We are used to the notion of classical harmonic oscillators; these are oscillators fluctuating coherently -this is, symmetrically- around their equilibrium position, experiencing a restoring force *F* proportional to the displacement *x* following the relationship *F* = –*kx*, being *k* a positive constant commonly known in the mechanics of ideal springs.

If *F* is the only force acting on the system (which means there is no friction with the environment) the system is called a **simple harmonic oscillator**, and it undergoes a sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency that does not depend on the amplitude.

In real life, for example in the case of a spring, we see a damped oscillation because it will decrease with time due to friction. So basically, the harmonic oscillation is a very useful idealization that allows to simplify many physical problems.

There is an equivalent quantum version that results from solving the famous Schrödinger equation of quantum mechanics for the quantum harmonic oscillator. In the animated image below, both types of systems, classical (cases A and B) and quantum mechanical, are shown.

^{Some trajectories of a harmonic oscillator according to Newton’s laws of classical mechanics (A–B), and according to the Schrödinger equation of quantum mechanics (C–H). In A–B, the particle (represented as a ball attached to a spring) oscillates back and forth. In C–H, some solutions to the Schrödinger Equation are shown, where the horizontal axis is position, and the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction. C, D, E, F, but not G, H, are energy eigenstates. H is a coherent state—a quantum state that approximates the classical trajectory. GBy Sbyrnes321 – Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=14059905}

One main difference between quantum and classical harmonic oscillators is that the energies that a quantum harmonic oscillator can have, happen in steps or quanta (instead of a continuous change) dictated by this equation below, known as the energy spectrum, that is quantized by a quantum harmonic oscillator:

where *n* is an integer that expresses the quantization of the harmonic oscillator (*n* = 0,1,2,3….) at each angular frequency *ω*. The expression above is the Einstein-Planck relation for energy written as E = *ħω* (in terms of the reduced Planck constant *ħ *and the angular frequency *ω*). The figure below shows these energies, increasing in steps or quanta: *E _{0 }* ,

*E*,

_{1 }*E*and so on as n increases, for a fixed frequency.

_{2}^{In the figure above we appreciate that the minimum energy that a quantum harmonic oscillator has (corresponding to the fundamental state of the quantum harmonic oscillator at n = 0) is not E}_{0 }^{ = 0 but E}_{0 }^{ = (ħω)/2, for any angular frequency ω. The value E}_{0 }^{ is known as zero-point energy (ZPE); the energy of the vacuum fluctuations at the quantum scale at each frequency. Image by Allen McC. at German Wikipedia – Eigene Darstellung, Public Domain, https://commons.wikimedia.org/w/index.php?curid=11542014 }

A critical consequence of the quantization of the harmonic oscillator, is a feature of having the lowest energy *E _{0 }*with nonzero value; this is one of the main contributions of quantum mechanics. It is the second distinctive difference between the classical and the quantum harmonic oscillators: for the classical harmonic oscillator at rest there is no displacement, and its energy (more precisely, kinetic energy) is zero (

*E*= 0), while the quantum harmonic oscillator is never at rest; it always has a residual vibration

_{x=0}*E*= (

_{0}*ħ ω*)/2 for each value of

*ω,*taken as the intrinsic energy of the quantum vacuum, also known as zero point field or quantum vacuum energy. This also relates to Heisenberg’s uncertainty principle in quantum mechanics.

Through the Casimir effect we have experimental proof of the existence of these zero point oscillations.

The important point here is that these quantum solutions corresponding to the harmonic case, are only expected to happen at quantum scale for systems at extremely low temperature, requiring sophisticated cooling methods to see these oscillators.

Using an organic semiconductor to produce polaritons, now a team of physicists lead by Dr. Hamid Ohadi, of the School of Physics and Astronomy at the University of St Andrews, have shown for the first time this kind of harmonic behavior at room temperature. The system is confined in a potential well created by four laser fields.

Polaritons are considered quasiparticles composed of half-light and half matter, which behaves as a Bose Einstein condensate or superfluid (fluids with no viscosity), even at room temperature, and they result from photons interacting with electron-hole pairs – called excitons – in a semiconductor. These excitons impose a dipole moment, which combined with the dipole of the electromagnetic field, couples strongly the excitons and the photons. The result is a polariton. In order for superfluidity to occur at room temperature, polaritons must be present.

When polaritons condense to form this type of quantum liquid, the condensate can be corralled within a pattern of laser beams to control its properties. This made the fluid oscillate with a series of harmonic frequencies. The shape of these quantized states of vibration, that were observed via the photoluminescence of the condensate, matched those of a “quantum harmonic oscillator”, as seen in the figure below.

^{Figure 2: Comparison between the measured photoluminiscence of an organic semiconductor to produce polaritons which behaves as a Bose Einstein condensate or superfluid, and the numercial simulation showing harmonic behavior at room temperature.}

Their results prove the capability of tailoring and manipulating room-temperature organic polariton condensates by simply configuring the pump geometry, which can be extended to polariton condensate lattices for future quantum applications [1]. The distance between the pumps controls the degree of harmonicity of the oscillations. At the optimal parameter, the harmonic behavior was found at room temperature.

## References

[1] Mengjie Wei et al, Optically trapped room temperature polariton condensate in an organic semiconductor, *Nature Communications* (2022). DOI: 10.1038/s41467-022-34440-0