Towards Absolute Zero
Discovery of Superconductivity and Super fl uidity
Recent advancements have allowed scientists to explore the fundamental behaviors of materials at extremely low temperatures, where electricity was anticipated to cease However, when liquid helium was used to cool mercury, Heike Kamerlingh Onnes discovered that its electrical resistance disappeared below 4 K, leading to the phenomenon known as superconductivity—where electrical current flows without resistance This remarkable property has since been observed in various materials at temperatures up to 130 K and has been applied in technologies such as medical MRI scanners, particle accelerators, and levitating maglev trains Additionally, Onnes and his team noted intriguing behaviors in liquid helium itself.
At around 2.2 K its heat capacity undergoes a discontinuous change, termed the
The "lambda" transition in helium is indicative of a phase change, leading to the classification of liquid helium into two distinct phases: helium I for temperatures above the critical temperature (T λ) and helium II for temperatures below T λ Subsequent experiments have shown that helium II exhibits remarkable properties, such as remaining liquid near absolute zero, flowing through extremely narrow pores, and resisting boiling These unique characteristics, along with the ability of helium to remain in a liquid state down to absolute zero at atmospheric pressure, result in a phase diagram for helium that significantly differs from that of conventional liquids.
In 1938, groundbreaking experiments by Allen and Misener, along with Kapitza, uncovered the remarkable property of helium II: its capacity to flow without viscosity This extraordinary internal mobility, similar to that of superconductors, prompted Kapitza to introduce the term “superfluid.” Subsequent observations revealed additional unusual phenomena, such as “fluid creep.”
Fig 1.2 Phase diagram of helium For a conventional substance (inset), there exists a triple point (TP), where solid, liquid and gas coexist.
The shaded area indicates the predicted occurrence of Bose–Einstein condensation in an ideal gas, particularly highlighting helium's unique ability to creep up vessel walls and overflow Additionally, the phenomenon known as the "fountain effect" is observed, where the application of heat to the liquid generates a continuous fountain.
Bose – Einstein Condensation
Superfluidity and superconductivity challenged classical physics, necessitating a new perspective In 1938, London revived Einstein's 1925 prediction to explain superfluidity, which posited that at low temperatures, a significant fraction of quantum particles would undergo Bose–Einstein condensation, forming a condensate while the rest behaved normally Although this concept initially stalled due to the required conditions being in the solid region of the pressure-temperature diagram, it laid the groundwork for further exploration Einstein's model, which predicts a discontinuity in heat capacity akin to that observed in helium, ultimately contributed to the development of the two-fluid model by Tizsa and Landau, which describes helium-II as a mixture of a viscosity-free superfluid and a viscous normal fluid.
Bose–Einstein condensation is a phenomenon that occurs with bosons, such as photons and helium-4 atoms, but not with fermions like protons, neutrons, and electrons due to the Pauli exclusion principle, which prohibits identical fermions from sharing the same quantum state However, in 1957, Bardeen, Cooper, and Schrieffer proposed that electrons could form pairs known as Cooper pairs, allowing these composite bosons to participate in Bose–Einstein condensation, thereby facilitating the flow of electrons in superconductivity.
Bose-Einstein condensation was significantly supported by the discovery of superfluidity in the fermionic helium isotope 3 He in 1972, observed at approximately 2 mK For additional details on superconductivity, refer to the relevant literature.
Superfluid helium and superconductors are examples of Bose-Einstein condensation, representing quantum fluids influenced by quantum mechanics While superconductors can be viewed as fluids of charged Cooper pairs, the complex interactions in liquids and solids make these systems significantly more intricate than Einstein's ideal-gas model It wasn't until the 1990s that an almost ideal state was successfully achieved.
Superfluidity is characterized by several key features, including the ability to flow without viscosity, a critical velocity beyond which superflow ceases, the formation of quantized vortices, persistent flow, and macroscopic tunneling exemplified by Josephson currents This book will explore these superfluid phenomena in detail, except for Josephson currents, which can be examined in other resources.
Ultracold Quantum Gases
Laser Cooling and Magnetic Trapping
Cryogenic refrigeration techniques have successfully cooled liquids and solids to milliKelvin and microKelvin temperatures, achieving a record low of 100 pK for nuclear spins in rhodium Meanwhile, the advancement of laser cooling in the 1980s significantly improved the cooling of gases In this process, laser beams impart momentum to gas atoms, slowing them down and enabling three-dimensional (3D) cooling This technique produced a gas at 240 μK in 1985, with average atom speeds of approximately 0.5 m/s, and later achieved 2 μK with speeds around 1 cm/s The resulting ultra-dilute vapors, with number densities of about 10^20 m^−3, transition from gas to solid very slowly at these temperatures Additionally, magnetic fields create traps to confine atoms, allowing for diverse configurations and manipulation of ultracold gases The significance of laser cooling and magnetic trapping techniques was recognized with the 1997 Nobel Prize in Physics.
Bose – Einstein Condensate à la Einstein
The development of ultracold gases brought Einstein's prediction of a gaseous condensate closer to reality, igniting a new wave of research According to Einstein's model, the condensate is expected to form at temperatures below a critical threshold, approximately T c ∼10 −19 n 2/3 However, the low gas densities used in experiments posed challenges to achieving this prediction.
To achieve temperatures colder than those possible with laser cooling alone, a method of evaporative cooling was implemented This technique involves selectively removing the hottest atoms, similar to how evaporation cools a cup of coffee, allowing for even lower temperatures to be reached.
In 1995, Cornell and Wieman achieved a groundbreaking milestone by cooling rubidium atoms to 200 nK, creating the first gaseous Bose–Einstein condensate (BEC) This experiment, illustrated in Figure 1.3, showcased the transition from a thermal gas with a broad speed distribution to a BEC characterized by a narrow speed distribution as the temperature dropped below the critical temperature (T_c) When the trap confining the gas was released, fast-moving atoms dispersed widely, while the cooled atoms accumulated into a state of minimal energy and speed, forming the BEC Shortly after, Ketterle independently produced a BEC of sodium atoms, marking a significant realization of Einstein's prediction nearly seventy years later For their pioneering work, Cornell, Wieman, and Ketterle were awarded the Nobel Prize in Physics in 2001.
The first observation of a gas Bose–Einstein condensate revealed the momentum distribution of a dilute ultracold gas of 87 Rb atoms confined in a harmonic trap As the temperature decreased, the gas transitioned from a broad, energetic thermal state to a narrower distribution, indicative of the formation of a condensate This significant finding is illustrated in an image sourced from the NIST Image Gallery.
There are now over 100 BEC experiments worldwide These gases are typically 10–100àm across (about the width of a human hair), exist in the temperature range
1 to 100 nK, contain 10 3 −10 9 atoms, and are many times more dilute than room temperature air BECs are most commonly formed with rubidium ( 87 Rb) and sodium
Various atomic and molecular species, including 23 Na atoms, have been successfully condensed into multi-component condensates that coexist These gases represent some of the purest quantum fluids, with around 99% of the atoms in a condensed state, making them suitable for first-principles modeling through the Gross–Pitaevskii equation, discussed in Chapter 3 Gaseous condensates exhibit extraordinary properties such as superfluidity, explored in Chapters 4 and 5, and unlike superfluid helium, the weak interaction between atoms aligns them closely with Einstein's ideal gas concept.
Degenerate Fermi Gases
As fermionic gases are cooled toward absolute zero, the Pauli exclusion principle prevents particles from occupying the same quantum state, leading them to fill available quantum states sequentially from the ground state upward, each state holding only one particle This phenomenon was notably observed in 1999 with the creation of a degenerate Fermi gas by cooling potassium-40 (40 K) atoms below this critical temperature.
At temperatures around 300 nK, the gas exhibits a wide distribution, reflecting a higher average energy compared to a Bose-Einstein condensate (BEC) The Pauli exclusion principle creates significant pressure that prevents further contraction, playing a crucial role in stabilizing neutron stars against collapse A notable experimental comparison reveals that as temperature decreases, the fermionic system's distribution remains unable to contract like that of the bosonic system Recent experiments have also investigated the formation of Cooper pairs within these gases.
Quantum Fluids Today
The discoveries of superfluid helium and atomic condensates have significant implications for various fields of science These breakthroughs have not only advanced our understanding of quantum mechanics but also opened new avenues for research and technology Currently, the field continues to evolve, with ongoing studies exploring the properties and potential applications of these fascinating states of matter.
Many-body quantum systems, particularly quantum fluids, exhibit unique quantum behavior on a macroscopic scale involving numerous particles, which leads to their extraordinary characteristics These fluids offer essential insights into the realm of quantum many-body physics Additionally, in the context of condensates, they further enhance our understanding of these complex systems.
As a 7 Li bosonic gas and a 6 Li fermionic gas are cooled towards absolute zero, the density profile changes significantly The bosonic gas condenses into a narrow distribution indicative of a low-energy state, while the fermionic gas maintains a broader distribution due to the outward Pauli pressure exerted by the fermions This phenomenon, illustrated in Fig 1.4, highlights the experimental capacity to manipulate system parameters such as interactions, dimensionality, and the presence of disorder and periodicity Such control enables the investigation of various many-body scenarios and the emulation of complex condensed matter systems, including superconductors.
Quantum fluids serve as an ideal model for studying nonlinear systems, characterized by the absence of viscosity and the presence of discrete, uniformly-sized vortices This unique behavior allows for the exploration of complex fluid dynamics, particularly turbulence, which is further examined in Chapter 5 Additionally, condensates present a controlled environment to investigate nonlinear phenomena, as atomic interactions create distinct nonlinearities that can be manipulated experimentally Notable nonlinear effects, such as solitons and four-wave mixing, have been the focus of experimental studies, with solitons discussed in Chapter 4.
Extra-terrestrial phenomena reveal that condensates resemble curved space-time, facilitating the study of analog black holes and Hawking radiation Both condensates and helium serve as models for the quantum vacuum thought to exist throughout the universe, playing a crucial role in its evolution since the Big Bang These cosmic phenomena, which are not directly observable on Earth, can be effectively simulated and investigated through controlled laboratory experiments.
Helium II's remarkable thermal conductivity makes it an ideal coolant, which is why it is utilized in various superconducting systems, including MRI machines in hospitals and the Large Hadron Collider at CERN.
Sensors are highly sensitive to external forces, including magnetic fields, gravity, and rotational forces, as demonstrated by various experiments Significant research efforts are focused on advancing these concepts into next-generation sensors, which have potential applications in testing fundamental physics, geological mapping, and navigation.
Since 2000, Bose-Einstein condensation has been successfully achieved in various new systems, including magnons in magnetic insulators and polaritons in semiconductor microcavities Notably, these advancements have led to the creation of quantum fluids of light, exhibiting superfluid properties, particularly in optical microcavities.
1 T Shachtman, Absolute Zero and the Conquest of Cold (Houghton-Mifflin, Boston, 2001)
2 J.F Annett, Superconductivity, Superfluids and Condensates (Oxford University Press, Oxford, 2004)
3 F Pobell, Matter and Methods at Low Temperatures, 3rd edn (Springer, Berlin, 2007)
4 H.J Metcalf, P van der Straten, Laser Cooling and Trapping (Graduate Texts in Contemporary Physics) (Springer, Berlin, 2001)
5 The Nobel Prize in Physics (1997) www.nobelprize.org/nobel_prizes/physics/laureates/1997/
6 C.J Pethick, H Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2008)
8 K.B Davis et al., Phys Rev Lett 75, 3969 (1995)
9 The Nobel Prize in Physics (2001).http://www.nobelprize.org/nobel_prizes/physics/laureates/ 2001/
11 K Levin, R.G Hulet, The Fermi Gases and Superfluids: Experiment and Theory, in Ultracold Bosonic and Fermionic Gases, ed by K Levin, A.L Fetter, D.M Stamper-Kurn (Elsevier,
12 A.G Truscott, K.E Strecker, W.I McAlexander, G.B Partridge, R.G Hulet, Science 291, 2570
Classical and Quantum Ideal Gases
Bose and Einstein's prediction of Bose-Einstein condensation stems from their exploration of quantum particle behavior in gases, rooted in Boltzmann's foundational statistical methods for classical particles This article traces the theoretical journey of Boltzmann, Bose, and Einstein, culminating in the derivation of Bose-Einstein condensation for an ideal gas and highlighting its essential characteristics.
Introduction
The air in our surroundings can be characterized by properties like temperature and pressure, which reflect our human sensitivity to these factors However, this gas is composed of tiny particles, including dust, molecules, and atoms, all in constant random motion Understanding how these microscopic behaviors give rise to the macroscopic properties we observe is a complex challenge.
An exact classical approach would proceed by solving Newton’s equation of motion for each particle, based on the forces it experiences For a typical room (volume
At room temperature and pressure, the air particle density is approximately 2×10²⁵ m⁻³, leading to the need to solve around 10²⁸ coupled ordinary differential equations, a task that is computationally impractical Since macroscopic properties are averaged over numerous particles, a detailed particle-by-particle analysis is overly complex Instead, statistical mechanics offers a more efficient approach by allowing us to describe the behavior of particles statistically By establishing rules for particle interactions and considering physical constraints such as boundaries and energy, we can deduce the most likely macroscopic state of the system.
We explore the behavior of an ideal gas composed of N identical, non-interacting particles at temperature T, confined within a volume V This isolated system does not exchange energy or particles with its surroundings, allowing us to predict its equilibrium state Initially, we analyze classical point-like particles before extending our findings to quantum particles, which introduces the phenomenon of Bose–Einstein condensation in an ideal gas This approach builds upon foundational principles in statistical mechanics.
1 In the formalism of statistical mechanics, this is termed the microcanonical ensemble. © The Author(s) 2016 9
Fig 2.1 Two different classical particle trajectories through 1D phase space
In classical mechanics, phase space is represented as a continuum of states, but it can be effectively visualized as a discretized system composed of finite-sized cells, defined by Δp_x and Δx This approach aligns with the foundational works of Boltzmann, Bose, and Einstein in statistical physics For a more comprehensive understanding, readers are encouraged to consult introductory textbooks on statistical physics.
Classical Particles
The state of a classical particle is defined by its position and momentum, requiring six coordinates in a 3D Cartesian system: (x, y, z, p_x, p_y, p_z) This can be visualized in a six-dimensional phase space, where the particle's instantaneous state is represented as a point that traces a trajectory over time For an N-particle gas, this translates to N points or trajectories within the same phase space The range of accessible phase space is influenced by spatial constraints, such as the box containing the gas, and the energy level, which sets the limit on the maximum momentum.
In classical mechanics, a particle's state, defined by its position and momentum, can be measured with arbitrary precision, resulting in a continuous phase space filled with an infinite number of accessible states This characteristic allows for the independent tracking of each particle, highlighting their distinguishable nature.
Ideal Classical Gas
Macrostates, Microstates and the Most Likely
The macroscopic equilibrium state of a gas is determined by how particles are distributed among available cells, with each cell having an equal likelihood of being occupied in the absence of energetic constraints For instance, when considering two distinguishable classical particles, A and B, and three cells, there are nine possible configurations known as microstates These configurations lead to six distinct occupancy combinations, or macrostates, represented as {N1, N2, N3} = {2,0,0}, {0,2,0}, {0,0,2}, {1,1,0}, {1,0,1}, and {0,1,1} Each macrostate can be realized through one or more corresponding microstates.
Particles are in constant motion, colliding and interacting randomly, which leads them to eventually explore all possible microstates over time, a phenomenon known as ergodicity Consequently, each microstate is visited by the particles during this process.
In a system of two classical particles, A and B, distributed across three equally-accessible cells with energy levels of 0, 1, and 2, only specific configurations are possible when the total energy is constrained to 1 The configurations {1,1,0}, {1,0,1}, and {0,1,1} are the most probable, each corresponding to two microstates, illustrating the principle of equal prior probabilities This concept connects the probabilistic nature of microstates to macrostates, where each macrostate reflects a distinct macroscopic property of a physical gas, such as temperature and pressure.
For a more general macrostate{N 1 ,N 2 ,N 3 , ,N I }, the number of microstates is,
Invoking the principle of equala prioriprobabilities, the probability of being in the jth macrostate is,
The probability Pr(j) is maximized when particles are evenly distributed across accessible cells However, energy factors influence the optimal distribution among these cells.
The Boltzmann Distribution
In the ideal-gas-in-a-box, each particle carries onlykinetic energy p 2 /2m=(p 2 x + p 2 y +p 2 z )/2m Having discretizing phase space, particle energy also becomes
In the discretized phase space (x, p_x) illustrated in Fig 2.3(a), the energy-momentum relation E = p²/2m results in the formation of distinct energy levels, as shown in Fig 2.3(b) These energy levels exhibit a degeneracy g, which is a key concept in quantum mechanics, highlighting the quantized nature of energy states This visualization emphasizes the relationship between phase space and energy levels in quantum systems, showcasing three specific energy levels.
E 1=0,E 2= p 2 1 /2mandE 3= p 2 2 /2m, are formed from the five momentum values
In two- and three-dimensional spaces, energy cells E_i are represented on circular and spherical surfaces, adhering to the equations p_x² + p_y² = 2mE_i and p_x² + p_y² + p_z² = 2mE_i, respectively The lowest energy level, E_1, is referred to as the ground state, while all higher energy levels are classified as excited states.
The total energy of the gasUis,
In the context of energy conservation, denoted as U, the energy of cell i (E_i) influences the distribution of microstates and macrostates By setting specific energy values, as illustrated in Fig 2.2, the range of permissible configurations is limited This results in a suppression of particle occupancy at higher energy levels, thereby shifting the distribution towards lower energy states.
In a system at thermal equilibrium with numerous particles, a single macrostate or a limited range of macrostates is predominantly favored This preferred macrostate can be accurately predicted by maximizing the number of microstates (W) in relation to the occupancy numbers {N1, N2, N3, , NI} For further details, refer to sources [1,2].
N i = f B (E i ), (2.3) where f B (E)is the famous Boltzmann distribution, f B (E)= 1 e ( E −μ)/ k B T (2.4)
The Boltzmann distribution describes the probable distribution of particle occupancy in an ideal gas based on energy levels, reflecting the thermodynamic equilibrium state It incorporates Boltzmann's constant (1.38×10 − 23 m² kg s − 2 K − 1) and temperature measured in Kelvin According to the equipartition theorem, each particle, on average, possesses kinetic energy of 3/2 k_B T, with 1/2 k_B T attributed to each direction of motion.
The Boltzmann distribution function f B is normalized to the number of particles,
N, as accommodated by the chemical potentialμ WritingA=e μ/ k B T gives f B A/e E / k B T , evidencing thatA, and therebyμ, controls the amplitude of the distribution function.
The Boltzmann distribution function f B (E)is plotted in Fig.2.4 Low energy states (cells) are highly occupied, with diminishing occupancy of higher energy states.
As temperature rises, the thermal energy increases, leading to a broader distribution of particle energy states This reflects the most probable distribution, as described by Boltzmann’s theory, which suggests that it's possible for all air molecules in a room to concentrate in a single high-energy state.
Fig 2.4 The Boltzmann distribution function f B ( E ) for 3 different temperatures
The phenomenon of energy, momentum, and position exhibiting an uneven distribution in a corner of the room, while statistically improbable, remains a possibility that has historically unsettled the scientific community.
When analyzing energy levels instead of states in phase space, it's essential to consider the concept of degeneracy, denoted as \( g_j \), which represents the number of states within a specific energy level This approach allows us to connect the Boltzmann results to the occupancy of the jth energy level.
Quantum Particles
A Chance Discovery
Quantum physics arose from the failure of classical physics to describe the emission of radiation from a black body in the ultraviolet range (the “ultraviolet catastrophe”).
In 1900, Max Planck formulated a groundbreaking equation that accurately represented energy distribution across all wavelengths, introducing the concept of energy being emitted in discrete quanta, expressed as h f, where h is Planck's constant and f denotes the frequency of radiation This revolutionary idea was further advanced by Albert Einstein in 1905, who proposed that light itself is also quantized.
The discovery of quantum particles occurred by chance when Indian physicist Satyendra Bose, during a 1920 lecture, aimed to illustrate the shortcomings of classical light theory A minor error in his argument instead led him to derive Planck’s empirical formula, based on two key assumptions: that radiation particles are indistinguishable and that phase space is divided into cells of size h³ Initially facing challenges in publishing his findings, Bose received support from Nobel Laureate Einstein, resulting in the publication of his paper “Planck’s law and the light quantum hypothesis” in 1924.
[3] Soon after Einstein extended the idea to particles with mass in the paper “Quan- tum theory of the monoatomic ideal gas” [4].
The division of phase space has long been a subject of intrigue, with Bose noting the lack of definitive insights into this subdivision, while Einstein acknowledged its elegance but found its essence elusive This concept is now recognized as a fundamental characteristic of particles, aligning with de Broglie's wave-particle duality, which suggests that particles are spread over a length scale defined by the de Broglie wavelength (λ dB = h/p), and Heisenberg's uncertainty principle, which states that there is an inherent uncertainty in a particle's position and momentum (ΔxΔyΔzΔp x Δp y Δp z = h³) Each cell within this phase space corresponds to a unique quantum state, highlighting the indistinguishability of particles, as it becomes impossible to differentiate between two overlapping particles in close proximity within this framework.
Bosons and Fermions
Quantum particles come in two varieties—bosonsandfermions:
Fermions, which include particles such as electrons, protons, and neutrons, are characterized by their half-integer spin Following the foundational work of Bose and Einstein, Fermi and Dirac established Fermi-Dirac statistics to describe the behavior of these particles A key principle governing fermions is the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state simultaneously.
1925), which states that two identical fermions cannot occupy the same quantum state simultaneously.
Bosons, which include particles like photons and the Higgs boson, follow Bose–Einstein statistics as established by Bose and Einstein Characterized by their integer spin, bosons can form composite particles from equal numbers of fermions, such as helium-4 (4 He), rubidium-87 (87 Rb), and sodium-23 (23 Na) Unlike fermions, multiple bosons can simultaneously occupy the same quantum state, highlighting their unique properties in quantum mechanics.
The indistinguishability of quantum particles significantly influences their statistical behavior, particularly through the distinct occupancy rules for bosons and fermions For example, when considering two indistinguishable quantum particles distributed across three cells, bosons exhibit six possible microstates, while fermions only have three, in contrast to the nine microstates available to classical particles This results in varying probabilities of paired versus unpaired states: classical particles have a ratio of 1:3, bosons 1:2, and fermions 0 Consequently, bosons tend to cluster together, demonstrating a strong inclination to group, whereas fermions exhibit a complete aversion to one another, reflecting their anti-social nature.
Fig 2.5 Possible configurations of two bosons (left) and two fermions (right) across three equally- accessible cells The classical case was shown in Fig 2.2
The Bose – Einstein and Fermi-Dirac Distributions
Boltzmann's approach to discretizing classical phase space is realized in the quantum realm, allowing for the calculation of distribution functions for indistinguishable particles like bosons and fermions, while adhering to their specific occupancy rules The Bose–Einstein and Fermi-Dirac distribution functions effectively characterize the average distribution of bosons and fermions across various energy states.
E in an ideal gas, are, f BE (E)= 1 e ( E −μ)/ k B T −1, (2.6) f FD (E)= 1 e ( E −μ)/ k B T +1 (2.7)
The rather insignificant looking−1/+1 terms in the denominators have profound consequences Figure2.6compares the Boltzmann, Bose–Einstein and Fermi-Dirac distributions.
We make the following observations of the distributions functions:
For a distribution function to be considered physical, it must meet the condition f ≥ 0 for all energy levels (E) This requirement leads to the conclusion that the chemical potential (μ) must be less than or equal to zero for the Bose-Einstein distribution In contrast, the Fermi-Dirac and Boltzmann distributions allow μ to assume any value, whether positive or negative.
Fig 2.6 The Boltzmann, Bose–Einstein and Fermi-Dirac distribution functions for a T 0 and b T ≈ 0
In the regime where (E−μ)/k_B T is much greater than 1, both the Bose-Einstein and Fermi-Dirac distributions converge to the Boltzmann distribution, indicating that the average occupancy of states is significantly less than one In this scenario, the effects of particle indistinguishability become minimal However, it is important to note that the classical limit condition should not be taken too literally, as it suggests that lower temperatures promote classical behavior, a result stemming from the complex temperature dependence of the chemical potential (μ).
• AsE→μfrom above, the Bose–Einstein distribution diverges, i.e particles accu- mulate in the lowest energy states.
• For Eμ, the Fermi-Dirac distribution saturates to one particle per state, as required by the Pauli exclusion principle.
• For decreasing temperature, the distributions develop a sharper transition about
E=μ, approaching step-like forms forT →0.
The Ideal Bose Gas
Continuum Approximation and Density of States
In this article, we analyze an ideal non-interacting gas of bosons confined within a box, where the occupation of energy levels follows the Bose–Einstein distribution For mathematical simplicity, we approximate the discrete energy levels as a continuum, which is valid when there is a substantial number of accessible energy levels By substituting the level variables with continuous quantities, we express the number of particles at energy E as a function of E, denoted as N(E).
N(E)= f BE (E)g(E)= g(E) e ( E −μ)/ k B T −1, (2.8) whereg(E)is thedensity of states The total number of particles and total energy follow as the integrals,
Fig 2.7 The volume of momentum space from p to p + d p is a spherical shell in
These are integrated in energy upwards from the E=0 (j =1) ground state. The density of statesg(E)is defined such that the total number of possible states in phase spaceN psis,
The expression N = g(E)dE g(p)dp illustrates the relationship between energy and momentum states in a system, with g(p)dp denoting the number of states between momenta p and p+dp These states occupy a six-dimensional volume in phase space, which is the product of their three-dimensional volume in position space, represented by the box volume V, and their three-dimensional volume in momentum space The momentum-space volume corresponds to a spherical shell defined by the range p to p+dp, with an inner radius of p and a thickness of dp, resulting in a volume of 4πp² dp, as depicted in Fig 2.7.
Hence the phase space volume is 4πp 2 Vdp Now recall that each quantum state takes up a volumeh 3 in phase space Thus the number of states betweenpandp+dpis, g(p)dp= 4πp 2 V h 3 dp (2.12)
Using the momentum-energy relation p 2 =2m E, its differential form dp √m/2EdE), and the relationg(E)dE=g(p)dp, Eq (2.12) leads to, g(E)= 2π(2m) 3 2 V h 3 E 1 2 (2.13)
The density of states for an ideal gas within a confined volume V shows that as energy approaches zero, the number of available states decreases, while higher energy levels correspond to an increasing number of states.
While the occupancy of a state goes like 1/(e ( E −μ)/ k B T −1) and diverges as
E →μ, theoccupancy of an energy levelgoes likeE 1 2 /(e ( E −μ)/ k B T −1)and dimin- ishes asE→0 (due to the decreasing amount of available states in this limit) These two distributions are compared in Fig.2.8a.
The occupancy of energy levels, represented by N(E) as a solid line, decreases to zero as energy E approaches zero, which is attributed to the declining density of states in this region In contrast, the Bose–Einstein distribution, depicted as a dashed line, provides a different perspective on particle distribution across energy levels.
Integrating the Bose – Einstein Distribution
Using Eqs (2.8,2.13) we can write the number of particles (2.9) as,
We seek to evaluate this integral To assist us, we quote the general integral, 2
0 t x −1 e − t dtis theGamma function 3 We have also defined a new function,g β (z)= ∞ p =1 z p p β ; an important case is whenz=1 for which it reduces to theRiemann zeta function, 4 ζ(β) ∞ p =1
2 This result can be derived by introducing new variables z = e μ/ k B T and x = E / k B T to rewrite part of integrand in the form ze − x /( 1 − ze − x ) , and then writing as a power series expansion.
3 Relevant values for us are Γ ( 3 / 2 ) = √ π/ 2 and Γ ( 5 / 2 ) = 3 √ π/ 4.
4 Relevant values for us are ζ( 3 / 2 ) = 2 612 and ζ( 5 / 2 ) = 1 341.
Takingα= 1 2 ,x=E/k B T andz=e μ/ k B T in the general result (2.15), we eval- uate Eq (2.14) as,
The equation N = (2πmk B T)^(3/2) V h^3 g^(3/2)(z) is derived using the result Γ(3/2) = √π/2 The variable z is constrained to the range 0 < z ≤ 1, where the lower limit ensures that z = e^(μ/k B T) > 0, and the upper limit prevents negative populations Additionally, within this range, it is important to note that μ must be less than or equal to 0 to comply with the Bose–Einstein distribution Figure 2.8b illustrates the function g^(3/2)(z) across this specified range.
Bose – Einstein Condensation
The prediction of Bose–Einstein condensation in the style of Einstein arises directly from Eq (2.16) Consider adding particles to the box, while at constant temperature.
An increase in N is accommodated by an increase in the functiong 3
2 (z)is finite, reaching a maximum value ofg 3
2 =ζ( 3 2 )=2.612 atz=1 In other words, the system becomessaturatedwith particles This critical number of particles, denotedN c, follows as,
Our derivation suggests a limit to the number of particles in the Bose–Einstein distribution, yet intuitively, more particles should always be added We realized a subtle error in our calculation of N, where we replaced the summation of discrete energy levels with an integral over a continuum, neglecting the ground state's population The density of states, g(E) ∝ E^(1/2), incorrectly indicates zero occupancy in the ground state Consequently, our prediction reflects the saturation of excited states, as any additional particles will occupy the ground state without energetic cost, leading to an anomalously large population when N exceeds N_c.
Bose–Einstein condensation, as described by Einstein, occurs when a growing number of atoms transitions to the ground quantum state, resulting in a separation where some atoms condense while others remain as a saturated ideal gas This phenomenon involves a condensation in momentum space, specifically the occupation of the zero momentum state When confined by a potential, condensation also occurs in real space, moving towards areas of lowest potential Unlike traditional phase transitions, which are driven by particle interactions, Bose–Einstein condensation is primarily influenced by particle statistics.
Based on the above hindsight, we note that the total atom numberNappearing inEqs (2.9), (2.14) and (2.16) should be replaced by the number in excited states,N ex
Critical Temperature for Condensation
When the number of particles and volume are fixed, a critical temperature, T_c, exists below which Bose-Einstein condensation occurs The distribution of excited particles at a specific temperature is described by Eq (2.16) For temperatures above T_c, the gas remains in a normal phase as it can accommodate all particles However, as the temperature decreases, the capacity for excited states diminishes Once the excited states can no longer hold all particles, Bose-Einstein condensation takes place The critical temperature is determined by setting z=1 in Eq (2.16) and solving for T.
As the temperature decreases, the number of particles in excited states diminishes, leading to an increasing number of particles occupying the ground state In the limit of absolute zero temperature (T → 0), all particles transition into the condensate, leaving no particles in excited states.
Condensate Fraction
The condensate fraction is a key metric for characterizing gas, defined as the ratio of particles in the condensate (N0) to the total number of particles (N) To analyze its behavior with temperature, we express the total number of particles as the sum of those in the condensate and those in the gas phase, represented by the equation N = N0 + Nex.
ForT ≤T c , the excited population N ex is given by Eq (2.16) withz=1, and the total population is given by Eq (2.14) withz=1 andT =T c Substituting both into the above gives,
ForT >T c , we expectN 0 /N ≈0 This behaviour is shown in Fig.2.9.
Fig 2.9 a Illustration of energy level occupations in the boxed ideal Bose gas At
T = 0 all particles lie in the ground state For
At temperatures below the critical temperature (0 < T < T c), certain particles occupy excited states while a significant number remain in the ground state However, when the temperature exceeds the critical threshold (T > T c), the occupation of the ground state becomes minimal This behavior is illustrated by the variation of the condensate fraction, represented as N 0 / N, in relation to temperature.
Particle-Wave Overlap
Bose–Einstein condensation occurs when N >N c, with N c given by Eq (2.17).
It is equivalent to write this criterion in terms of the number density of particles, n =N/V, as, n>ζ
According to de Broglie, particles behave like waves, with a wavelengthλ dB=h/p.
For a thermally-excited gas, the particle wavelength isλ dB= h
√2πmk B T Employ- ing this, the above criterion becomes, nλ 3 dB >ζ
Upon noting that the average inter-particle distance d =n − 1 3 andζ( 3 2 ) 1 3 ∼1 we arrive at, λ dB d (2.23)
Bose–Einstein condensation occurs when particle waves overlap, resulting in individual particles merging into a single, large wave of matter known as the condensate.
The transition from a classical gas to a Bose–Einstein condensate is characterized by distinct temperature phases At high temperatures, the gas behaves as a thermal gas composed of point-like particles As the temperature decreases and approaches the critical temperature (T c), the de Broglie wavelength (λ dB) becomes significant but remains smaller than the average particle spacing (d) At the critical temperature (T c), the matter waves overlap, indicating the beginning of Bose–Einstein condensation, where λ dB becomes comparable to d.
Internal Energy
The internal energy of a gas, denoted as U, is derived solely from its excited states, as the ground state is assigned zero energy Consequently, U can be calculated by integrating the energies of the excited state particles.
Upon evaluating this integral below and aboveT cwe find,
The result aligns with the classical equipartition theorem for ideal gases, indicating that each particle averages 1/2 k_B T of kinetic energy per motion direction Additionally, the differing behavior observed for temperatures below T_c confirms the existence of a unique state of matter.
Pressure
The pressure of an ideal gas is defined by the equation P = 2U/3V For temperatures above the critical temperature (T > T_c), the relationship between pressure, temperature, and volume follows the standard ideal gas law, where P is proportional to T/V However, for temperatures below the critical temperature (T < T_c), it is observed that pressure is proportional to T raised to the power of 5/2, with T_c being inversely related to V^(2/3) Notably, at absolute zero, the pressure of the condensate reaches zero and remains independent of the box's volume, leading to the conclusion that the condensate possesses infinite compressibility.
Heat Capacity
The heat capacity of a substance is the energy required to raise its temperature by unit amount At constant volume it is defined as,
A detailed analysis of the dependence of specific heat at intermediate temperatures is provided in Reference [6] Figure 2.11 illustrates the specific heat, C V (T), which exhibits a cusp-like behavior near the critical temperature, T c Typically, abrupt changes in the gradient of C V (T) indicate phase transitions between different states of matter.
Fig 2.11 Left Heat capacity C V of the ideal Bose gas as a function of temperature T Right
Experimental heat capacity data for liquid helium, particularly around the λ-point of 2.2 K, reveals a distinct cusped structure in both measured curves This similarity between predicted and observed heat capacity curves serves as crucial evidence connecting helium II to the phenomenon of Bose-Einstein condensation.
Ideal Bose Gas in a Harmonic Trap
2.5.10.1 Critical Temperature and Condensate Fraction
In typical experiments, atomic Bose–Einstein condensates are confined by harmonic (quadratic) potentials, rather than boxes, 5 with the general form,
The atomic mass (m) and the trap frequencies (ω x , ω y , and ω z) define the strength of the trap in three dimensions, affecting the density of states The density of states is expressed as g(E) = E² / (2³ ω x ω y ω z), which influences the critical temperature in the system.
, (2.29) and for the condensate fraction to vary with temperature as,
The predictions align closely with experimental measurements of harmonically-trapped atomic Bose-Einstein condensates (BECs), as illustrated in Fig 2.12, even though these atomic BECs are not ideal and exhibit substantial interactions among the atoms.
The density profile of a non-interacting condensate in a harmonic trap can be derived from the ground quantum state, which corresponds to the ground harmonic oscillator state In a spherically-symmetric trap where the angular frequencies are equal (ω x = ω y = ω z ≡ ω r), the ground state for a single particle is determined by solving the time-independent Schrödinger equation within this harmonic potential The resulting ground harmonic oscillator wavefunction is expressed as ψ(r) = m ω / π.
|ψ(r)| 2 represents the probability of finding the particle at positionr For a condensate of N 0such particles, withN 01, the particle density profile will follow as,
5 Box-like traps [8, 9] are also possible, and allow the condensate to have uniform density, facilitating comparison with the theory of homogeneous condensates.
Fig 2.12 Variation of condensate fraction N 0 / N with temperature for a harmonically-trapped BEC, with the ideal-gas predictions (solid line) compared to experimental measurements from Ref [10]
(circles), with T c = 280 nK n(r)=N 0|ψ| 2 =N 0 ( 2 r ) −3/2 e − r 2 / 2 r , (2.31) where we have introduced the harmonic oscillator length r =/mω r which char- acterises the width of the density distribution.
We can also deduce the density profile of the thermal gas Taking the classical limit, the atoms will be distributed over energy according to the Boltzmann distribution
The relationship between energy and temperature is described by the equation N(E)∝e − E / k B T, where the trapping potential V(r) enables the mapping of energy to spatial distribution This results in a particle density distribution given by n(r)=N ex ( 2 r ,th ) −3/2 e − r 2 / 2 r,th, with r,th defined as 2k B T/mω 2 r, which indicates the thermal gas width As temperature increases, atoms gain average energy, causing them to ascend the trap walls and resulting in a broader spatial profile While both the ideal condensate and thermal gas exhibit Gaussian profiles, their widths differ; specifically, the thermal gas width is temperature-dependent, unlike that of the condensate.
The formation of a Bose-Einstein Condensate (BEC) typically begins with cooling a warm gas to near absolute zero As the temperature drops below the critical temperature (T_c), the gas transitions from a broad thermal distribution to a narrower condensate distribution, resulting in a bimodal density profile With continued cooling, the condensate profile expands while the thermal gas diminishes, becoming negligible once the temperature falls below T_c Although atomic interactions can alter the exact shapes of these density profiles, this general process is consistently observed in BEC experiments.
Ideal Fermi Gas
The behavior of an ideal Fermi gas is characterized by the restriction that identical fermions can occupy only one state each, preventing Bose-Einstein condensation As the temperature approaches absolute zero (T → 0), the Fermi gas exhibits distinct properties, with the Fermi-Dirac distribution simplifying to a step function at T = 0.
All states are filled up to the Fermi energy (E F), which corresponds to the chemical potential at absolute zero temperature (T = 0) This simplified distribution allows for easy integration of the total number of particles present.
The continuum approximation for the density of states is applicable for a large number of fermions, as the occupation of the ground state remains negligible By rearranging the equation, we can express the Fermi energy in relation to the particle density, where \( n = \frac{N}{V} g(E) \).
From this we define the Fermi momentum p F =k F wherek F =(6π 2 n) 1 / 3 is the Fermi wavenumber In momentum space, all states are occupied up to momentum p F , termed theFermi sphere.
Similarly, the total energy of the gas atT =0 is,
From the pressure relation for an ideal gas, P=2U/3V, the pressure of the ideal Fermi gas atT =0 is,
Unlike Bose and classical gases, the pressure in fermions remains finite even at absolute zero temperature (T = 0) and is not a result of thermal agitation Instead, this pressure, known as degeneracy pressure, arises from the arrangement of particles in energy levels dictated by quantum rules This degeneracy pressure plays a crucial role in preventing extremely dense stars, like neutron stars, from collapsing under their own gravitational forces.
At absolute zero temperature (T = 0), an ideal Fermi gas exhibits full occupation of energy states up to the Fermi energy level As the temperature rises to the Fermi temperature (T = TF), there is a noticeable excitation of energy states near the Fermi energy (E = EF).
T T F , the system approaches the classical limit, with particles occupying many high-energy states
As the temperature rises from absolute zero, the Fermi-Dirac distribution broadens around the Fermi energy (E_F), indicating that some high-energy particles gain enough energy to exceed E_F This phenomenon can be quantified by introducing the Fermi temperature (T_F), defined as E_F divided by the Boltzmann constant (k_B) At low temperatures, approximately equal to T_F, only particles in energy states near the Fermi level are significantly populated.
At high temperatures (T > T_F), particles in the system experience significant excitation, leading to thermal effects that dominate and the system approaches classical Boltzmann behavior The Fermi temperature marks the onset of degeneracy, where quantum effects begin to prevail These distinct regimes are illustrated in Figure 2.13.
In a harmonic trap, the behavior of a Fermi gas varies with temperature; at temperatures above the Fermi temperature (T_F), it exhibits a broad, classical profile As the temperature decreases, the profile narrows but eventually saturates below T_F due to degeneracy pressure At absolute zero, the width of the Fermi gas is proportional to N^(1/6) r, indicating that for large N, the Fermi gas cloud is significantly wider than its classical and Bose gas counterparts This behavior is supported by experimental images.
Summary
In 1925, Einstein predicted Bose–Einstein condensation in ideal gases, identifying hydrogen, helium, and electron gas as potential candidates However, hydrogen and helium no longer behave as gases at the necessary densities, while electron gas is fermionic For many years, Bose–Einstein condensation was considered largely theoretical and too fragile to manifest in real gases due to their finite size and particle interactions The concept gained traction in 1938 when Fritz London noted parallels between the heat capacity curves of helium and the superfluid phase It took decades to solidify this connection with microscopic theory, but today, Bose–Einstein condensation is recognized as fundamental to superfluid helium-4 and helium-3, superconductors, and ultracold atomic Bose gases, which we will explore in the next chapter.
In a system with 6 classical particles and a total energy of 6, distributed across 7 energy cells with values 0, 2, 3, 4, 5, and 6, we analyze the cell populations for each macrostate and calculate the statistical weighting (W) for each macrostate The average population per cell, denoted as N¯(E), is derived by averaging over the different macrostates The most probable macrostate is identified based on the highest statistical weighting A plot of N¯(E) versus E reveals that the average distribution closely resembles the Boltzmann distribution, highlighting the system's behavior even with a limited number of particles.
2.2 Consider a system withN classical particles distributed over 3 cells (labelled
1,2,and 3) of energy 0, and 2 The total energy isE=0.5N.
(a) Obtain an expression for the number of microstates in terms ofN andN 3 , the population of cell 3.
As the number of microstates is plotted against N², which represents the macrostate, significant changes in distribution are observed for N values of P, 0, and 500 It is anticipated that as N increases to much larger values, the distribution will converge towards a more defined shape, reflecting the underlying statistical mechanics principles.
2.3 Consider an ideal gas of bosons in two dimensions, confined within a two- dimensional box of volumeV 2D.
(a) Derive the density of statesg(E)for this two-dimensional system.
(b) Using this result show that the number of particles can be expressed as,
1−ze − x d x, where z=e μ/ k B T and x=E/k B T Solve this integral using the substitution y=ze − x
(c) Obtain an expression for the chemical potentialμand thereby show that Bose– Einstein condensation is possible only atT =0.
The internal energy of a boxed 3D ideal Bose gas varies with temperature, as outlined in Equation (2.25) For temperatures below the critical temperature (T < T_c), where the fugacity z equals 1, the internal energy can be derived, while for temperatures above the critical temperature (T > T_c), where z exceeds 1, a different expression applies Additionally, these findings can be extended to obtain the heat capacity expressions detailed in Equation (2.27).
2.5 Bose–Einstein condensates are typically confined in harmonic trapping poten- tials, as given by Eq (2.28) Using the corresponding density of states provided in Sect.2.5.10.1:
(a) Derive the expression for the critical number of particles.
(b) Derive the expression (2.29) for the critical temperature.
(c) Determine the expression (2.30) for the variation of condensate fractionN 0 /N withT/T c.
In a groundbreaking Bose–Einstein condensation experiment, a gas composed of 40,000 Rubidium-87 atoms, each with an atomic mass of 1.45×10 − 25 kg, achieved condensation at an extremely low temperature of 280 nK The experiment utilized a spherically-symmetric harmonic trap with a frequency of ω r 30.
To calculate the critical temperature for an ideal Bose gas, we use the formula \( T_c = \frac{h^2}{2\pi k_B m} \left( \frac{N}{V} \right)^{2/3} \), where \( h \) is Planck's constant, \( k_B \) is Boltzmann's constant, \( m \) is the mass of the atoms, \( N \) is the number of particles, and \( V \) is the volume For an atomic density of \( 2.5 \times 10^{18} \, \text{m}^{-3} \), the critical temperature can be derived and compared to the results obtained for a boxed gas The findings reveal important insights into the behavior of Bose-Einstein condensation under varying conditions.
2.6 The compressibilityβ of a gas, a measure of how much it shrinks in response to a compressional force, is defined as, β= −1
Determine the compressibility of the ideal gas forT