Quantum optics studies the **interaction of light and matter in resonant environments**, most commonly provided by *cavities*, where particular modes of the electromagnetic field are enhanced with respect to the empty space. When we speak of matter in this context, we have in mind a simple atomic transition between two levels, near-resonant with the cavity mode. This ``atom'' is an **exceptionally nonlinear entity. **It can be construed as a linear oscillator with an equidistant spectrum from which we have sequestered only the first two levels and discarded the rest; such a truncation sets an upper bound to the available excitation, quantified by the so-called population *inversion* of the atom.

The light mode, on the other hand, retains the character of a **linear oscillator**. The two-level ``atom'' absorbs and re-emits *quanta* of light as it interacts with the resonant mode, which are significant in the dynamical evolution. Statistical physics, on the other hand, deals traditionally with large systems consisting of a large number of macroscopic parts for which collective attributes can be defined. *Entropy* comes into play here, obeying the fundamental thermodynamic law of increase during a set process or being constant at equilibrium. **Physical quantities which describe a macroscopic body are, in the vast majority of cases, very close to their mean values.** If the uncertainty of entropy is very small compared to unity, then small deviations from the mean values, called *fluctuations*, obey Gaussian statistics and can be treated thermodynamically. Things are different, however, **in systems driven out of equilibrium**, where we most commonly talk of a *steady state* which can be the outcome of untamed **fluctuations departing from the Gaussian law**. When discussing critical phenomena in statistical physics, such as phase transitions, to get a feeling of how large the involved quantities are alongside their fluctuations, one scales them with the so-called system-size parameters which depend on the internal couplings between the constituents of a system and the external couplings between the system and its environment. **For us, what links the two frames of quantum optics and statistical physics is quantum coherence**, i.e., the ability to produce non-classical interference effects in very small scales. When speaking about coherence, the laser - invented in the 1960s, prevailing since in scientific and everyday life - readily pops up as a driven device producing a coherent state of light with high excitation. The laser, however, is essentially a classical a many-body apparatus which, when operated above threshold, generates a quasi-constant output with just a bit of noise representing the fluctuations about the mean. Strong coupling between light and matter is not the focus of attention here: the relevant system-size parameter gets larger with weaker coupling. This defines a

*weak-coupling limit*. What are we though to expect when we have one sole atom at our disposal, strongly coupled to resonant radiation? This is where another kind of limit arises.

## The strong-coupling limit

**The strong-coupling limit is the one where fluctuations do not vanish when the system size grows, since it sends the light-matter coupling strength to infinity.** This property has profound implications in the way we understand the dynamics involved in very small scales, revealing the

**distinct nature of quantum fluctuations as opposed to thermal noise**: the former are ultimately a consequence of the statistical interpretation of the wavefunction, or the density matrix, as we get a chance to visit the foundations of quantum theory in experimentally feasible systems. In the macroscopic world, on the other hand, we usually picture fluctuations as slight deviations from the establishment of an equilibrium pattern amenable to the standard thermodynamic description. The inconsequential role of such deviations is asserted by attaining a familiar weak-coupling limit, as in the laser operation, where quantum dynamics loses its special character.

In our recently published article, Strong-coupling limit of the driven dissipative light-matter interaction, we plunge deep into the microcosmos and focus on light and matter as they interact in a resonant environment, modeled by the *Jaynes-Cummings Hamiltonian*, provided by a leaky cavity connected to a reservoir, which is also pumped by a compensating steady drive - both being communications with the macrocosmos. Thus, we take the light-matter coupling we described in the beginning and impose an input to such an interaction, through driving with a coherent field, and monitoring the output, through coupling to a radiative reservoir, allows us to trace the behavior of the quantum system in the language of phase transitions as part of statistical mechanics. What happens when the photon loss rate to the reservoir gets smaller and smaller in comparison to the light-matter coupling strength which sets the scale of the quantum energy level structure? This is where we need to consider only one single two-level atom in order to organize the strong-coupling limit leading us to a variety of new phenomena as the two worlds, the macrocosmos and the microcosmos, interlace. Bistability, i.e. the existence of two metastable states with lifetimes significantly larger than the relaxation times set by dissipation, is a key player along the way. We find it even for the very weak external drives, in a craggy landscape of photons progressively blocking the absorption of an additional one, due to the formation of new effectively resonant structures. The persistence of photon blockade in the thermodynamic limit has no analog in the classical world. This inherently quantum feature wanes nevertheless, to produce an intense radiation that is eventually unable to couple to the atom when the external drive imposes its presence on resonance.

## Contact information

Themistoklis Mavrogordatos, email: themis.mavrogordatos@fysik.su.se

**Link to the published article:**

Strong-coupling limit of the driven dissipative light-matter interaction

Th. K. Mavrogordatos

Phys. Rev. A 100, 033810 – Published 10 September 2019