Hotter than any temperature there is

Temperature is introduced as a measure of how much energy a system has, and on that reading a negative one is nonsense. Temperature is more usefully defined as how much the entropy changes when energy is added — and on that reading a negative one is an ordinary state of an ordinary system, reachable in a laboratory and first demonstrated in 1951.

The requirement is a ceiling. A system whose energy can be increased without limit has an entropy that rises without limit too, and its temperature stays positive for ever. A system with a highest possible energy does not.

The definition that has to be used #

The thermodynamic definition of temperature is

$$\frac{1}{T} = \frac{\partial S}{\partial E}$$

and everything in this essay follows from taking it seriously rather than as a formal restatement of something more intuitive.

Entropy is a count of microstates, so this says: temperature measures how rapidly the number of available arrangements grows as energy is added. For a gas that growth never stops — more energy means faster molecules, means a larger region of velocity space, means more arrangements — so the derivative is always positive and so is $T$.

The definition that has to be used is the thermodynamic one — temperature as the reciprocal of how entropy changes with energy — and everything here follows from taking it literally. The number of arrangements of a two-state system, plotted against how many are in the upper state, is a binomial coefficient: largest in the middle, falling to one at each end. Its logarithm is the entropy, so the entropy has a maximum rather than growing without limit, and past that maximum adding energy makes the entropy go down. A quantity whose reciprocal is negative there is not a paradox; it is what the definition says.

For a set of two-level systems the count is a binomial coefficient. With all the units down there is one arrangement; with half up there are as many as there can be; with all up there is one again. The entropy therefore rises, peaks and falls, and past the peak adding energy reduces the number of arrangements.

Past that peak $\partial S/\partial E$ is negative, so $T$ is negative. Nothing has gone wrong; the definition has been applied to a system it was not usually applied to.

Why it is hot rather than cold #

The unsettling part is the ordering, and the ordering is decided by a criterion that has nothing to do with the sign of anything.

Put two systems in thermal contact and energy flows in whichever direction increases the total entropy. If system A has a larger $\partial S/\partial E$ than system B, then moving energy from B to A raises the total — which is the ordinary statement that heat flows from hot to cold, with “hot” meaning “small $\partial S/\partial E$”.

A system at negative temperature has a negative $\partial S/\partial E$, which is smaller than any positive value. So energy flows out of it into anything at all, however hot that anything is. It is hotter than every positive temperature there is.

The clean way to say all of this is that the right variable is $1/T$, usually written $\beta$. It runs smoothly from large and positive at low temperature, through zero at infinite temperature, to negative on the far side, and the ordering by hotness is simply the ordering by decreasing $\beta$. Nothing is discontinuous. The temperature, being one over a quantity that passes through zero, has a discontinuity at infinity that is an artefact of the coordinate.

That is a good reason to regard $\beta$ as the physical quantity and $T$ as its awkward reciprocal — which is also how it appears in the Boltzmann factor, where $\beta$ multiplies the energy and $T$ has to be divided by it.

What the populations look like #

The Boltzmann distribution runs backwards at negative temperature, and that is what the state consists of.

At positive $T$ the population of a level falls exponentially with its energy, so the lower level is more occupied. At infinite $T$ the exponential is flat and both are equally occupied. At negative $T$ the exponential rises with energy, so the upper level is more occupied — a population inversion.

The populations make the same statement more directly. Raising the temperature flattens the Boltzmann distribution towards equality between the two states, and perfect equality is what infinite temperature means — there is nowhere further to go by heating. Going beyond it requires the exponential to turn upwards, putting more in the upper state than the lower, and that is all a negative temperature is. The sequence runs from cold, through hot, through infinite, to negative — and negative temperatures sit at the hot end of that list rather than below zero on it.

The five populations in the earlier figure make this concrete. Five per cent up is 0.34 in units of the level splitting over Boltzmann’s constant; forty-two per cent up is 3.1; eighty per cent up is −0.72. Nothing has been assumed about any of them: the temperature was read out of the populations by the Boltzmann ratio and independently off the slope of the entropy, and the two agree.

Equal populations is the awkward case, and it is worth naming why. It is a perfectly ordinary state of the system — as ordinary as any other — described by an infinite number. The generator that drew these refuses a population within two per cent of equal for exactly that reason: the state is fine and the number is unprintable.

The system it was done on #

Purcell and Pound demonstrated this in 1951 on the nuclear spins of lithium fluoride, and the method is worth describing because it explains what “a system with a ceiling” means in practice.

A nucleus with spin one-half in a magnetic field has exactly two states and therefore exactly two energies. Its energy has a ceiling by construction. Prepare the spins in equilibrium in a strong field, so that most are in the lower state, and then reverse the field faster than the spins can follow. The populations are unchanged and the level ordering has swapped, so what was the lower level is now the upper one, and the system is inverted.

There is an alternative preparation that makes the point even more directly and is used in modern demonstrations: apply a radio pulse of exactly the right duration to rotate every spin through 180°. That takes a thermal population and turns it upside down in a few microseconds, and the resulting state is a negative-temperature one whose magnitude is the mirror of the temperature it started at. A pulse that rotates through 90° gives equal populations, which is infinite temperature — so a single knob on a spectrometer sweeps a spin system’s temperature from room temperature through infinity to minus room temperature and back, and the whole range is traversed in a time shorter than any relaxation.

Sorting spins with an analyser turns the algebra into counts, and it is how the population inversion is actually confirmed. What is being asserted about the sample is a ratio between two occupancies, and a beam split by a field into two paths makes that ratio a matter of counting arrivals rather than of inference.

The measurement that showed it was the sign of the nuclear magnetic resonance signal, which reverses: an inverted population emits rather than absorbs at the resonant frequency.

What makes the experiment work is a separation of timescales. The spins exchange energy among themselves in about $10^{-5}$ seconds and with the crystal lattice in about five minutes, so for several minutes the spin system is internally in equilibrium at a well-defined negative temperature while the lattice it sits in is at room temperature. The system in question is the spins, and it is a genuine thermodynamic system for as long as that separation holds.

The laser #

Every working laser has a population inversion in it, so every working laser contains a subsystem at a negative temperature while it is lasing.

A laser is the engineering consequence, and the only one most people meet. Stimulated emission outruns absorption exactly when the upper state is more populated than the lower — that is what “gain” means — so a laser medium is a system held at a negative temperature for as long as the pump keeps it there. The population inversion is not a metaphor borrowed from thermodynamics; it is the same inequality, in the same variables.

That is not a colourful way of describing an inversion; it is the definition applied to the populations. It also explains why an inversion cannot be produced by heating. Heating drives populations towards equality, which is infinite temperature, and no amount of it takes them past equality — which is exactly why a two-level system cannot be inverted by pumping it at its own transition frequency, and why every laser has at least three levels.

The three-level scheme gets round it by pumping into a level that decays quickly into the upper laser level, so the population arrives somewhere it was not pumped. That is a non-thermal process, which is the point: an inversion is a state no thermal source can produce, and the machinery for producing one is the whole engineering of a laser.

It also explains why a laser is not a heat engine and cannot be analysed as one. A heat engine converts a flow of energy between two positive temperatures into work, and its efficiency is bounded by the ratio. A laser’s active medium is not at a positive temperature at all while it works, so the bound does not apply — which is why a laser’s efficiency is limited by pumping and quantum defect rather than by anything Carnot-shaped. The engine ceiling has a condition, and this is a system that fails it.

There is a thermodynamic reading of the same fact. A negative-temperature reservoir is a heat source hotter than anything else, so an engine running between it and a positive-temperature sink has an efficiency that exceeds the ordinary Carnot expression — and it can, in principle, exceed one, because “efficiency” then means work out over heat in from the negative-temperature side, and heat also flows in from the cold side. The ceiling on every engine is derived for two positive temperatures and does not apply.

The maser that sorted its molecules #

The laser was described above as producing its inversion by pumping through a third level. The first device to achieve one did something more direct, and it is the clearest possible demonstration that a negative-temperature state is prepared rather than heated.

An ammonia molecule is a pyramid with the nitrogen above a triangle of hydrogens, and the nitrogen can tunnel through to the other side. That tunnelling splits what would be one level into two, separated by a frequency of about 23.9 gigahertz, and the two states differ in how they respond to an electric field.

Townes and his colleagues in 1954 exploited that difference mechanically. A beam of ammonia was sent along the axis of an electrostatic quadrupole, whose field is weak on the axis and strong off it. Molecules in the upper state have their energy lowered by a strong field and are pulled outward; molecules in the lower state are pushed toward the axis. Run the beam through such a focuser and what emerges down the middle is almost entirely upper-state molecules.

No energy was added to any molecule and nothing was pumped. The inversion was produced by sorting — the same trick a Stern–Gerlach magnet performs on spins, applied to a molecular level and used to build a population that no thermal process could produce.

Fed into a resonant cavity, that inverted beam amplified: stimulated emission exceeded the cavity’s losses, the thing oscillated, and the output was more monochromatic than any oscillator that had existed. It was called a maser, it shared the 1964 Nobel Prize, and the laser is the same idea at a wavelength six thousand times shorter.

The state of affairs in space is stranger and larger. Molecular clouds and the envelopes of evolved stars contain regions where hydroxyl, water, silicon monoxide or methanol molecules are pumped into inverted populations by infrared radiation or by collisions, over path lengths of astronomical units. They amplify whatever passes through, and the resulting emission is fantastically bright: a water maser’s brightness temperature can reach $10^{15}$ kelvin, in gas whose kinetic temperature is a few hundred.

Those are two temperatures describing one object and neither is the gas’s. The radiation is characterised by an enormous positive brightness temperature; the level populations producing it are at a negative one; and the molecules’ motion is at a few hundred kelvin. Which number is meant has to be said, and the fact that all three are legitimately called temperatures is a fair measure of how much work the word is doing.

The spins that have their own temperature #

The Purcell and Pound experiment worked because the spins reach equilibrium among themselves much faster than with the crystal around them. That separation is not a special trick for one experiment; it is the basis of a working quantity used every day.

In a solid, nuclear spins exchange energy with one another through their magnetic dipole coupling in tens of microseconds, and with the lattice’s vibrations in seconds, minutes or hours. Between those two times the spin system is internally in equilibrium and thermally isolated from everything else, so it has a temperature of its own — the spin temperature — which can be entirely different from the temperature of the material it sits inside.

That is a genuine thermodynamic statement rather than a manner of speaking. The spin system has an energy, a heat capacity, and an entropy; it can be heated, cooled, and brought into contact with other reservoirs; and the whole apparatus of thermodynamics applies to it for as long as the separation of timescales holds. Solid-state magnetic resonance is largely the business of manipulating it.

The most consequential manipulation runs the temperature far below the lattice’s rather than through infinity. An electron’s magnetic moment is about six hundred and sixty times a proton’s, so at a given field and temperature an electron spin system is that much more strongly polarised than a nuclear one. Irradiating a sample at the right microwave frequency transfers that polarisation from the electrons to the nuclei — which is to say, it cools the nuclear spin system by a factor of hundreds, while the sample as a whole stays where it was.

Dynamic nuclear polarisation of that kind, performed at a couple of kelvin and then followed by rapidly dissolving the frozen sample, produces nuclear polarisations ten thousand times the thermal value. The enhanced signal survives for a minute or so, which is long enough to inject the material into a living subject and watch its metabolism by magnetic resonance in real time — a measurement that is impossible at the thermal polarisation and routine at a spin temperature of a few millikelvin.

So the same conceptual move underlies both halves of this subject. A subsystem with its own fast internal equilibrium has its own temperature; that temperature can be pushed below the lattice’s, which is a technique, or past infinity, which is this essay.

Where the confusion comes from #

It is worth asking why “negative temperature is hotter than infinite temperature” reads as a paradox when it follows from one derivative, because the answer says something about how temperature is usually learned.

The confusion is worth naming because it is entirely a matter of the word. “Negative” suggests below zero, and the ordering of temperatures here is not the ordering of the numbers: it runs $+0$, through the positive values, to $+\infty$, which is the same state as $-\infty$, and then up through the negative values to $-0$. A system at $-1$ K is hotter than any positive temperature whatever, and will give heat to anything it touches. Reading the scale as a number line is what makes it sound impossible.

Temperature is first met as a property of ordinary matter, where it is monotonically related to energy, and that relation gets absorbed as the definition. On that reading, more energy means hotter, so a system with the most energy possible ought to be hottest and one with negative temperature ought to be below the bottom.

The relation is a property of unbounded systems, not of temperature. What temperature actually measures is a rate — how the count of arrangements responds to energy — and the count’s behaviour depends on whether there is a ceiling.

There is a second habit doing damage, which is thinking of the Kelvin scale as a line with a floor at zero and nothing below it. That picture is right for $T$ and wrong for hotness. The hotness scale is $\beta$, it runs from $+\infty$ to $-\infty$ with no privileged point, and the Kelvin scale covers only the positive half of it — folded so that both ends of the line map onto the point at infinity. A negative-temperature state is not off the end of the Kelvin scale; it is on a part of the hotness scale the Kelvin scale does not reach.

Once that is granted, everything else in this essay follows without any further oddity, which is a fair test of whether an apparent paradox was a paradox or a coordinate problem.

What cannot do this #

The requirement is stringent and it excludes almost everything.

What cannot do this is anything with an unbounded energy. A gas of moving molecules has no ceiling — there is always a faster molecule available — so its entropy grows without limit as energy is added, the slope never turns over, and no negative temperature is reachable even in principle. That is why every demonstration of this effect has used a spin system or something like one: the requirement is not exotic matter but a finite ladder of states with a top to it.

Anything that can move is excluded, because kinetic energy has no upper bound. A gas, a liquid, a solid’s vibrations — all have entropies that rise without limit, so their temperatures are positive at every energy. That is why negative temperatures are not encountered in ordinary thermodynamics and why the concept sounds impossible.

A system must also be internally in equilibrium, on a timescale short compared with its contact with everything else. Without that there is no temperature at all, negative or otherwise, and the phrase “population inversion” is all that can honestly be said.

The exceptions that qualify are therefore all systems with a small number of internal states: nuclear and electronic spins, the internal levels of atoms in a trap, and — since 2013 — the motional states of atoms in an optical lattice, where the band structure supplies the ceiling that free motion lacks. That last one is the most interesting, because it produces a negative-temperature state of a system’s motion, which had been thought impossible for exactly the reason above.

What the pictures cannot show #

Everything is a two-level system. More levels give a more complicated entropy curve, and the temperature is defined by its slope at the energy in question; the qualitative statement survives as long as the energy is bounded above.

The system is isolated from everything else. Real spins are in a lattice, real laser levels are in a crystal, and the negative-temperature state has a lifetime set by the coupling. Nothing here is an equilibrium state of the whole apparatus, and it cannot be.

Volume and pressure are absent. A negative-temperature system in a container would have a strange equation of state, and the usual thermodynamic relations need care about which variables are held fixed; the arguments here are for a system whose only variable is its energy.

The entropy plotted is per unit and the count is the arrangements of many. Nothing about a single two-level system has a temperature; the temperature is a property of an ensemble of them large enough that the binomial coefficient means something, which is the usual condition for any statistical quantity and is met easily in a crystal.

And the third law is not violated. Unreachability of absolute zero is a statement about approaching $T = 0$ from above, and a negative temperature is not below zero on the ordering that matters — it is above infinity. Going from positive to negative does not pass through zero; it passes through infinity.

The ladder from here #

Later rungs on this anchor: the thermodynamics of a bounded system generally, and which of the usual relations survive; the negative-temperature engine and what “efficiency” means when one reservoir is on the other side of infinity; the 2013 optical-lattice realisation and what it took to bound a kinetic energy; and the argument over whether the Gibbs entropy, which never gives a negative temperature, is the correct definition — a dispute about which of two definitions of entropy is right, still unsettled.

The neighbouring ladders are entropy is a count, which is the definition this essay differentiates, the exponential that decides everything, which is the Boltzmann factor run backwards here, and the staircase that never reaches the floor, which is the same spin system used to get cold instead.