1. Statistical methods for bosons Lecture 2. 9 th January, 2012

2. 2 Short version of the lecture plan: New version 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 Thomas-Fermi approximation Infinite systems: Bogolyubov-de Gennes theory, BEC and symmetry breaking, coherent states Time-dependent GPE. Cloud spreading and interference. Linearized TDGPE, Bogolyubov-de Gennes equations, physical meaning, link to Bogolyubov method, parallel with the RPA Dec 19 Jan 9

3. Reminder of Lecture I. Offering many new details and alternative angles of view

4. 4 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

5. 5 Trap potential 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

6. 6 Ground state orbital and the trap potential level number 200 nK 400 nK characteristic energy characteristic length

7. 7 BEC observed by TOF in the velocity distribution Qualitative features:  all Gaussians  wide vs.narrow  isotropic vs. anisotropic

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

9. 9 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

10. Today: BEC for interacting bosons Mean-field theory for zero temperature condensates Interatomic interactions Equilibrium: Gross-Pitaevskii equation Dynamics of condensates: TDGPE

11. Inter-atomic interaction Interaction between neutral atoms is weak, basically van der Waals. Even that would be too strong for us, but at very low collision energies, the efective interaction potential is much weaker.

12. 12 Are the interactions important? Weak interactions In the dilute gaseous atomic clouds in the traps, the interactions are incomparably weaker than in liquid helium. Perturbative treatment 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 thermalization the atomic collisions take care of thermalization mean field 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

13. 13 Fortunate properties of the interactions The atomic interactions in the dilute gas favor formation of long lived quasi-equilibrium clouds with condensate 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 almost elastic and spin independent: they only weakly 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.

14. I. Interacting atoms

15. 15 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.

16. 16 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

17. 17 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 

18. 18 Experimental data asas

19. 19 Experimental data for “ordinary” gases nm 3.4 4.7 6.8 8.7 10.4  (  K) 5180 940 --- 73 ---

20. 20 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")

21. 21 Experimental data  (  K) 5180 940 --- 73 --- nm -1.4 4.1 -1.7 -19.5 5.6 127.2 "well behaved”; decrease increase monotonous seemingly erratic, very interesting physics of Feshbach scattering resonances behind for “ordinary” gasesVLT clouds asas nm 3.4 4.7 6.8 8.7 10.4

22. II. The many-body Hamiltonian for interacting atoms

23. Many-body Hamiltonian

24. At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential

25. Many-body Hamiltonian At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential Experimental data for “ordinary” gasesVLT clouds asas

26. Many-body Hamiltonian At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential Experimental data for “ordinary” gasesVLT clouds asas

27. Many-body Hamiltonian At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential Experimental data for “ordinary” gasesVLT clouds asas NOTES weak attraction ok weak repulsion ok weak attraction intermediate attraction weak repulsion ok strong resonant repulsion

28. Many-body Hamiltonian: summary At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential Experimental data for “ordinary” gasesVLT clouds asas NOTES weak attraction ok weak repulsion ok weak attraction intermediate attraction weak repulsion ok strong resonant repulsion

29. III. Mean-field treatment of interacting atoms In the mean-field approximation, the interacting particles are replaced by independent particles moving in an effective potential they create themselves

30. 30 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

31. 31 HARTREE APPROXIMATION ~ many electron systems  mean field approximation Most of the interactions absorbed into the mean field explicit quantum correlation corrections remain Many-body Hamiltonian and the Hartree approximation self-consistent system

32. 32 HARTREE APPROXIMATION ~ many electron systems  mean field approximation Most of the interactions absorbed into the mean field explicit quantum correlation corrections remain Many-body Hamiltonian and the Hartree approximation self-consistent system ?

33. 33 HARTREE APPROXIMATION ~ many electron systems  mean field approximation Most of the interactions absorbed into the mean field explicit quantum correlation corrections remain Many-body Hamiltonian and the Hartree approximation

34. 34 HARTREE APPROXIMATION ~ many electron systems  mean field approximation Most of the interactions absorbed into the mean field explicit quantum correlation corrections remain Many-body Hamiltonian and the Hartree approximation self-consistent system

35. 35 On the way to the mean-field Hamiltonian ADDITIONAL NOTES

36. 36 On the way to the mean-field Hamiltonian ADDITIONAL NOTES  First, the following exact transformations are performed eliminates SI (self-interaction) particle density operator

37. 37 On the way to the mean-field Hamiltonian ADDITIONAL NOTES  Second, a specific many-body state is chosen, which defines the mean field: Then, the operator of the (quantum) density fluctuation is defined: The Hamiltonian, still exactly, becomes

38. 38 On the way to the mean-field Hamiltonian ADDITIONAL NOTES  In the last step, the third line containing exchange, correlation and the self-interaction correction is neglected. The mean-field Hamiltonian of the main lecture results: REMARKS Second line … an additive constant compensation for double- counting of the Hartree interaction energy In the original (variational) Hartree approximation, the self-interaction is not left out, leading to non-orthogonal Hartree orbitals The same can be done for a time dependent Hamiltonian substitute back and integrate

39. 39 Many-body Hamiltonian and the Hartree approximation self-consistent system only occupied orbitals enter the cycle HARTREE APPROXIMATION particle-particle pair correlations neglected the only relevant quantity... single-particle density

40. 40 Hartree approximation for bosons 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.

41. 41 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 Hartree approximation for bosons at zero temperature

42. 42 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

43. 43 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

44. 44 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

45. 45 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

46. 46 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

47. 47 Gross-Pitaevskii equation – "Bohmian" form For a static condensate, the order parameter has CONSTANT PHASE. Then the Gross-Pitaevskii equation becomes Bohm's quantum potential the effective mean-field potential

48. 48 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).

49. 49 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).

50. 50 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

51. 51 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

52. 52 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 in thermodynamic limit

53. 53 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

54. IV. Interacting atoms in a homogeneous gas

55. 55 Gross-Pitaevskii equation – homogeneous gas The GP equation simplifies For periodic boundary conditions in a box with

56. 56 The simplest case of all: a homogeneous gas

57. 57 The simplest case of all: a homogeneous gas

58. 58 The simplest case of all: a homogeneous gas If, it would be and the gas would be thermodynamically unstable.

59. 59 The simplest case of all: a homogeneous gas If, it would be and the gas would be thermodynamically unstable. impossible!!!

60. V. Condensate in a container with hard walls

61. 61 A container with hard walls A semi-infinite system … exact solution of the GP equation

62. 62 A container with hard walls A semi-infinite system … exact analytical solution of the GP equation Particles evenly distributed Density high Density gradient zero Internal pressure due to interactions prevails

63. 63 A container with hard walls A semi-infinite system … exact solution of the GP equation Particles pushed away from the wall Density low Density gradient large Quantum pressure prevails Particles evenly distributed Density high Density gradient zero Internal pressure due to interactions prevails

64. 64 A container with hard walls A semi-infinite system … exact solution of the GP equation Particles pushed away from the wall Density low Density gradient large Quantum pressure prevails Particles evenly distributed Density high Density gradient zero Internal pressure due to interactions prevails

65. 65 A container with hard walls A semi-infinite system … exact solution of the GP equation Particles pushed away from the wall Density low Density gradient large Quantum pressure prevails Particles evenly distributed Density high Density gradient zero Internal pressure due to interactions prevails

66. 66 A container with hard walls no interactions Interactions gradually flatten out the density in the bulk Near the walls there is a depletion region about wide

67. VI. Interacting atoms in a parabolic trap

68. 68 Parabolic trap with interactions GP equation for a spherical trap ( … the simplest possible case) Where is the particle number N? ( … a little reminder)

69. 69 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

70. 70 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

71. 71 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

72. 72 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" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is

73. Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is 73 Importance of the interaction unlike in an extended homogeneous gas !!

74. 74 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" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is

75. Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is 75 Importance of the interaction

76. 76 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" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is

77. 77 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" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is

78. 78 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" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is

79. VII. High density limit – Thomas-Fermi approximation The idea: Use the quasi-classical property of the condensate at high particle densities

80. 80 Thomas – Fermi approximation IDEA: a spherical harmonic trap with large interaction energy per particle large extended cloud, smooth variation of the density  quantum pressure small & local approximation i.e. zero for bosons corresponds to the static solution of the Euler equation without the quantum potential Solution 0

81. 81 Thomas – Fermi approximation IDEA: a spherical harmonic trap with large interaction energy per particle large extended cloud, smooth variation of the density  quantum pressure small & local approximation i.e. zero for bosons Solution Comments It only makes sense for dense systems, where the interaction is dominant The equation applies in the classically accessible region, outside it holds n 0 = 0 The TF solution fails near the transition points between both classical regions It must be completed by finding the chemical potential 0

82. 82 Thomas – Fermi approximation Radius of the TF condensate in a spherical harmonic trap Normalization yields the chemical potential 0

83. 83 This picture shown for the last time 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, shows the TF behavior (warning, this is the columnar density!)

84. VIII. Variational approach to the condensate ground state Idea: Use the variational estimate of the condensate properties instead of solving the GP equation

85. 85 General introduction to the variational method  VARIATIONAL PRINCIPLE OF QUANTUM MECHANICS The ground state and energy are uniquely defined by In words, is a normalized symmetrical wave function of N particles. The minimum condition in the variational form is  HARTREE VARIATIONAL ANSATZ FOR THE CONDENSATE WAVE F. For our many-particle Hamiltonian, the true ground state is approximated by the condensate of non-interacting particles (Hartree Ansatz, here identical with the symmetrized Hartree-Fock)

86. 86 Here, is a normalized real spinless orbital. It is a functional variable to be found from the variational condition Explicit calculation yields the total Hartree energy of the condensate Variation of energy with the use of a Lagrange multiplier: This results into the GP equation derived here in the variational way: eliminates self-interaction General introduction to the variational method

87. Restricted search as upper estimate to Hartree energy 87 Start from the Hartree energy functional Evaluate for a suitably chosen orbital having the properties Minimize the resulting energy function with respect to the parameters This yields an optimized approximate Hartree condensate

88. Application to the condensate in a parabolic trap

89. 89 Reminescence: The trap potential and the ground state level number 200 nK 400 nK ground state orbital

90. 90 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. Rescaling the ground state for non-interacting gas SCALING ANSATZ

91. 91 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. Rescaling the ground state for non-interacting gas SCALING ANSATZ variational parameter b

92. 92 Scaling approximation for the total energy Introduce the variational ansatz into the energy functional

93. 93 Scaling approximation for the total energy 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 Introduce the variational ansatz into the energy functional dimension-less energy per particle dimension-less orbital size self-interaction

94. 94 Scaling approximation for the total energy 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

95. 95 Minimization of the scaled energy 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

96. 96 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 in agreement with TFA 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

97. 97 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. Next, the total energy is calculated for this orbital The solution (for) is Evaluation of the total energy ADDITIONAL NOTES SCALING ANSATZ

98. 98 Evaluation of the total energy ADDITIONAL NOTES dimension-less energy per particle dimension-less orbital size For an explicit evaluation, we (have used and) will use the identities: The integrals, by the Fubini theorem, are a product of three: Finally, This expression is plotted in the figures in the main lecture.

99. Now we will be interested in dynamics of the BE condensates described by

100. Time dependent Gross-Pitaevskii equation

101. 101 Time dependent Gross-Pitaevskii equation Condensate dynamics is often described to a sufficient precision by the mean field approximation. It leads to the Time dependent Gross-Pitaevskii equation Intuitively, Today, I proceed following the conservative approach using the mean field decoupling as above in the stationary case Three steps: 1. Mean field approximation by decoupling 2. Assumption of the condensate macrosopic wave function 3. Transformation of the TD Hartree equation to TDGPE

102. 102 1. Mean-field Hamiltonian by decoupling -- exact part

103. 103 1. Mean-field Hamiltonian by decoupling -- approximation

104. 2. Condensate assumption & 3. order parameter Assume the Hartree condensate Then and a single Hartree equation for a single orbital follows define the order parameter The last equation becomes the desired TDGPE:

105. 105 Physical properties of the TDGPE 1.In the MFA, the present  coincides with the order parameter defined from the one particle density matrix and (at zero temperature) all particles are in the condensate 2.TDGPE is a non-linear equation, hence there is no simple superposition principle 3.It has the form of a self-consistent Schrödinger equation: the same type of initial conditions, stationary states, isometric evolution 4.The order parameter now typically depends on both position and time and is complex. It is convenient to define the phase by 5.The MFA order parameter (macroscopic wave function) satisfies the continuity equation

106. 106 Physical properties of the TDGPE Stationary state of the condensate for time-independent Make the ansatz Then  satisfies the usual stationary GPE This is a real-valued differential equation and has a real solution All other solutions are trivially obtained as

107. 107 Physical properties of the TDGPE Continuity equation The TDGPE and its conjugate: Just like for the usual Schrödinger equation, we get Only complex fields are able to carry a finite current (quite general) Employ the form The current density becomes

108. 108 Physical properties of the TDGPE Continuity equation The TDGPE and its conjugate: Just like for the usual Schrödinger equation, we get Only complex fields are able to carry a finite current (quite general) Employ the form The current density becomes phase r, t dependent

109. 109 Physical properties of the TDGPE Continuity equation The TDGPE and its conjugate: Just like for the usual Schrödinger equation, we get Only complex fields are able to carry a finite current (quite general) Employ the form The current density becomes phase gradient superfluid velocity field

110. 110 Physical properties of the TDGPE Continuity equation The TDGPE and its conjugate: Just like for the usual Schrödinger equation, we get Only complex fields are able to carry a finite current (quite general) Employ the form The current density becomes phase gradient superfluid velocity field The superfluid flow is irrotational (potential): But only in simply connected regions – vortices& quantized circulation!!!

111. 111 Physical properties of the TDGPE Isometric evolution Three formulations  The norm of the solution of TDGPE is conserved: This follows from the continuity eq.  The condensate particle number is conserved (condensate persistent)  Finally, isometric evolution I avoid saying "unitary", because the evolution is non-linear

112. A glimpse at applications 1.Spreading of a cloud released from the trap 2.Interference of two interpenetrating clouds

113. 113 1. 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

114. 114 1. 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

115. 115 1. 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

116. 116 1. 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

117. 117 1. Example of calculation Even without interactions and at zero temperature, the cloud would spread quantum-mechanically The result would then be identical with a ballistic expansion, the width increasing linearly in time The repulsive interaction acts strongly at the beginning and the atoms get "ahead of time". Later, the expansion is linear again From the measurements combined with the simulations based on the hydrodynamic version of the TDGPE, it was possible to infer the scattering length with a reasonable outcome Castin&Dum, PRL 77, 5315 (1996) To an outline of a simplified treatment 

118. 118 2. Interference of BE condensate in atomic clouds Sodium atoms form a macroscopic wave function Experimental proof: Two parts of an atomic cloud separated and then interpenetratin again interfere Wavelength on the order of hundreth of millimetre experiment in the Ketterle group. Waves on water

119. 119 Basic idea of the experiment 1. Create a cloud in the trap 2.Cut it into two pieces by laser light 3. Turn off the trap and the laser 4. The two clouds spread, interpenetrate and interfere

120. 120 2. Experimental example of interference of atoms Two coherent condensates are interpenetrating and interfering. Vertical stripe width …. 15  m Horizontal extension of the cloud …. 1,5mm

121. 121 2. Calculation for this famous example It is an easy exercise to calculate the interference of two spherical non-interacting clouds Surprisingly perhaps, interference fringes are formed even taking into account the non-linearities – weak, to be sure To account for the interference, either the full Euler eq. including the Bohm quantum potential – OR TD GPE, which is non-linear: only a numerical solution is possible (no superposition principle) Theoretical contrast of almost 100 percent seems to be corroborated by the experiments, which appear as blurred mostly by the external optical transfer function A.Röhrl & al. PRL 78, 4143 (1997)

122. 122 2. Calculation for this famous example It is an easy exercise to calculate the interference of two spherical non-interacting clouds Surprisingly perhaps, interference fringes are formed even taking into account the non-linearities – weak, to be sure To account for the interference, either the full Euler eq. including the Bohm quantum potential – OR TD GPE, which is non-linear: only a numerical solution is possible (no superposition principle) Theoretical contrast of almost 100 percent seems to be corroborated by the experiments, which appear as blurred mostly by the external optical transfer function A.Röhrl & al. PRL 78, 4143 (1997) Idealized theory Realistic theory Experim. profile

123. Linearized TDGPE

124. 124 Small linear oscillations around equilibrium I. Stationary state of the condensate Stationary GPE All solutions are obtained from the basic one as I

125. 125 Small linear oscillations around equilibrium I. Stationary state of the condensate Stationary GPE All solutions are obtained from the basic one as II. Small oscillation about the stationary condensate Substitute into the TDGPE and linearize (autonomous oscillations, not a linear response)

126. 126 where Small linear oscillations around equilibrium III. Linearized TD GPE for the small oscillation

127. 127 where Small linear oscillations around equilibrium III. Linearized TD GPE for the small oscillation

128. 128 where IV. Ansatz for the solution Substitute into the linearized TDGPE and its conjugate to obtain coupled diff. eqs. for (which are just Bogolyubov – de Gennes equations) Small linear oscillations around equilibrium III. Linearized TD GPE for the small oscillation

129. 129 precisely identical with the eqs. for the Bogolyubov transformation generalized to the inhomogeneous case (a large yet finite system with bound states). For a large box, go over to the k- representation to recover the famous Bogolyubov solution for the homogeneous gas Small linear oscillations around equilibrium V. Bogolyubov – de Gennes equations VI. Bogolyubov excitations of a homogeneous condensate

130. Excitation spectrum sound region quasi-particles are collective excitations high energy region quasi-particles are nearly just GP particles cross-over defines scale for k in terms of healing length asymptotically merge

131. More about the sound part of the dispersion law Entirely dependent on the interactions, both the magnitude of the velocity and the linear frequency range determined by g Can be shown to really be a sound: Even a weakly interacting gas exhibits superfluidity; the ideal gas does not. The phonons are actually Goldstone modes corresponding to a broken symmetry The dispersion law has no roton region, contrary to the reality in 4 He The dispersion law bends upwards  quasi-particles are unstable, can decay

132. 132 VII. Interpretation of the coincidence with Bogolyubov theory Except for Leggett in review of 2001, nobody seems to be surprised. Leggett means that it is a kind of analogy which sometimes happens. A PARALLEL Small linear oscillations around equilibrium Electrons (Fermions)Atoms (Bosons) JelliumHomogeneous gas Hartree Stationary GPE Linearized TD HartreeLinearized TD GPE equivalent with RPAequivalent with Bogol. plasmonsBogolons (sound)

133. The end

134. 134 Useful digression: energy units