Schrödinger's paper (1926) on Bose–Einstein condensation was submitted for publication on 15 December, 1925, immediately preceding the first of his papers on wave mechanics. Historians of science (Hanle, 1977; Klein, 1964) have seen ‘an organic connection’ between the two. It came between the three papers of Einstein (1924, 1925 a, b ) on the boson gas and Uhlenbeck's doctoral thesis (Uhlenbeck, 1927), and did not receive the attention it deserved, although, with hindsight, it is seen to contain the gem of the idea which was later to resolve the conflict between Einstein and Uhlenbeck. Einstein based his prediction (Einstein, 1925 a ) of condensation in the free boson gas on a combination of Bose–Einstein statistics and the classical density of states in phase-space; this leads to an expression for the mean particle number density as a function of kinetic energy which saturates as the total density increases so that there is a maximum possible density of particles having nonzero kinetic energy. Einstein claimed that, if the particle density is increased above this maximum, the excess goes into a state with zero kinetic energy. This is the phenomenon of condensation. Uhlenbeck (1927) objected that the result holds only ‘when the quantization of translational motion is neglected’. The situation remained unclear until the Amsterdam meeting to celebrate the centenary (23 November 1937) of the birth of Johannes van der Waals; here Kramers pointed out the importance of the thermodynamic limit for the sharp manifestatioon of phase transitions.
Schrödinger's paper (1926) on Bose–Einstein condensation was submitted for publication on 15 December, 1925, immediately preceding the first of his papers on wave mechanics. Historians of science (Hanle, 1977; Klein, 1964) have seen ‘an organic connection’ between the two. It came between the three papers of Einstein (1924, 1925 a, b ) on the boson gas and Uhlenbeck's doctoral thesis (Uhlenbeck, 1927), and did not receive the attention it deserved, although, with hindsight, it is seen to contain the gem of the idea which was later to resolve the conflict between Einstein and Uhlenbeck. Einstein based his prediction (Einstein, 1925 a ) of condensation in the free boson gas on a combination of Bose–Einstein statistics and the classical density of states in phase-space; this leads to an expression for the mean particle number density as a function of kinetic energy which saturates as the total density increases so that there is a maximum possible density of particles having nonzero kinetic energy. Einstein claimed that, if the particle density is increased above this maximum, the excess goes into a state with zero kinetic energy. This is the phenomenon of condensation. Uhlenbeck (1927) objected that the result holds only ‘when the quantization of translational motion is neglected’. The situation remained unclear until the Amsterdam meeting to celebrate the centenary (23 November 1937) of the birth of Johannes van der Waals; here Kramers pointed out the importance of the thermodynamic limit for the sharp manifestatioon of phase transitions.
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01
In 1995 two different research groups observed for the first time the Bose-Einstein condensation (BEC) in ultracold gases. When the confining magnetic trap was turned off the gas was left free to expand, and the velocity of the particles showed a clear peak: most of the particles were occupying the same single particle state, the one of lowest energy. The Bose-Einstein condensation had been predicted in 1925 by Einstein, written by inspiration of a work on the statistic of the photons by Bose (1924). In this work Bose described the behavior of an ensemble of photons, treating them as massless particles, with no number conservation associated. Einstein extended this approach to particles with a mass and with fixed number, creating what is now called the Bose-Einstein distribution. The particles that follow such a description are called ``bosons'', as opposed to the ``fermions'' of the Fermi-Dirac statistics. Einstein predicted that in a gas of bosons - under a critical temperature - a finite fraction of the total number of particles would have been in the ground state, and act as a single entity. This amusing theoretical discovery found its utility a few years later. In the late thirties, new techniques allowed to cool Helium-4 at few Kelvins above the absolute zero. The properties of the resulting liquid were a puzzlement to the scientific community: among others, it could flow without experiencing friction. The liquid was called a ``superfluid''. A first explanation was given by London in 1938, which linked the superfluid behavior to the presence of a BEC among the bosonic Helium particles. The fermions cannot condense by themselves. On the other hand, they can form bound pairs and act as bosons, as it happens in a metal at low temperature. Using this approach, in 1957 Bardeen, Cooper and Schrieffer created a successful model of superconductivity by describing a superconductor as a superfluid in a charged system. During the course of these years we explored the superfluid properties of Bosons and Fermions in different settings. The original contributions of the thesis are described starting from the third chapter, where we speak about the generation and stability of solitons in a periodic optical lattices, both fixed or in motion. In the fourth chapter we study the generation of giant vortices in cold fermions, by using a generalized hydrodynamical approach. In chapter 5 we study the effect of a quasiperiodic lattice and the glassy phase it produces on a gas of bosons. Finally, we study the interaction of normal matter and superfluids with gravitational waves. While this interaction is seen to be extremely small, we believe that the resulting formalism is interesting by itself.
The field of atomic quantum gases has seen rapid and sometimes surprising developments since its beginnings in 1995. In this chapter, I summarize, highlight, and comment on selected topics. Introduction Bose-Einstein condensation (BEC) in atomic gases was first observed in 1995, and has changed the face of atomic physics. It is for atoms or matter waves what the laser is for photons: a macroscopically occupied quantum state. It was regarded as an elusive goal until it was discovered in 1995. Although BEC was immediately viewed as a major accomplishment, its impact has far exceeded expectations. Now, twenty years on, there is no question that the field remains exciting. It is impossible to give a review over the developments during those twenty years. Instead, in this chapter, I want to illustrate how often predictions or expectations changed in the pursuit of Bose-Einstein condensation. There were many surprises, and some advances and breakthroughs happened although they were predicted to be impossible. A lesson we can learn from this is that we should always carefully read the fine print when something is proven or assumed impossible, and try to figure out if the assumptions can be circumvented! Since the early history of Bose-Einstein condensation provides several examples for such “impossibility theorems,” I digress into those earlier developments in the first part of this chapter. Early Theoretical Questions Validity of the Prediction of Bose-Einstein Condensation Einstein predicted Bose-Einstein condensation in 1924 in the second of two papers where he generalized Bose's treatment from photons to massive particles [1]. Using statistical arguments introduced by Bose, he found that below a critical temperature, bosonic particles condense in the lowest energy state of the system. In contrast, at very low temperature photons can simply disappear. However, for about a decade, this prediction was not taken at face value. Einstein himself wrote to Ehrenfest, “From a certain temperature on, the molecules ‘condense’ without attractive forces; that is, they accumulate at zero velocity. The theory is pretty, but is there some truth in it?” [2]. In 1927, George E. Uhlenbeck concluded that the predicted BEC phase transition was an artifact of the replacement of the summation over states by an integral and wrote that no “splitting into two phases” would occur [3].
It is natural to think that a phase changes because particles attract one another. But the condensation of bosons happens without any attraction at all. The condensation temperature of an ideal Bose gas is computed in three cases, and what fixes that temperature is examined. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──condensation of the ideal Bose gas (Einstein 1925), the thermal de Broglie wavelength, and the value of the Riemann zeta function at 3/2 are all standard. Interacting condensates are not treated──scattering lengths, mean-field equations and the shift of the condensation temperature by interactions are not included. Only the ideal gas is examined. Experiments are not discussed──how the gas is cooled (evaporative cooling, magneto-optical traps) and how the condensate is observed are not the subject. Superconductivity is not explained──condensation that requires pairing is not entered into. The separator "does it disappear when the interaction is switched off" is placed, and no more. Finite systems are not treated──the counting is in the thermodynamic limit, and the rounding that comes with small particle numbers is not included. Relation to earlier papers: Paper 482 showed that the words "critical dimension" name two different numbers above and below, and stated that it did not treat condensation itself, the existence proof of a phase transition, behaviour in finite systems, or the interacting case──this paper takes that seat and counts, for the same ideal Bose gas, the condition for condensation. What is added is showing from the contents of the formula that no interaction enters the condensation temperature, confirming to 10 digits that three different cases return the same phase-space density, deriving the ratio of wavelength to mean spacing, and placing the separator on switching off the interaction. First, no interparticle potential appears in the formula for the condensation temperature──only the Planck constant, the Boltzmann constant, the mass of the particle and the number density enter it; no quantity measuring the strength of an interaction appears (Section 2). Second, the temperature is nevertheless definite──Rb87 at 10^20 per cubic metre gives 398.3307 nK, and at the density of liquid helium (2.181607x10^28 particles) 3.134378 K, the latter being 1.439902 times the measured lambda point of 2.1768 K (Section 2). Third, and this is the core. The condition is a single dimensionless number──at the condensation temperature, the number density times the cube of the thermal de Broglie wavelength is exactly 2.6123753487, agreeing to 10 digits across three cases of different mass and density (Section 3). Fourth, the boundary is where the wavelength overtakes the spacing──for Rb87 the mean spacing is 2.154435x10^-7 m while the thermal de Broglie wavelength at the condensation temperature is 2.967189x10^-7 m, or 1.377247 times as long (Section 3). Fifth, raising the density does not raise the temperature in step──ten times the density gives only 4.641589 times the temperature, which is ten to the two thirds (Section 2). Sixth, the separator is whether switching off the interaction makes it disappear──the computation here contains no interaction and still condenses, so switching one off changes nothing. A condensation that requires pairs to form does disappear (Section 4). Condensation does not happen because particles attract one another. The formula for the condensation temperature of an ideal Bose gas contains only constants, the mass and the number density, and it still returned definite temperatures: 398.3307 nK for Rb87 and 3.134378 K at the density of liquid helium. At those temperatures, three cases of widely different mass and density returned the same phase-space density, 2.6123753487──the condition is not the strength of an attraction but the thermal de Broglie wavelength overtaking the spacing between particles, by a factor of 1.377247 in the example here. The separator is whether switching off the interaction makes it disappear. Placed among the earlier papers──in the seat Paper 482 left open by not treating condensation itself, the condition for it was counted. To be honest──the factor of 1.439902 against the lambda point is the work of interactions, and its size is not explained here. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: Bose-Einstein condensation, ideal Bose gas, phase-space density, thermal de Broglie wavelength, critical temperature, Bose statistics. ----- 相が変わるのは、粒子どうしが引き合うからだと考えるのは自然である。だがボース粒子の凝縮は、引力を一つも入れなくても起きる。理想ボース気体の凝縮温度を三つの場合で数え、何がその温度を決めているのかを見る。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──理想ボース気体の凝縮(アインシュタイン 1925)、熱的ド・ブロイ波長、リーマンのゼータ関数の 3/2 での値は、いずれも既知である。相互作用のある凝縮体を扱わない──散乱長、平均場の方程式、相互作用による凝縮温度のずれは入れない。理想気体だけを見る。実験を論じない──どう冷やすか(蒸発冷却、磁気光学トラップ)、どう観測するかは主題ではない。超伝導を説明しない──対をつくる凝縮に本稿は立ち入らない。分離子として「相互作用を切ったら消えるか」を置くにとどめる。有限の系を扱わない──熱力学極限で数えており、粒子数が少ないときの丸みは入れない。既刊との関係:論文482 は「臨界次元」という一語が上と下で別の数を指すことを示し、射程注記で「凝縮そのものを論じない──相転移の存在証明、有限系での振舞い、相互作用のある場合は主題ではない」と明記した──本稿はその席に座り、同じ理想ボース気体で、凝縮の条件そのものを数える。加えたのは、凝縮温度の式に相互作用が入っていないことを式の中身で示したこと、三つの場合で位相空間密度が同じ定数になることを 10 桁で確かめたこと、波長と粒子間距離の比を出したこと、分離子を「相互作用を切ったら消えるか」に置いたことである。 第一に、凝縮温度の式には、粒子間のポテンシャルが一つも入っていない──入っているのはプランク定数、ボルツマン定数、粒子の質量、数密度の四つだけで、相互作用の強さを表す量は現れない(第2節)。 第二に、それでも温度は具体的に決まる──Rb87 を 1 立方メートルあたり 10^20 個にすると 398.3307 nK、液体ヘリウムの密度(2.181607x10^28 個)なら 3.134378 K で、後者は実測のラムダ点 2.1768 K の 1.439902 倍だった(第2節)。 第三に、これが本稿の芯である。条件は一つの無次元の数である──凝縮温度では、数密度に熱的ド・ブロイ波長の三乗を掛けた量がちょうど 2.6123753487 になり、質量も密度も違う三つの場合で 10 桁一致した(第3節)。 第四に、波長が粒子の間隔を越えるところが境目である──Rb87 の例では、粒子間の平均距離 2.154435x10^-7 m に対して、凝縮温度での熱的ド・ブロイ波長は 2.967189x10^-7 m で、1.377247 倍だった(第3節)。 第五に、密度を上げても温度は同じようには上がらない──密度を 10 倍にしても凝縮温度は 4.641589 倍にしかならず、これは 10 の 2/3 乗である(第2節)。 第六に、分離子は、相互作用を切ったら消えるかである──本稿の計算は相互作用を一つも含まずに凝縮を出しており、切っても消えない。対をつくる必要がある凝縮は、切れば消える(第4節)。 凝縮は、粒子どうしが引き合うから起きるのではない。理想ボース気体の凝縮温度の式に入っているのは定数と質量と数密度だけで、それでも Rb87 で 398.3307 nK、液体ヘリウムの密度で 3.134378 K という具体的な温度が出た。その温度では、質量も密度も大きく違う三つの場合が、同じ位相空間密度 2.6123753487 を返した──条件は引力の強さではなく、熱的ド・ブロイ波長が粒子間の距離を越えること(本稿の例では 1.377247 倍)である。分離子は、相互作用を切ったら消えるかである。既刊との位置──論文482 が「凝縮そのものは論じない」と空けた席で、その条件を数えた。正直に言えば──ラムダ点との 1.439902 倍の差は相互作用の効きであり、本稿はその大きさを説明していない。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:ボース・アインシュタイン凝縮、理想ボース気体、位相空間密度、熱的ド・ブロイ波長、臨界温度、ボース統計。
This thesis is devoted to the rigorous study of dilute Bose gases with two-body and three-body interactions at zero temperature in two and three dimensions. Bose gases are many-body quantum systems composed of bosons, which notably do not obey the Pauli exclusion principle. As a consequence, at extremely low temperatures, a macroscopic fraction of the particles can occupy the same one-body quantum state. This phenomenon, known as Bose–Einstein condensation, was predicted in the 1920s by Bose and Einstein for noninteracting Bose gases, and first realised experimentally by the groups of Cornell and Wieman, and Ketterle, earning them the Nobel Prize for Physics in 2001. However, proving the occurrence of Bose–Einstein condensation in general interacting bosonic systems remains a major open problem and constitutes a central theme in this thesis. Another important property of bosonic matter is that it is generally more unstable than fermionic matter. In particular, for systems interacting via the Coulomb force, the Pauli exclusion principle is essential for explaining the stability of the second kind, which states that the energy and volume scale linearly with the number of particles. Nonetheless, bosonic systems with attractive but less singular interactions can also be stable, and the rigorous proof of this fact forms a second focus of this thesis. In three dimensions, we consider predominantly repulsive interactions. Firstly, we prove a leading order upper bound on the thermodynamic ground state energy of Bose gases with hard-core three-body interactions, which matches an existing lower bound. In addition, we obtain a simpler proof for integrable potentials. Secondly, we derive the leading order of the thermodynamic ground state energy of Bose gases with combined two-body and three-body interactions. More specifically, we show that, at the leading order, the ground state energy is entirely characterised by the scattering length of the two-body potential and the scattering hypervolume of the three-body potential. Thirdly, we study the time evolution of Bose–Einstein condensates with three-body interactions in the so-called Gross–Pitaevskii regime. We show that Bose–Einstein condensation is preserved under many-body evolution and that the condensate wavefunction evolves according to the quintic Gross–Pitaevskii equation. In particular, we show that the effective coupling constant is universal in the sense that it depends only on the scattering hypervolume of the interaction potential. As a by-product of our proof, we derive a stability of the second kind result for Bose gases with attractive two-body interactions and repulsive three-body interactions. In two dimensions, we consider trapped Bose gases with mean-field-type interactions, including the purely attractive case. We prove the stability of the second kind of the system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger energy functional. As a consequence of the stability of the second kind, we also obtain Bose–Einstein condensation for the many-body ground states.