# Quantum Dynamics of Kerr Optical Frequency Combs

below and above Threshold:

Spontaneous Four-Wave-Mixing, Entanglement and Squeezed States of Light

###### Abstract

The dynamical behavior of Kerr optical frequency combs is very well understood today from the perspective of the semi-classical approximation. These combs are obtained by pumping an ultra-high whispering-gallery mode resonator with a continuous-wave laser. The long-lifetime photons are trapped within the torus-like eigenmodes of the resonator, where they interact nonlinearly via the Kerr effect. In this article, we use quantum Langevin equations to provide a theoretical understanding of the non-classical behavior of these combs when pumped below and above threshold. In the configuration where the system is under threshold, the pump field is the unique oscillating mode inside the resonator, and triggers the phenomenon of spontaneous four-wave mixing, where two photons from the pump are symmetrically up- and down-converted in the Fourier domain. This phenomenon can only be understood and analyzed from a fully quantum perspective as a consequence of the coupling between the field of the central (pumped) mode and the vacuum fluctuations of the various sidemodes. We analytically calculate the power spectra of the spontaneous emission noise, and we show that these spectra can be either single- or double peaked depending on the value of the laser frequency, chromatic dispersion, pump power, and spectral distance between the central mode and the sidemode of interest. We also calculate as well the overall spontaneous noise power per sidemode, and propose simplified analytical expressions for some particular cases. In the configuration where the system is pumped above threshold, we investigate the phenomena of quantum correlations and multimode squeezed states of light that can occur in the Kerr frequency combs originating from stimulated four-wave mixing. We show that for all stationary spatio-temporal patterns, the side-modes that are symmetrical relatively to the pumped mode in the frequency domain display quantum correlations that can lead to squeezed states of light under some optimal conditions that are analytically determined. These quantum correlations can persist regardless the dynamical state of the system (rolls or solitons), regardless of the spectral extension of the comb (number sidemodes), and regardless of the dispersion regime (normal or anomalous). We also explicitly determine the phase quadratures leading to photon entanglement, and analytically calculate their quantum noise spectra. For both the below- and above-threshold cases, we study with particular emphasis the two principal architectures for Kerr comb generation, namely the add-through and add-drop configurations. It is found that regardless of the configuration, an essential parameter is the ratio between out-coupling and total losses, which plays a key role as it directly determines the efficiency of the detected spontaneous noise or squeezing spectra. We finally discuss the relevance of Kerr combs for quantum information systems at optical telecommunication wavelengths, below and above threshold.

###### pacs:

03.65.Ud, 42.50.Dv, 42.50.Lc, 42.65.Sf^{†}

^{†}thanks: Email address: yanne.chembo@femto-st.fr

## I Introduction

Kerr optical frequency combs are sets of equidistant spectral lines that are generated after pumping a whispering gallery mode (WGM) or ring resonator with a continuous-wave (cw) laser Vahala_PRL_2004 (); Maleki_PRL_LowThres (); DelhayeKipp (). When the bulk resonator has both an ultra-high quality factor and a Kerr nonlinearity, it can at the same time trap the pump photons for a significantly long time in the torus-like eigenmodes of the resonator, and host the nonlinear interactions amongst them. When the pump power is sufficiently low, the intra-cavity photons remain in a single cavity-mode and their frequency essentially remains the same as the one of the pump laser. However, above a certain threshold, these confined and long-lifetime pump photons are transferred through four-wave mixing (FWM) to neighboring cavity-modes, provided that phase-matching, energy and momentum conservation conditions are fulfilled. This process can be further cascaded and yield a frequency comb with all-to-all coupling, and involving up to several hundred modes over several THz. In comparison to optical frequency comb generators based on femtosecond mode-locked lasers, Kerr comb generators are fairly simple, compact, robust and energy-efficient: they are expected to be a core photonic systems for many applications, such as integrated photonics, metrology, aerospace and communication engineering Lipson_NatPhot (); Review_Kerr_combs_Science (); Nature_Ferdous (); PRL_Vahala_2012 (); PRL_NIST (); Nat_Gaeta_rev (); PfeifleNatPhot (); NIST_Optica (); Self_injection_NIST (); Kipp_Soliton (); PRL_WDM (); Nanophotonics ().

Beyond these potential applications, which have been a very powerful drive, Kerr comb are also actually an ideal test-bench systems for fundamental physics, and particularly, for quantum optics. In fact, understanding Kerr comb generation is strikingly simple when one considers the photon picture and describes the process as the photonic interaction , where two input photons labelled and interact coherently via the Kerr nonlinearity to yield two output photons and . Without further analysis, this interpretation already suggests that purely quantum phenomena based on the non-classical nature of light can eventually arise in Kerr combs.

From a theoretical point of view, it is well known today that in the semi-classical limit, Kerr combs can be described using either a set of coupled ordinary differential equations (one equation per mode YanneNanPRL (); YanneNanPRA (); YanneNanOL ()), or using a single partial differential equation (one equation for the sum of the modes Matsko_OL_2 (); PRA_Yanne-Curtis (); Coen ()). It is also well known that both formalisms are in fact perfectly equivalent PRA_Yanne-Curtis (), with the first one emphasizing the spectro-temporal dynamics of the system, while the second emphasizes the spatio-temporal dynamics. It is important to note here that these Kerr comb models are singularly accurate: the comparison between the numerical power spectra obtained using the models and the experimental ones is excellent across a dynamical range that can be as large as dB YanneNanPRL (); IEEE_PJ (); OL_phaselocking (); Chaos_paper ().

The spatio-temporal formalism is generally known as the Lugiato-Lefever equation (LLE), and was introduced for the first time by Lugiato and Lefever in the context of ring resonators where the semi-classical cavity fields where subjected to Kerr nonlinearity and diffraction LL ().
In the approximation of 1D-diffraction, some of the key dynamical properties of this optical system
had also been derived in the same article, such as for example the super- and sub-critical Turing instability leading to roll patterns.
The LLE used to model Kerr combs has an essential dissimilarity with the one initially introduced by Lugiato and Lefever:
diffraction is replaced by dispersion.
This difference is of no importance from the mathematical point of view.
However, from the physical standpoint, the difference is significant.
On the one hand, Kerr comb generation is genuinely 1D, originates from a small bulk cavity (from m- to mm-size), and involves guided fields: the system is experimentally compact, simple, low-power, versatile, controllable, and its behavior can be described by the LLE with high accuracy as emphasized above despite its high dimensionality (from three to up to several hundred modes).
On the other hand, in the initial system, the approximation of 1D diffraction is rather poor (the 2D approximation is much better), the fields are propagating freely, and the cavity is set up with mirrors: the system is experimentally very complex and the LLE is a rather simplistic model, even though the number of interacting modes is always very limited
(rarely more that ).

In the scientific literature, several researchers have explored the quantum properties of optical resonators with Kerr nonlinearity when pumped under or above threshold.

In the case of a resonator pumped below threshold, the classical viewpoint assumes that the pump field is the unique oscillating mode inside the resonator, while all the sidemodes have zero power (hence, there is technically no comb in this case). From a quantum standpoint, the pump field is actually at the origin of spontaneous four-wave mixing where two pump photons are symmetrically up- and down-converted in the Fourier domain, thereby leading to the simultaneous and spontaneous generation of signal and idler photons, respectively. This phenomenology corresponds to the photonic interaction , where , and are the pump, idler and signal angular frequencies, respectively. The phenomenon of spontaneous FWM (which is also sometimes referred to as parametric fluorescence) can only be understood and analyzed from a fully quantum perspective, because it results from the coupling between the intracavity pump photons and the vacuum fluctuations of the various sidemodes. This topic is the focus of a very large body of literature, particularly related to the generation of correlated pairs of entangled photons with chip-scale and integrated ring-resonators (see for example refs. Sharping_OE (); Clemmen_OE (); Helt_OL (); Chen_OE (); Azzini_OE (); Helt_scale_JOSAB (); Azzini_OL (); Camacho_OE (); Takesue_SciRep (); Reimer_OE (); Vernon_Arxiv (); Engin_OE (); Grassani_Optica (); Qubit_entanglement (); Wakabayashi () and references therein).

When the system is pumped above threshold, the photonic interaction becomes steadily sustained: from a classical perspective, the signal and idler sidemodes are correlated twin beams in the frequency domain, yielding a roll pattern in the spatial domain. By analogy to laser theory, it is considered that this phenomenon corresponds to stimulated four-wave mixing Agrawal ().
In ref. Lugiato_Castelli (), Lugiato and Castelli have pioneered investigations on the quantum properties of the paradigmatic system described in LL () when pumped above threshold in the approximation of 1D-diffraction.
In that work, they have demonstrated that the intensity difference exhibits fluctuations below the standard quantum noise limit (QNL). This important result, which for the first time predicted squeezing in optical systems ruled by the LLE, was obtained in the three-mode approximation (central pumped mode and two sidemodes), and therefore, was only valid close to the threshold leading to the rolls in the super-critical case. Zambrini et al. numerically showed later on that the squeezing behavior when certain additional degrees of freedom are accounted for is still consistent with the one of the reduced three-mode truncation Zambrini_PRA ().
Further research on the quantum properties of optical systems ruled by the LLE was performed with the more realistic case of 2D-diffraction. However, in that case, the roll pattern is unstable and instead, the simplest non-trivial solution is
an hexagonal structure which emerges through a sub-critical bifurcation. As a consequence, the number of modes involved in the dynamics increases significantly because of the hexagonal structure itself (the smallest order truncation now involves modes, instead of for the roll pattern), and because of its sub-critical nature (the higher-order sidemodes can not be legitimately discarded anymore, even close to threshold, so that even the -modes truncation is not very accurate).
However, using that lowest-order truncation, Grynberg and Lugiato had shown very early that these hexagons
can display four-fold mode squeezing in a lossless cavity close to threshold Grynberg_Lugiato (), while Gatti and Mancini have extended the results and shown that squeezing and multimode entanglement persists even in the presence of losses, and even far above threshold as long as the -mode truncation remains a good approximation Gatti_Mancini ().
In view of these preceding results, it could therefore be foreshadowed that Kerr combs, which can be described with great accuracy by the LLE in the semi-classical limit, can display a non-classical behavior as well. In this regard,
an elegant demonstration of the theoretical prediction of Lugiato and Castelli has been achieved recently:
in the research work reported in ref. Arxiv_squeezing_Cornell (), squeezing in a Kerr comb is experimentally demonstrated between the two-sidemodes of a th-order roll pattern.
Most important, this experiment is also the very first demonstration of squeezing in Kerr optical frequency combs, to the best of our knowledge.

From a purely technical point of view, other important parameters to consider are the central frequency of the comb, its spectral span, and the frequency separation between the comb lines. Even though some works have shown that the combs can be obtained with a pump close to the lower and upper limits of the near-infrared range ( nm Nature_Selectable_freq () and nm Nature_Mid_IR ()), the overwhelming majority of Kerr combs are generated today with laser pumps around nm. Since this wavelength corresponds to the well known telecom spectral window, there is a plethora of commercial off-the-shelf optical components (lasers, photodiodes, narrow filters, amplifiers, phase shifters, etc.) that are available for the manipulation of the photons around that wavelength, even at the single-photon level. It is also noteworthy that many nonlinear amorphous and crystalline materials have low dispersion and losses in that wavelength window, and these are two features that are of extreme importance in Kerr comb generation.

Moreover, Kerr combs originate from stimulated FWM which is an hyper-parametric process: hence, the frequency separation between the spectral lines generally ranges from GHz to THz for the Kerr combs of interest, instead of THz for parametric processes. Hence, in Kerr combs, the photo-detected signals fall into the microwave range where there is a very wide variety of technological solutions for the careful handling of low-noise signals.

For the above reasons, Kerr combs have many singular advantages for quantum optics experiments, powered
by the possibility to manipulate the photons in the optical frequency domain, and
measure their slowly-varying attributes (amplitude and phase) in the microwave frequency domain.
They also have the potential to play a major role in compact or integrated quantum-information systems
at optical telecommunication wavelengths OL_QKD_Bloch (); PRA_Merolla_2010 (); PRA_Merolla_2014 ().

Despite the aforementioned theoretical works in the context of quantum phenomena of LLE-based systems, and despite the promising technological opportunities highlighted above, several critical problems remain wide open for the understanding of the quantum properties of spontaneous and stimulated FWM combs in WGM resonators.

The first topic of interest is the analysis of the spontaneous FWM comb spectra when the system is pumped below threshold. Many groups have investigated experimentally the main characteristics of this phenomenon, but a coherent theoretical basis explaining the influence of the various parameters of the system (dispersion, frequency detuning, etc.) on the output spectra is still lacking.

A second challenge is that in the literature, the available research results to this date only consider minimally truncated expansions, whose validity is automatically restricted to a parameter range close to threshold. However, Kerr combs are generally operated far above threshold, and can be very large – up to several hundreds of modes. They can also correspond to different kinds of spatiotemporal patterns such as rolls (super- and sub-critical) or solitons (bright and dark), for example. It is therefore important to investigate in detail the quantum correlations in the case of Kerr combs where spectrum amplitude, size and span restrictions do not apply.

A third issue is related to the sources of quantum noise in the system. Previous theoretical works on LLE-based systems focused on gedanken experiments were the unique source of losses was the semi-reflecting mirror used to couple the light in and outside the cavity (the intrinsic losses were null). The corresponding equations therefore included only one vacuum fluctuation term. However, in the case or Kerr combs, the resonators are bulk, and then, necessarily lossy. This introduces an extra term related to vacuum fluctuations induced by these intrinsic losses. Actually, the in- and out-coupling processes might also be distinct (like in the add-drop configuration, for example), so that overall, we might have up to three vacuum fluctuation terms, instead of just one. In order to remain close to the experimental reality, it is therefore necessary to understand the effect of all these intrinsic and extrinsic vacuum fluctuations at the quantum level.

The fourth open point is the explicit determination of the quadratures that can potentially lead to multimode squeezing. The conjugate variable of the photon number operator is the phase operator comment_phase_operators (), so that when the squeezing occurs for a linear combination of modal intensities, there exists is necessarily a corresponding linear combination of correlated phase quadratures in the system. In Kerr combs, the large number of modes and the complexity of the all-to-all coupling amongst them allows for a large variety of phase-locking patterns in the semi-classical limit: the determination of the equivalent quantum correlations in terms of phase quadratures is therefore of particular relevance.

Our objective is to provide answers to the four open points highlighted above, and the article is therefore organized as follows. In the next section, we present a brief overview of the mean-field models used to model the dynamics of Kerr combs in the semi-classical limit. Important physical considerations such as orders of magnitudes and system architecture will be discussed in detail. In Sec. III, we build the quantum models for Kerr combs, using both the canonical quantization and the Hamiltonian formalism. Particular emphasis will be laid on the various sources of quantum noise that have to be accounted for depending on the in- and out-coupling configuration. The dynamics of the system below threshold is investigated in Sec. IV, where the spontaneous FWM spectra are explicitly calculated as a fonction of the system’s parameters. Quantum correlations and squeezing for the photon numbers is investigated in Sec. V, where we will explain why the squeezing properties of the comb are degraded as the size of the comb increases. Section VI is devoted to the study of the quantum correlations and squeezing behavior in both the amplitude and phase quadratures, after the explicit derivation of the relevant quantum Langevin equations. Particular emphasis is laid on the analysis of squeezing in rolls and solitons (bright and dark), which are the most prevalent spatiotemporal patterns in Kerr comb generation, and their squeezing spectra will be investigated in Sec. VII. We sum up our main results in the last section, which concludes this article.

## Ii Semi-classical models for Kerr optical frequency combs

We provide here a brief overview of the semi-classical models for Kerr combs, which are useful to gain a deep understanding of the quantum models that will be developed in the next section, and which are also needed to introduce the key macroscopic parameters needed to describe the system.

### ii.1 Modal expansion model

WGM resonators, as well as ring-resonators, generally have several families of longitudinal (azimuthal) modes SelTop_Matsko_I (); SelTop_Matsko_II (); Feron_Review (). Let us consider that only one family is involved in our case, and without loss of generality, we also consider that it is the fundamental family (torus-like modes). In that case, the modes of interest, which are sometimes referred to as azimuthal, can be unambiguously defined by a single integer wavenumber , which characterizes each member’s angular momentum. In the case of WGM resonators, this number can be considered as equal to the total number of reflections that a photon undergoes during one round trip in the cavity (ray-optics interpretation). Let us also consider that the eigennumber of the mode that is pumped by the external laser is . In the spectral neighborhood of , the eigenfrequencies of the resonator can be expanded in a Taylor series, following

(1) |

where is the eigenfrequency at and is the order of truncation for the expansion.

For a disk resonator with main radius , the parameter stands for the free-spectral range (FSR), with being the velocity of light and the group-velocity refraction index at . This intermodal angular frequency is, of course, linked to the round-trip period of a photon through the resonator as . The parameter stands for the second-order group-velocity dispersion of the eigenmodes (normal GVD for , and anomalous GVD when ). We recall that is generally the sum of two contributions, namely the geometrical dispersion (normal) and the material dispersion (normal or anomalous). The parameters for stand for higher-dispersion terms and in this study, these terms will be considered as uniformly null. Note that perfect equidistance for the eigenfrequencies is achieved when for all . More details can be found in refs. YanneNanPRA (); YanneNanPRL (); PRA_Yanne-Curtis (); Jove (), for example.

The resonator is also characterized by its losses, which can be internal or external. For each mode, the internal losses (bulk absorption, surface scattering, etc.) are quantified by the linewidth . On the other hand, the external losses are here considered to be induced by both the in- or out-coupling processes of the optical fields. The total losses are just defined as the sum of the two aforementioned contributions following . The loaded (or total) factor for each mode can be defined as , and the modal photon lifetime is .

The total electric field (in V/m) inside the cavity can be expanded as

(2) |

where is the time, is the complex-valued slowly-varying amplitude of the -th mode, is the corresponding spatial mode profile (units of m), is the permittivity of vacuum, is the refraction index at the laser pump wavelength, and c.c. stands for the “complex conjugate” of all the preceding terms YanneNanPRA (). It is important to note that in Eq. (2), and the fields have been normalized such that is equal to the number of photons in the -th mode.

It has been shown in ref. YanneNanPRA () that the slowly varying envelopes of the modes are governed by the following system of equations:

where is the Kronecker delta-function that equals when and equals zero otherwise. In the above equation, the Kronecker functions indicate that only the mode is pumped, and that the allowed four-wave mixing interactions will be those for which the total angular momentum of the interacting photons is conserved, following .

The four-wave mixing gain is , where is Planck’s constant, is the Kerr coefficient at , and is the effective mode volume of the pumped mode. The parameter is an intermodal coupling tensor which weights the spatial overlap amongst the various modes. The laser pump field is characterized by the detuning between its angular frequency and the resonance frequency of the pumped mode, and by which stands for the external pumping field, with representing in-coupling losses only.

Equation (II.1) can be further simplified and rewritten in a more convenient form, suitable for the canonical quantization. The first step is to introduce the reduced eigennumber , so that the pumped mode is now , while the various sidemodes symmetrically expand as , with “” and “” standing respectively for higher and lower frequency sidemodes. The modes , and in the four-wave mixing sum will now be simply replaced by their reduced counterpart as . The second step is to consider that the spectral extension of the comb is narrow enough to consider that the modes are quasi-degenerate in space and frequency (), and that the modal losses are quasi-degenerate as well, with . The last step is to replace the fields in Eq. (II.1) by , so that explicit time dependence is removed in Eq. (II.1). From a physical viewpoint, this latter transformation corresponds to setting the frequency reference at the laser frequency instead of the cold-cavity resonance of the pumped mode, and to express the modal frequencies with respect to the equidistant (FSR-spaced) frequency grid, instead of the dispersion-detuned eigenfrequency grid PRA_Unified ().

After implementing these mathematical transformations, it can be shown that the new modal fields obey the following set of autonomous, nonlinear and coupled ordinary differential equations:

(4) | |||||

where the overdot indicates the time derivative. Note that higher-order dispersion at arbitrary order can be accounted for by replacing by which is obtained from Eq. (1). Without loss of generality, we can arbitrarily consider the phase of the external pump field as a reference and set it to zero, so that this field becomes real-valued and can be written as

(5) |

It is important to recall the normalization in the semi-classical Eqs. (4) is such that is a number of photons (cavity fields), while is a number of photons per second (propagating fields). This normalization is physically the most appropriate at the time to perform the canonical quantization.

### ii.2 The two configurations under study

Two configurations are routinely used to generate Kerr optical frequency combs, as displayed in Fig. 1. It therefore important to identify precisely all the loss terms as well as the out-coupled fields in each case, because as we will see later on, the vacuum quantum noise terms are closely related to these losses and out-coupling configurations.

In the first architecture, that we call add-through comment_AT (), a single coupler is used to pump the cavity and to retrieve the comb signal, which is detected at the through port. This architecture allows for limited coupling losses (and therefore, low threshold power for Kerr comb generation). However, a disadvantage of this architecture is that the output signal is a superposition of the intra-cavity and a portion of the pump which is directly passing through the coupling waveguide IEEE_PJ (). In this add-through configuration, the total and external linewidths in Eq. (4) can be written as

(6) | |||||

(7) |

while the modal output fields obey

(8) |

with standing for the coupling losses in the through port.

In the second architecture, referred to as add-drop, two different couplers are used to perform in- and out-coupling tasks. The comb is therefore retrieved at the drop port. This double-coupling has the disadvantage to increase the overall losses (thereby increasing the threshold for Kerr comb generation), but however, at the opposite of the precedent case, the output signal is proportional to the intra-cavity field and provides an unambiguous representation of the physical processes that are taking place inside the resonator. For the add-drop configuration, the linewidths in Eq. (4) are explicitly defined as

(9) | |||||

(10) |

and the modal output fields simply obey

(11) |

where stands for the coupling losses in the drop port JSTQE_AurYanne ().

In all cases, the various linewidths are related to their corresponding quality factors by . A technique routinely used to determine the various quality factors at the experimental level is the cavity-ring-down method Feron_CRD ().

### ii.3 Spatiotemporal formalism

Several studies on the quantum properties of self-organized dissipative optical structures are performed on systems that are ruled by the LLE. In the case of Kerr combs, it has be shown in ref. PRA_Yanne-Curtis () that the above modal expansion model is exactly equivalent to the following normalized LLE

(12) | |||||

where is the total intra-cavity field and is the azimuthal angle along the circumference of the resonator. Higher-order dispersion can be accounted for by replacing by where the dispersion coefficients exactly correspond to those used in fiber optics. The total number of intracavity photons is therefore , while the output field is in the add-through configuration, and in the add-drop configuration. In several theoretical studies, Eq. (12) is further normalized to

(13) |

where is the dimensionless intra-cavity field, and is the dimensionless time. The dimensionless parameters of this normalized equation are the frequency detuning , the cavity second-order dispersion , and the external excitation . In the context of Kerr comb generation, the LLE has been extensively investigated in several articles since the pioneering works of refs. Matsko_OL_2 (); PRA_Yanne-Curtis (); Coen ().

In ref. PRA_Unified (), an exhaustive study of the various dynamical regimes of the LLE has been performed, and the stability basin of the various solutions has been determined. In the anomalous dispersion regime, the stationary solutions are rolls (super- and sub-critical), bright solitons (isolated or coexisting), and soliton molecules (isolated or coexisting). In the case of normal dispersion, the stationary solutions can be rolls, dark solitons (isolated or coexisting), and non-smooth dark solitons (sometimes referred to as platicons, see ref. platicons ()). For all these stationary solutions, the Kerr comb is perfectly symmetric in the semi-classical limit, and we will see in Sec. V that this symmetry opens the way for multimode squeezing when quantum noise is accounted for.

### ii.4 Orders of magnitude in experimental systems

In order to facilitate comparisons between theory and experiments, it is important to link the normalized parameters and variables to their counterparts in SI units. In particular, knowing the power levels involved provides key information at the time to choose the low-noise, high sensitivity components needed to perform experiments with non-classical light Bachor_Ralph_book ().

In our Eq. (4), the dispersion parameter is linked to the parameter used in fiber optics by (in sm), where is the group velocity. The coefficient can be converted to the nonlinear coefficient (in Wm) which is also well known in fiber optics, where is the effective area, and is the effective volume. For a spherical resonator of radius , an approximation of the effective volume of a WGM of azimuthal eigenumber and polar eigennumber is given in ref. Braginski_PLA () as . Since for the WGMs of interest, the effective area can therefore be approximated as for a spherical WGM resonator, and this is generally a higher bound estimate for WGM disks or ring resonators. Finally, the intra-cavity and output dimensionless intensities and can be converted in watts following and .

The theory based on the stability analysis of the normalized LLE indicates that Kerr combs can scarcely be generated when the normalized intra-cavity power and external pump power are inferior to . Therefore, the condition leads the following absolute minimum pump power (in watts) to trigger Kerr comb generation

(14) |

which correspond to an absolute minimum photon flux of . On the other hand, the condition yields the following formula for the minimum intra-cavity power (in watts)

(15) |

which corresponds to a minimal intra-cavity number of photon equal to . The above values are therefore absolute minima (necessary but not sufficient for comb generation), that can be reached when the laser is accurately detuned to in the anomalous dispersion regime (see refs. YanneNanPRA (); PRA_Unified ()). For any other detuning, and in both dispersion regimes, the threshold pump power for Kerr comb generation will necessarily be higher than , up to a factor . However, the threshold number of intra-cavity threshold number of photons will still be equal, or very close, to the minimal value PRA_Unified ().

Therefore, for mm-size crystalline resonator with GHz free-spectral range ( ps), Wkm, , and at nm in the add-through configuration, the absolute minimum threshold power can be as low as mW. Such low pumping power has already been demonstrated experimentally, like in ref. Comb_2mW () where a threshold power of mW was sufficient to trigger Kerr comb generation. On the other hand, for an integrated silicon nitride resonator with GHz repetition rate, Wkm, , and quality factors at nm in the add-through configuration, the absolute minimum threshold pump power is rather W.

## Iii Quantum model for Kerr optical frequency combs

The construction of quantum models for Kerr combs is required in order to understand the spatio- and spectro-temporal behavior of the system when it is in a dynamical state like one of those displayed in Fig. 2. The determination of this dynamical behavior at the quantum level can be performed through the canonical quantization of the semi-classical model, or by defining an Hamiltonian operator ruling the relevant interactions in the system. The first approach has the advantage to be more intuitive, while the second is generally helpful at the time to establish conservation rules (which are closely related to commutators involving the Hamiltonian). In the present article we will use both formalisms, which will be introduced in this section to derive the temporal behavior of the Kerr comb.

### iii.1 Canonical quantization

The canonical quantization permits to derive the quantum counterpart of a semi-classical model, and in our case it consists in three steps Gardiner (); Grynberg_Aspect_Fabre_book (): (i) replace all the fields and their complex conjugates by annihilation and creation operators and , respectively notational_convention (); (ii) introduce vacuum fluctuation operators for every loss mechanism (intrinsic or extrinsic) in the optical system; (iii) introduce vacuum fluctuation operators at both the in- and out-coupling ports.

The creation and annihilation operators obey the following boson commutation rules

(16) | |||

(17) |

The semi-classical photon number , which was a measure of the intra-cavity optical energy for each mode, is now represented by its quantum counterpart, which is the photon number operator

(18) |

It is useful to recall that the ordering of the operators and can not be arbitrarily swapped, as these two operators do not commute. We adopt here the so-called normal ordering which consists in placing the creation operators on the left and the annihilation operators on the right.

The vacuum fluctuations associated with losses and coupling can be explicitly introduced in each mode using the vacuum operators for the intrinsic losses, for the coupling losses in the through port, and for the coupling losses in the drop port. These free-field operators have zero-mean value and obey the commutation rules

(19) |

where s, s’ = t (through), i (intrinsic), or d (drop). The vacuum fluctuations, which are necessary to avoid a violation of the Heisenberg uncertainty principle, have following correlation properties

(20) | |||

(21) | |||

(22) |

The pumping field is now defined as a coherent state

(23) |

which is the sum a semi-classical contribution (this is a shorthand for , where is the identity operator), and a vacuum fluctuation that will be inserted in the through port. Its commutation rules is therefore

(24) |

and it then, has the same quantum-noise properties as a vacuum fluctuation.

The canonical quantization can be now be performed by transforming the semi-classical Eqs. (4), (8) and (11) into their quantum counterparts.

Let us first introduce the following notation for the sake of conciseness:

(25) | |||||

(26) | |||||

(27) | |||||

(28) |

For the add-through configuration, the quantum model explicitly reads

(29) |

with

(30) | |||

(31) |

On the other hand, for the add-drop configuration, the quantum model is

(32) |

where the losses and the output field operator obey

(33) | |||

(34) |

Note that because of the normal ordering, the creation operator in the nonlinear interaction terms is always placed on the left. Also, in the canonical quantization procedure, the pump fields have not been explicitly replaced by the operator , since the related vacuum fluctuation is already accounted for in the generic term .

### iii.2 Hamiltonian formalism

The theoretical understanding of the quantum properties of Kerr optical frequency combs can also be achieved through an Hamiltonian formalism, and in our case, the total Hamiltonian of the system has three contribution.

The first contribution corresponds to the propagation of the fields, following

(35) |

The second contribution originates from the external pump field, and reads

(36) |

The third and last contribution comes from the interactions related to the Kerr nonlinearity:

(37) | ||||

For the physical understanding of the quantum phenomena in Kerr media, it is sometimes useful to decompose the interaction Hamiltonian itself into three contributions following

(38) |

where

(39) |

is the self-phase modulation (SPM) contribution (a single mode is involved in the interaction),

(40) |

is the cross-phase modulation (CPM) contribution (two distinct modes are involved), while the four-wave mixing (FWM) term gathers all the remaining monomials of , which necessarily involve three or four distinct interacting modes.

The total Hamiltonian is therefore

(41) |

and it is interesting to note that this Hamiltonian can be very large for Kerr combs. In earlier studies related to quantum correlations in systems ruled by the LLE, the Hamiltonian was always truncated to a maximum of few tens of monomials. However, in our case, if we consider a comb with (that is, a comb with modes), then the interaction Hamiltonian has exactly monomials: this number therefore grows in a cubic polynomial fashion with the number of modes, and for a comb with modes, there is already monomials in the Hamiltonian.

The Hamiltonian can now be used to track the temporal dynamics of the quantum Kerr comb, as it permits to obtain an explicit equation for the annihilation operator following

(42) |

where the index s runs across the various loss terms corresponding to the configuration under study, that is

(43) |

The term stands for the total losses [see Eqs. (6) and (9)], and represent the vacuum fluctuations corresponding to these losses. On the other hand, the output field is

(44) |

where the index stands for the output port following

(45) |

Equation (42) is identical to Eqs. (29) and (32), and the output field operators defined in Eq. (44) in the add-through and add-drop configurations obey the same relationships as in Sec. III.1. The commutator generates exactly monomials, and accordingly, Eq. (42) includes a large number of terms as well. We also note that this formalism is close to the one adopted by Matsko et al. to investigate the temporal dynamics of Kerr combs in the deterministic regime, that is, when all the vacuum noise terms are uniformly set to zero Matsko_Normal_Comb ().

Another approach is to study the following Master Equation Lugiato_Castelli ():

(46) |

where is the density operator for the comb, and is a Liouvillian explicitly defined as

(47) |

In this article, we will however only consider the Hamiltonian in the context of Eq. (42), which yields a set of equations that are formally identical to those obtained through the canonical quantization in Sec. III.1.

### iii.3 Spatiotemporal formalism

The quantum form of the spatio-temporal LLE for Kerr comb generation is

(48) | |||||

where is the total intra-cavity annihilation operator. The quantum equation in the case where higher-order dispersion is accounted for is straightforwardly obtained by replacing by . The multimode vacuum fluctuation operators are analogously defined as , and the output field annihilation operator reads . Quantum versions of the LLE for other physical systems have previously been investigated by several researchers in one and two transverse spatial dimensions (see for example refs. Lugiato_Castelli (); Zambrini_PRA (); Grynberg_Lugiato (); Gatti_Mancini (); Hoyuelos_pumpmeter_PRA ()).

## Iv System under threshold: Spontaneous four-wave mixing

When the system is pumped under threshold (this is always the case when ), only the pumped mode is excited from the semi-classical standpoint, that is, and for . However, from a quantum perspective, there are quantum fluctuations in all modes, which are allowing for the spontaneous photonic interaction . The objective of this section is to determine the power spectra of all the sidemodes and their eventual correlations as a function of pump power, dispersion, detuning and nonlinear gain. In the scientifc literature, the topic of quantum dynamics of nonlinear optical systems pumped under threshold has been the focus of several research works, essentially in the context of parametric down conversion Lugiato_spatial_PRL (); Gatti_quantum_images_PRA (); Lugiato_Marzoli (); Gatti_Langevin_OPO_PRA (); Gatti_Multiphoton_PRA (); Zambrini_Polar_PRA (); Fabre_J_Phys (). or for spontaneous FWM Garcia_Ferrer_JQE (); Brainis_PRA (). A convenient method to determine consists in establishing the linearized time-domain equation for the quantum fluctuations, and then calculate their Fourier spectra.

### iv.1 Quantum Langevin equations

In order to understand the effect of these quantum fluctuations, let us consider that under threshold, the annihilation operator in the various modes of the resonator can be explicitly rewritten as

(49) |

where the operators stand for the quantum fluctuations in a given mode . By inserting Eq. (49) into Eq. (42), it appears that the quantum dynamics of the system is decomposed under the form of a nonlinear algebraic equation

(50) |

for the central mode , while we have the set of differential equations

(51) |

for the quantum fluctuations in the sidemodes , with

(52) | |||||

(53) |

being complex-valued parameters. Equations (51) can be rewritten under the form of independent sets of quantum-noise driven linear flows, following

(54) |

where

(55) |

is a Jacobian matrix. It is interesting to note that the quantum fluctuations are mutually coupled, and are independent from the other modes of order .

### iv.2 Spontaneous emission spectra

In the Fourier domain, we transform the operators as

(56) |

and we find that in the spectral domain, Eq. (54) can be rewritten as