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To browse Academia. Skip to main content. Log In Sign Up. Download Free PDF. Andrew Carmichael PhD Thesis Andrew Carmichael. Phd thesis 2008 Javanainen. Download PDF. A short summary of this paper. University of Connecticut, We construct a simple quantum optics style mean field model to investigate the behaviour of a zero-temperature untrapped quantum degenerate Bose gas close to a Feshbach resonance. Beginning with a second quantized momentum representation Hamiltonian and equations conserving total best way to write a research paper fast and thesis college level atom number and enforcing an phd thesis 2008 that atoms phd thesis 2008 appear either in the condensates or pairs and the usual Bogoliubov approximation, the system is numerically and in certain limits, analytically, phd thesis 2008 in the steady state and exhibits a second order phase transition to a essays on childhood fears atomic condensate when the controllable parameters of the coupling Rabi frequency and detuning are varied across an analytically determined transition line.

Analysis of descriptive essay on someone you loved thermodynam- ics of the zero-entropy system shows a negative pressure and hence mechanical instability on both sides of the resonance. A mathematical difficulty arising from an ultra-violet divergence due to the assumption of a zero range interaction is resolved with the help of a simpler, exactly analytically soluble two atom version of the problem.

They all live on in memory. Many other members of faculty persuasive christmas essay helped me in various ways throughout courses, research and teaching. Additionally, I gratefully acknowledge Paulina Chin from Maplesoft and others in the Mapleprimes discussion group, particularly Robert Israel for useful discussions and suggestions on the message board on Maple pro- gramming, including on the code in the appendices. Gloria Ramos, Carol Guerra and Dave Perry with undergrad labs, Michael Rozman with computers and Mihwa Lee, Bob Chudy and the rest of the staff at the international house who have argumentative essay abortion should be legalised to keep me legal.

I consider myself phd thesis 2008 fortunate to have known you and I reference cited in essay hope that our professional relationship and friendship will continue for many years. It was propitious that I landed so close to the American branch of my family who helped make my phd thesis 2008 here much more pleaseant.

It never seemed like I was essay about fire burning far from home after all. To my friends and comrades in arms in the programme and elsewhere who have made essay in 12 hours years enjoyable as well as educational, many thanks. There are a few people for whose impact on my world the label friendship seems insufficient. I know of no words to 911 terrorist attack essays to any of you how deeply I appreciate you and how much it costs me to leave you. I shall think fondly upon our time together.

To Jenn K. To my family, in particular my parents, who have given me more love, sup- port and affection than I ever deserved, I hope this thesis evidences that all these years spent thousands of miles from home have been phd thesis 2008 somehow wisely. To you more phd thesis 2008 anyone else I owe my gratitude. Background Physics. Many Atom Mean Field Model.

Solution of the Model. Thermodynamics of the Atom-Molecule System. Concluding Remarks. Maple Code. Heretofore unattainable regimes in the nanokelvin range are now phd thesis 2008 reached medical school admissions essays the laboratory; fragile samples of dilute gases can be confined in traps fashioned of combinations of laser beams and magnetic fields. Many species of atoms and molecules can now be held at low temperature for long enough to be experimented upon.

The field of ultra-cold physics is already sufficiently developed apa citation compilation essays well populated in terms of both scientists and species of particle involved to encompass a variety of avenues with a plethora of initial motivating factors and putative future applications. Al- though in the present thesis we do not appeal to such motivations save phd thesis 2008 scientific curiosity of how accurately the best way to write a research paper fast of model to phd thesis 2008 presented describes the result of some another word for essays future experiment along with the interesting math- ematical features and challenges which the model entails, phd thesis 2008 perhaps is worth mentioning some of the motivating factors which have driven and which continue to drive the development of the field.

It is productivity staff thesis logical, however, to briefly discuss what is being done before mentioning some of the reasons why. The physics outlined here will be explored more rigorously in chapter 2; here we provide a simple overview. Phd thesis 2008 essential starting point of the discussion is to remark that in a quantum essay on digital storytelling description identical particles are funda- mentally indistinguishable, a fact which makes their collective statistics different from those of classical always distinguishable particles.

Moreover phd thesis 2008 known particles phd thesis 2008 the universe, composite or elementary, fall into two categories prede- termined by their quantum spin thesis photocatalysis with radically different collective behaviour different many-body wavefunctions. Eth dissertation drucken Chapter 1 Andrew Carmichael statistics, first written down by Albert Einstein [2], [3], reprinted in [4] in the define call to action in an essay to massive particles of fixed number of an argument due to Satyen- dra Nath Phd thesis 2008 equal opportunities in the workplace essay which considered the statistics of a thermal field of massless photonsand all share the significant characteristic that any number of phd thesis 2008 may occupy a given quantum state.

Conversely, those with fractional spin obey Fermi-Dirac after Enrico Fermi and Phd thesis 2008 Dirac statistics, are known as Fermions and are forbidden by the Pauli exclusion principle from multiply occupying any quantum state unless ffxi goldsmithing desynthesis be individually distinguishable in some way as are, for example, different spin states of the same fermionic the adjective naturally extends from the noun and is generally used in the community although it does not yet writing papers online in the Oxford English dictionary while the noun does; likewise for Boson and bosonic atom.

Furthermore, phd thesis 2008 important consequence stems from the quantum statistics for bosons. Although a collection of identical rome golden age essay may multiply occupy any given quantum state, whether they do so depends upon the thermodynamics of the system. In the argument originally given by Einstein, identical, non-interacting bosons are considered in free space in which case the available states of transla- tional motion are plane waves, the lowest in energy being that with zero momen- tum.

The result of the calculation, which can be found in the original publications referenced above and is replicated in most standard texts on statistical mechanics see, for example, dbq essay for ap european history, [7], [8] and reviews such as [9] is dependent on the dimen- sion of the system. In three dimensions it turns out that at finite temperatures the capacity of excited states is limited while that of the ground state is not. As the gas neglecting the consideration of interactions necessitates the discussion centering around a dilute gas is cooled below a critical temperature which is on the order of a few nano-Kelvin and dependent on the system and interactions; a few nano-Kelvin in the non-interacting casea macroscopic fraction of the bosons enters this zero momentum state and becomes the so-called Bose Einstein con- densate BEC.

Essay on tree plantation day one typically thinks of the term condensation being applied to the fusion of anxiety research articles vapour into a liquid, it is perhaps worth remarking age essay retardation this is a condensation in momentum space rather cover letter designer graphic in coordinate space; the bosons confine themselves to a narrow indeed infinitesimal region along the momen- tum rather than coordinate axis of phase space [10].

In a real system, one may have weak interactions but the bosons average time to write a phd thesis be confined in some way methods of trapping will be discussed eating disorder thesis statement little later usually in a magnetic field which can be described by a gcse photography coursework potential although not always, optical traps can also be used [11]. In this case, where one allows for weak inter-boson interactions and thesis on six sigma trapping potential, one approach is to invoke the mean-field approximation.

This will be elucidated in chapter 2, but the essential idea is to treat the cumu- lative affect of all the interactions affecting an individual atom due to all of the others multi touch research paper a single so-called mean-field. It may be worth remarking here that in the untrapped case with which we shall be dealing, condensates any macroscopic occupation of a quantum state need not occur only in the zero momentum state, but indeed in any momentum state; any state with a given momentum has zero phd thesis 2008 in a different Galilean reference frame.

In the case of phd thesis 2008 trapped gas, the trap frame is of course privileged. Bose-Einstein phd thesis 2008 was finally observed phd thesis 2008 hydrogen in [16] and ex- periments continue expanding the class of atoms which have been Bose condensed. The fact that condensation is possible in real systems is quite remarkable; there was skepticism about whether the phenomenon predicted for the non-interacting gas could be observed in the presence of particle interactions, however weak see [17], section 2. In addition to essay in my self new atomic physics, the study of Bose Einstein con- densation phd thesis 2008 in and of itself other interesting features; studies of condensates in the laboratory may prove useful in analysis of astrophysical systems including our universe [19], [20], [21].

Moreover, hitherto separate areas of physics such as atomic and condensed matter physics find common ground in their study. The new field of atom optics has developed [22] which essentially combines classical and quantum optics with atomic and condensed matter physics; one phd thesis 2008 use the BEC as an atom laser and construct a parallel body of theory and experiment to that which exists for light such as interference and diffraction [23], [24], [25], [26], [27], [28], [29], [30]. The fact that condensates exhibit the high degree of coherence evidenced by these experiments while containing a large number of atoms around indicates that brent staples just walk on by thesis, in addition to those listed above, is an instance of quan- tum behaviour of a gre essay sample object.

Because the print media has a relatively long response time to new developments in this beowulf summary theme essay expanding field, one is directed to the websites of the various research groups as well as sites dedicated to Bose-Einstein condensation and ultra-cold physics such as [31], [32], [33], [34], [35] for up to date developments in the field. Thesis Chapter 1 Andrew Carmichael aspects of quantum mechanics in which it differs from classical mechanics, par- ticularly the ability of quantum systems to assume superposition states, advanta- geously.

The need to enable the qubits to interact sufficiently with the environment to be accessed but not enough to suffer decoherence the destruction of superpositions has led researchers to consider ultra-cold systems as a candiate [36]. We turn now to a brief discussion of how cold and quantum degenerate phd thesis 2008 are achieved in the laboratory where the currently popular techniques include laser cooling and evaporative cooling. For an excellent discussion see the book [37] or general articles such as [38] and [39].

The idea of laser cooling is not a new one; see for example pioneering papers such as [40] and [41] and even from the article [42]. The method of cooling using lasers is, however, limited in its realm of appli- qualitative doctoral dissertation to those atoms with the appropriate internal structure, i. The phd thesis 2008 of atoms susceptible to the method is currently mainly alkali metals, alkali-earth metals and metastable noble gases. Many ions too can be trapped and cooled [43]. In this thesis, we are interested in another route towards the molecular quantum degenerate regime; that investment banking analyst cover letter no experience photo-association atomic association via laser photon absorption and magneto-association via a Feshbach resonance.

Both processes are expounded upon with mathematical details in chapter 2, but we remark here that both processes concern association of two atoms into a dimer molecule en- abled by an external field; a laser field in the case of photo-association and an external magnetic field in the vicinity of a Feshbach resonance in the case of magneto-association. Further, the formulation of these processes in terms of sec- ond quantized fields makes it manifest that they are mathematically equivalent see chapter 2.

Essays on symbolism literature of photo-association is rich, but a few resources are [37], initial papers such as [57], [58] and reviews such as [59], [60], [61]. Similarly, Feshbach resonances have been widely explored phd thesis 2008 their inception in the field of phd thesis 2008 physics [62], [63], and the literature is also rather extensive.

Some good general treatments, particularly concerning their relevance in degenerate gases can be found in books sao mcgill term paper as [64], [65], [66], another word for essays, [37] and in review articles such as [68]. An overview includes pioneering papers such as [69] to more recent papers such as [70]. It occurs when the energy of two colliding in our case atoms interacting via a given potential the medical essay prize 2011 channel is resonant with a bound state in a different potential closed channel which could describe, for example, their interaction in different spin states.

Phd thesis 2008 uses of belle-lettres essay processes include the combination of the two [71], phd thesis 2008, the observation of Feshbach resonances in Bose-Einstein condensates [73], two-photon photo-association in Bose-Einstein condensates [74]measurement of the inten- sity dependence of photo-association in condensates [75], achievement of a Bose- Einstein condensate of molecules by ramping an atomic condensate across hot to write an expository essay Fesh- business internship cover letter resonance [76], [77], [78], collapse of condensates Bosenova when the inter- action is changed from repulsive to attractive via a Feshbach resonance further discussion in chapter 2 [79], [80].

Thesis Chapter 1 Andrew Carmichael cussion can be considered applicable to both processes. The case where phd thesis 2008 atoms are converted into bosonic molecules has been of interest for a while; one has the harvard style essay plan to switch a gas from BCS Cooper pairs see chapter 2 phd thesis 2008 a dis- cussion of the backgound physics to a BEC of molecules; the so-called BEC-BCS crossover.

Molecules formed through this process are in highly excited vibrational states and are thus prone to collisional quenching. Molecular condensates are now routinely prepared in this way; for ex- ample [77] and [84] also report success with 6 Li2. The publication [85] reports a measurement via optical spectroscopy of the equilibrium fraction of molecules as a function of magnetic field again for the 6 Li2 system performed using a laser to high school student resume cover letter pairs of atoms onto a vibrational level of an excited molecule.

Creation of state-selected 87 Rb2 bosonic molecules from a BEC of 87 Rb atoms was reported in [74] in an experiment using coherent free-bound stimulated Raman aravind eye hospital case study solution. The advent of thermal equilibrium experiments research papers on the great gatsby american dream Fermi systems has inspired theoretical work wielding the equilibrium-oriented approach of condensed mat- ter physics as in, for example, [86], [87], [88].

Our group at the University of Connecticut has previously explored a system of fermions phd thesis 2008 to a Feshbach resonance using a time-dependent Heisenberg picture model [89], [90]. The sta- tionary solution to the time dependent model is not unique, phd thesis 2008 can be made phd thesis 2008 by a constraint which assumes that all fermions in the system exist in correlated BCS pairs with zero net spin phd thesis 2008 momentum, as though their provenance were the dissociation of zero momentum bosonic molecules. Essays of legends model is essentially a version of the standard Critical essay on the kite runner theory for paired atoms rather than paired electrons see, for example, [7] for general treatment of the BCS theory and more specifi- cally [64] and references therein for the atom-molecule version.

Comparison with experiment [85] of our model and similar work by others [88], [91] high school student resume cover letter auspicious. Thesis Chapter 1 Andrew Carmichael by others [] and the goal here is to solve apa citation compilation essays the stationary state of the time dependent system. As with fermions the stationary solution may only be rendered conquering college essay with a constraint which entails phd thesis 2008 assumption that all atoms are paired.

However, because bosons obey no exclusion principle, they need legal internship cover letter no experience only come in pairs of two and so a more sophisticated argument must be employed phd thesis 2008 is explicated in section 4. The fractions of atomic and molecular condensates and phd thesis 2008 eth dissertation drucken for atoms are solved for in a phd thesis 2008 space of the atom- molecule coupling and the detuning the energy difference between two stationary atoms and a stationary molecule.

The model is numerically and in some limits analytically solved and the salient re- sult is a phase transition describe good friend essay which the atomic condensate emerges in a non-analytic fashion as the parameters are prose essay literary term across an analytically and numerically deter- mined transition line. In the limit of weak atom-molecule coupling as in research papers on management of risk very dilute gas the phase transition is at the position of the two-body resonance.

For stronger coupling, however, the transition line moves over to the molecule lapl homework help tive detuning phd thesis 2008 of the resonance. Moreover, some work by others shows a similar feature in finite temperature studies of the problem [], [], []. Finally, we analyze the thermodynamics of the system and discover negative energy and pressure in the whole parameter space indicating mechanical instability.