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Absolute zero

Absolute zero is the lowest possible temperature that can be theoretically achieved, corresponding to 0 kelvin (K) on the absolute temperature scale, where the kinetic energy of particles in a system reaches its minimum and thermal motion effectively ceases.^[1]^[2] This temperature equates to −273.15 °C on the Celsius scale and −459.67 °F on the Fahrenheit scale, marking the point at which a system's entropy approaches a minimum value for a perfectly ordered state.^[3]^[4] The concept of absolute zero emerged in the 19th century through studies of gas behavior, with William Thomson, later known as Lord Kelvin, proposing it in 1848 based on extrapolations from Charles's law, which describes how the volume of an ideal gas decreases linearly with temperature at constant pressure.^[3]^[2] Kelvin's work established the Kelvin scale, an absolute thermodynamic temperature scale starting at this zero point, which became the international standard for scientific measurements.^[4] This scale underpins the third law of thermodynamics, formulated by Walther Nernst in 1906–1912^[5], which states that the entropy of a perfect crystalline substance is zero at absolute zero, implying that reaching this temperature would require an infinite number of steps and is thus unattainable in practice.^[6]^[7] In modern physics, absolute zero serves as a foundational reference for understanding quantum phenomena and low-temperature behaviors, such as Bose-Einstein condensates, where atoms cooled to within billionths of a kelvin above absolute zero behave as a single quantum entity.^[2] Techniques like laser cooling and evaporative cooling have approached but never reached this limit, with the closest laboratory achievements around 38 picokelvin (pK) in 2021 using ultracold atomic gases.^[8] These efforts highlight absolute zero's role in advancing technologies like atomic clocks and quantum computing, while reinforcing the thermodynamic barriers to its attainment.^[2]

Definition and Fundamentals

Thermodynamic Basis

Absolute zero is defined as the temperature of 0 kelvin (K), equivalent to -273.15 °C or -459.67 °F, representing the lowest conceivable temperature where the thermal energy of particles is at its minimum and the entropy of a system achieves its lowest possible value.^[1]^[9] In thermodynamic equilibrium at absolute zero, all classical degrees of freedom of the system occupy their ground state, implying the cessation of thermal motion and the absence of random kinetic energy associated with temperature.^[10] The third law of thermodynamics, also known as the Nernst heat theorem, states that the entropy S of a perfect crystalline substance is exactly zero at 0 K.^[6]^[11] This principle implies that the heat capacity C of the system approaches zero as the temperature T approaches absolute zero, since C = T \left( \frac{\partial S}{\partial T} \right)_V, and with S \to 0 and \frac{\partial S}{\partial T} \to 0, no additional heat is required to change the temperature further. A key implication arises in the minimization of Gibbs free energy G, defined as

G = H - TS,

where H is the enthalpy and T is the temperature. At absolute zero, with T = 0 K and S = 0 from the third law, the equation simplifies to G = H, indicating that the free energy equals the enthalpy as entropy contributions vanish.^[12]^[6]

Relation to Gas Laws

The concept of absolute zero emerged prominently from the study of gas behavior in the late 18th century, particularly through Charles's law, formulated by French physicist Jacques Charles in 1787. Charles observed that, at constant pressure, the volume V of a fixed amount of gas is directly proportional to its temperature T measured on the Celsius scale, expressed as \frac{V}{T} = \constant, or more precisely, V = V_0 (1 + \alpha T), where V_0 is the volume at 0°C and \alpha is the thermal expansion coefficient, approximately \frac{1}{273} per degree Celsius for many gases.^[13] This linear relationship, later confirmed and publicized by Joseph Louis Gay-Lussac in 1802, implied that extrapolating the volume-temperature plot backward would yield zero volume at approximately -273°C, suggesting a natural lower limit to temperature below which gases could not exist in their expanded state.^[13] This empirical insight from Charles's law finds a deeper foundation in the ideal gas law, PV = nRT, where P is pressure, n is the number of moles, R is the universal gas constant, and T is absolute temperature in kelvins. Rearranging gives T = \frac{PV}{nR}, indicating that temperature approaches zero when either pressure or volume does, corresponding to the cessation of molecular motion. From the kinetic-molecular theory underlying the ideal gas law, the average translational kinetic energy of gas molecules is \frac{3}{2} kT per molecule (with k as Boltzmann's constant), so at T = 0 K, this thermal kinetic energy averages to zero, representing the point where random molecular agitation vanishes.^[14]^[15]^[16] Experimental measurements on dry air and other gases, conducted by plotting volume or pressure against temperature at constant conditions, consistently extrapolate to the same limit of -273.15°C, remarkably independent of the specific gas used, as long as it behaves nearly ideally at accessible temperatures.^[17]^[18] However, real gases deviate from this ideal behavior near absolute zero due to intermolecular forces and finite molecular volumes, often liquefying well above 0 K—for instance, air components like nitrogen at 77 K—preventing direct observation of the extrapolated limit but reinforcing the conceptual validity of the gas laws for defining the scale.^[19]^[20]

Temperature Scales and Measurements

Absolute Scales

Absolute scales of temperature are thermodynamic scales that set absolute zero as the lower fixed point, representing the theoretical absence of thermal energy, thereby ensuring all temperatures are non-negative. These scales facilitate precise calculations in physics and engineering by aligning directly with fundamental laws without requiring offsets or adjustments for negative values. The two primary absolute scales are the Kelvin scale, which is the international standard, and the Rankine scale, commonly used in certain engineering contexts. The Kelvin scale (K) defines 0 K as absolute zero and assigns 273.16 K to the triple point of water, where water exists in equilibrium as solid, liquid, and gas phases at a pressure of 611.657 pascals.^[21] As the SI base unit of thermodynamic temperature, the Kelvin scale is offset from the Celsius scale such that a temperature interval of 1 K equals 1 °C, with the relation T(K) = t(°C) + 273.15 for the ice point.^[22] This definition originated from the 10th General Conference on Weights and Measures in 1954, which fixed the Kelvin to the triple point of water for enhanced precision in temperature measurements.^[21] Although the modern definition (since 2019) ties the Kelvin to the fixed value of the Boltzmann constant (k = 1.380649 × 10^{-23} J/K), the triple point remains a key reference at exactly 273.16 K.^[21] The Rankine scale (°R), an absolute counterpart to the Fahrenheit scale, similarly places absolute zero at 0 °R, equivalent to 0 K. On this scale, the boiling point of water at standard atmospheric pressure is 671.67 °R, reflecting the Fahrenheit boiling point of 212 °F plus the 459.67 °R offset from absolute zero.^[23] Conversion between the scales follows the relation °R = K × 9/5, preserving the degree size of the Fahrenheit scale while anchoring to absolute zero.^[24] A key advantage of absolute scales like Kelvin and Rankine is their avoidance of negative temperatures in thermodynamic processes, which simplifies interpretations of heat transfer and entropy. They enable direct application in fundamental equations, such as the ideal gas law (PV = nRT), where temperature must represent proportional kinetic energy without arbitrary zero points; this arises from extrapolating gas behavior to the point where volume or pressure approaches zero at absolute zero.^[24] In contrast, relative scales require constant adjustments, potentially leading to errors in scientific computations.^[25]

Historical and Comparative Scales

Early temperature scales, such as the Réaumur scale developed by René Antoine Ferchault de Réaumur in 1730, were based on arbitrary reference points tied to observable phenomena like the freezing and boiling of liquids, without incorporating a theoretical lower limit.^[26] On the Réaumur scale, the freezing point of water was set at 0°Ré and the boiling point at 80°Ré, using alcohol as the thermometric fluid, but it provided no framework for temperatures below freezing beyond extrapolation, lacking a defined absolute zero.^[26] Similar limitations applied to other pre-19th-century scales, which prioritized practical calibration over physical principles, leading to inconsistencies in scientific applications across regions.^[3] The concept of an absolute lower temperature limit emerged in the late 17th century through experimental work on gas behavior. In 1703, French physicist Guillaume Amontons constructed an air thermometer and observed that gas pressure decreases linearly with cooling, extrapolating that pressure would reach zero at a temperature approximately 240 degrees below the freezing point of water on contemporary scales, providing the first rough estimate of absolute zero around -240°C.^[27] This approximation, though imprecise, highlighted the need for scales anchored to thermodynamic reality rather than empirical fixed points.^[27] The Celsius scale, proposed by Anders Celsius in 1742 and later inverted, defines 0°C as the freezing point of water at standard atmospheric pressure, with absolute zero occurring at -273.15°C.^[22] The conversion between Celsius and the Kelvin absolute scale is given by the equation:

T(K) = t(°C) + 273.15

where T(K) is the temperature in kelvins and t(°C) is the temperature in degrees Celsius.^[22] This places absolute zero as a negative value on the Celsius scale, reflecting its origin in water-based references rather than an absolute thermodynamic zero.^[22] Likewise, the Fahrenheit scale, introduced by Daniel Gabriel Fahrenheit in 1724, sets 0°F as the temperature of a saturated brine solution (a mixture of ice, water, and ammonium chloride), chosen for its reproducibility in laboratory conditions, with absolute zero at -459.67°F.^[28]/12:_Temperature_and_Kinetic_Theory/12.2:_Temperature_and_Temperature_Scales) The conversion to kelvins is:

T(K) = \frac{t(°F) + 459.67}{1.8}

illustrating the scale's finer divisions (180 between water's freezing and boiling points) and its offset from absolute zero due to historical calibration.^[22]/12:_Temperature_and_Kinetic_Theory/12.2:_Temperature_and_Temperature_Scales) The transition to absolute scales gained momentum in the 19th century, driven by advances in thermodynamics and the need for consistent measurements in engineering and physics. William Thomson (Lord Kelvin) formalized the absolute scale in 1848, proposing a system starting at absolute zero (-273.15°C) to align with the ideal gas law and Carnot's efficiency principles, marking a shift from arbitrary empirical scales to physics-based ones for greater scientific precision.^[3] This evolution addressed the shortcomings of earlier scales like Réaumur, Celsius, and Fahrenheit, which, while practical for everyday use, could not adequately describe phenomena near the lower temperature limit without negative values or extrapolation.^[3]

Thermodynamic Implications

Unattainability Principle

The third law of thermodynamics implies that absolute zero is theoretically unattainable through any finite number of thermodynamic processes, as reaching it would require removing all thermal energy from a system, but the heat capacity of materials approaches zero as temperature nears 0 K, necessitating an infinite sequence of cooling steps to extract the remaining heat.^[29] This limitation arises because, while entropy reaches a minimum (typically zero for a perfect crystal), the infinitesimal heat transfers required become progressively smaller, preventing complete extraction in finite time or steps.^[30] Nernst's heat theorem, a foundational statement of the third law, further elucidates this unattainability by asserting that the entropy change (ΔS) for any reversible process approaches zero as the temperature approaches 0 K, rendering isothermal processes at absolute zero impossible without external work input, as no heat can be exchanged without violating the second law.^[31] This theorem implies that the system's entropy is fixed at its minimum value near 0 K, blocking any finite process from achieving exact equilibrium at absolute zero, as even ideal reversible operations would require infinite precision and effort.^[32] In practical terms, the constraints of thermodynamic cycles, such as the Carnot refrigeration cycle, demonstrate escalating difficulty: the minimum work required to extract a fixed amount of heat Q_c from a cold reservoir at temperature T_c to a hot reservoir at T_h is given by

W \geq Q_c \left( \frac{T_h}{T_c} - 1 \right),

which diverges to infinity as T_c \to 0 K, since the coefficient of performance approaches zero, demanding ever-greater energy input for diminishing cooling effects.^[33] Thus, no classical or quantum thermodynamic process can reduce a system's temperature to exactly 0 K in finite time, as both the third law and cycle efficiencies enforce this fundamental barrier.^[32]

Low-Temperature Properties

As temperatures approach absolute zero, the heat capacity of insulators exhibits a characteristic dependence described by the Debye model, where the molar heat capacity at constant volume follows C_v \propto T^3.^[34] This behavior arises because, at low temperatures, only low-frequency phonon modes contribute significantly to the thermal energy, with the density of states for these modes scaling as the square of the frequency, leading to the cubic temperature dependence after integration over the phonon spectrum.^[34] Similarly, the coefficient of thermal expansion \alpha in insulators under the Debye model also varies as \alpha \propto T^3 at low temperatures, reflecting the close coupling between lattice vibrations and volume changes via the Grüneisen parameter.^[35] Near absolute zero, many materials undergo striking phase transitions that reveal macroscopic quantum effects. In certain metals, superconductivity emerges below a critical temperature T_c, where electrical resistance drops to zero and the material expels magnetic fields in the Meissner effect, behaving as a perfect diamagnet./University_Physics_II_-Thermodynamics_Electricity_and_Magnetism(OpenStax)/09%3A_Current_and_Resistance/9.07%3A_Superconductors) For example, mercury becomes superconducting at T_c = 4.2 K./University_Physics_II_-Thermodynamics_Electricity_and_Magnetism(OpenStax)/09%3A_Current_and_Resistance/9.07%3A_Superconductors) Liquid helium-4, when cooled below the lambda point of 2.17 K, transitions to a superfluid state characterized by zero viscosity, allowing it to flow without dissipation through narrow channels and climb container walls against gravity. The third law of thermodynamics implies that entropy S approaches zero as temperature nears absolute zero, rendering exact attainment unattainable in finite steps./05%3A_The_Third_Law_of_Thermodynamics/5.01%3A_The_Third_Law_of_Thermodynamics) Consequently, the Gibbs free energy G = U + PV - TS simplifies to G \approx U_0 + PV at sufficiently low temperatures, where U_0 is the ground-state internal energy, emphasizing the thermodynamic stability of the lowest-energy phase./05%3A_The_Third_Law_of_Thermodynamics/5.01%3A_The_Third_Law_of_Thermodynamics) This form underscores how low-temperature phases, such as superconductors and superfluids, represent the global minimum in free energy under constant pressure./05%3A_The_Third_Law_of_Thermodynamics/5.01%3A_The_Third_Law_of_Thermodynamics)

Quantum and Exotic Phenomena

Zero-Point Energy

In quantum mechanics, absolute zero cannot eliminate all motion in bound systems due to the Heisenberg uncertainty principle, which states that the product of the uncertainties in position and momentum must satisfy \Delta x \Delta p \geq \frac{\hbar}{2}.^[36] This inequality implies that a particle confined to a finite region (small \Delta x) must have a non-zero uncertainty in momentum (\Delta p > 0), resulting in a minimum kinetic energy even at 0 K.^[37] Consequently, systems cannot reach a state of complete rest, leading to the concept of zero-point energy as the irreducible ground-state energy.^[38] Zero-point energy is exemplified by the quantum harmonic oscillator, a model for vibrational modes in atoms and molecules, where the energy levels are quantized as E_n = \hbar \omega \left(n + \frac{1}{2}\right), with n = 0, 1, 2, \dots and \omega the angular frequency.^[39] For the ground state (n=0), this yields E_0 = \frac{1}{2} \hbar \omega > 0, representing the lowest possible energy despite no classical excitation.^[40] This zero-point energy arises directly from the wave-like nature of particles, ensuring the ground state wavefunction spreads sufficiently to satisfy the uncertainty principle.^[41] In molecules, vibrational zero-point energy manifests as residual oscillations in bonds, such as the ground-state stretching in diatomic species like H_2, contributing to the overall internal energy even at absolute zero.^[42] These quantum vibrations lead to non-zero entropy in imperfect crystals, where disorder from defects or frozen-in configurations prevents the third law's ideal zero-entropy state, as multiple microstates persist due to unresolvable vibrational ground states. A macroscopic example is the Casimir effect, where the attractive force between uncharged conducting plates arises from modifications to the electromagnetic field's zero-point energy in the vacuum between them.^[43] In quantum field theory, zero-point energy extends to the vacuum state, where each field mode contributes \frac{1}{2} \hbar \omega_k summed over all frequencies k, though observable effects are regulated in simple systems like the harmonic oscillator.^[44]

Negative Temperatures

In systems with a bounded spectrum of energy levels, such as two-level spin systems in a magnetic field, negative temperatures on the absolute Kelvin scale can occur when population inversion takes place, with more particles occupying the higher-energy state (spins aligned against the field) than the lower-energy state. This inversion leads to an effective temperature that ranges from negative infinity to zero Kelvin, connecting continuously through positive infinity in the thermodynamic formalism. Such states are possible only in isolated subsystems where the total energy can exceed half the maximum possible energy, inverting the usual monotonic increase of entropy with energy.^[45] The concept is formalized through the Boltzmann distribution, where the probability of occupying a state with energy E is proportional to e^{-E / kT}, with k being Boltzmann's constant and T the temperature. For negative T, the exponent becomes positive for increasing E, resulting in higher-energy states being more probable than lower-energy ones, which inverts the distribution compared to positive temperatures. This inversion implies that negative-temperature systems are hotter than any positive-temperature system, including infinite temperature, because heat flows from them to positive-temperature reservoirs, and adding energy to a negative-temperature system drives T from -\infty toward 0 K, increasing its "hotness." These states do not violate the second law of thermodynamics, as the full system (including the magnetic field or external reservoir) maintains overall entropy increase, but the isolated spin subsystem alone exhibits the negative T.^[45] A classic example is the nuclear spin system in lithium fluoride (LiF) subjected to a strong magnetic field, where rapid radio-frequency pulsing achieved population inversion, yielding a negative temperature of approximately -100 K as measured by the spin polarization. More recent experiments extended this to motional degrees of freedom in a Bose gas of potassium-40 atoms, using optical lattices and magnetic field tuning to create a stable negative-temperature state below -273.15°C (nanokelvin scale), demonstrating inverted momentum distributions and negative pressures. In this setup, the atoms effectively "cooled" to negative temperatures via tailored interactions, forming a condensate-like state hotter than infinite temperature.^[46]^[47] Further advances in 2023 demonstrated negative optical temperatures in photon gases through nonlinear photon-photon interactions, enabling observations of thermodynamic processes such as isentropic expansion-compression and Joule expansion.^[48] In the same year, light waves in multimode optical fibers were observed to thermalize to negative-temperature Rayleigh-Jeans equilibrium states.^[49]

Historical Development

Early Ideas on Cold Limits

In ancient Greek philosophy, Aristotle conceptualized cold as one of the four primary qualities of matter—alongside heat, wetness, and dryness—that governed the transformation of elements, but he did not articulate a quantifiable minimum limit to cold, viewing it instead as a relative property balanced against heat in natural processes.^[50] Early attempts at thermometry, such as those using alcohol or air expansion in the 16th and 17th centuries, lacked an absolute zero point, relying instead on arbitrary fixed points like the melting of ice or human body temperature, which reflected subjective perceptions rather than a theoretical cold boundary. During the 17th and 18th centuries, experimental advancements began to hint at a potential lower limit to temperature. In 1703, French physicist Guillaume Amontons constructed an air thermometer based on pressure changes and extrapolated that air pressure would cease at a point approximately -240°C below the freezing point of water, interpreting this as an "extreme cold" where the spring of air would vanish.^[51] Similarly, in 1724, German instrument maker Daniel Gabriel Fahrenheit introduced his mercury thermometer scale, setting zero at the lowest temperature he could achieve with an ice-salt-ammonium chloride mixture (around -18°C), an arbitrary choice driven by practical reproducibility rather than any philosophical or physical absolute.^[52] Philosophical speculations further shaped these notions, with René Descartes proposing in the mid-17th century that cold represented the absence of heat rather than a distinct substance, aligning with a growing view of cold as a privation of motion in matter.^[53] His vortex theory of the universe, which explained celestial and terrestrial motions through swirling ethereal matter, implicitly suggested a cold limit in states of minimal agitation, where heat—generated by friction and motion—would be absent.^[54] In the 1740s, French natural philosophers, including Jean-André Deluc, conducted experiments with frigorific mixtures of ice, salt, and acids, achieving temperatures down to about -40°C and speculating on a fundamental lower bound beyond which further cooling might be impossible, based on observations of fluid behaviors and evaporation limits. These efforts, though imprecise, foreshadowed the transition to 19th-century gas law investigations that would formalize absolute zero.

Kelvin's Contributions and Scale

In 1848, William Thomson, later ennobled as Lord Kelvin, proposed an absolute temperature scale in his seminal paper titled "On an Absolute Thermometric Scale founded on Carnot's Theory of the Motive Power of Heat, and calculated from Regnault's Observations."^[55] This work established a thermodynamic foundation for temperature measurement independent of arbitrary fixed points, defining temperatures relative to an absolute zero where no further decrease is possible.^[55] Thomson's scale resolved longstanding issues in heat engine analysis by avoiding negative temperature values inherent in scales like Celsius, which complicated efficiency computations.^[3] Central to Thomson's proposal was Sadi Carnot's 1824 theory of the motive power of heat, as elaborated by Émile Clapeyron in 1834.^[55] Carnot demonstrated that the efficiency \eta of an ideal heat engine operating reversibly between a hot reservoir at temperature T_h and a cold reservoir at T_c is \eta = 1 - \frac{T_c}{T_h}.^[55] Thomson recognized that this formula implies a maximum efficiency of 1 when T_c = 0, establishing absolute zero as the theoretical lower bound of temperature on an absolute scale calibrated such that temperature intervals correspond to equal changes in this efficiency metric.^[55] This thermodynamic definition provided a universal basis, applicable to any substance, unlike empirical scales tied to specific materials.^[3] To assign numerical values, Thomson integrated Carnot's thermodynamic insights with empirical observations from gas thermometry, particularly the linear relationship between gas volume and temperature at constant pressure—now known as Charles's law.^[55] He posited that an ideal air thermometer, where temperature is proportional to the volume of dry air under constant pressure, could realize this absolute scale practically.^[55] Extrapolating the linear volume-temperature relation to the point of zero volume yielded absolute zero, providing a concrete reference.^[3] Thomson's calculations relied heavily on the precise experimental data of French physicist Henri Victor Regnault, whose 1841–1842 studies on the thermal expansion of gases confirmed the near-linearity of the volume-temperature relation across various gases like air, hydrogen, and carbon dioxide under moderate pressures.^[55] Regnault's measurements of expansion coefficients, accurate to within 0.1% for air, allowed Thomson to determine that absolute zero corresponds to approximately -273°C on the air-thermometer scale, a value remarkably close to the modern -273.15°C.^[55] This integration of theory and experiment solidified the scale's predictive power, forecasting that all thermal motion ceases at 0 on the absolute scale.^[3] The scale proposed by Thomson in 1848, initially termed the absolute thermometric scale, was retrospectively named the Kelvin scale; it received formal recognition in 1954 when the International Committee for Weights and Measures adopted the "degree Kelvin" (°K) as the SI unit, later simplified to "kelvin" (K) in 1967.^[3] By setting 0 K as the absolute zero of temperature, Kelvin's framework not only advanced thermodynamics but also provided a consistent reference for subsequent low-temperature research.^[55]

Modern Pursuit of Low Temperatures

The pursuit of temperatures approaching absolute zero advanced significantly in the 20th century through innovative cryogenic techniques that extended beyond earlier liquefaction methods. In the 1930s, adiabatic demagnetization emerged as a pivotal approach, exploiting the magnetocaloric effect in paramagnetic salts to achieve millikelvin (mK) ranges. Developed by William F. Giauque, this method involved magnetizing a sample at low temperature, thermally isolating it, and then adiabatically demagnetizing to reduce entropy and thus temperature, reaching as low as 0.001 K (1 mK) in early experiments. This single-shot cooling technique opened the millikelvin regime for solid-state physics studies, confirming third-law predictions without violating the unattainability of exact zero. Building on this, the mid-1960s introduced dilution refrigeration using mixtures of helium-3 (³He) and helium-4 (⁴He), providing continuous cooling to even lower temperatures. Pioneered by researchers like H.E. Hall at the University of Manchester, the process leverages the phase separation of ³He into a dilute phase within superfluid ⁴He at around 0.3 K, followed by forced circulation of ³He across the phase boundary, which absorbs heat via the endothermic dilution effect.^[56] Initial systems achieved base temperatures below 10 mK, with modern iterations routinely reaching 2-4 mK and records as low as 1.75 mK, enabling long-duration experiments in superconductivity and quantum materials.^[57] The late 20th century revolutionized low-temperature physics with optical methods, particularly laser cooling and evaporative cooling, which targeted dilute atomic gases rather than bulk matter. In 1985, laser cooling using Doppler shifts slowed neutral atoms to microkelvin (μK) velocities, earning Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips the 1997 Nobel Prize in Physics for their foundational work on magneto-optical traps. Combining this with evaporative cooling—selectively removing hot atoms from a magnetic trap—Eric Cornell and Carl Wieman at JILA achieved the first Bose-Einstein condensate (BEC) in 1995 with rubidium-87 atoms at 170 nK, a milestone that demonstrated macroscopic quantum coherence. BEC formation requires temperatures below the critical threshold, typically under 1 μK for dilute gases, allowing atoms to occupy a single quantum state and facilitating quantum simulations of complex systems like superfluids. These techniques have pushed records progressively closer to absolute zero, with ongoing refinements in the 21st century. A 2021 experiment at the University of Bremen using adiabatic demagnetization on a rhodium sample reached 38 picokelvin (pK), the lowest artificial temperature recorded to date. As of 2025, this record holds, though experiments such as laser cooling of positronium in 2024 have approached similar limits for exotic systems.^[8]^[58] In parallel, space-based applications have employed laser cooling for atomic clocks, such as China's 2016 Space Cold Atom Clock on the Tiangong-2 station, which cooled cesium atoms to around 100 μK in microgravity to enhance precision for navigation and relativity tests.^[59] BECs at nanokelvin scales continue to underpin quantum computing efforts, where coherent atomic ensembles simulate qubit operations and enable scalable quantum information processing.^[60]

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Debye Model For Specific Heat - Engineering LibreTexts
Sep 7, 2021 · This model correctly explains the low temperature dependence of the heat capacity, which is proportional to and also recovers the Dulong-Petit ...
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[PDF] Temperature dependence of thermal expansion - arXiv
Because the Debye model could fit better to the measurement at low 4 Page 5 temperature, the Debye approximation will be used in subsequent discussions. The ...
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nglos324 - zeropointenergy
This zero point energy results from Heisenberg's Uncertainty Principle which places limits on the product of the uncertainties in position and momentum.
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Quantum Harmonic Oscillator - HyperPhysics
The ground state energy for the quantum harmonic oscillator can be shown to be the minimum energy allowed by the uncertainty principle.Missing: mechanics | Show results with:mechanics
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[PDF] Lecture 13: March 10, 2021 Heisenberg's Uncertainty Principle
(I) Zero Point Energy: According to Heisenberg uncertainty principle, even at absolute zero temperature, particles cannot be at rest when confined in a finite ...
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[PDF] 9. Harmonic Oscillator - MIT OpenCourseWare
E = ω(a†a + 1 ) = ω(N + 1 ) and the minimum energy hω/2 is called the zero point energy. ... energy level n when the system is in a quantum coherent state: n n.
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qm_harmonic_oscillator.html - UNLV Physics
The vertical direction represents the energy of the quantized energy levels. The allowed energy levels obey the formula. E = (n+1/2)ħω ,. where n is an integer ...
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[PDF] Linear Harmonic Oscillator
En = hω (n +. 1. 2. ), n = 0, 1, 2, ... (4.37). It is most remarkable that the energy 1. 2 hω of the ground state is larger than the lowest classical- ly ...
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Vibrational Zero-Point Energies (ZPE) - CCCBDB
The vibrational zero-point energy is the energy difference between the lowest point on the potential energy surface and the energy of the vibrationless energy ...
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[PDF] the casimir effect - Physics Courses
The attractive force can be considered as arising due to the change in the zero-point energy of the electromagnetic field when the plates are brought into ...
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Qu,Thomas1.html
In quantum field theory, zero-point energy refers to the lowest possible energy of a field, such as the electromagnetic field; the zero-point energy of the ...
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Thermodynamics and Statistical Mechanics at Negative Absolute ...
The circumstances under which negative absolute temperatures can occur are discussed, and principles of thermodynamics and statistical mechanics at negative ...
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A Nuclear Spin System at Negative Temperature | Phys. Rev.
A nuclear spin system at negative temperature. EM Purcell and RV Pound Department of Physics, Harvard University, Cambridge, Massachusetts.
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Negative Absolute Temperature for Motional Degrees of Freedom
Negative Absolute Temperature for Motional Degrees of Freedom. S. Braun, J. P. ... DOI: 10.1126/science.1227831 · PREVIOUS ARTICLE. Alignment of Magnetized ...
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Aristotle's Natural Philosophy
May 26, 2006 · Aristotle had a lifelong interest in the study of nature. He investigated a variety of different topics, ranging from general issues like motion, causation, ...
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(PDF) Guillaume Amontons - ResearchGate
He experimented with an air-thermometer and pointed out that the extreme cold of such a thermometer would be that which reduced the spring of the air (pressure) ...
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Daniel Gabriel Fahrenheit | Biography, Thermometer, & Facts
Sep 12, 2025 · He originally took as the zero of his scale the temperature of an equal ice-salt mixture and selected the values of 30 and 90 degrees for the ...
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René Descartes - Stanford Encyclopedia of Philosophy
Dec 3, 2008 · ... Descartes considers that cold may instead simply be the absence of heat and so isn't a quality of its own. Accordingly, this case should be ...
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Descartes' Physics - Stanford Encyclopedia of Philosophy
Jul 29, 2005 · Descartes' vortex theory attempts to explain celestial phenomena, especially the orbits of the planets or the motions of comets.
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Mercury - Cavendish - MPRL - Max-Planck-Gesellschaft
He carried out experiments on freezing mixtures and on the freezing of mercury at his house, but he was content to limit his public contribution to planning, ...
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[PDF] Vol. 33
Within the limits of natural temperature in ordinary climates, the density of saturated vapour is actually found by Regnault (Etudes Flyg1·ometriques in the ...
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celebrating-50-years-of-dilution-refrigeration - Nanoscience
Mid 1960s: The first 3He-4He dilution refrigerator is built at University of Manchester by Henry Hall and the first commercial dilution refrigerator is ...
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Development of dilution refrigerators—A review | Request PDF
Aug 8, 2025 · The lowest temperature achievable nowadays by a conventional dilution refrigeration is 1.75 mK, and largest cooling power at 100 mK is 2 mW. In ...<|separator|>
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New record set for lowest temperature—38 picokelvins - Phys.org
Oct 13, 2021 · Prior research has suggested that the lowest possible temperature that can exist is absolute zero—0 Kelvin. Prior research has also shown ...
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Development of a space cold atom clock - PMC - PubMed Central
We developed a space cold atom clock (SCAC), which was launched into orbit with the Space Lab TG-2 in 2016. Before it deorbited with TG-2 in 2019.
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Classical analog of qubit logic based on a magnon Bose–Einstein ...
Aug 1, 2022 · We present a classical version of several quantum bit (qubit) functionalities using a two-component magnon Bose–Einstein condensate (BEC) formed at opposite ...