Hydrodynamics of the Bose-Einstein Condensate According to the Gross-Pitaevskii Theory

Hydrodynamics of the Bose-Einstein condensate according to the Gross-Pitaevskii theory

Author: Agisilaos Papatheodorou

Supervisor: Ch.C. Moustakidis, Assosiate Professor

Department of Physics, Aristotle University of Thessaliniki

February 13, 2020 Abstract

We consider a dilute and ultra-cold bosonic gas of particles. Within the framework of statistical physics, we derive the thermodynamic equations that dictate the gas’s behavior for non-interacting particles. With the use of quantum mechanics the addition of inter- particle interaction through s-wave scattering is possible obtain- ing the Gross-Pitaevskii equation. By investigating the weakly- interacting boson gas’s characteristics we deﬁne it’s size and shape. Lastly we draw a connection from the quantum mechanical equations to those of ﬂuid dynamics and we utilize these hydrodynamic properties to examine the gas’s behavior and attributes.

i Περίληψη

Ο σκοπός αυτής της εργασίας είναι η αναπαραγωγή της στα- τιστικής θεωρίας και της θεωρίας Gross-Pitaevskii για την πε- ριγραφή των ιδιοτήτων ενός συμπυκνώματος Bose-Einstein. Τα συμπυκνώματα Bose-Einstein είναι μία μορφή ύλης αποτελούμε- νη από σωματίδια με ακέραια τιμή κβαντικού αριθμού ιδιοστρο- φορμής, λεγόμενα μποζόνια, τα οποία έχουν ψυχθεί σε θερμο- κρασίες κοντά σε αυτή του απόλυτου μηδενός. Τα σωματίδια αυτά, καθώς η θερμοκρασία τείνει να μηδενιστεί, ωθούνται μα- ζικά στην βασική τους ενεργειακή κατάσταση γεγονός όπου καταστεί εύφορη μία πληθώρα κβαντικών φαινομένων υπαίτια για την αλλαγή των ιδιοτήτων του υλικού. Μελετώντας την αλληλεπίδραση σε υποατομική κλίμακα των σωματιδίων όπου αποτελούν το μποζονικό αέριο σε χαμηλές θερμοκρασίες στο- χεύουμε στην παραγωγή των αξιωμάτων που διέπουν τις ιδιότη- τες του συμπυκνώματος Bose-Einstein. Αυτή η διαδικασία απο- δίδει μοντέλα τα οποία διακρίνονται σε αέρια μποζονίων τα οπο- ία δεν αλληλεπιδρούν μεταξύ τους και σε αέρια μποζονίων όπου αλληλεπιδρούν μεταξύ τους. Το μοντέλο μη-αλληλεπιδρώντων σωματιδίων με ακέραια ιδιοστροφορμή, το οποίο είναι το πρώτο όπου αναπαράγουμε σε αυτή την εργασία, αναπτύχθηκε από τους Satyendra Nath Bose και Albert Einstein εν έτη 1924, μέσω της θεωρίας της στατιστικής φυσικής. ΄Εχει ως στόχο την περιγραφή της συμπεριφοράς των μη-αλληλεπιδρώντων και μη-διακριτών μποζονίων κατά την αλλαγή της θερμοκρασίας του αερίου. Χρησιμοποιώντας στατιστικές εξισώσεων για την περιγραφή της ενεργειακής κατανομής των μποζονίκων σω- ματιδίων, είναι δυνατή η εξαγωγή των θερμοδυναμικών εξι- σώσεων που περιγράφουν ένα συμπύκνωμα αποτελούμενο α- πο τέτοια σωματίδια. Η εξάρτηση των μακροσκοπικών ποσο- τήτων του συμπυκνώματος από την θερμοκρασία παρέχει μια πρώτη εικόνα για την φύση του συστήματος ωστόσο παράγει και ασυνέχειες στο σημείο μετάβασης του αερίου από την φυ- σιολογική του κατάσταση σε αυτή του συμπυκνώματος μπο- ζονίων. Με σκοπό την ένταξη των σωματιδιακών αλληλεπι-

ii δράσεων, γεγονός όπου εδράζεται στην πραγματικότητα, α- ναπτύσουμε μέσα από κβαντομηχανικές εξισώσεις την θεωρία Gross-Pitaevskii και την συνονόματη εξίσωση. Το δυναμικό αλ- ληλεπίδρασης που χρησιμοποιούμε είναι αυτό της σκέδασης δύο σωματιδίων για μηδενική στροφορμή, το οποίο και υπολογίζου- με για τις αλληλεπιδράσεις μεταξύ κάθε μποζονίου του συμπυ- κνώματος με κάθε αλλό στο ίδιο σύστημα σωματιδίων. Η λύση της μη-γραμμικής διαφορικές εξίσωσης Gross-Pitaevskii, που προκύπτει, καθιστά περίπλοκη και δυσνόητη την περαιτέρω α- νάλυση της αναλυτικής λύσης του συμπυκνώματος. Για να ξεπεράσουμε αυτό το εμπόδιο χρησιμοποιούμε προσσεγγίσεις όπως η Thomas-Fermi προσέγγιση, γεγονός που επιτρέπει τον προσδιορισμό των χαρακτηριστικών του συμπυκνώματος όπως το μέγεθος και την συνεκτικότητα στον χώρο του. Τα αξιώμα-

iii τα αυτής της προσέγγισης βασίζονται στον αυξημένο αριθμό σωματιδίων εντός του συμπυκνώματος όπως και την ελαχι- στοποιημένη θερμοκρασία για να απλοποιήσουν την εξίσωση Gross-Pitaevskii αφαιρώντας τον όρο της κινητικής ενέργειας των μποζονικών σωματιδίων. ΄Εχοντας εξάγει τις σχέσεις που καθορίζουν τα βασικά χαρακτηριστηκά μεγέθη ενός αερίου αλ- ληπεδρώντων μποζονίων, παράγουμε τις απαραίτητες υδροδυ- ναμικές εξισώσεις για την περιγραφή της κίνησης ενός συμπυ- κνώματος στον χώρο ως ρευστό, μέσω της κυματοσυνάρτη- σης του συστήματος σωματιδίων όπου το αποτελούν. Βασι- ζόμενοι στα παραπάνω μελετούμε την διάδοση των μηχανικών κυμάτων εντός του συμπυκνώματος, όπου παρατηρούμε φαι- νόμενα υπέρρευστης συμπεριφοράς. Το αποτέλεσμα το οποίο

μας οδηγεί σε αυτό το συμπέρασμα είναι η μεταβολή της ενερ- γειακής εξάρτησης σε σχέση με το κυματοδιάνυσμα της ορμής των σωματιδίων από τετραγωνική εξάρτηση σε γραμμική. Το συμπέρασμα αυτό βρίσκεται σε συμφωνία με τις πειραματικές μετρήσεις, όσον αφορά την υπερρευστότητα ψυγμένων μποζο- νίων, με τα πρώτα πειράματα σε σχέση με αυτό το φαινόμενο να είναι αυτό του Pyotr Kapitsa και του John F. Allen οι οπο-

iv ίοι ξεχωριστά παρατήρησαν υπέρρευστη συμπεριφορά σε υγρό ήλιο, εν έτη 1938. Με την ανάλυση Gross-Pitaevskii αιτιολογε- ίται αυτή η υπέρρευστη μακροσκοπικής κλίμακας συμπεριφορά αρχίζοντας από υποατομικής κλίμακας σωματιδιακές αλληλε- πιδράσεις προσθέτοντας βασικό θεωρητικό υπόβαθρο για την πλήρη κατανόηση του φαινομένου. Τέλος αξιολογούμε την θε- ωρία Gross-Pitaevskii ως εργαλείο μελέτης συμπυκνωμάτων α- ντιπαραθέτοντάς την με προγενέστερες όπως και μεταγενέστε- ρες θεωρίες και αναδίδοντας την συμβολή της στην θεωρητική και πειραματική φυσική.

v Introduction

The Bose-Einstein condensate is a state of matter of bosonic particles cooled to near absolute zero temperature. These conditions drive a large fraction of the subatomic particles to the ground energy state and perplex their wavefunctions. These microscopic quantum phenomena translate into different macroscopic properties as superconductivity and superfluidity. The Gross-Pitaevskii theory, developed by Eugene P. Gross and Lev Petrovich Pitaevskii in 1961, is a theoretical model based on the Hartee-Fock approximation which approximately describes a Bose-Einstein condensate of interacting bosons in their ground state through a pseudopo- tential. This theory is able to produce a non-linear differential equation called the Gross-Pitaevskii equation for the whole condensate. This allows for hydrodynamic equations to be extracted and further theoretical analysis of the phenomenon.

vi CONTENTS

Contents

Abstract i

Περίληψη ii

Introduction vi

1 General Information 1 1.1 Historical Context ...... 1 1.2 Bose-Einstein condensation criteria ...... 3 1.3 Bose-Einstein distribution ...... 3 1.4 Generalized density of states ...... 5 1.4.1 Free particle density of states ...... 5 1.4.2 Harmonic-Oscillator density of states ...... 6 1.5 Bosonic interaction ...... 7 1.5.1 Scattering length ...... 7 1.5.2 Eﬀective interaction ...... 9 1.6 Hartee-Fock approximation ...... 11 1.7 Fermion condensates - Cooper pairs ...... 11

2 Bose-Einstein condensation statistics 13 2.1 Transition temperature ...... 13 2.2 Condensed fraction ...... 14 2.3 Thermodynamic Quantities ...... 15 2.3.1 Condensed phase ...... 15 2.3.2 Normal phase ...... 17 2.3.3 Continuity close to the transition temperature . . . . . 18

3 Gross-Pitaevskii theory 20 3.1 Gross-Pitaevskii equation ...... 20 3.2 Cloud of trapped bosons in their ground state ...... 23 3.2.1 Thomas-Fermi approximation ...... 23 3.2.2 Healing of the condensate ...... 26 3.3 Non-Zero temperatures ...... 28

4 Hydrodynamics of the condensate 30 4.1 Hydrodynamic equations ...... 30 4.2 Time-dependent Gross-Pitaevskii equation ...... 30

vii LIST OF FIGURES

4.3 Derivation of the Hydrodynamic equations ...... 31 4.3.1 Continuity equation ...... 31 4.3.2 Momentum conservation equation ...... 32 4.4 Condensate sound-waves ...... 33

5 Conclusion 37

List of Figures

1 (Provided by: Wikipedia)Electrical impedance of mercury in low temperatures...... 2 2 (Provided by: Wikimedia Commons)Distribution functions of Fermi-Dirac and Bose-Einstein statistics...... 5 3 (Vladimir V. Meshkov, Andrey V. Stolyarov, Robert J Le Roy, Rapid, accurate calculation of the s-wave scattering length, 2011) The wave-function in comparison to the two particles distance during s-wave scattering...... 9 4 (Karmela Padavić,Cooper Pairs, Superconductivity and Flash Mobs, 2018) Electrons forming a Cooper pair through a crystalline lattice...... 12 5 (Matthew J. Davis: Microcanonical temperature for a classical field: Application to Bose-Einstein condensation, 2003)The specific heat C as a function of T , for different values of a. . . 19 Tc 6 (Cornell group, 2010) BEC under rapid rotation according to the Gross-Pitaevskii equations...... 24 7 (Carlo F. Barenghi, Introduction to quantised vortices and turbulence) The distance from the rigid wall in comparison to the wave-function, where the healing or coherence length is denoted a0 instead of ξ as in our calculations...... 28 8 (Clement Jones, Paul H. Roberts, Motions in a Bose condensate. IV. Axisymmetric solitary waves, 1982) Energy to momentum-wave vector spectrum, where the dotted line is the small value ~q contribution and the continuous one the high value ~q contribution...... 35

viii 1 General Information

1.1 Historical Context The first experimental evidence of BEC1 came from Kammerlingh Onnes2 of Holland’s Leiden University. Onnes was the first to liquify Helium at 4.14 kelvin in 1908 following his famous tutor’s, Johannes Van der Waals, work on the equation of state of real gasses. He later did further studies involving different properties of metals such as mercury, tin and lead in extremely low temperatures, of around 1 degree kelvin, and especially aimed his attention at electrical impedance which was thought to tend to infinity as the temperature drops to zero. This was based on the hypothesis that electrons move through the conductor with kinetic energy dependent on the temperature of the material. In 1911 he successfully measured their electrical impedance but instead of the expected result he observed a drop to practically zero impedance at temperatures lower than a certain value [Figure 1]. This phenomenon was later called superconductivity. The next step in understanding the nature of BEC came in 1938 by Pyotr Kapitsa and John F. Allen who independently discovered a shift in properties in Helium-4 at extremely low temperatures around 2.1 degrees kelvin. The super-cooled Helium-4 below that certain temperature, which they named lambda point, appeared to be in a different state of matter where it behaves like a liquid with zero vis- cosity, subjected only to it’s own inertia, a super-fluid state. At the same time progress was being made in the theoretical aspect of things with Indian physicist Satyendra Nath Bose of University of Dhaka publishing a paper in 1924 concerning Planck’s quantum radiation law deriving from counting same state identical particles, which peaked Albert Einstein’s interest. Einstein ex- tended Nath Bose’s idea generalizing the work done on photons by Bose to a statistical distribution for identical particles with integer spin that were later named bosons. This was the birth of the Bose-Einstein statistics. One of the characteristics of this model was that bosonic particles cooled to low temperatures tend to occupy the lowest energy state in overwhelming majority and several quantum phenomena take place with the greatest being particle wave-function interference, resulting in a new state of matter. In 1938 the German physicist Fritz London of Duke University combined the theoretical findings to the experimental data proposing BEC as the mechanism for su-

1Bose-Einstein condensation 2Awarded with the physics Nobel prize in 1913

1 1.1 Historical Context

Figure 1: (Provided by: Wikipedia)Electrical impedance of mercury in low temperatures. perconductivity and super-ﬂuidity. At a later date two theoretical physicists Eugene P. Gross of Brandeis University with a ﬁeld interest on quantum liq- uids and Lev Petrovich Pitaevskii working on quantized vortices as a PhD student of Lev Landau separately presented what is now known as the Gross- Pitaevskii theory of Bose-Einstein condensates. This theory uses a quantum mechanic description of a weakly interacting system of bosons and is widely used to describe the physics of ultracold bosonic gasses today. The resulting Gross-Pitaevskii equation is a non-linear Hamiltonian describing the ground state of the bosonic system. It is worth mentioning that similar work was done by Vitaly Lazarevich Ginzburg and Lev Landau for superconductivity although without taking into account the microscopic properties of the phenomenon. Several additions were made to the Gross-Pitaevskii equation to make it whole such as the Thomas-Fermi approximation, the Bogoliubov transformations for excited states, the Popov approximation for the mixing of particle-like excitations, rotating condensates and rotating traps.

2 1.2 Bose-Einstein condensation criteria

1.2 Bose-Einstein condensation criteria As it was made already obvious the nature of a bosonic gas changes depending on it’s temperature. The point in which the gas changes from a classic phase to a condensed phase in a temperature axis is called critical or transition temperature. At lower temperatures the bosonic gas can develop supercon- ductive or superﬂuid properties whereas higher than the critical temperature it has no distinction to a regular non-bosonic gas [6]. Knowing that the condensate properties depend on the particle’s wave function interference there is an easy way to discover the point at which the quantum eﬀects come into play. That is by determining the thermal de Broglie wavelength and comparing it to the interparticle spacing. The interparticle distance is of order − 1 of n 3 , n being the numerical density of the condensate, and the thermal de Broglie wavelength is calculated by taking the 1-dimensional box partition function of length L for a continuous energy spectrum with energy values of h2n2 3 4 En = 8mL2 [1].

Z 8 Z 8 2 2 r −En −h n 2πmkT kT 8mL2kT Z = e dn = e dn = 2 L (1) 0 0 h L The partition function is also equal to Z ≈ λ , λD being the de Broglie wavelength, so h λD = √ (2) 2πmkT

The equation (2) shows that when the temperature is high λD is small in value and the gas behaves classically where as in low temperatures it’s value − 1 rises and can be comparable to n 3 .

1.3 Bose-Einstein distribution According to the grand canonical ensemble of statistical physics a particle in a system of N non-identiﬁable, non-interacting particles has a possibility of achieving a certain state with a certain energy level of [6]

(µ−E ) ni e i kT pi(ni) = (3) Zi

3 −34 −34 m2kg Planck’s constant denoted h = 6.62607015 ∗ 10 Js ,¯h = 1.0545718 ∗ 10 s 4Bolzman’s constant denoted k = 1.38064852 ∗ 10−23m2kgs−2K−1

3 1.3 Bose-Einstein distribution

5 where µ is the chemical potential , Zi a constant named partition function and ni the number of particles occupying a certain state. By deﬁnition the sum of all possibilities equals to one and the sum of all particles occupying diﬀerent states equals to the total number of particles N in the system.

ni X X X (µ−Ei) N = ni pi(ni) = 1 Zi = e kT (4)

i=1 ni ni Particles with half-integer spin, that follow the Fermi-Dirac statistics, are forced through the Pauli exclusion principle to be limited to maximum one particle per energy state. Bosons that have integer spin and follow the Bose- Einstein statistics don’t have such restrictions, so their partition functions become

(µ−Ei) F − D : Zi = 1 + e kT (5) ni X (µ−Ei) B − E : Zi = e kT (6)

ni Considering that the number of particles N is extremely high, the bosonic partition function takes the form of a geometric series that converges for chemical potential values of µ < Ei and since E1 < E2 < ... < Ei < .. we conclude that µ < E1 and the partition functions becomes 1 B − E : Z = (7) i (µ−Ei) 1 − e kT We calculate the average bosonic distribution with the sum of the number of occupied states times their possibility of coming true.

X dlnZi n¯ = n p (n ) = kT ( ) (8) i i i i dµ ni 1 n¯ = (9) i (Ei−µ) e kT − 1 It is apparent from the distribution formula (9) that as the energy approaches that of the ground state, getting closer to the chemical potential value, the exponential tends to one and the average distribution raises drastically as shown [Figure 2].

5Energy required to change the particle number by one

4 1.4 Generalized density of states

Figure 2: (Provided by: Wikimedia Commons)Distribution functions of Fermi-Dirac and Bose-Einstein statistics.

1.4 Generalized density of states 1.4.1 Free particle density of states A basic procedure of statistical physics is the counting of different microscopic states of particles to export a profile for the density of states of a macroscopic condition. Assuming that all particles are in one particular state we calculate that one free particle in three dimensions can have a magnitude of momentum 4 3 less than p in a region equal to 3 πp which is a sphere of radius p. The next step is to divide this outcome with the phase space volume that is equal to h 3 ( L ) , L being the length of each side of the three dimensional phase space box. With these calculations we find the the function of the total number of p2 states depending on the momentum p. Knowing that E = 2m we can find

5 1.4 Generalized density of states it’s dependence on the energy value. 4πp3 G(p) = (10) h 3 3( L ) 3 V 4π(2mE) 2 G(E) = (11) 3h3 where V is the volume of the system. We attempt to ﬁnd the density of the states between energy values of (E,E+dE), therefore

3 1 dG(E) m 2 E 2 g(E) = = V √ (12) dE 2π2h¯3 Doing the same procedure for diﬀerent dimensions we generalize the density a −1 of states g(E) ∝ E 2 [1] in a a dimension system.

1.4.2 Harmonic-Oscillator density of states Considering a particle in an anisotropic harmonic-ocsillator potential of the form 1 V (x, y, z) = m(ω2x2 + ω2y2 + ω2z2) (13) 2 1 2 3 with ωi being the oscillator’s frequencies we can derive the energy levels depending on three integer numbers n1, n2, n3 with values greater or equal to zero. 1 1 1 E(n , n , n ) = (n + )¯hx + (n + )¯hy + (n + )¯hz (14) 1 2 3 1 2 2 2 3 2

For high energies that don’t compare with thehω ¯ i value we can treat the ni P3 variable as a continuous one and the energy equal to Ei =hω ¯ iniE = i=1 Ei. For a region of energy values lower that E we consider the total number of states G(E) as the integral

1 Z E Z E−E1 Z E−E1−E2 E3 G(E) = 3 dE1dE2dE3 = 3 (15) h¯ ω1ω2ω3 0 0 0 6¯h ω1ω2ω3 And the density of states dG(E) E2 g(E) = = 3 (16) dE 2¯h ω1ω2ω3

6 1.5 Bosonic interaction

Calculating the same result for diﬀerent dimensions we ﬁnd the general form of the density of states [1] for a a dimension oscillator.

Ea−1 g(E) = Qa (17) (a − 1)! i=1 hω¯ i

1 Or by setting a constant of Ca = Qa the density of states takes a (a−1)! i=1 ¯hωi a−1 simpliﬁed form of g(E) = CaE . Typical values are a = 3 for an three 3 dimensional harmonic oscillator and a = 2 for a free particle in a three dimensional box.

1.5 Bosonic interaction 1.5.1 Scattering length To describe the interaction mechanics for the Gross-Pitaevskii theory between bosons we use the quantum scattering theory for two particles of the same mass that interact via te potential V (r), where r = x1 − x2, that vanishes for large distances. The wave-function is dependent by the Schrodinger¨ equation

h¯2∇2 [− − V (r)]Ψ(r) = EΨ(r) (18) m where m is the reduced mass and E the energy in the center of mass system. To describe the scattering between two particles we have to only take into ¯h2k2 account the solutions with positive energy eigenvalues E = 2Ek, Ek = 2m or else the particles would be bound to the potential. Since the inter-particle

potential is of short range we estimate that for r → 8 the solution is given by a superposition of an outgoing and an incoming plain wave. This solution is a sum of the homogeneous and the specific differential solution. The homogeneous solution is that of a free traveling wave and is found by the Schrodinger¨ differential equation [2]

2 2 (∇ + k )Ψ1(r) = 0 (19) i~k~r Ψ1(~r) = e (20)

7 1.5 Bosonic interaction

6 Expanding the result with Legendre polynomials Pl(cosθ) [9].

8 l ~ X (2l + 1)i lπ eik~r = sin(kr − )P (cosθ) (22) kr 2 l l=0 furthermore considering a form of outgoing spherical wave and adding it the wave-function for large r is

eik0r Ψ(~r) = ei~k~r + f(~k, k~0) (23) r where f(~k, k~0) is the scattering amplitude. In order to solve for the partial outgoing wave we expand the wave-function and scattering amplitude with the Legendre polynomials

X8 Ψ2(r, θ) = AlPl(cosθ)Rkl(r) (24) l=0 8 ~ ~0 X f(k, k ) = fl(k)Pl(cosθ) (25) l=0

, Rkl(r) is the radial wave-function which satisﬁes the Schrodinger¨ equation with an angular momentum l

2 l(l + 1) 2m R00 (r) + R0 + [k2 − − V (r)]R (r) = 0 (26) kl r kl r2 h¯2 kl Taking the result from this diﬀerential equation and comparing it to the plane incoming wave form we conclude that

X8 eikr Ψ (r, θ) = (2l + 1)(ei2δl − 1)P (cosθ) (27) 2 l 2ikr l=0

6Legendre polynomials according to the Rodrigues formula

1 dn(x2 − 1)n P (x) = (21) n 2nn! dxn

8 1.5 Bosonic interaction

Where δl is a phase-shift parameter dependent on the angular momentum. It is now apparent that

X8 (2l + 1) f(~k, k~0) = (e2iδl − 1)P (cosθ) (28) 2ik l l=0 For ultra-cold particles the scattering is dominated by the l = 0 term (s- wave scattering). Additionally for low energies where k → 0 the scattering

Figure 3: (Vladimir V. Meshkov, Andrey V. Stolyarov, Robert J Le Roy, Rapid, accurate calculation of the s-wave scattering length, 2011) The wave- function in comparison to the two particles distance during s-wave scattering. amplitude takes a constant value f(0, 0) = −a. The constant a is called scattering length and the wave-function takes the form of a Ψ(r) = 1 − (29) r

1.5.2 Effective interaction To find the effective interaction during the scattering of two particles it is required to shift to a momentum representation [2]. By Fourier transforming 7 [9] the wave-function it becomes equal to

0 3 ~0 ~ ~0 Ψ(k ) = (2π) δ(k − k) + Ψsc(k ) (31)

7A function F (r) and it’s Fourier transform F (q) are related by

1 Z F (r) = F (q)eiqrdq (30) (2π)3

9 1.5 Bosonic interaction

8 This wave-function satisﬁes the Schrodinger¨ equation, which is

h¯2k2 h¯2k02 1 X ( − )Ψ c(k~0) = U(k~0,~k) + U(k~0, k~00)Ψ (k~00) (32) m m s V sc k00

¯h2k2 ~0 ~00 ~0 ~00 where m = E the energy eigenvalue and U(k , k ) = U(k − k ) the Fourier transform of the particle-particle interaction. The equation can be written as h¯2k2 h¯2k02 1 X Ψ (k~0) = ( − + iδ)−1(U(k~0,~k) + U(k~0, k~00)Ψ (k~00)) (33) sc m m V sc k00 (34) where an inﬁnitesimal imaginary part iδ has been added to ensure that only the scattered wave is present regarding the outgoing waves. This expression can also be written as h¯2k2 h¯2k02 h¯2k2 Ψ (k~0) = ( − + iδ)−1T (k~0,~k; ) (35) sc m m m

~0 ~ ¯h2k2 where T (k , k; m ) is called the Lippmann-Schwinger scattering matrix. For zero energy scattering, meaning k = 0, E = 0, we Fourier transform the equation (36) producing mT (0, 0; 0) Ψ (r) = − (36) sc 4πh¯2r Using the Born approximation9 and the results from equation (29) we conclude that

Ψsc = −a T (0, 0; 0) = U(0) (37)

4πh¯2a U(0) = (38) m 8Kronecker’s delta function with the property of

Z 8 δ(x − y)f(x)dx = f(y)

− 8

9The Born approximation considers the incident field in place of the total field as the driving field at each point in the scattering process which is accurate if the scattered field is small compared to the incident field

10 1.6 Hartee-Fock approximation

1.6 Hartee-Fock approximation A condensate of bosons can be described by quantum mechanic equations using the Hartee-Fock or mean-ﬁeld approximation [1]. This dictates that a N particle system containing symmetric single-particle wave-functions in the same single-particle state can be represented by the wave-function

N Y Ψ(~r1, ~r2, ..., ~rn) = φ(~ri) (39) i=1 where φ(~ri) is the single-particle wave-function which is normalized as Z 2 |φ(~ri)| dr = 1 (40) and Ψ(~r1, ~r2, ..., ~rn) the whole N particle wave-function. To describe the condensate conveniently we use the condensate wave-function for our calculations which is normalized as Z 2 |ψ(~r1, ~r2, ..., ~rn)| dr = N (41)

From the equations (41) and (42) we can derive a general expression con- necting the total wave-function with the single-particle one (43) as well as N the equation of the density of particles n = V (44).

1 2 ψ(~r) = N 2 φ(~r) (43) n(~r) = |ψ(~r)| (44)

1.7 Fermion condensates - Cooper pairs Particles with a half-integer spin, called fermions, by nature are restricted by the Pauli exclusion principle and can’t form a Bose-Einstein condensate. This comes in contrast with the experimental evidence of superconductivity, a strictly BEC related property, in metals where the prevailing particle is the electron, a fermion. The explanation for this contradiction between theory and experiment came from John Bardeen, Leon Cooper and John Robert Schrieffer10 in 1957 who formed the BCS11 theory that describes superconductivity as a microscopic effect [3]. Leon Cooper was the first to observe

10All three shared the 1972 Nobel prize in Physics for their theory 11Bardeen–Cooper–Schrieﬀer

11 1.7 Fermion condensates - Cooper pairs an attraction between electrons that are weakly bound to the crystalline lattice of metals at extremely low temperatures despite their Coulomb repelling forces. It was later proved that this unexpected attraction is due to an

Figure 4: (Karmela Padavić,Cooper Pairs, Superconductivity and Flash Mobs, 2018) Electrons forming a Cooper pair through a crystalline lattice. electron-phonon or electron-lattice interaction where the Coulomb forces of the electron locally distort the crystalline structure creating an area of positive charge following the electron’s path. Other negatively charged electrons are attracted to that area, indirectly being attracted to the passing electron as well. This phenomenon forms weakly interacting pairs of electrons called Cooper pairs12. The interaction is of long range resulting in two electrons being in a Cooper pair even hundred nanometers apart although it’s energy is comparably small, of the order of 10−3eV . A Cooper pair has mass equal to the reduced mass of the two electrons and spin equal to the sum or difference of the electrons spins. This results in a new composite particle of spin equal to zero or one. The integer spin of Cooper pairs classifies them as bosons and inherently make them plausible candidates for condensation. During the re- duction of thermal energy electrons in a crystalline lattice become entangled in pairs forming composite bosons that condense to their ground state be- coming able of flowing freely into the material. Similar predictions of Cooper pairs forming from electron-exciton or electron-plasmon interactions as well as Cooper pairs made from protons or neutrons have been made although without any experimental verification as of yet.

12Named after Leon Cooper

12 2 Bose-Einstein condensation statistics

The topic of this section is describing a uniform non-interacting Bose-Einstein condensate of bosons using statistical mechanics. This semi-classical process allows for the calculation of equilibrium properties and thermodynamic characteristics of the many-particle system. In order to be able to export these results from the statistical condensate model some approximations are necessary. Firstly we consider the separation of neighboring energy levels to be insigniﬁcant compared to the thermal energy, making the energy spectrum a continuum and replacing sums by integrals [6]. The contribution of the lowest energy state is inaccurately accounted for due to this fact and so it is separately measured. Secondly we examine the condensed and normal phase separately below and above the transition temperature respectively.

2.1 Transition temperature Miscopicaly the transition temperature is deﬁned by the highest energy in which the occupation of the lowest energy state is present. Considering our N particle system in an anisotropic harmonic-oscillator (13) the lowest energy is given by the equation (14). It bares an insigniﬁcantly small contribution to our calculations, so it’s value is considered to be equal to zero. The number of particles in excited states is given by integrating the distribution function (9) times the density of states (17) over the energy spectrum [1].

Z 8 Nex = g(E)¯nBE(E)dE (45) 0 The number of excited particles achieves it’s greatest value when there is a minimum of particles occupying the lowest energy state. This condition applies for a system in it’s critical temperature. It is also apparent from the distribution function equation that there is a correlation between the excited particle number and the chemical potential µ. For µ = 0, Nex achieves it’s maximum value which is equal to N assuming that the total number of particles can be accommodated in the excited states.

Z 8 (a−1) CaE N = Nex(Tc, µ = 0) = E dE (46) 0 e kTc − 1

Where Tc is the transition or critical temperature and a the dimensions of the harmonic-oscillator. For the dimensionless variable x = E this integral kTc

13 2.2 Condensed fraction becomes

Z 8 (a−1) a x a N = Ca(kTc) x dx = Ca(kTc) Γ(a)ζ(a) (47) 0 e − 1 Where Γ(a) is the gamma function13 and ζ(a) the Riemann zeta function14 1 [1]. It is clearer to understand the connection of the zeta function with ex−1 if we make the the common denominator e−x, using that for the gamma 1 function, and geometrically expanding the remaining quantity to 1−e−x =

−x −2x P 8 −nx 1 + e + e + ... = n=0 e . The integral takes the form 8 8 8 8 X Z 8 X 1 Z X 1 x(a−1)e−(n+1)xdx = z(a−1)e−zdz = Γ(a) (n + 1)a n0a n=0 0 n=0 0 n0=1 (48) where z = (n + 1)x and n0 = n + 1. From the equation (45) we derive the transition temperature as

1 1 N a Tc = (49) k CaΓ(a)ζ(a)

2.2 Condensed fraction In this subsection we calculate the number of particles in the ground state in comparison to the total number of particles and the temperature. Below the transition temperature the number of excited particles Nex is given by the equation (43). While keeping the chemical potential value still and equal to zero we calculate the excited particle number as

a Nex(T ) = Ca(kT ) Γ(a)ζ(a) (50) Combining this equation (48) to the equation of the transition temperature related to the total number of particles (45) it is clear that the excited particle number is a percentage of the total particle number dictated by the fraction of the temperature to the transition temperature. T a Nex(T ) = N (51) Tc

13 R 8 (a−1) −x One of the gamma function properties is Γ(a) = 0 x e dx

14 P 8 −a ζ(a) = n=1 n

14 2.3 Thermodynamic Quantities

The remaining percentage is of course the non-excited number of particles N0 that are occupying the ground state. T a N0 = N − Nex(T ) = N 1 − (52) Tc

2.3 Thermodynamic Quantities In an attempt to realize the underlying physics of the Bose-Einstein condensate we calculate it’s thermodynamic properties such as it’s energy, entropy and speciﬁc heat. It is apparent that for the statistical theory on ideal Bose gasses to be precise these quantities should follow the law’s set by classic thermodynamics as we will test. This analysis is split into three chapters, one for temperatures under the transition temperature, one for higher temperatures and one for the continuity of the two close to the transition temperature.

2.3.1 Condensed phase The math used for the energy below the transition temperature are similar to that of the excited states one (43) as their internal energy15 is calculated as

Z 8 Z 8 (a−1) CaE E E = g(E)¯nBE(E)EdE = E dE (53) 0 0 e kTc − 1 Solving this integral (51) with the gamma and zeta function method16, with the only diﬀerence being the power which the energy variable is raised, we ﬁnd the expression of the internal energy in comparison to the system’s temperature.

(a+1) (a+1) ζ(a + 1) T E = Ca(kT ) Γ(a + 1)ζ(a + 1) = Nka a (54) ζ(a) Tc 15It is worth reminding that the system ground state is considered to be the zero-energy state which has no impact on the total energy 16To simplify this result we use one additional property of the gamma function which is Γ(a + 1) = aΓ(a)

15 2.3 Thermodynamic Quantities

The speciﬁc heat17 C is found by the derivative of the internal energy with respect to the system’s temperature.

dE E ζ(a + 1) T a C = = (a + 1) = Nka(a + 1) (55) dT T ζ(a) Tc Another thermodynamic quantity to be calculated is the system’s entropy dS C which is given as dT = T . ζ(a + 1) T a S = Nk(a + 1) (56) ζ(a) Tc To check the validity of our statistical calculations we compare them to these from classical thermodynamics. For high temperatures where both models are eﬀective the equations of energy (51) and particle number (43) become

Z 8 a µ−E E = Ca E e kT dE (57) 0

Z 8 (a−1) µ−E N = Ca E e kT dE (58) 0 In these conditions the equation of particle number (56) and energy (55) become identical to the ones from the Maxwell-Boltzmann distribution. Solving the energy integral, we obtain

E = aNkT (59) and for the high temperature speciﬁc heat

C = aNk (60)

For a homogeneous gas in three dimensions the dimension variable a takes 3 3 the value 2 and the speciﬁc heat becomes C = 2 Nk. For a three dimensional harmonic-oscillator potential a is equal to 3 and the speciﬁc heat C = 3Nk. These results come into agreement with the equipartition theorem validating once more our calculations [6].

17We consider the parameters of the trap unchanged or for free particles in a box the volume of the box to be left unchanged

16 2.3 Thermodynamic Quantities

2.3.2 Normal phase In the normal phase the chemical potential is considered to have a non-zero value, transforming the general expressions for the particle number (43) and internal energy (51) to

Z 8 E(a−1) N = Ca dE (61) ( E−µ ) 0 e kT − 1

Z 8 Ea E = Ca dE (62) ( E−µ ) 0 e kT − 1

E−µ For high temperatures it is expected for a new variable equal to x = kT 1 to diminish in value but in reality the mean occupation numbernBE ¯ = ex−1 becomes insigniﬁcantly small indicating a raise in the value of x. For large x we can consider the quantity e−2x ≈ 0, from which we derive the expansion 1 −x −2x ex−1 ≈ e + e of the mean occupation number. The particle number integral becomes

2µ0 8 8 µ Z e kT Z 0 kT (a−1) (a−1) −z a 0a −z 0 N ≈ Cae (kT ) z e dz + Ca a (kT ) z e dz (63) 0 2 0

E 0 2E where z = kT and z = kT . The integrals are then transformed into gamma functions whose properties in combination with the equation (47) of the transition temperature help us derive the internal energy equation for the normal phase

ζ(a) T a E ≈ aNkT 1 − c (64) 2a+1 T

It is worth mentioning that the chemical potential is eliminated by blending the particle number (59) to the internal energy (60) equations. The speciﬁc heat is given by

ζ(a) T a C ≈ aNk 1 − (a − 1) c (65) 2a+1 T

These approximations are valid for temperatures even slightly above the transition temperature Tc.

17 2.3 Thermodynamic Quantities

2.3.3 Continuity close to the transition temperature We investigate the continuity of our calculations at higher and lower temperatures than Tc by examining the behavior of the speciﬁc heat with respect to temperature as well as the dimension variable a. We ﬁrst consider our system to be of constant volume and particle number and the energy a function of the chemical potential µ and the temperature. These assumptions give us the following equations for the total derivative of the particle number and energy [6].

dE dE DE = dT + dµ (66) dT dµ dN dN DN = dT + dµ = 0 (67) dT dµ

It is apparent that any possible discontinuity close to Tc is attributed to the dE dE dµ term and not the dT one due to the fact that the dT term remains the same just above and just below the transition temperature where as the dµ 18 dE dE dµ term changes . Taking into account that C = dT = dµ dT we conclude that near Tc dµ ∆C = aN (68) dT T =Tc+

dE where we have equated dµ = aN. We determine the dependency of µ to T close to the transition temperature using the equation (65) and the derivative in question becomes

dµ dµ dN dN −1 dN = − = − (69) dT dN dT dµ dT

dN From the equation of the particle number (59) we calculate that dµ = ζ(a−1) N and dN = a N which yield ζ(a) kTc dT Tc

dµ ζ(a) = −ak (70) dT ζ(a − 1)

18The chemical potential is considered to have zero value below and non-zero value above the transition temperature

18 2.3 Thermodynamic Quantities

and for infinitesimal differences in temperature T − Tc << Tc the chemical potential and specific heat are calculated.

ζ(a) µ ≈ −ak (T − T ) (71) ζ(a) c ζ(a) ∆C = −a2Nk (72) ζ(a − 1)

For higher values of the dimension variable a the discontinuity raises near

Figure 5: (Matthew J. Davis: Microcanonical temperature for a classical field: Application to Bose-Einstein condensation, 2003)The specific heat C as a function of T , for different values of a. Tc the transition temperature.

19 3 Gross-Pitaevskii theory

In this section we examine the Bose-Einstein condensate in the presence of inter-particle interactions. We accomplish that by deriving the Gross- Pitaevskii equation for a non-uniform gas at zero temperature comprised of identical bosons, of scattering length a, in the same single-particle wave- function. A condensate of N particles in the same exact state follows the Hartee-Fock approximation (39) with it’s wave-function being the product of all the bosons that it consists of

N Y Ψ(~r1, ~r2, ..., ~rn) = φ(~ri). i=1 Furthermore the interaction between two particles is described by the s-wave 4π¯h2a scattering theory [2] with eﬀective interaction potential equal to U(0) = m in momentum space and U = U(0)δ(~r − r~0) in coordinate space, with ~r and r~0 being the position of the two particles.

3.1 Gross-Pitaevskii equation We build the Gross-Pitaevskii equation starting from the eﬀective Hamilto- nian for our N particle system [4].

N 2 N N X ~pi X X H = + V (~r ) + U δ(~r − ~r ) (73) 2m i 0 i j i=1 i=1 i6=j where V (~ri) is the external potential. The energy values for the wave-function Ψ are given by

hΨ| H |Ψi E = (74) hΨ|Ψi

We break down the separate terms of the Hamiltonian and calculate the numerator of (72). First for the kinetic energy

N 2 N 2 Z X ~pi X h¯ hΨ| |Ψi = − φ(~r )∇2φ(~r )d~r (75) 2m 2m i i i i=1 i=0

20 3.1 Gross-Pitaevskii equation

At this point we use Green’s identity [9].

¯ ¯ ¯ 2 ∇[φ(~ri)∇φ(~ri)] = ∇φ(~ri)∇φ(~ri) + φ(~ri)∇ φ(~ri) (76) where the left side of the equation, upon integration, is equal to zero giving the result

¯ ¯ 2 ∇φ(~ri)∇φ(~ri) = −φ(~ri)∇ φ(~ri) (77)

This transforms the ﬁrst term of the numerator as

N 2 N 2 Z 2 Z X ~pi X h¯ h¯ hΨ| |Ψi = ∇φ(~r )∇φ(~r )d~r = N ∇φ(~r)∇φ(~r)d~r 2m 2m i i i 2m i=1 i=0 (78)

The second term involving the external potential with simple calculations becomes N N Z Z X X ¯ ¯ hΨ| V (~ri) |Ψi = φ(~ri)V (~ri)φ(ri)dri = N φ(~ri)V (~r)φ(r)dr (79) i=1 i=1 To describe the energy term for the interaction of particles we consider the expression

N N ZZ X X X X ¯ ¯ hΨ| U0 |Ψi = U0 φ(~ri)φ(~rj)δ(~ri − ~rj)φ(~ri)φ(~rj)dridrj i=1 i6=j i=1 i6=j (80)

Using the Kronecker’s delta function properties this double integral can be reduced to a single one

21 3.1 Gross-Pitaevskii equation

Taking into account that every one of the N bosons can interact with N − 1 other bosons from the system and dismiss double interactions [1] [4] we arrive at the conclusion that the interaction energy is described by

N X X 1 Z hΨ| U δ(~r − ~r ) |Ψi = U N(N − 1) φ¯(~r)φ(~r)|φ(~r)|2d~r (83) 0 i j 2 0 i=1 i6=j Combining all the resulting terms into one we ﬁnd the energy of the system

Z h¯2 1 E(φ, φ¯) = ∇φ¯(~r)∇φ(~r) + V (~r)φ¯(~r)φ(~r) + U (N − 1)φ¯(~r)φ(~r)φ¯(~r)φ(~r)d~r 2m 0 2 (84) where the quantity hΨ|Ψi, which is equal to N, has been simpliﬁed by the numerator.For a large particle number N >> 1 we approximate N ≈ N − 1 and by using the equation (43) we can simplify this term by equating N|φ(~r)|2 = |Ψ(~r)|2 = n(~r). We calculate the free energy our N particle system to be equal to F = E − µN [6] and minimize this quantity. We do this by using the Euler-Lagrange equation19 [9] twice ﬁrstly considering φ(~r) as the function of r and the second time considering φ¯(~r) as the function of r. We symbolize as a function L with variables L(~r, φ(~r), φ¯(~r), φ0(~r), φ¯0(~r)) the quantity

h¯2 N ∇φ¯(~r)∇φ(~r) + V (~r)φ¯(~r)φ(~r) + U φ(~r)2φ¯(~r)2 − µφ¯(~r)φ(~r) (85) 2m 0 2 with the last term describing the chemical potential of the free energy. By applying the L function to the Euler-Lagrange equation ﬁrst for the function φ(~r)

2 dL ¯h ¯0 d dφ0 dL d 2m φ (~r) − = − V (r)φ¯(~r) − U Nn(~r)φ¯(~r) + µφ¯(~r) (86) d~r dφ d~r 0

19The Euler-Lagrange equation dictates that for the integral of an equation L with variables L(r, q(r), q0(r)) in the form of R L(r, q(r), q0(r))dr with r being real, the minimum and maximum values are found by the equation

dL d dq0 dL − = 0 dr dq

22 3.2 Cloud of trapped bosons in their ground state

Knowing that the free energy minimizes at zero we extract the Gross-Pitaevskii equation for the conjugate wave-function

h¯2 − ∇2φ¯(~r) + V (r)φ¯(~r) + U Nn(~r)φ¯(~r) = µφ¯(~r) (87) 2m 0 Now we denote a new wave-function for the Gross-Pitaevskii equation that is a little easier to handle which is the condensate wave-function (44).

h¯2 − ∇2ψ¯(~r) + V (r)ψ¯(~r) + U n(~r)ψ¯(~r) = µψ¯(~r) (88) 2m 0 Doing the same procedure using the Euler-Lagrange equation with φ¯(~r) as the function in question we derive the Gross-Pitaevskii equation for the normal wave-function h¯2 − ∇2ψ(~r) + V (r)ψ(~r) + U n(~r)ψ(~r) = µψ(~r) (89) 2m 0 This equation has the form of a Schrœdinger equation although the eigenvalues that can be extracted from it are those of the chemical potential and not the energy. The main difference in solving it, rather than a typical Schrœdinger equation, is the addition of the non-linear term U0n(~r) that takes into account the mean field produced by the interaction of bosons. This difference induces a raise in value for the chemical potential equal to [1]

∆µ = U0n(~r) (90) which agrees with the thermodynamic relation of the chemical potential with dE the energy µ = dN .

3.2 Cloud of trapped bosons in their ground state 3.2.1 Thomas-Fermi approximation Using the Gross-Pitaevskii equation we examine the radius and equilibrium of the forming cloud of bosons in their ground state inside a isotropic oscil- 2 ~r2 lator potential V (~r) = mω0 2 . For a non-interacting boson gas the terms contributing to the system’s energy is the potential which is analogous to

23 3.2 Cloud of trapped bosons in their ground state

Figure 6: (Cornell group, 2010) BEC under rapid rotation according to the Gross-Pitaevskii equations.

2 1 ¯h R and the kinetic energy R2 , since the momentum is of order of R from the Heisenberg principle [1] [5]. The value for the cloud’s radius is equal to

1 h¯ 2 aosc = (91) mω0 which is also the length scale of the harmonic-oscillator. Taking into account 4π¯h2a 1 the interaction term U0n(~r) = m n(~r) where n(~r) R3 it is apparent that for small values of R or large particle numbers N or scattering length a this term prevails over the kinetic energy. To calculate the radius in which the cloud achieves equilibrium we neglect the kinetic energy term since its contribution is insigniﬁcant and equate the two remaining terms in regards to the resulting radius from the non-interacting gas (88) giving us the result

1 Na 5 R ∼ aosc (92) aosc This result shows the inﬂuence of the interaction over the cloud’s radius. It 1 5 is worth mentioning that the term Na in experiments involving atomic aosc

24 3.2 Cloud of trapped bosons in their ground state condensates with repulsive interaction takes values much larger that unity. To export a better and mathematically more accurate description we solve the Gross-Pitaevskii equation and neglect the kinetic energy from the beginning. This approximation is called the Thomas-Fermi approximation and it results in an equation of the form

[V (r) + U0n(r)] ψ(r) = µψ(r) (93) and consequently

[µ − V (r)] n(r) = (94) U0 For the boundary of the condensate the particle density becomes equal to zero for r = R~ so for r ≥ R

V (r) = µ (95)

This result (92) means that the energy required to add a particle to the system remains the same. For a external potential in the form of the harmonic- 2 r2 oscillator V (r) = mω 2 the radius R of the condensate is 2µ R2 = (96) mω2 Knowing that the chemical potential for zero kinetic energy non-interacting particles is equal to zero from the equation (87) we can extract relation of the chemical potential to the particle number by integrating over the space of the system

3 2 8π 3 µ 8π 2µ µ N = R = 2 (97) 15 U0 15 mω U0 and consequently with the help of equation (88)

2 2 15 5 Na 5 µ = hω¯ (98) 2 aosc 1 5 1 Na R = 15 5 aosc (99) aosc

25 3.2 Cloud of trapped bosons in their ground state

This results come in accordance with our rough estimates on equation (89) with a slight numeric diﬀerence. We can also extract a relation between the dE total energy and the chemical potential if we take into account that dN = µ [5] and we integrate the expression (95) over N, giving 5 E = µN (100) 7 Integrating the two energy terms given by equation (82) over space with the use of equation (92) we can ﬁnd their separate relation the the chemical potential. The inter-particle interaction energy term is

Z Z R 1 2 2 1 2 2 2 Z 1 02 2 1 2 ( 2 mω R − 2 mω r ) 2 1 2 4 4 (1 − r ) 02 0 U0n (~r)d~r = r dr = m ω R r dr 2 0 2U0 2 0 2 (101)

0 r where r = R . And for the potential energy term

Z Z R 1 1 1 1 Z 1 V (~r)n(~r)d~r = ( mω2R2 − mω2r2) mω2r4dr = m2ω4R4 (1 − r02)r04dr0 0 4 2 2 4 0 (102)

0 r using the same variable transformation r = R . To simplify the result we derive the ratio of the two terms [1] [5].

R 1 (1−r2)2 2 Eint 0 2 r dr 2 = 1 = (103) Epot R 2 4 3 0 (1 − r )r dr This result is the equivalent to the Virial theorem, for the condensate. We conclude that each energy term and the chemical potential are related by 5 2 3 E = µN E = µN E = µN (104) 7 int 7 pot 7

3.2.2 Healing of the condensate In the previous chapter we generated the equations necessary to describe the bosonic cloud’s radius and equilibrium. In contrast, this segment is dedi- cated to the cloud’s healing length and it’s behavior when the wave-function is restricted in a conﬁned box and not in an equilibrium state. Opposite to

26 3.2 Cloud of trapped bosons in their ground state our previous approximation, that of dismissing the kinetic energy, we now consider the energy contributions of the inter-particle interaction term and the kinetic energy term leaving aside the external potential term. The dynamic between the two remaining terms in the Gross-Pitaevskii equation will show the behavior of the wave-function as it approaches the rigid wall [1]. The Gross-Pitaevskii equation is now written as h¯2 d2ψ(r) − + U |ψ(r)|2ψ(r) = µψ(r) (105) 2m d~r2 0 Using the equation (87) for regions far from the wall where the kinetic energy is negligible and ∆µ = µ − 0 = µ we equate the chemical potential to the 2 wave-function µ = |ψ0| U0. The resulting equation is a known diﬀerential equation of second degree h¯2 d2ψ(r) − = −U (|ψ |2 − |ψ(r)|2)ψ(r) (106) 2m dr2 0 0 This diﬀerential equation can be solved analytically if we consider a point in the radius coordinate r = ξ, where the kinetic energy is equal to the interaction energy U0n(ξ). Considering the restrictions that for r = 0,ψ(0) =

0 and for r → 8 , ψ(r) = ψ0, meaning that the start of our space axis is located at the wall, we conclude that r ψ(r) = ψ0tanh √ (107) 2ξ The ξ constant can be determined by the condition of the two energy terms being equal. Since the kinetic energy per particle is of order of h¯2 = n(ξ)U (108) 2mξ2 0 Solving for ξ yields h¯2 1 ξ = = (109) 2mn(ξ)U0 8πn(ξ)a

4π¯h2a where we replaced the eﬀective interaction term with U0 = m . This new constant ξ is referred to as the coherence length [1]of the condensate and describes the distance over which the wave-function approaches it’s bulk value ψ0.

27 3.3 Non-Zero temperatures

Figure 7: (Carlo F. Barenghi, Introduction to quantised vortices and turbulence) The distance from the rigid wall in comparison to the wave-function, where the healing or coherence length is denoted a0 instead of ξ as in our calculations.

3.3 Non-Zero temperatures A BEC in non-zero temperatures is bound to have a percentage of it’s particles in excited states. For temperatures far lower than the transition temperature the excited particle percentage drops making the contribution of their interactions insigniﬁcant. In equilibrium the particle number remains the same, despite the addition of the excited states particles. This allows for a description of the condensate through the Bose-Einstein distribution without the chemical potential term. 1 fp = (110) e(Ep/kT )−1 Making the calculation of the thermal energy contribution simple Z E E(T ) − E(T = 0) = V p d~p (111) (2πh¯)3(e(Ep/kT ) − 1)

For temperature equal or lower than a speciﬁc value T∗ which obeys the 2 equation kT∗ = ms , where s is the the mechanical wave velocity in the condensate, the excitations formed in the boson gas are phonon-like having

28 3.3 Non-Zero temperatures

energy equal to Ep = sp. For a uniform gas in a three dimensional box, 3 where the dimension variable is equal to a = 2 , the contribution of these excitations in the total energy is calculated to be of a factor of T 4, using the equation (109), instead of the T 5/2 factor calculated by using statistical mechanics (52).

29 4 Hydrodynamics of the condensate

4.1 Hydrodynamic equations To examine the behavior of any liquid or gas a set of equations that elegantly introduce the conservation laws into the movement of the fluid is required [8]. The first equation portraits the conservation of mass using the density as a variable. It states that the rate of change in mass contained in a certain volume equates to the rate at which mass is been exchanged through the volume’s walls. dn + ∇(n~v) = 0 (112) dt where ~v is the velocity of the fluid. The second equation, needed to describe the motion of the fluid Euler’s equation of momentum conservation. It states that any change in the fluid’s velocity within a certain volume is attributed to the net flow of momentum into and out of that volume and the sum of all external forces. d~v ∇p = F~ − (113) dt n where F~ is the sum of all external forces and p is the pressure on the condensate’s surface.

4.2 Time-dependent Gross-Pitaevskii equation Repeating term in the hydrodynamic equations (110) and (111) is the time variable. Looking back at the Gross-Pitaevskii equation for the normal wave- function (87), the time variable is missing. So, in order to describe the hydrodynamic properties of the condensate a time-dependent Gross-Pitaevskii equation is necessary. To produce such equation we make the assumption that the condensate’s time dependent wave-function must develop in time −iµ t as a factor of e h¯ [1]. To test if this assumption produces a consistent time-dependent Gross-Pitaevskii equation with the time-independent one we compare the ground-state of the condensate of N particles to that of N − 1 particles.

ψ(~r, t) = hN − 1| ψ(~r) |Ni (114)

30 4.3 Derivation of the Hydrodynamic equations

Considering that the time-dependent wave-function of a quantum mechanical −iE t system is related to the spacial wave-function as ψ(~r, t) = ψ(~r)e h¯ , deriving from the Shœdinger equation, we conclude that

−i(E −E ) t ψ(~r, t) ∼ e N N−1 h¯ (115)

For large numbers of particles N the diﬀerence in the ground-state energy dE levels becomes dN which is equal to the chemical potential µ. The relation between the chemical potential and the condensate’s wave-function becomes.

dψ(~r, t) ih¯ = µψ(~r, t) (116) dt And for the conjugate wave-function

dψ¯(~r, t) −ih¯ = µψ¯(~r, t) (117) dt This produces the new time-dependent Gross-Pitaevskii equations for the normal and conjugate wave-functions.

h¯2 dψ(~r) − ∇2ψ(~r) + V (r)ψ(~r) + U n(~r)ψ(~r) = ih¯ (118) 2m 0 dt h¯2 dψ¯(~r) − ∇2ψ¯(~r) + V (r)ψ¯(~r) + U n(~r)ψ¯(~r) = −ih¯ (119) 2m 0 dt 4.3 Derivation of the Hydrodynamic equations 4.3.1 Continuity equation To derive the continuity equation we multiply the normal-wave function equation (115) with the conjugate wave-function and the conjugate -wave-function equation with the normal wave-function [7], which yields

h¯2 ψ(~r) − ∇2ψ(~r)ψ¯(~r, t) + V (r)ψ(~r)ψ¯(~r, t) + U n(~r)ψ(~r)ψ¯(~r, t) = ih¯ψ¯(~r, t) 2m 0 dt (120) h¯2 ψ¯(~r) − ∇2ψ¯(~r)ψ(~r, t) + V (r)ψ¯(~r)ψ(~r, t) + U n(~r)ψ¯(~r)ψ(~r, t) = −ihψ¯ (~r, t) 2m 0 dt (121)

31 4.3 Derivation of the Hydrodynamic equations

Subtracting these two equations together produces d(ψ(~r, t)ψ¯(~r, t)) h¯2 ih¯ + [∇2ψ(~r, t)ψ¯(~r, t) − ∇2ψ¯(~r, t)ψ(~r, t)] = 0 (122) dt 2m ¯h2 ¯ Which by subtracting and adding the quantity 2m ∇ψ(~r, t)∇ψ(~r, t) and some simplifying calculations becomes d(|ψ(~r, t)|2) h¯ + ∇ [∇ψ(~r, t)ψ¯(~r, t) − ∇ψ¯(~r, t)ψ(~r, t)] = 0 (123) dt 2mi This is the continuity equation (110) of the condensate if we consider the density equal to n(~r, t) = |ψ(~r, t)|2 (124) an assumption already made by the Hartee-Fock approximation (44) and the velocity equal to h¯ ∇ψ(~r, t)ψ¯(~r, t) − ∇ψ¯(~r, t)ψ(~r, t) ~v = (125) 2mi |ψ(~r, t)|2

4.3.2 Momentum conservation equation To make the derivation of the momentum conservation equation we consider the condensate’s normal and conjugate wave-functions to have the form ψ = feiφ (126) ψ¯ = fe−iψ (127) Transforming the equations (124) and (125) into n = f 2 (128) for the density and h¯ [f∇f + if 2∇φ − (f∇f − if 2∇φ)] h¯ v = = ∇φ (129) 2mi f 2 m for the condensate’s velocity20. We now insert the new wave-function (126) to the time-dependent Gross-Pitaevskii equation (120). Taking into consid- eration −∇2ψ = [−∇2f + (∇φ)2f − i∇2φf − 2i∇φ∇f]eiφ (130) dψ df dφ i = i eiφ − feiφ (131) dt dt dt 20For a non-singular wave-function phase φ any rotational movement for the condensate h¯ becomes zero due to the velocity being a gradient of a scalar quantity ∇×v = m ∇×∇φ = 0

32 4.4 Condensate sound-waves and separating the real from the imaginary parts, the Gross-Pitaevskii equation yields two equations d(f 2) h¯ = ∇(f 2∇φ) (132) dt m for the imaginary part, which is an expression of the continuity equation (123) we previously produced with the addition of the new wave-function parameters, and dφ h¯2 1 −h¯ = ∇2f + mv2 + V + U f 2 (133) dt 2mf 2 0 for the real part. Taking the gradient of equation (133), so as to introduce the velocity, and reverting the amplitude f and phase φ of the condensate’s wave- function to known variables reveals the momentum conservation equation. √ d~v h¯2∇2 n 1 m = −∇(V + nU − √ + mv2) (134) dt 0 2m n 2 In order to simplify it’s form and make it’s match to the expected equation (113) apparent we use the Giibbs-Duhem relation [1] of dp = ndµ, which with the help of equation (90) transforms into dp = nU (135) dn 0 2 resulting in a pressure value equal to p = n U0. Simpliﬁed the momentum conservation equation becomes √ d~v 1 1 h¯2∇2 n 1 = − ∇p − ~v∇~v + ∇( √ ) − ∇V (136) dt mn m 2m n m

4.4 Condensate sound-waves To examine the propagation of mechanical waves at low temperatures through a condensate we introduce a small variation of the density value around a equilibrium density, which we name neq, into the hydrodynamic equations we previously produced from the time-dependent Gross-Pitaevskii equation [8] [7]. For the new density n = neq + δn the density continuity equation (123) becomes dδn + ∇(n ~v) = 0 (137) dt eq

33 4.4 Condensate sound-waves

considering that δn is a topical, time dependent variable and neq is time independent. The momentum conservation equation (126) transforms into √ d~v 1 h¯2∇2 n 1 m = −~v∇~v + ∇( √ ) − ∇V (138) dt m 2m n m in which we have eliminated the pressure gradient term due to it’s small contribution at low temperatures. Using the time independent√ Gross-Pitaevskii ¯h2∇2 n 1 √ 1 equation (89) we replace the term −~v∇~v + m ∇( 2m n ) − m ∇V with −∇µ, simplifying the equation (138) to d~v m = −∇µ (139) dt Combining the equations (139) and (137) results in the diﬀerential equation that describes the density ﬂuctuations. d2n m = ∇(n ∇µ) (140) dt2 eq To simplify the solution we exclude the external potential making the assumption of a uniform boson gas of free particles. The propagating wave solution should be proportional to ei~q~r−iωt, where ~q is the wave vector and ω the frequency of the oscillating wave. This transforms the equation the gradient of the chemical potential to h¯2q2 δµ = (U + )δn (141) 0 4mn and consequently the equation (140) to

h¯2q4 mω2 = nU q2 + (142) 0 4m

The mechanical wave’s energy is Eq=ωh¯ , which is more convenient to use than the wave’s frequency. For simpliﬁcation reasons we use the free-particle 0 ¯h2q2 energy Eq = 2m as a variable to our equations q 0 0 2 Eq = 2nU0Eq + (Eq ) (143)

This equation contains terms that transfuse both a linear and a quadratic correlation between the wave’s energy and it’s wave vector. For small ~q values

34 4.4 Condensate sound-waves

Figure 8: (Clement Jones, Paul H. Roberts, Motions in a Bose condensate. IV. Axisymmetric solitary waves, 1982) Energy to momentum-wave vector spectrum, where the dotted line is the small value ~q contribution and the continuous one the high value ~q contribution.

0 2 we can eliminate the (Eq ) term leaving the energy equation r U n E ≈ 0 hq¯ (144) q m q U0n In which the term m equals to the sound velocity inside the condensate r U n s = 0 (145) m This result comes in agreement with the calculated hydrodynamic sound 2 dp nU0 velocity equation s = dn = m . The linear behavior of the energy spectrum comes in agreement with the observations from super-ﬂuidity experiments and is the one responsible for the super-ﬂuidity of the condensate [7]. For high values of ~q the relation of the wave vector to the wave’s energy becomes quadratic with the energy being analogous to the square of the wave vector. ¯h2q2 This occurs when the kinetic energy term 2m has a larger contribution to the energy than that of the inter-particle energy term nU0 [Figure 8]. By equating these two term we conclude that this transition in√ the behavior 2mnU0 of the sound wave takes place for wave vector values of q ∼ ¯h . This value is the inverse of the coherence length ξ (109), meaning that beyond the

35 4.4 Condensate sound-waves coherence length, where the wave-function takes it’s bulk value, the particles move collectively in a super-ﬂuid state whereas at regions where the wave- function value is diminished they behave as free particles.

36 5 Conclusion

The Gross-Piatevskii theory is the culmination of the combined work of statistical and quantum physics due to it’s unique ability of scaling an microscopic phenomenon, such as quantum particle scattering, to macroscopic magnitude. Previous models, as the statistical physics one, failed to describe the state of the condensate accurately enough to predict super-fluid or super- conducting properties that have been observed experimentally in the past. This ability of the Gross-Pitaevskii theory justifies this extreme behavior of the condensate through inter-particle interaction and also gives insight regarding the size, shape and healing length of the condensate, making it an important tool for experimentalists. Another beneficial characteristic of the Gross-Pitaevskii theory is that it is not confined only between bosonic condensates but extends to super-cooled atomic clouds and even fermionic clouds or boson-fermion mixtures. The connection between small and large scales is achieved by linking the single particle wave-function to the whole condensate’s wave-function trans- lating it’s properties to the larger scale by the Gross-Pitaevskii equation. The solution to this equation, while just in some instances can be calculated analytically, give raise to new phenomena in need of research regarding the condensate.This new approach allows for further development beyond the Gross-Pitaevskii theory such as the dynamic coupling of the condensate to the thermal cloud or Zaremba, Nikuni, Griffin (ZNG) theory (1999), presence of local quantum depletion and anomalous average density with the Lee-Huang-Yang correction and many experiments on this subject as the T.Fukuzawa experiment of quasiparticle condensate using excitons, the Eric A. Cornell, Wolfgang Ketterle and Carl E. Wieman experiment on alkali atoms, which earned them the 2001 Nobel Prize in physics and more recently many coherently guided matter-wave interferometry experiments. In contrast, what the Gross-Pitaevskii theory lacks in detail and adapt- ability, comparing to latter work, it gains in teaching prowess as it describes comprehensively intricate phenomena, as super-fluidity, without being per- plexed and unapproachable.

37 REFERENCES

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