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Frequency-filtered photon correlations, or adding the colour degree of freedom in quantum optics, not only provides clear and new insights into the quantum dynamics and level structure of an emitter, it also enables one to access, tailor and optimize different regimes of operation, simply by tuning the frequency and bandwidth of detection. One can easily switch in this way between different types of quantum emission, from single-photon to thermal, N-photon, entangled or squeezed light, using the same source. This is achieved by isolating specific transitions with sought statistical properties, to distil and purify the emission into only those photons that present certain desirable characteristics.

One of the new class of correlations made available by this paradigm is leapfrog N-photon emission, which can then be harvested by Purcell-enhancement with a cavity. Beyond such direct transitions, tunable interferences open new prospects for N-photon squeezing. In both cases, the paradigm wide-opens multi-photon physics with no restriction of principle on the number of photons.

With the objective of investigating the filtered correlation properties of light emitted by any given quantum system (such as molecules, atoms or quantum dots), and exploit them for quantum technology applications, and also to delve deeper into the exciting new types of emission that they unveil, in 2012 we developed a very efficient theoretical method that is both conceptually and technically simpler to implement than the direct definition, being mathematically equivalent and completely general, for any open quantum system (matter, light, phonons, etc.) [1].

We include the frequency-sensitive detectors or sensors in the dynamics, modelled as simple harmonic oscillators, by coupling them weakly to the field of interest. This can be done with a Hamiltonian coupling term with strength ε, in which case one must make sure that it is small enough not to perturb the original system dynamics in any way. Then, the sensors can also be modelled by N two-level systems (truncated harmonic modes), which, furthermore, converge in the case of identical frequencies to autocorrelations from a single N-level system.

A second equivalent option is to use Gardiner’s cascaded formalism, that ensures unidirectional coupling even for a non-negligible population of the sensors and, therefore, is ideal to implement montecarlo simulations [2]. The key in any case is that the detectors are passive objects performing the Fourier transform (and all the messy integrals) dynamically for us. The sensor decay (included as a Lindblad term in the master equation formalism) naturally provides the frequency resolution of the measurement, Γ, and its inverse, 1/ Γ, the corresponding time resolution, as required by Heisenberg uncertainty principle.

With this, the time- and frequency-resolved N-photon coherence function is computed as a single intensity cross correlator between sensor-population operators. This can be done for any kind of dynamics, allowing, for instance, pulsed or continuous excitation, of a coherent or incoherent nature. In the case of zero-delay correlations, in the steady state under continuous excitation, that we call N-photon spectrum (NPS) [3], one does not even need to apply the quantum regression formula but simply compute the resulting mean values.

Some intrinsic features of any 2PS measurement, that must be pinpointed in order to make accurate and meaningful interpretations or predictions, are:

- At large Γ, the standard Glauber g(2) is recovered.
- At small Γ, the uncorrelated boson statistics is recovered. That is, for incoherent excitation, 2! when photons are indistinguishable (ω1=ω2), converging to the frequency-blind Hanbury-Brown and Twiss effect, and 1 if they are not (ω1≠ω2). These limits are not always met when exciting with a laser that is considered fully classical (of zero linewidth) [3]. For instance, if detection is at the laser frequency itself (ω1=ω2=ωL), we recover the Poissonian value of 1.
- At intermediate Γ, the statistics of indistinguishable photons manifests in the 2PS as tendency towards bunched correlations, on the diagonal line.
- Meaningful correlations between photons emitted from transitions between the system levels (such as cascaded processes or single-photon generation) are found when Γ is of the order of the corresponding peaks in the emission spectrum (1PS). These spots are accompanied by vertical and horizontal lines at the frequencies of all allowed single-photon transitions.
- When ω1+ω2 matches the energy of a two-photon transition in the system, bunching antidiagonal lines appear (that we call leapfrogs), corresponding to pairs of virtual photons with strong correlations and other interesting quantum properties such as classical inequality violation [4].
- If two different de-excitation processes produce photons at equal ω1, ω2, interference between them occurs, leading to unexpected values in the 2PS [2].
- Correlations undergo oscillations in the delay time between photons if the frequencies do not match the spectral peaks. Beatings appear due to the interplay between the different frequencies.
- If the measurement is done through various devices, worsening the resolution in one of the variables (e.g. in frequency, Γ), this corresponds to cascading further the measurement, coupling the output to more sensors, or simply integrating in that variable over a wider range (Ω), with the other variable resolution remaining the same (1/Γ).

Thanks to the passive nature of the filters, these functions also evidence how an emitter will perform when it is integrated in a larger set up or circuit, by jointly unveiling the temporal (or statistical) and spectral domains, which are two sides of the same dynamics.

Time and frequency resolved correlations are a powerful tool to characterize any quantum system, allowing to distill and exploit precious properties for quantum technologies such as interferences, squeezing, entanglement or bundle emission. They can be easily computed with the sensor method, expanding the power spectrum into a two-photon spectrum.

See the paper

*Photon correlations in both time and frequency*, E. del Valle, J. C. López Carreño, F. P. Laussy, arXiv:1802.04540.

for more details, at a divulgative level.

[1] *Theory of frequency-filtered and time-resolved N-photon correlations*, E. del Valle, A. Gonzalez-Tudela, F. P. Laussy, C. Tejedor and M. J. Hartmann. Phys. Rev. Lett. 109, 183601 (2012). (arXiv:1203.6016). Also, the erratum.

[2] *Frequency-resolved Monte Carlo*, J. C. López Carreño, E. del Valle, F. P. Laussy, Sci. Rep. 8, 6975 (2018) (arXiv:1705.10978).

[3] *Two-photon spectra of quantum emitters*, A. Gonzalez-Tudela, F. P. Laussy, C. Tejedor, M. J. Hartmann, E. del Valle. New J. Phys. 15, 033036 (2013) (arXiv:1211.5592)

[4] *Violation of classical inequalities by frequency filtering*, C. Sánchez Muñoz, E. del Valle, C. Tejedor, F. P. Laussy. Phys. Rev. A 90, 052111 (2014) (arXiv:1403.6182). See a summary and a related video.

10. *Emitters of N-photon bundles*, C. Sánchez-Muñoz, E. del Valle, A. González-Tudela, S. Lichtmannecker, K. Müller, M. Kaniber, C. Tejedor, J.J. Finley and F.P. Laussy. Nature Photonics 8, 550 (2014) (arXiv:1306.1578). Also see the News and Views
titled *Cavity quantum electrodynamics: A bundle of photons, please* by Dmitry V. Strekalov.