Design and Characterization of Quadrupole-Ioffe (QUIC) trap for Bose-Einstein Condensation

TABLE OF CONTENTS

TABLE OF CONTENTS

LIST OF FIGURES

CHAPTER I. Introduction

CHAPTER II. A Route to Achieve BEC in Dilute Gases

CHAPTER III. Magnetic Trapping of Atoms
3.1 Optically - Plugged Quadrupole Trap
3.2 TOP Trap
3.3 Ioffe Trap
3.4 QUIC Trap

CHAPTER IV. Calculation and Simulation of Magnetic Field For QUIC Trap
4.1 Overview
4.2 Calculation of Magnetic Field due to Quadrupole Coils
4.3 Calculation of Magnetic Field of the Ioffe Coil
4.4 Magnetic Field Simulation Through Matlab

CHAPTER V. Design of QUIC Trap
5.1 Design of QUIC Trap Coils
5.2 Design of QUIC Trap with Cooling Jacket

CHAPTER VI. Observations and Results

REFERENCES

APPENDIX

VITA

ACKNOWLEDGEMENT

I would like to address my respect and appreciation to President, Pro -President, Registrar, Dean, Director, HOD and Department of Electronics and Communication Engineering of Manipal University Jaipur, Jaipur, Rajasthan, India, for providing me each and every help and support during book writing.

I would like to address my respect and appreciation to my former Ph.D., Research Assistant and M. Tech. (Master of Technology) Supervisors Prof. Myoungsik Cha (Pusan National University, Busan South Korea), Prof. Y.-C. Chen (Institute of atomic and molecular sciences, academia sinica, Taipei, Taiwan) and Prof. S. R. Mishra (Raja Ramanna Centre for Advanced Technology, Indore, India).

I would like to address my respect and appreciation to my former teachers (Master of Technology degree) of school of physics Devi Ahilya Vishwavidyalya, Indore, India, who opened my eyes and encouraged me in the study of optics and photonics, especially Prof. A. K. Dutta, Prof. B. Kumar, Prof. K. P. Maheshwari, and more.

Most importantly, this work would have been impossible without the love and patience of my parents, sisters, brother, my jija jee, Juhi, nieces and nephews. They have been a constant source of love, concern, support and strength throughout my life. I would like to express my heart-felt gratitude to my family.

I would like to thank my wife Ragini for her understanding and love during the past few years. Her support and encouragement was in the end what made this book possible.

Lastly, my two sweet daughters Manaviya and Arohi have given me so much happiness and made me to keep moving forward overcoming all obstacles.

Devotion to the duty is not a sacrifice but it is a

Justification to your existence in this world...

LIST OF FIGURES

Fig. 1.1. Criterion for Bose – Einstein Condensation.

Fig. 2.1 BEC chamber Layout.

Fig. 3.1 Magnetic field due to quadrupole coils.

Fig. 3.2 (a) Schematic diagram of Time orbiting potential. (b) Time averaged potential of TOP trap.

Fig. 3.3 Magnetic field due to QUIC trap.

Fig. 4.1 Schematic diagram of a QUIC trap configuration. The y - axis is directed positively into page.

Fig. 4.2 In Cartesian coordinate system one of the quadrupole coils placed at z = -l axis.

Fig. 4.3 In Cartesian coordinate system the Ioffe coil placed along x = -l axis.

Fig. 5.1 Design of QUIC coils.

Fig. 5.2 Design of QUIC Trap with cooling jacket.

Fig. 6.1 Variation of magnetic field with x - axis for anti - Helmholtz coil at I = 20 Amp coil parameters are R1 = 23 mm, N1 = 46, R2 = 27 mm, N2 = 46, R3 = 31 mm, N3 = 46, R4 = 35 mm, N4 = 46. where Ra1, Ra2, Ra3 and Ra4 are radius of four segments of anti - Helmholtz coil and Na1, Na2, Na3 and Na4 are no of turns in anti - Helmholtz coil in four segments. Where no of total turns in anti - Helmholtz coil N = N1 + N2 + N3 + N4. Distance of anti - Helmholtz coils from common center = 39 mm.

Fig. 6.2 Variation of magnetic field with x - axis for anti - Helmholtz coil at I = 25 Amp coil parameters are R1 = 23 mm, N1 = 46, R2 = 27mm, N2 = 46, R3 = 31 mm, N3 = 46, R4 = 35 mm, N4 = 46. where Ra1, Ra2, Ra3 and Ra4 are radius of four segments of anti - Helmholtz coil and Na1, Na2, Na3 and Na4 are no of turns in anti - Helmholtz coil in four segments. where total no of turns in anti - Helmholtz coil N = N1 + N2 + N3 + N4. Distance of anti - Helmholtz coils from common center = 39 mm.

Fig. 6.3 Variation of magnetic field with x - axis for anti - Helmholtz coil at I = 30 Amp coil parameters are R1 = 23 mm, N1 = 46, R2 = 27 mm, N2 = 46, R3 = 31 mm, N3 = 46, R4=35 mm, N4=46. where Ra1, Ra2, Ra3 and Ra4 are radius of four segments of anti - Helmholtz coil and Na1, Na2, Na3 and Na4 are no of turns in anti - Helmholtz coil in four segments. where total no of turns in anti - Helmholtz coil N = N1 + N2 + N3 + N4. Distance of anti - Helmholtz coils from common center = 39 mm.

Fig. 6.4 Magnetic field of Ioffe coil along x - axis at I = 20 Amp where R1, R2, R3 and R4 are radius of four segments of Ioffe coil and N1, N2, N3 and N4 are no of turns in Ioffe coil in four segments. where total no of turns in Ioffe coil N = N1 + N2 + N3 + N4. R1 = 5 mm, N1 = 64, distance of Ioffe coil from common center l1 = -53. R2 = 9 mm, N2 = 58, distance of Ioffe coil from common center l2 = -58. R3 = 13 mm, N3 = 48, distance of Ioffe coil from common center l3 = -63. R4 = 17 mm, N4 = 40, distance of Ioffe coil from common center l4 = -68.

Fig. 6.5 Magnetic field of Ioffe coil along x - axis at I = 25 Amp where R1, R2, R3 and R4 are radius of four segments of Ioffe coil and N1, N2, N3 and N4 are no of turns in Ioffe coil in four segments. where total no of turns in Ioffe coil N = N1 + N2 + N3 + N4. R1 = 5 mm, N1 = 64, distance of Ioffe coil from common center l1 = -53. R2 = 9 mm, N2 = 58, distance of Ioffe coil from common center l2 = -58. R3 = 13 mm, N3 = 48, distance of Ioffe coil from common center l3 = -63. R4 = 17 mm, N4 = 40, distance of Ioffe coil from common center l4 = -68.

Fig. 6.6 Magnetic field of Ioffe coil along x - axis at I = 30 Amp where R1, R2, R3 and R4 are radius of four segments of Ioffe coil and N1, N2, N3 and N4 are no of turns in Ioffe coil in four segments. where total no of turns in Ioffe coil N = N1 + N2 + N3 + N4. R1 = 5 mm, N1 = 64, distance of Ioffe coil from common center l1 = -53. R2 = 9 mm, N2 = 58, distance of Ioffe coil from common center l2 = -58. R3 = 13 mm, N3 = 48, distance of Ioffe coil from common center l3 = -63. R4 = 17 mm, N4 = 40, distance of Ioffe coil from common center l4 = -68.

Fig. 6.7 Absolute value of the magnetic field for QUIC trap along the x - axis at I = 20 Amp. All parameters for four segments as follow in above anti - Helmholtz and Ioffe coil.

Fig. 6.8 Absolute value of the magnetic field for QUIC trap along the x - axis at I = 25 Amp. All parameters for four segments as follow in above anti - Helmholtz and Ioffe coil.

Fig. 6.9 Absolute value of the magnetic field for QUIC trap along the x - axis at I = 30 Amp. All parameters for four segments as follow in above anti - Helmholtz and Ioffe coil.

Fig. 6.10 Absolute value of the magnetic field for QUIC trap along x - axis at given below all parameter. Iah = 15 Amp, Iioffe = 45 Amp, Distance for anti - Helmholtz coil from origin = 39 mm. Radius and number of turns for anti - Helmholtz coil segment 1 Ra1 = 25 mm, Na1 = 46. Radius and number of turns for anti - Helmholtz coil segment 2 Ra2 = 28 mm, Na2 = 46. Radius and number of turns for anti - Helmholtz coil segment 3 Ra3 = 31 mm, Na3 = 46. Radius and number of turns for anti - Helmholtz coil segment 4 Ra4 = 34 mm, Na4 = 46. For Ioffe coil Radius, distance from origin and number of turns for Ioffe coil segment 1 Ra1 = 5 mm, l1 = -48 mm, Na1 = 64. Radius, distance from origin and number of turns for Ioffe coil segment 2 Ra2 = 9 mm, l2 = -53 mm, Na1 = 58. Radius, distance from origin and number of turns for Ioffe coil segment 3 Ra3 = 13mm, l3 = -58 mm, Na3 = 48. Radius, distance from origin and number of turns for Ioffe coil segment 4 Ra4 = 17 mm, l4 = -63 mm, Na4 = 40.

Fig. 6.11 Absolute value of the magnetic field for QUIC trap along x - axis at given below all parameter. Iah = 25 Amp, Iioffe = 30 Amp, Distance for anti - Helmholtz coil from origin = 39 mm. Radius and number of turns for anti-Helmholtz coil segment 1 Ra1 = 25 mm, Na1 = 46. Radius and number of turns for anti - Helmholtz coil segment 2 Ra2 = 28 mm, Na2 = 46. Radius and number of turns for anti - Helmholtz coil segment 3 Ra3 = 31 mm, Na3 = 46. Radius and number of turns for anti - Helmholtz coil segment 4 Ra4 = 34 mm, Na4 = 46. For Ioffe coil Radius, distance from origin and number of turns for Ioffe coil segment 1 Ra1 = 6 mm, l1 = -35 mm, Na1 = 64. Radius, distance from origin and number of turns for Ioffe coil segment 2 Ra2 = 10 mm, l2 = -40 mm, Na1 = 58. Radius, distance from origin and number of turns for Ioffe coil segment 3 Ra3 = 14 mm, l3 = -45 mm, Na3 = 48. Radius, distance from origin and number of turns for Ioffe coil segment 4 Ra4 = 17 mm, l4 = -50 mm, Na4 = 40. 38

CHAPTER I

Introduction

Bose - Einstein condensation (BEC) is a purely quantum statistical phenomenon which was predicted by A. Einstein in 1925. Though its manifestations were observed earlier in superfluid helium, it was experimentally achieved in true sense in 1995 at JILA in dilute vapor of RbAbbildung in dieser Leseprobe nicht enthalten. At present there is tremendous work worldwide is being carried in this exciting field. Because of novel properties of Bose condensate samples, it finds scope in many areas of research and development as following:

- Atom optics: - Atom Laser and related applications, atomic wave - guides, nonlinear wave mixing, matter - wave solitons, atom interferometry etc.
- Basic physics: Fluid physics, ultra - cold interactions and many body physics, basic quantum physics (tunneling, entanglement, squeezing, quantum computation etc.).
- Precision measurements: gravity and gravity - gradiometry, atom – gyroscope, surface physics, Atomic clocks etc.

For non - interacting bosons in thermodynamic equilibrium, the mean occupation number of the single particle state i is given by the Bose distribution function 1.

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Einstein observed that integral decreases with T, at low T values. To keep N conserved, Einstein modified the equation as, the decreased number in integral at low T goes to N0, means gas gets condensed. When temperature of the gases is decreased, below a certain temperature, known as critical temperature (Tc), atoms start accumulating in the ground state. The critical temperature for uniform Bose gas in a three-dimensional box of volume V is given by 1,

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Physically, Bose - Einstein condensation occurs when the distance between atoms is comparable to their de Broglie wavelength. The exact criteria for a BEC to occur for a three - dimensional gas in free space turns out to be 2,

Abbildung in dieser Leseprobe nicht enthalten

Here m is the mass of atom and T is the temperature.

In Fig.1.1, it is shown that at High temperatures a weakly interacting gas can be treated as a system of hard spheres. In a simplified quantum description, the atoms can be regarded as wave packets with an extension lAbbildung in dieser Leseprobe nicht enthalten. At the BEC transition temperature, lAbbildung in dieser Leseprobe nicht enthalten becomes comparable to the distance between atoms, and a Bose condensate forms. As the temperature approaches zero, the thermal cloud disappears leaving a pure Bose condensate 3.

Abbildung in dieser Leseprobe nicht enthalten

Fig. 1.1 Criterion for Bose – Einstein Condensation.

CHAPTER II.

A Route to Achieve BEC in Dilute Gases

BEC is difficult to observe because it involves reaching very low temperatures. Required number density and temperature for BEC are n = 10 12 - 1013 /cc and T = 100 - 200 nK for Rb87. Although the process of making a Bose Condensate is complex, the methods involved are based on the familiar concepts such as evaporative cooling. For reaching the low temperature as above involves a wide range of techniques and methods that include laser cooling, magnetic trapping, RF evaporative cooling. Besides that other techniques such as optical detection and ultrahigh vacuum are also involved. BEC in alkali atoms can be achieved through combination of laser cooling and the evaporative cooling techniques.

A simple diagram as following describes the route to reach BEC:

Abbildung in dieser Leseprobe nicht enthalten

The creation of a dense sample of ultra - cold atoms is necessary to attain BEC. This is normally done using a double MOT setup as in Fig. 2.1. The upper part consists of a Rubidium magneto- optical trap (MOT) at a vapor pressure of about 10Abbildung in dieser Leseprobe nicht enthaltenTorr from where the atoms are later directed into a second chamber (10Abbildung in dieser Leseprobe nicht enthalten to 10Abbildung in dieser Leseprobe nicht enthalten Torr) where they are trapped in a second MOT. After atoms being captured in the second MOT, they are optically pumped to a suitable hyperfine level. Then atoms are then transferred to magnetic trap for evaporative cooling 2.

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Fig. 2.1 BEC chamber Layout.

CHAPTER III.

Magnetic Trapping of Atoms

An atom behaves as a small magnetic dipole. In a magnetic field, B, it feels a potential, U B = g F m F m B Bz 4. By spatially varying the magnetic field, it is possible to build a potential that traps the atoms. For the 87Rb ground state 5 2S1/2, the Lande factors are g F = -1/2 and g F = 1/2. This means that the states ÷2, +2>, ÷2, +1> ÷1, +1> are low field seeking states in that they will seek out a field minimum to reduce their potential energy. Since it is not possible to make a static magnetic field with a field maximum, traps must be made with a field minimum. Therefore atoms in low field seeking states are trappable atoms. The simplest form of trap is a quadrupole trap, which is created using a pair of coils operated in the anti - Helmholtz configuration. It produces a linearly varying magnetic field as shown in Fig.3.1. These have been very useful for trapping and compressing atoms having a magnetic moment. A limitation, however is that the spin can flip, undergoing a Majorana transition in the region where field is zero, to an untrapped magnetic hyperfine state inducing a loss and a possible heating mechanism.

Abbildung in dieser Leseprobe nicht enthalten

Fig. 3.1 Magnetic field due to quadrupole coils.

This difficulty was overcome using several different trap configurations that permitted the attainment of BEC, which are as follows,

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3.1 Optically - Plugged Quadrupole Trap

To minimize the losses at zero field point Ketterle’s group 5 used a tightly focused Ar - ion laser beam at the trap centre. The optical force from the laser beam kept the atoms out of this region, and since the laser frequency was very far from the resonance frequency of the atoms, it did not cause any absorption or heating. The disadvantage of this trap is the directional stability of the laser beam, which caused the plug to move away from the exact centre and not plug the hole completely.

3.2 Time Orbiting Potential (TOP) Trap

To minimize the losses at zero field point JILA group [5,6] superimposed a rotating homogeneous magnetic field onto the quadrupole field which is known as the time-orbiting potential (TOP) trap Fig.3.2. This superimposed bias field moves the magnetic field zero away from the trap centre. By rotating the bias field, the magnetic field zero can be moved on a circle in the plane around the centre of the trap. This rotation was done so quickly that the atoms in the trap centre did not have time to re - centre at the shifted magnetic field zero and trap losses were minimized. But this approach is limited in its applications due to the low trap depth and the peculiarities arising from the rotating field.

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Fig. 3.2 (a) Schematic diagram of Time orbiting potential. (b) Time averaged potential of TOP trap.

3.3 Ioffe Trap

The majority of the current BEC experiments use magneto-static traps of the Ioffe type geometry. This geometry is characterized by non - zero magnetic field minimum, which gives rise to a potential that is harmonic and cylindrically symmetric 6.

The Ioffe trap consists of 4 bars and two pinch coils each bar carries a current whose direction opposes that of its nearest neighbours. The four bars produce a two - dimensional confinement of the atoms. The magnetic field produced by the pinch coils confines the atom along the third dimension. An atom approaching a pinch coil experiences an increasing magnetic field and is repelled towards the trap centre .The direction of the current through both pinch coils is the same, so that the minimum of the trapping potential has a non - zero magnetic field. The radial confinement of the atoms can be Increased by applying a homogeneous bias field, which is oriented parallel to the 4 bars. This field is produced by two additional coils with large diameters than the pinch coils. The resulting trapping potential has the shape of a cigar, where the radial confinement is due to the 4 bars. The major drawback of these traps is that they typically dissipate – kilowatts of power and are operated at several hundred amperes. This causes considerable cooling, stabilization and switching problems. Another Experimental difficulty in the operation of the Ioffe – type traps is the alignment of the centre of the Magneto - optical trap with the centre of the magnetic trap.

3.4 QUIC Trap

The quadrupole Ioffe configuration (QUIC) trap, first designed by the group of Theodor W. Hansch 7. It consists of a quadrupole trap made a pair of anti - Helmholtz coils and a third coil known as the Ioffe coil. The advantage of QUIC trap is simpler magnetic coil configuration, much lower current required for the trap and the coils can remain outside the vacuum. A quadrupole trap is formed when the current flows through the quadrupole coils. This configuration is used to load atoms from a magneto - optical trap into the magnetic trap. By turning on the current through the Ioffe coil, the trap centre moves towards the Ioffe coil and the trapping potential is converted into a Ioffe – type geometry.

Field curvature produced by the Ioffe coil which scales as I Ioffe /RAbbildung in dieser Leseprobe nicht enthalten, with R being the radius of the coil and I Ioffe the current through the Ioffe coil. Since the minimum of the trapping potential is close to the Ioffe coil, a small radius R can be chosen so that the atoms are tightly confined even for a low current I Ioffe.

A current Iq through the quadrupole coils produces a spherical quadrupole trap in the central region of the two coils, Where is the field gradient along the axial direction of the quadrupole coils. By increasing the current I Ioffe the magnetic zero of the quadrupole is shifted towards the Ioffe coil. By further increasing a second zero appears in the magnetic field, resulting in a second quadrupole trap in the vicinity of the Ioffe coil. When the current I Ioffe = I q the two spherical quadrupole traps, which are perpendicular to each other, merge and an Ioffe trap is formed. The magnetic field generated by the QUIC trap (quadrupole and Ioffe coils together) near the center can described by

Abbildung in dieser Leseprobe nicht enthalten

Where B0 is the minimum magnetic field, α is the field gradient in the radial direction and β is the field curvature in the axial direction. Thus the potential in Fig. 3.3 for the atoms in such a trap can be given by

Abbildung in dieser Leseprobe nicht enthalten

Where w r and w z are the oscillation frequencies of atoms in the radial and axial directions respectively and m is the mass of the rubidium atom 8.

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Fig. 3.3 Magnetic field due to QUIC trap.

CHAPTER IV

Calculation and Simulation of Magnetic Field for QUIC Trap

4.1 Overview

After laser cooling of atom, the. Evaporative cooling is then necessary to reach the BEC transition temperature. In order to perform evaporative cooling, atoms must be trapped in QUIC trap. The coil configuration for this trap consists of two quadrupole coils along z - axis with opposing currents and a tapered conical solenoid placed with its x - axis at right angles to the axis of symmetry of the quadrupole coils (see Fig. 4.1). All coils can be placed outside the vacuum chamber, reducing experimental difficulties. The tapered conical solenoid allows optical access for the trapping beams although the edge of the cone is placed close to the center of the MOT 9.

Abbildung in dieser Leseprobe nicht enthalten

Fig. 4.1 Schematic diagram of a QUIC trap configuration. The y - axis is directed positively into page.

After laser cooling of atom, the. Evaporative cooling is then necessary to reach the BEC transition temperature. In order to perform evaporative cooling, atoms must be trapped in QUIC trap. The coil configuration for this trap consists of two quadrupole coils along z - axis with opposing currents and a tapered conical solenoid placed with its x - axis at right angles to the axis of symmetry of the quadrupole coils (see Fig. 4.1). All coils can be placed outside the vacuum chamber, reducing experimental difficulties. The tapered conical solenoid allows optical access for the trapping beams although the edge of the cone is placed close to the center of the MOT 10.

4.2 Calculations of Magnetic Field Due to Quadrupole Coils

Consider the coil in the 2D in X - Y plane. The coordinate of coil on the rim is [Rcos(s), Rsin(s), 0]. When this coil is placed in 3D the coordinate of general point G (x, y, z). The coordinate of A on the rim when coil is placed at z = -l in 3D is [Rcos(s), Rsin(s), -l], where R is radius and l is the distance of the one quadrupole coil from origin.

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