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P.G. DEPARTMENT OF PHYSICS UNIVERSITY OF KASHMIR DISSERTATION SUBMITTED IN THE PARTIAL FULFILLMENT FOR THE DEGREE OF M.SC IN PHYSICS

JAHENGIR AHMAD SHAH

SHAH AARIF UL ISLAM

TASADUQ HUSSAIN MIR

SYED PEERZADA ROUOOF AHMAD SHAH

Bose-Einstein Condensation in External Traps

Under the Supervision of Dr. Sekh Golam Ali by

2014

U N I V E R S I T Y O F K A S H M I R , H A Z R A T B A L S R I N A G A R - 1 9 0 0 0 6

P0ST GRADUATE DEPARTMENT OF PHYSICS UNIVERSITY OF KASHMIR, HAZRATBAL-SRINAGAR-190006, J & K. ____________________________________

CERTIFICATE This is to certify that the project entitled Bose Einstein Condensation in External Traps ,, submitted by Jahengir Ahmad Shah(12069100003), Shah Aarif Ul Islam (12069100007) , Tasaduq Hussain Mir (12069100009) and Syed Peerzada Rouoof Ahmad Shah (12069100027) in the partial fulfillment for the degree of M.Sc in Physics by University of Kashmir, Hazratbal, Srinagar has been carried out by them under my consistent guidance and supervision. Dr. Manzoor A. Malik Dr. Sekh Golam Ali (Head of The Department ) ( Supervisor )

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Chapter 1

Introduction Inclusion of quantum nature of particles in Statistical Mechanics is one of the most important developments of Modern Physics. It is first started by S N Bose in 1924 in order to derive Planck's radiation formula which can explain black-body spectrum. In particular, he considers the radiation as a bunch of photons and finds their distribution in different energy cells with the assumptions, (i) photons are indistinguishable and (ii) a cell can contain zero to any number of particles. This distribution of mass-less particles (photons) is generalized by A Einstein in 1925. The generalized distribution formula, which includes massive as well as mass-less particles, is known as Bose-Einstein distribution and, the particles which follow this distribution are called bosons. The statistics behind this distribution is termed as B-E statistics. The fundamental property of BE statistics, which tells us that a state can accommodate any number of particles, leads to the concept of Bose-Einstein condensation when only available state is the lowest energy state. If a dilute gas of bosons is cooled to temperatures very close to absolute zero (that is, very near 0 K or273.15 C) then a large fraction of the bosons occupy the lowest quantum state at which point quantum effects become apparent on a macroscopic scale. This macroscopic quantum phenomenon is the so-called Bose-Einstein condensation. Bose-Einstein condensation is based on the indistinguishibility and wave nature of particles, which are both basic concepts of quantum mechanics. In a simplified picture, atoms in a gas may be regarded as quantum mechanical wave packets which have an extent on the order of a thermal de-Broglie wavelength, . In particular, 1 T , where T is the absolute temperature. At high temperature, the weakly interacting gas can be treated as a system of Billiard balls. The lower the temperature, the longer is the de-Broglie wavelength. When atoms are cooled to the point where the thermal de-Broglie wavelength is comparable to the interatomic separation, then the atomic wave packets overlap and indistinguishibility of the

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particles becomes important. Fig.1.1.At this moment bosons undergo a phase transition and form a Bose-Einstein Condensate, the dense and coherent cloud of atoms all occupying the same quantum mechanical state. The relation between the transition temperature and the peak density n, can be simply expressed as = 2.612, where the thermal de-Broglie wavelength is defined as = ( 2 ) 1/2 And m is the mass of the atom. Fig.1.1. Formation of Bose-Einstein condensation As we mentioned above, that Bose-Einstein condensation, is predicted in 1924. However, it is realized experimentally in 1995[1,2]. For this great achievement a Nobel Prize in Physics is also awarded in 2001. Clearly, it takes 70 years to realize BEC experimentally. One of the main reasons behind this delay is the unavailability of cooling techniques to reach near

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absolute zero temperature. This difficulty is overcome with the invention of laser cooling and evaporative cooling techniques. Our objective in this dissertation is to give a review on the basic concept of Bose-Einstein condensation starting from Black-body spectrum. In the second Chapter, we discuss black-body spectrum and Bose's derivation of Planck's formula. In Chapter 3, we discuss Einstein's generalization of Bose statistics and his prediction of B-E condensation for ideal gas where there is no trapping. Here, we also calculate transition temperature and critical number of atoms for BEC to occur. In chapter 4, we consider BEC in a magnetic trap and derive explicitly the expression for transition temperature and condensate fraction. In addition we present some cooling and trapping techniques to realize BEC experimentally. We also briefly outline some properties of trapped BECs and its possible applications. Finally, we give concluding remarks in chapter 5.

Black-body spectrum and Bose statistics for massparticles

Explanation of black-body spectrum is one of the On the basis of classical physics Rayleigh (1900) and Jeans (1905) derive a formula, called Rayleigh-Jeans formula, that can explain the spectrum in long wavelength region. However, in the short wavelength region it fails. The formula which can explain whole region of spectrum is derived by Max Planck in 1900. Here, he considers quantum nature of radiation in his calculation but does not pay attention to the statistics radiation quanta (photons). In 1924, S N Bose derives Planck's radiation formula considering statistical distribution of photons and, as a consequence the Bose statistics is invented.

2.1 Black body radiation spectrumBlack-body radiation is the type ofthermodynamic equilibrium with its environment, or emitted by anon-reflective body) held at constant, uniform temperature. The radiation has a specific spectrum and intensity that depends only on the tempera

Fig.2.1: Black

body spectrum and Bose statistics for mass

body spectrum is one of the major challenges in late nineteenth century. On the basis of classical physics Rayleigh (1900) and Jeans (1905) derive a formula, called

Jeans formula, that can explain the spectrum in long wavelength region. However, ion it fails. The formula which can explain whole region of

spectrum is derived by Max Planck in 1900. Here, he considers quantum nature of radiation in his calculation but does not pay attention to the statistics radiation quanta (photons). In

Bose derives Planck's radiation formula considering statistical distribution of photons and, as a consequence the Bose statistics is invented.

2.1 Black body radiation spectrum is the type of electromagnetic radiation within or surrounding a body in

thermodynamic equilibrium with its environment, or emitted by a black bodyreflective body) held at constant, uniform temperature. The radiation has a specific

spectrum and intensity that depends only on the temperature of the body[3]

: Black-body radiation spectrum

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Chapter 2

body spectrum and Bose statistics for mass-less

major challenges in late nineteenth century. On the basis of classical physics Rayleigh (1900) and Jeans (1905) derive a formula, called

Jeans formula, that can explain the spectrum in long wavelength region. However, ion it fails. The formula which can explain whole region of

spectrum is derived by Max Planck in 1900. Here, he considers quantum nature of radiation in his calculation but does not pay attention to the statistics radiation quanta (photons). In

Bose derives Planck's radiation formula considering statistical distribution of

surrounding a body in black body (an opaque and

reflective body) held at constant, uniform temperature. The radiation has a specific [3] (Fig 2.1).

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Black-body radiation has a characteristic, continuous frequency spectrum that depends only on the body's temperature. The spectrum is peaked at a characteristic frequency that shifts to higher frequencies with increasing temperature, and at room temperature most of the emission is in the infrared region of the electromagnetic spectrum. As the temperature increases past about 500 degrees Celsius, black bodies start to emit significant amounts of visible light. Viewed in the dark, the first faint glow appears as a "ghostly" grey. With rising temperature, the glow becomes visible even when there is some background surrounding light: first as a dull red, then yellow, and eventually a "dazzling bluish-white" as the temperature rises[4]. When the body appears white, it is emitting a substantial fraction of its energy as ultraviolet radiation. The Sun, with an effective temperature of approximately 5800 K, is an approximately black body with an emission spectrum peaked in the central, yellow-green part of the visible spectrum, but with significant power in the ultraviolet as well.

2.2 Rayleigh-Jean formula To explain the black-body spectrum, Rayleigh and Jeans begins the classical calculation by considering a blackbody as a radiation filled cavity at the temperature T (Fig 2.2)

Fig.2.2: A hole in the wall of a hollow object is an excellent approximation of a blackbody.

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Fig.2.3: EM radiation in a cavity whose walls are perfect reflectors consists of standing waves that have nodes at the walls, which restricts their possible wavelengths. Shown above are three possible wavelengths when the distance between the opposite walls is L.

Since the cavity walls are assumed to be perfect reflectors, the radiation must consist of standing electromagnetic waves, as in (Fig 2.3). In order for a node to occur at each wall, the path length from wall to wall, in any direction, must be an integral number j, of half-wavelengths. If the cavity is a cube, having edge of length L, this condition means that for standing waves in the x, y, and z directions respectively, the possible wavelengths are such that

= , = , = . (2.1) Here , and represent the number of half-wavelengths in the x,y and z directions respectively and they can take any integer value, e.g. 1,2,3 and so on. For a standing wave in any arbitrary direction, it must be true that in order the wave terminate in a node at its ends

+ + = . (2.2) Here, = 0,1,2 , = 0,1,2 , and = 0,1,2

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(Of course, if = = = 0, there is no wave, though it is possible for any one or two of the " to equal 0.) To count the number of standing waves #()& within the cavity whose wavelengths lie between and + &, what we have to do, is to count the number of permissible sets , , values that yield wavelengths in this interval. Lets imagine a j-space whose coordinate axis are , and . Fig.2.4 shows part of the plane of such a space. Each point in the j-space corresponds to a permissible set of , , values and thus correspond to a standing wave.

Fig.2.4: Each point in the j-space corresponds to a possible standing wave.

If j is a vector from the origin to a particular point , , its magnitude is = ( + + (2.3) The total number of wavelengths between and + & is the same as the number of points in j-space whose distance from the origin lie between and + &. The volume of the spherical shell of radius and thickness & is 4&, but we are only interested in the octant of this shell that includes non negative values of , and . Also, for each standing wave counted in this way, there are two perpendicular directions of polarization, hence the no, of independent standing waves in the cavity is

#()& = (2) *+ (4&) = &. (2.4) What we really want is the no. of standing waves in the cavity as a function of their frequency , instead of as a function of . From equations (2.2) and (2.3) we have

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= = -. /& & = . &,. (2.5) Therefore, no. of standing waves

#(,)&, = -. . &, = +01.1 ,&,. (2.6) The cavity volume is 2, which means that the number of independent standing waves per unit volume or the density of standing waves in a cavity is

3(,)&, = *1 #(,)&, = +0-4-.1 . (2.7) Equation (2.7) is independent of the shape of the cavity, even though we used a cubical cavity to facilitate the derivation. The higher the frequency, the shorter the wavelength and greater the number of standing waves that are possible. The next step is to find the average energy per standing wave. According to the classical theorem of equipartition of energy, as already mentioned, the average energy per degree of freedom of an entity that is part of a system of such entities in thermal equilibrium at the temperature T is 54. Each standing wave is a radiation-filled cavity corresponds to two degrees of freedom, for a total 6 of , because each wave originates in an oscillator in the cavity wall. The energy per unit volume, 7(,)&, in the cavity, in the frequency interval from , to , + &, is therefore, according to classical physics, 7(,)&, = 63(,)&, = 3(,)&, = +0-489-.1 , (2.8) which is Rayleigh-Jeans formula[5]. The Rayleigh-Jeans formula, which has the spectral energy density of blackbody radiation increasing as , without limit, is obviously wrong. Not only does it predicts a spectrum different from the observed one (see Fig.2.5), but integrating (2.8) from , = 0 to , = give the total density as infinite at all temperatures.

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Fig.2.5: Comparison of Rayleigh-Jeans Formula for the spectrum of the radiation from a blackbody at 1500K with the experimentally observed spectrum. The discrepancy is known as Ultraviolet catastrophe because it increases with increasing frequency.

This discrepancy between theory and observation was at once recognized as fundamental. This is the failure of classical physics that led Max Plank in 1900 to discover that only if light emission is considered as a quantum phenomenon, can correct the formula for 7(,)&, be obtained. And thus we arrive at the Planks radiation formula.

2.3 Planks Radiation Formula Plank found that he had to assume that the oscillators in the cavity walls were limited to energies of 6: = ,, where = 0,1,2, . He then used the Maxwell-Boltzman distribution law to find that the number of oscillators with the energy 6: is proportional to ? 89 at the temperature T. In this case the average energy per oscillator (and so per standing wave in the cavity) is

6 = @-ABC DE =* (2.9) Instead of the energy-equipartition average which Rayleigh and Jeans had used. . Here is where Classical and Quantum physics diverge. The energy in the interval , to , + &, now becomes

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7(,)&, = 63(,)&, = +0@.1 -1-ABC DE =* (2.10) which is Planks Radiation formula[5], that agrees with the experimental findings. To check, how does Planks formula explain experimental results, we consider different cases, as:

(i) When , , the term

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Let the radiation be enclosed in a volume V and its total energy be E. Let their be different species of quanta each characterized by the no. Ns and energy ,M (" = 0 " = ).The total energy E is then

L = OM,MM = P I-&, (2.14) The solution of our problem requires then the determination of the numbers OM which determine I-. If we can state the probability for any distribution characterized by an arbitrary set of OM ,then the solution is determined by the requirement that the probability be a maximum provided auxiliary condition (2.14) is satisfied. It is this probability which we now intend to find. The quantum has a moment of magnitude ,M K in the direction of its forward motion. The instantaneous state of the quantum is characterised by its coordinates x, y, z and associated momenta R, R, R. These six quantities can be interpreted as point coordinates in a six dimensional space; they satisfy the relation

R + R + R = , K (2.15) By virtue of which the above mentioned point is forced to remain on a cylindrical surface which is determined by the frequency of the quantum. In this sense the frequency domain &,M is associated with the phase space domain

&S&T&U&R&R&R = P4(, K ) &, K = 4(, K )P&, (2.16) If we subdivide the total phase-space volume into cells of magnitude , then the no.of cells belonging to the frequency domain &, is 4P(, K )&, (2.17) Concerning the kind of subdivision of this type nothing definitive can be said. However, the total no. Of cells must be interpreted as the no. of the possible arrangements of one quantum in the given volume. In order to take into account the polarization, it appears mandatory to multiply this no. by the factor 2 so that the no. of cells belonging to an interval &, becomes 8P(, K )&, (2.18) It is now very simple to calculate the thermodynamic probability of a macroscopically defined state. Let OM be the no. of quanta belonging to the frequency domain &,M. In how

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many different ways can we distribute these quanta over those cells which belong to the frequency interval &,M? Let RWM be the no. of vacant cells, R*M the no. of those cells which contain one quantum, RM the no. of cells containing two quanta, etc; then the no. of different distributions is

XM!RWM! R*M! (2.19) where XM = (8, K )&,M and OM = 0RWM + 1R*M + 2RM + is the no. of quanta belonging to the interval &,M. The probability of the state which is defined by all the R\ M is obviously ^_!`a_ !`5_ !M In view of the fact that we can look at the R\ M as large numbers, we have ln d = XM ln XM R\M ln R\M\MM (2.20) where XM = R\M\ This expression should be maximum satisfying the auxiliary condition L = OM,M ;M OM = fR\M\ (2.21) Carrying out the variation gives the condition gR\M(1 + ln R\M) = 0, gOM,M = 0,M\M (2.22) gR\M = 0, gOM = fgR\M\\ (2.23) It follows that

gR\M(1 + ln R\M + M) + *h ,MM fgR\M\ = 0\M (2.24) From (2.24), we can write

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R\M = iM

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Chapter 3

Bose-Einstein Condensation for an ideal gas Bose's derivation of Plank's formula includes statistics of photons which are massless. It fails to include statics of massive particles having same properties of photons. Just after Bose's work, Einstein gives a general statistics to include massive particles also. This statistics is known as Bose-Einstein statistics and the particles behaving in accordance to Bose's statistics are today called bosons. Bosons in three-dimensional configuration start to condense in the lowest energy state when they are cooled below a critical temperature .

3.1 Bose-Einstein Statistics In order to discuss Einsteins generalization of Bose statistics, we consider a system of N bosons with total energy U. Suppose that the system has an energy level st with degeneracy #t, containing t bosons. The states may be represented by #t 1 lines, and the bosons by t circles; distinguishable microstates correspond to different orderings of the lines and circles. For example, with 9 particles in 8 states corresponding to a particular energy, a particular microstate might be considered as shown in Fig.3.1:

Fig.3.1: Distribution of particles in different states The number of distinct orderings of lines and circles using 2.19 is

ut = (t + #t 1)!t! (#t 1)! (3.1) A particular distribution has a specified number of particles t within each of the possible energy levels st. The total number of microstates for a given distribution is therefore:

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u({t}) = y (t + #t 1)!t! (#t 1)!t . (3.2) Let us assume that each state has a high degeneracy, i.e.#t 1. Then we can make the approximation:

u({t}) y (t + #t)!t! #t!t . (3.3) To find the most probable distribution, we maximise (3.3):

u({t}) = y (t + #t)!t! #t!t subject to the constraint on the total number of particles: z tt = O (3.4) and the constraint on the total energy: z sttt = { (3.5) As usual, rather than maximise t directly, we maximise ln u. If we assume that both #t and t are large enough for Stirlings approximation to hold for ln #t! and ln t! we find that ln u is given by: ln u z[(t + #t) ln(t + #t) #t ln #t &t ln t]t . (3.6) The change in ln u resulting from changes &t in each of the populations t is then: & ln u z[(t + #t)&t ln t&t]t (3.7) From the constraints (3.4) and (3.5), we find: z &t = 0t , z st&tt = 0 (3.8)

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Combining (3.7) and (3.8) with Lagrange multipliers and , we have:

& ln u z ln t + #tt + + pstt &t. (3.9) For appropriate values of and , equation (3.9) is true for all &t, hence we have ln t + #tt + + pst = 0. (3.10) We then find that the most probable distribution can be written as t = #t

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3.2 Bose-Einstein Condensation Equation (3.12) shows that t is directly proportional to #t. As a result t#t = 1i< 89 1 (3.14) may be interpreted as the most probable number of particles per energy level in the i th cell. It is important to note that the final result (3.14) is totally independent of the manner in which the energy levels of the particles are grouped into cells so long as the number of levels in each cell is sufficiently large. Also, the entropy of the gas is given by l = zt( + pst) #t ln1

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becomes infinitely high. Since the number of particles is conserved, the chemical potential enters in the distribution function (3.17). The chemical potential is determined as a function of N and T by the condition that the total number of particles be equal to the sum of the occupancies of the individual levels. It is sometimes convenient to work in terms of the quantity =

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Fig. 3.2: The Bose distribution function W = 1/( =*

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In terms of thermal de Broglie wave length

= 2 , (3.21) Combining (3.20) and (3.21), we can write OP = U (3.22) From (3.22), for z to be less than unity, we must have OP 1. (3.23) The quantity n (n = N/V ) or fugacity is an appropriate parameter in terms of which the various physical properties of the system can be addressed. For example, we consider three cases. (i) n0. In that case, 0 and the particle aspect of the gas molecules or atoms dominates over the wave aspect. Thus the system is classical. (ii) 1 > n> 0. We can now expand all physical quantities as a power series in this parameter and investigate how the system tends to exhibit non-classical or quantum behaviour. (iii) n 1. The system becomes significantly different from the classical one and typical quantum effects begin to dominate. From (3.21)

n = :@1(089)1 4 (3.24) This expression clearly shows that the system is more likely to display quantum behaviour when it is at a relatively low temperature or has a relatively high density of particles. Moreover, for smaller particle mass, the quantum effects will be more prominent. From (3.17) the total number of particles N in the system is obtained as

O = z = z 1U=*

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I(s)&s = 2P (2) s* &s (3.26) so that OP = 2 (2) s* U=*

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The equality in (3.34), gives the maximum number of particles in the excited states. If the actual number of particles N of the system exceeds this limiting values, then N0 number of particles given by

OW = O P 2 (3 2 ) (3.35) will be pushed into the ground state. Since N0 = z/(1 z ), the precise value of z can be determined using

U = OWOW 1 1. (3.36) For z to be one, the chemical potential must be zero. Thus from (3.17),

= 1exp(s ) 1. (3.37) This result shows that for large N, there is no limitation to the number of particles that can go into the ground states s = 0. This curious phenomenon of a macroscopically large number of particles accumulating in a single particle states s = 0 is referred to as Bose-Einstein condensation. It is purely of quantum mechanical origin, even in the absence of inter-particle forces. It takes place in the momentum space. The condition for the onset of Bose-Einstein condensation is N > Ne (3.38) which gives a critical value of temperature as[9]

. = 2 OP(3 2 ) . (3.39) For given values of N and V, Bose-Einstein condensation takes place when temperature T of the gas is less than ..

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Chapter 4 Bose-Einstein Condensation in External Traps In the 1980s laser based techniques were developed to trap and cool neutral atoms (Chu, Cohen Tannoudji, Phillips). Technically, trapping and cooling in this approach go by the names, magneto-optical trapping (MOT) and laser cooling. Alkali metal atoms are well suited to laser based methods because their optical transitions can be excited by available lasers and because they have a favourable internal energy-level structure for cooling to very low temperatures. Once they are trapped, their temperature can be lowered further by evaporative cooling to observe Bose-Einstein Condensation. 4.1 Effect of trapping With a view to calculate effects of trapping, we first calculate density of state for both the free and trapped gas. While calculating density of states for a trapped BEC, we shall assume that all particles are in one particular internal (spin) state, and therefore we generally suppress the part of the wave function referring to the internal state. In most cases the confining traps are well approximated by harmonic potentials, and therefore, we consider only particles in harmonic trap. In three dimensions, for a free particle in a particular internal state, there is on average one quantum state per volume (2) of phase space. The region of momentum space for which the magnitude of the momentum is less than p has a volume 4R/3 equal to that of a sphere of radius p and, since the energy of a particle of momentum p is 6` = R/2, the total number of states G(6) with energy less than 6 is given by[8] G(6) = P 0 (>)1/4(0)1 = P 5/104 (>)1/41 (4.1) where V is the volume of the system. Quite generally, the number of states with energy between 6 and 6+d 6 is given by g(6)d 6, where g(6) is the density of states. Therefore g(6) = (> )> (4.2)

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which, from Eq. (4.1), is thus

g(6) = 1/45/4041 6*/. (4.3) For a free particle in d dimensions, the density of states is independent of energy for a free particle in two dimensions. Let us now consider a particle in the anisotropic harmonic-oscillator potential (trap)

V (r) =* S + T + U (4.4) which we will refer to as a harmonic trap. Here the quantities t( = S, T, U) denote the three force constants, which are generally unequal. The corresponding classical oscillation frequencies i are given by t = t and we shall therefore write the potential as

V (r) =* S + T + U (4.5) The energy levels, 6 , , , are then 6, , =( + *) + + * + + * (4.6) where the numbers t assume all integer values greater than or equal to zero. We now determine the number of states 3(6) with energy less than a given value 6 . For energies large compared with t, we may treat the t as continuous variables and neglect the zero-point motion. We therefore introduce a coordinate system defined by the three variables 6t = tt, interms of which a surface of constant energy (4.6) is the plane 6 = 6 + 6 + 6. Then 3(6) is proportional to the volume in the first octant bounded by the plane,

3(6) = *1 &6 &6 &6 = A11>=>=>W>=>W>W (4.7) Since#(6) = &3/&6, we obtain a density of states given by

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#(6) = >41 (4.8) For a d-dimensional harmonic-oscillator potential with frequenciest, the analogous result is #(6) = >5(=*)! . (4.9) We thus see that in many contexts the density of states varies as a power of the energy, and we shall now calculate thermodynamic properties for systems with a density of states of the form #(6) = 6=* (4.10) where is a constant. In three dimensions, for a gas confined by rigid walls, is equal to 3/2. The corresponding coefficient may be read off from Eq. (4.3), and it is

/ = 1/45/4041 (4.11) Understandably, (4.3) for d=3, coincides with (4.10) for =3/2. The coefficient for a three-dimensional harmonic-oscillator potential ( = 3), which may be obtained from Eq. (4.8), is

= *1 (4.12) 4.1.1 Critical temperature The transition temperature . is defined as the highest temperature at which the macroscopic occupation of the lowest-energy state appears. When the number of particles, N, is sufficiently large, we may neglect the zero-point energy in (4.6) and thus equate the lowest energy 6t: to zero, the minimum of the potential (4.5). The number of particles in excited states is given by

OA = &(6)#(6)W(6)W . (4.13) Where W(6) = (6), as given in eq. (3.37) This achieves its greatest value for = 0, and the transition temperature . is determined by the condition that the total number of particles can be accommodated in excited states, that is

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O = OA(. , = 0) = &

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is the geometric mean of the three oscillator frequencies. The result (4.18) may be written in the useful form

. 4.5 *WW O51, (4.20) where = /2. For a uniform Bose gas trapped in a three-dimensional box of volume V the index is 3/2. Using the expression (4.11) for the coefficient /, one finds for the transition temperature the relation

. = 0[(/)]4/1 4:4/1 3.31 4:4/1 (4.21) . = 2[(3 2 )] . (4.22)

where = O/P is the number density. For a uniform gas in two dimensions, is equal to 1, and the integral in (4.15) diverges. Thus BoseEinstein condensation in a two-dimensional box can occur only at zero temperature. However, a two-dimensional Bose gas can condense at non-zero temperature if the particles are confined by a harmonic-oscillator potential. In that case = 2 and the integral in (4.15) is finite. When the number of particles is extremely high we can neglect the zero point energy in the harmonic trap. This is, however, not true when the system consists of finite number of atoms. The finiteness of the number of particles calls for zero point energy to be taken into account. This reduces the critical temperature by an amount Tc such that .. = (2)2[(3)] O=* , (4.23) where = * (*)* .Clearly from (4.23) 99 0, as O . Thus we see that one of the effects of trapping is to lower the critical temperature by confining a finite number of atoms. Besides finiteness of the system, trapping makes the Bose gas inhomogeneous such that density variations occur on a characteristic length scale, /@ = ( () , provided by the frequency of the trapping oscillator. This is a major difference with respect to other systems like the super fluid helium where the effects of in-homogeneity take place on a microscopic scale in the coordinate space. In-homogeneity of super fluid helium, in fact cannot be detected in the coordinate space such that all observations are made in the

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momentum space. As opposed to this, the in-homogeneity of the Bose gas is such that both coordinate and momentum spaces are equally suitable for observations. 4.1.2 Condensate Fraction Below the transition temperature the number OA of particles in excited states is given by OA() = &66=*W *A/DE=* , (4.24) provided the integral converges, that is > 1. We may write this result as OA = ()()(), (4.25) where is a constant. In three dimensions, for a gas confined by rigid walls, is equal to 3/2, so that

/ = P/2*/ (4.26) Note that this result does not depend on the total number of particles. However, if one makes use of the expression for ., it may be rewritten in the form

OA = O . (4.27) The number of particles in the condensate is thus given by

OW() = O OA() (4.28) or

OW = O j1 99k (4.29) For particles in a box in three dimensions, is 3/2, and the number of excited particles per unit volume, obtained from Eqs. (4.26) and (4.27) is given by

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A = = (3/2) 8904/ (4.30) The condensate fraction is therefore given by

OW = O1 (/.)/ (4.31) For a three-dimensional harmonic-oscillator potential ( = 3), the condensate fraction is given by

OWO = 1 . (4.32) Comparing 4.31 and 4.32 one can see the variation of condensate fraction in the two cases.

4.2 Method of Cooling and Trapping Basically laser and evaporative cooling techniques are used to produce BEC in the Laboratory. In order to trap the gas, one can use magnetic as well as electric fields. Here we consider only magnetic trap. 4.2.1 Laser cooling Laser beams are often used to pre-cool the atomic vapour and the method used goes by the name laser cooling. The physical mechanism by which the collision between photons and atoms reduces the temperature of the atomic vapour can be visualized as follows. If an atom travels toward the laser beam and absorbs a photon from the laser it will be slowed down by the photon impact. Understandably, totality of such events will lower the temperature. On the other hand, if the atom moves away from the photon, the latter will speed up resulting in the increase of temperature. Thus it is necessary to have more absorption from head on photons if our goal is to slow down the atoms with a view to lower the temperature. One simple way to accomplish this in practice is to tune the laser slightly below the resonance absorption of the atom.

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Suppose that the laser beam is propagating in a definite direction. An atom in the gaseous system can move towards the beam or it may move away from the beam. In both cases the frequency of the photon will be Doppler shifted. In the first case the frequency of the laser beam will increase while in the other case the frequency will be decreased. In the case of head on collision the photon will be absorbed by the atom via resonance only when the original laser beam is kept below the frequency of atomic resonance absorption. When the atom and photon travel in the opposite direction, there cannot be momentum transfer from the photon to the atom because Doppler shift in this case produces further detuning of the already detuned laser beam. The minimum temperature achieved so far by laser cooling technique is less than 10-6K.

Fig.4. 1: Laser cooling 4.2.2 Magnetic Trapping Magnetic traps are used to confine low-temperature atoms produced by laser cooling. These traps use the same principle as that in the Stern-Gerlach experiment. Otto Stern and Walter Gerlach used the force produced by a strong inhomogeneous magnetic field to separate the spin states in a thermal atomic beam as it passes through the magnetic field. But for cold atoms the force produced by a system of magnetic coils bends the trajectories right around so that low energy atoms remain within a small region close to centre of the trap. This can be realized as follows. A magnetic dipole moment in a magnetic field i has energy P = . i . (4.33) For an atom in a hyperfine state |i, V corresponds to a Zeeman energy P = #i, (4.34) where = Bohr magneton and

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# # ( + 1) + ( + 1) l(l + 1)2( + 1) (4.35) The magnetic force along z-direction is = # &i&U (4.36) From (4.33) the energy depth of the magnetic trap is determined by ti. The atomic magnetic moment t is of the order of Bohr magneton which in temperature units 0.67 Kelvin/Tesla. Since laboratory magnetic fields are generally considerably less than 1 Tesla, the depth of magnetic traps is much less than a Kelvin, and therefore atoms must be cooled in order to be trapped magnetically. For confinement, Zeeman energy must have a minimum. We can consider two different cases for (4.34). Case 1 : # > 0. Here the Zeeman energy can be minimum, if B has a local minimum. Case 2 : # < 0. In this case V can have a local minimum if B has a local maximum. Maxwells equations do not allow a maximum of a static field. As a result the trapping of atoms for # < 0 is not allowed. In view of the above one can trap atoms only in a minimum of a static magnetic field. This explains how magnetic field can be used to trap atoms. For detail we refer the book in [10]. 4.2.3 Evaporative cooling and condensates density profile The basic idea of the evaporative cooling is simple. First, let's consider atoms trapped by the magnetic potential as shown in the panel (a) of the picture below. The most energetic (i.e. fastest, hottest) atoms, in the course of their movement within the trap, can go much higher up the potential walls. Now, we lower the walls of the trap potential. As a result, the most energetic atoms will likely fall out of the trap. The remaining atoms will collide with each other, exchanging momentum and reaching a new, lower temperature equilibrium; this is called re-thermalization. After the atoms have been allowed to re-thermalize, we lower the walls of the trap potential again, and allow the atoms to rethermalize again. This procedure is repeated until the critical temperature of a BEC is reached[11,12].

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Fig. 4.2: Evaporative cooling 4.3 Experimental Observation of BEC An observation Bose-Einstein condensation in the laboratory for different values of temperatures, namely, below, at and above critical temperatures are shown in Fig. 4.3.

Fig.4.3: Velocity distribution of Rubidium-87 condensates. The false colors indicate the number of atoms at each velocity, with red being the fewest and white being the most. The areas appearing white and light blue are at the lowest velocities. The Bose-Einstein condensate is characterized by its slow expansion observed 6 ms after the atom trap was turned off. The left picture shows an expanding cloud cooled just above the critical temperature; middle: just after the condensate appeared; right: after further evaporative cooling has left an almost a pure condensate. In the first observations [13], four features were used to identify the formation of a Bose-Einstein condensate: (1) The sudden increase in the density of the cloud.

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(2) The sudden appearance of a bimodal cloud consisting of a diffuse normal component (3) The velocity distribution of the condensate was anisotropic in contrast to the isotropic expansion of the normal (non-condensed) component. (4) The good agreement between the predicted and measured transition temperatures Possible applications of Bose-Einstein Condensate: Atomic BEC systems are in basic research areas. It is supposed that the BEC will have a lot of applications in near future. (i) Bose condensates can be used to simulate condensed matter systems. For example, In the presence of "optical lattice", a periodic potential created by interference pattern of multiple laser beams[14], the atoms behave like electrons in crystal lattices. The big advantage of BEC optical lattice systems over real condensed matter systems is that lattice parameters are more easily tunable. (ii) A possible application of BEC is its use in precision measurement. (iii) Researchers are looking for ways to use BEC systems for quantum information processing. (iv) It is also hoped to use of BECs in atomic clocks and atom lasers [15].

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Chapter 5

Conclusion Studies of Bose-Einstein condensation in external potentials have received a great deal of attention. Since the experimental realization of BEC in 1995, a huge amount of work has been done in this direction[16]. In view of this, we review in this dissertation some interesting and important topics related to Bose-Einstein condensation. We start from Black-body spectrum and discuss how does Bose's concept of photon distribution gives the Planck's radiation formula. We also discuss how Einstein's generalization leads to BE statistics. One of the important consequences of BE statistics is occurrence of BE condensation at an extremely low temperature. We consider Bose gases in the presence and absence of external potentials separately and give a detail review. Here we clearly mentioned effects of trapping on the critical temperature and number of atoms. In addition, we show by simple calculations that an external trapping allows condensation to occur even in low dimensional systems. We also briefly discuss different cooling and trapping techniques that are required to realize BEC in the laboratory. Characteristics of ultra-cold dilute Bose gas after the onset of BECs are also pointed out. We conclude by noting that the dissertation serves as a good introductory review in a very straightforward manner.

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Bibliography: [1] K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, W. Ketterle,BoseEinstein condensation in a gas of sodium atoms, Physical Review Letters 75,3969 (1995)

[2] M.R. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Observation of Bose-Einstein Condensation in a Dilute Atomic Vapour, Science 269, 198 (1995)

[3] P. Theodore Landsberg, Thermodynamics and statistical mechanics, (Reprint of Oxford University Press 1978 ed.) Courier Dover Publications. (1990).

[4] J.W. Draper, On the production of light by heat, London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 30: 345 (1847). [5] A. Beiser, S. Mahajan and S.R. Choudhury, Concepts of Modern Physics , McGraw Hill Education (India) ,2013.

[6] S.N. Bose, Planks Law and Light Quantum hypothesis, Zeitschrift fr Physik , 26,178(1924)

[7] F. Reif, Fundamentals of Statistical and thermal Physics, McGraw Hill Book Company

[8] C. J. Pethick and H. Smith, BoseEinstein Condensation in Dilute Gases, Cambridge University Press, Cambridge, 2001.

[9] R.K. Pathria and P. D.Beale, Statistical Mechanics, Elsevier, 2011. [10] H. J. Metcalf, P.van der Straten, Laser Cooling and Trapping, Springer New York,1999. [11] K.B. Davis, M.-O. Mewes, W. Ketterle, An analytical model for evaporative coolingof atoms, Appl. Physics. B 60, 155 (1995). [12] K.B. Davis, M.-O. Mewes, M.A. Joffe, M.R. Andrews, W. Ketterle, Evaporative coolingof sodium atoms, Physical Review Letters 74,5202 (1995).

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[13] W. Ketterle, M.R. Andrews, K.B. Davis, D.S. Durfee, D.M. Kurn, M.-O. Mewes, N.J.van Druten, Bose-Einstein condensation of ultracold atomic gases, Phys. Scr. T66, 31(1996). [14] I. Boch, Ultracold Quantum gases in Optical lattices, Nature Physics 1,23 (2005). [15] M.H. Anderson, J.R. Ensher, M.R. Mathews, C.E. Wieman, E.A. Cornel, Observation of Bose-Einstein Condensation in a dilute Atomic Vapour, Science 269, 198(1995). [16] S. G. Ali, M. Salerno, A. Saha and B.Talukdar, Displaced dynamics of binary mixtures in linear and non-linear optical lattices, Physical Review A 85,023639(2012).

Chapter 1Introduction