The joys of theoretical physics

Have you always thought mathematics is dull and complicated? You are certainly not alone. But there is a lot of beauty hidden in it and in the way it describes our world.

Theoretical physics is all about using maths to describe nature. As the universe we live in is vast and filled with myriads of phenomena — starting with the universe itself expanding due to dark energy, galaxies held together thanks to dark matter, new stars being born and dying, planets and asteroids orbiting these start and colliding with one another; through processes happening on and inside those planets including the miracle of life; down to the perplexing world of molecules, atoms, and subatomic particles — so the mathematical language in which these processes are described uses a lot of tools, often very complex. And yet, there is a surprising level of similarity between different systems.

For a theoretical physicist, there is no difference between an oscillating pendulum, a vibrating string, and a propagating beam of light. Heat transfer and particle diffusion are equivalent because they obey the same mathematical law. According to quantum field theorists, every type of particle (be it a proton, an electron, or a photon) can be seen as a harmonic oscillator and there is almost no qualitative difference between them.

Some physicists are trying to take this idea one step further and find a single physical theory encompassing all physics as we know it. Thus, the Grand Unified Theory was developed which unifies three of the four fundamental interactions in nature — electromagnetic, weak, and strong. Including the fourth one — gravitational — is a feat that has not yet been achieved. Some even doubt that such a Theory of Everything will ever be formulated.

Many theoretical physicists (like me, for example) do not pursue such noble quests but focus on smaller, albeit not less meaningful tasks: How does X work? Can it be used for something worthwhile? What is the best way to do it? These are not important questions on the global scale (compared to questions such as ‘How did the universe come to be?’) but the more important for technological progress. Such development is ultimately the domain of experimental physicists and engineers but finding ways of using new bits of physics in ways humanity can benefit from is a part of theoretical physicist’s work.

Such a process can be illustrated on a problem that is occupying many a scientific mind: building the quantum computer.  It will, of course, be experimental physicists who will build the first functioning prototype (assuming we ever develop one) but theoretical physicists examine how such a device should be built. Should we use atoms as the information carriers? Photons? Something more exotic? Those are some of the questions quantum information theorists are trying to answer.

There is a lot of mathematical beauty in solving such tasks, too. After all, theoretical physicist sees a quantum computer as a large register of quantum bits on which an arbitrary operation can be performed, which can be stored for a long time in a quantum memory, and which can be sent to another quantum computer via a quantum internet channel. The need for investigating various platforms comes from the experimental realisation — each potential platform has its own unique advantages and disadvantages that have to be carefully weighed when finding the optimal architecture for a successful quantum computer.

All that said, there are many more surprises hidden in quantum information theory. One often finds other, unexpected connections between the weirdest parts of the theory. And finding them is always one of the biggest delights working with theoretical physics can bring.

All this mathematical beauty can, actually, also be useful. If two different systems behave in a similar way, we can use one to simulate dynamics of the other. This is more and more often used in quantum physics where complicated systems (especially those that cannot be observed directly in a laboratory) can be simulated using much simpler systems. This way, we can relatively easily learn a lot about the elaborate system which cannot be simulated on classical computers. (Due to their nature, it is possible to simulate only small quantum systems on classical computers.)

The field of quantum simulations (i.e., using simple quantum systems to simulate evolution of more complicated systems) is still in its infancy. But it will probably not take long before we can simulate systems that too difficult to solve for regular computers. We can then expect better understanding of many physical and chemical processes such as high-temperature superconductivity, quantum phase transitions, dynamics of chemical reactions, or photosynthesis. And all that thanks to the incredibly rich and intriguing structure of the mathematical language we use to describe our world.

How to measure time

Precise timekeeping is crucial for many of our daily activities. High-speed communication (on the internet or in a mobile phone network), satellite navigation, and many other tasks require time synchronisation over long distances to work properly. But how is time measured? And can quantum physics help reach better accuracies?

The basic idea behind measuring time is simple and similar to any other measurement — you simply compare the time duration with a reference. The reference has to be an event that regularly repeats itself so that a single repetition is the basic time unit and the number of repetitions gives the overall time. An example of such  process can be the Sun rising and setting every day. Its regular movement in the sky defines one of the most fundamental units of time: a day. It is a very simple and natural way to measure time but it has one disadvantage — it is long. If we want to measure shorter times, we need a better reference — a process that repeats itself faster than once a day.

For this, we can use a pendulum and let it swing. Its movement will be periodic and counting the number of swings, we can measure time. Since the period of the oscillations depends on the length of the pendulum, we can even tune it and choose how fast our reference should be. If the length of the pendulum is about 25 centimetres, its period will be 1 second. (Clocks normally use pendulums that are about 99.4 centimetres long resulting in a period of two seconds, or a half-period of one second.) Using a system of gears, the periods can be counted and transformed into movements of hands that then show time on a clock.

There is one problem with using pendulum, though. The exact period of the swinging depends on the local gravitational field which varies on different places on Earth, depending on their latitude and altitude. A pendulum that oscillates with 1 second frequency on the Equator will have a period of 997 milliseconds on the North Pole. That might not seem like such a big difference but in a single day, the North Pole clock will be faster by more than four minutes! Clearly, if we want a more precise time measurement, we need something that oscillates even faster.

Quartz crystal for a wristwatch or a clock.
Quartz crystal for a wristwatch or a clock. (Public domain, source.)

The most commonly used oscillator in today’s clocks and watches is a quartz crystal. It can be made very small and due to its mechanical properties, it can vibrate at much larger frequencies. Typically, the crystal vibrates more than 32 thousand times in a single second and is therefore much more precise than pendulum clocks. The accuracy is improved from about 15 seconds per day to half a second per day — an improvement by a factor of 30. (The improvement is not larger because quartz clocks — especially wristwatches — suffer from many technical imperfections that are not so strong in pendulum clocks.)

We can use even faster processes to further improve the accuracy of timekeeping. But it is difficult to make mechanical oscillators — pendulums, vibrating crystals, or anything else — that can oscillate at such high frequencies. We therefore need some natural oscillator with a very high frequency. For that, we can use atoms because their internal energy can only have certain discrete values and an energy difference between two levels corresponds to a certain frequency of electromagnetic field that can be emitted or absorbed by this energy transition.

On-chip atomic clock.
On-chip atomic clock. (Public domain, source.)

Atomic clocks have two advantages: they are natural oscillators (not human-made) so that atoms of a given species will always oscillate at the same frequency, and they oscillate very fast — billion times a second. There is a price to pay for this precision because, naturally, it is extremely difficult to count individual periods of a system oscillating so fast. It can still be done, though, and such clocks are now the most precise time standards we have — their error is about one second in 100 million years.

Some atomic transitions have even higher frequencies than a few gigahertz which are used in atomic clocks now. Transitions in the optical domain (in contrast to microwave transitions for gigahertz frequencies) oscillate million billion times in a single second. Those oscillations are, of course, even more challenging to count than the oscillations in current atomic clocks. Clocks based on the optical transitions — called optical clocks — are nevertheless being developed and promise incredible accuracy. With optical clocks, it is possible to measure the age of the universe (about 14 billion years) with error smaller than one second!

What are such highly precise measurements good for? Without well synchronised time across the Earth, internet communication (and any other form of high-speed communication, including mobile phone networks or TV and radio signals) would be much slower. Navigation systems (such as GPS) would not work with a few-metre precision. GPS receivers measure time delay in signals from satellites and determine the position from the delay and positions of the satellites. More precise time means better accuracy of the navigation system.

There are also many scientific applications. With precise time measurements, we can, for instance, test one of the predictions of Einstein’s general relativity which states that the flow of time is affected by a gravitational field. In a strong gravitational field, time passes slower than when the gravity is weaker. The effect is very weak in the conditions on Earth but it still has to be taken into account for satellite systems. Current atomic clocks are, in fact, so precise that this difference in passage of time can be measured in two places that are about ten centimetres above one another.