The end

149. 149

150. V. Interacting atoms

151. 151 Are the interactions important? In the dilute gaseous atomic clouds in the traps, the interactions are incomparably weaker than in liquid helium. That permits to develop a perturbative treatment and to study in a controlled manner many particle phenomena difficult to attack in He II. Several roles of the interactions the atomic collisions take care of thermalization the mean field component of the interactions determines most of the deviations from the non-interacting case beyond the mean field, the interactions change the quasi-particles and result into superfluidity even in these dilute systems

152. 152 Fortunate properties of the interactions 1.Strange thing: the cloud lives for seconds, or even minutes at temperatures, at which the atoms should form a crystalline cluster. Why? For binding of two atoms, a third one is necessary to carry away the released binding energy and momentum. Such ternary collisions are very unlikely in the rare cloud, however. 2.The interactions are elastic and spin independent: they do not spoil the separation of the hyperfine atomic species and preserve thus the identity of the atoms. 3.At the very low energies in question, the effective interaction is typically weak and repulsive … which enhances the formation and stabilization of the condensate.

153. 153 Fortunate properties of the interactions 1.Strange thing: the cloud lives for seconds, or even minutes at temperatures, at which the atoms should form a crystalline cluster. Why? For binding of two atoms, a third one is necessary to carry away the released binding energy and momentum. Such ternary collisions are very unlikely in the rare cloud, however. 2.The interactions are elastic and spin independent: they do not spoil the separation of the hyperfine atomic species and preserve thus the identity of the atoms. 3.At the very low energies in question, the effective interaction is typically weak and repulsive … which enhances the formation and stabilization of the condensate. Quasi-Equilibrium

154. 154 Interatomic interactions For neutral atoms, the pairwise interaction has two parts van der Waals force strong repulsion at shorter distances due to the Pauli principle for electrons Popular model is the 6-12 potential: Example:  corresponds to ~12 K!! Many bound states, too.

155. 155 Interatomic interactions minimum  vdW radius For neutral atoms, the pairwise interaction has two parts van der Waals force strong repulsion at shorter distances due to the Pauli principle for electrons Popular model is the 6-12 potential: Example:  corresponds to ~12 K!! Many bound states, too.

156. 156 Interatomic interactions The repulsive part of the potential – not well known The attractive part of the potential can be measured with precision Even this permits to define a characteristic length

157. 157 Interatomic interactions The repulsive part of the potential – not well known The attractive part of the potential can be measured with precision Even this permits to define a characteristic length rough estimate of the last bound state energy compare with 

158. 158 Scattering length, pseudopotential Beyond the potential radius, say, the scattered wave propagates in free space For small energies, the scattering is purely isotropic, the s-wave scattering. The outside wave is For very small energies,, the radial part becomes just This may be extrapolated also into the interaction sphere (we are not interested in the short range details) Equivalent potential ("Fermi pseudopotential")

159. 159 Experimental data asas

160. 160 Useful digression: energy units

161. 161 Experimental data nm -1.4 4.1 -1.7 -19.5 5.6 127.2 NOTES weak attraction ok weak repulsion ok weak attraction intermediate attraction weak repulsion ok strong resonant repulsion "well behaved; monotonous increase seemingly erratic, very interesting physics of scattering resonances behind for “ordinary” gasesVLT clouds asas nm 3.4 4.7 6.8 8.7 10.4

162. 162 Experimental data  (  K) 5180 940 --- 73 --- nm -1.4 4.1 -1.7 -19.5 5.6 127.2 NOTES weak attraction ok weak repulsion ok weak attraction intermediate attraction weak repulsion ok strong resonant repulsion "well behaved; monotonous increase seemingly erratic, very interesting physics of scattering resonances behind for “ordinary” gasesVLT clouds asas nm 3.4 4.7 6.8 8.7 10.4

163. 163 Experimental data nm 3.4 4.7 6.8 8.7 10.4 nm -1.4 4.1 -1.7 -19.5 5.6 127.2 NOTES weak attraction ok weak repulsion ok weak attraction intermediate attraction weak repulsion ok strong resonant repulsion "well behaved; monotonous increase seemingly erratic, very interesting physics of scattering resonances behind for “ordinary” gasesVLT clouds asas

164. VI. Mean-field treatment of interacting atoms

165. 165 This is an educated way, similar to (almost identical with) the HARTREE APPROXIMATION we know for many electron systems. Most of the interactions is indeed absorbed into the mean field and what remains are explicit quantum correlation corrections Many-body Hamiltonian and the Hartree approximation We start from the mean field approximation. self-consistent system

166. 166 Hartree approximation at zero temperature Consider a condensate. Then all occupied orbitals are the same and Putting we obtain a closed equation for the order parameter: This is the celebrated Gross-Pitaevskii equation. This is a single self-consistent equation for a single orbital, the simplest HF like theory ever.

167. 167 Hartree approximation at zero temperature Consider a condensate. Then all occupied orbitals are the same and Putting we obtain a closed equation for the order parameter: This is the celebrated Gross-Pitaevskii equation. This is a single self-consistent equation for a single orbital, the simplest HF like theory ever. single self-consistent equation for a single orbital

168. 168 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and Putting we obtain a closed equation for the order parameter: This is the celebrated Gross-Pitaevskii equation. single self-consistent equation for a single orbital

169. 169 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and Putting we obtain a closed equation for the order parameter: This is the celebrated Gross-Pitaevskii equation. has the form of a simple non-linear Schrödinger equation concerns a macroscopic quantity  suitable for numerical solution. single self-consistent equation for a single orbital

170. 170 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and Putting we obtain a closed equation for the order parameter: This is the celebrated Gross-Pitaevskii equation. has the form of a simple non-linear Schrödinger equation concerns a macroscopic quantity  suitable for numerical solution. The lowest level coincides with the chemical potential single self-consistent equation for a single orbital

171. 171 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and Putting we obtain a closed equation for the order parameter: This is the celebrated Gross-Pitaevskii equation. has the form of a simple non-linear Schrödinger equation concerns a macroscopic quantity  suitable for numerical solution. The lowest level coincides with the chemical potential How do we know?? single self-consistent equation for a single orbital

172. 172 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and Putting we obtain a closed equation for the order parameter: This is the celebrated Gross-Pitaevskii equation. has the form of a simple non-linear Schrödinger equation concerns a macroscopic quantity  suitable for numerical solution. The lowest level coincides with the chemical potential How do we know?? How much is it?? single self-consistent equation for a single orbital

173. 173 Gross-Pitaevskii equation – "Bohmian" form For a static condensate, the order parameter has ZERO PHASE. Then The Gross-Pitaevskii equation becomes Bohm's quantum potential the effective mean-field potential

174. 174 Gross-Pitaevskii equation – "Bohmian" form For a static condensate, the order parameter has ZERO PHASE. Then The Gross-Pitaevskii equation becomes Bohm's quantum potential the effective mean-field potential

175. 175 Gross-Pitaevskii equation – "Bohmian" form For a static condensate, the order parameter has ZERO PHASE. Then The Gross-Pitaevskii equation becomes Bohm's quantum potential the effective mean-field potential

176. 176 Gross-Pitaevskii equation – variational interpretation This equation results from a variational treatment of the Energy Functional It is required that with the auxiliary condition that is which is the GP equation written for the particle density (previous slide).

177. 177 Gross-Pitaevskii equation – variational interpretation This equation results from a variational treatment of the Energy Functional It is required that with the auxiliary condition that is which is the GP equation written for the particle density (previous slide). LAGRANGE MULTIPLIER

178. 178 Gross-Pitaevskii equation – chemical potential This equation results from a variational treatment of the Energy Functional It is required that with the auxiliary condition that is which is the GP equation written for the particle density (previous slide). From there

179. 179 Gross-Pitaevskii equation – chemical potential This equation results from a variational treatment of the Energy Functional It is required that with the auxiliary condition that is which is the GP equation written for the particle density (previous slide). From there chemical potential by definition

180. 180 Gross-Pitaevskii equation – chemical potential This equation results from a variational treatment of the Energy Functional It is required that with the auxiliary condition that is which is the GP equation written for the particle density (previous slide). From there chemical potential by definition

181. 181 Gross-Pitaevskii equation – energetics summary Energy Functional – three components of the total energy Gross-Pitaevskii equation written for the particle density Multiply by n and integrate:

182. 182 Gross-Pitaevskii equation – energetics summary Energy Functional – three components of the total energy Gross-Pitaevskii equation written for the particle density Multiply by n and integrate:  ADDITIONAL NOTES

183. Interacting atoms in a constant potential

184. 184 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies

185. 185 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies

186. 186 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies The repulsive interaction increases the chemical potential If, it would be and the gas would be thermodynamically unstable. The repulsive interaction increases the chemical potential

187. 187 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies The repulsive interaction increases the chemical potential

188. 188 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies The repulsive interaction increases the chemical potential

189. 189 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies The repulsive interaction increases the chemical potential If, it would be and the gas would be thermodynamically unstable.

190. 190 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies The repulsive interaction increases the chemical potential If, it would be and the gas would be thermodynamically unstable. impossible!!!

191. The end

192. VII. Interacting atoms in a parabolic trap

193. 193 Reminescence: The trap potential and the ground state level number 200 nK 400 nK

194. 194 Reminescence: The trap potential and the ground state level number 200 nK 400 nK characteristic energy characteristic length

195. 195 Parabolic trap with interactions GP equation for a spherical trap ( … the simplest possible case)

196. 196 Parabolic trap with interactions GP equation for a spherical trap ( … the simplest possible case) Where is the particle number N? ( … a little reminder)

197. 197 Parabolic trap with interactions GP equation for a spherical trap ( … the simplest possible case) Where is the particle number N? ( … a little reminder) Dimensionless GP equation for the trap

198. 198 Parabolic trap with interactions GP equation for a spherical trap ( … the simplest possible case) Where is the particle number N? ( … a little reminder) Dimensionless GP equation for the trap

199. 199 Importance of the interaction – synopsis Without interaction, the condensate would occupy the ground state of the oscillator (dashed - - - - -) In fact, there is a significant broadening of the condensate of 80 000 sodium atoms in the experiment by Hau et al. (1998), perfectly reproduced by the solution of the GP equation

200. 200 Importance of the interaction – synopsis Without interaction, the condensate would occupy the ground state of the oscillator (dashed - - - - -) In fact, there is a significant broadening of the condensate of 80 000 sodium atoms in the experiment by Hau et al. (1998), perfectly reproduced by the solution of the GP equation guess: internal pressure due to repulsion competes with the trap potential

201. 201 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is

202. 202 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is

203. 203 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is

204. 204 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is unlike in an extended homogeneous gas !!

205. 205 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is

206. 206 Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is Importance of the interaction

207. 207 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is

208. 208 Importance of the interaction weak individual collisions collective effect weak or strong depending on N can vary in a wide range Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is

209. 209 Importance of the interaction weak individual collisions collective effect weak or strong depending on N can vary in a wide range realistic value 0 Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is

210. The end

211. Variational method of solving the GPE for atoms in a parabolic trap

212. 212 Reminescence: The trap potential and the ground state level number 200 nK 400 nK

213. 213 Reminescence: The trap potential and the ground state level number 200 nK 400 nK ground state orbital

214. 214 Reminescence: The trap potential and the ground state level number 200 nK 400 nK ground state orbital

215. 215 The condensate orbital will be taken in the form It is just like the ground state orbital for the isotropic oscillator, but with a rescaled size. This is reminescent of the well-known scaling for the ground state of the helium atom. Variational estimate of the condensate properties SCALING ANSATZ

216. 216 The condensate orbital will be taken in the form It is just like the ground state orbital for the isotropic oscillator, but with a rescaled size. This is reminescent of the well-known scaling for the ground state of the helium atom. Variational estimate of the condensate properties SCALING ANSATZ variational parameter b

217. 217 Importance of the interaction: scaling approximation Variational ansatz: the GP orbital is a scaled ground state for g = 0

218. 218 Importance of the interaction: scaling approximation Variational estimate of the total energy of the condensate as a function of the parameter Variational parameter is the orbital width in units of a 0  ADDITIONAL NOTES Variational ansatz: the GP orbital is a scaled ground state for g = 0 dimension-less energy per particle dimension-less orbital size self-interaction

219. 219 Importance of the interaction: scaling approximation Variational estimate of the total energy of the condensate as a function of the parameter Variational parameter is the orbital width in units of a 0  ADDITIONAL NOTES Variational ansatz: the GP orbital is a scaled ground state for g = 0 dimension-less energy per particle dimension-less orbital size self-interaction

220. 220 Importance of the interaction: scaling approximation Variational estimate of the total energy of the condensate as a function of the parameter Variational parameter is the orbital width in units of a 0  ADDITIONAL NOTES Variational ansatz: the GP orbital is a scaled ground state for g = 0 dimension-less energy per particle dimension-less orbital size self-interaction

221. 221 Importance of the interaction: scaling approximation Variational estimate of the total energy of the condensate as a function of the parameter Variational parameter is the orbital width in units of a 0 Variational estimate of the total energy of the condensate as a function of the parameter Variational parameter is the orbital width in units of a 0

222. 222 Importance of the interaction: scaling approximation Variational estimate of the total energy of the condensate as a function of the parameter Variational parameter is the orbital width in units of a 0 Variational estimate of the total energy of the condensate as a function of the parameter Variational parameter is the orbital width in units of a 0

223. 223 Importance of the interaction: scaling approximation Variational estimate of the total energy of the condensate as a function of the parameter Variational parameter is the orbital width in units of a 0 The minimum of the curve gives the condensate size for a given. With increasing, the condensate stretches with an asymptotic power law Variational estimate of the total energy of the condensate as a function of the parameter Variational parameter is the orbital width in units of a 0

224. 224 Importance of the interaction: scaling approximation Variational estimate of the total energy of the condensate as a function of the parameter Variational parameter is the orbital width in units of a 0 The minimum of the curve gives the condensate size for a given. With increasing, the condensate stretches with an asymptotic power law For, the condensate is metastable, below, it becomes unstable and shrinks to a 'zero' volume. Quantum pressure no more manages to overcome the attractive atom-atom interaction Variational estimate of the total energy of the condensate as a function of the parameter Variational parameter is the orbital width in units of a 0

225. The end