Contexts, Systems and Modalities: a new ontology for quantum mechanics Alexia Auffèves (1) and Philippe Grangier (2) (1): Institut Néel, BP 166, 25 rue des Martyrs, F38042 Grenoble Cedex 9, France (2): Institut d'Optique, 2 avenue Augustin Fresnel, F91127 Palaiseau, France

In this article we present a possible way to make usual quantum mechanics fully compatible with physical realism, defined as the statement that the goal of physics is to study entities of the natural world, existing independently from any particular observer's perception, and obeying universal and intelligible rules. Rather than elaborating on the quantum formalism itself, we propose to modify the quantum ontology, by requiring that physical properties are attributed jointly to the system, and to the context in which it is embedded. In combination with a quantization principle, this non-classical definition of physical reality sheds new light on counter-intuitive features of quantum mechanics such as the origin of probabilities, non-locality, and the quantum-classical boundary.

I. HISTORICAL REMINDERS

It is well known that physicists, while they all agree about how to use Quantum Mechanics (QM), still all disagree about what it means, and even more about "the real stuff" it describes: that is, its ontology. However, let us take as a basis for an agreement the two statements that (i) the quantum formalism is correct, and (ii) the quantum ontology must agree with physical realism, which states that the goal of physics is to study entities of the natural world, existing independently from any particular observer's perception, and obeying universal and intelligible rules 1 . Then the disagreement quoted above finds its roots in the difficulty to make (i) and (ii) compatible, as it has been much debated in the literature [START_REF] Einstein | Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?[END_REF][START_REF] Bohr | Can Quantum-Mechanical Description of Physical Reality be Considered Complete?[END_REF][START_REF] Heisenberg | The Physical Principles of the Quantum Theory[END_REF][START_REF] Bell | On the Einstein-Podolski-Rosen paradox[END_REF][START_REF] Bell | Against measurement[END_REF][START_REF] Aspect | Bell's inequality test: more ideal than ever[END_REF][START_REF] Mermin | Is the moon here when nobody looks? Reality and the quantum theory[END_REF][START_REF] Grangier | Contextual objectivity: a realistic interpretation of quantum mechanics[END_REF][START_REF] Grangier | Contextual objectivity and the quantum formalism[END_REF][START_REF] Mermin | Quantum mechanics: Fixing the shifty split[END_REF][START_REF] Pusey | On the reality of the quantum state[END_REF], giving rise to many different interpretations of QM [START_REF] Laloë | Do we really understand quantum mechanics[END_REF].

The purpose of this article is to propose and discuss a way to have the quantum formalism (i) and physical realism (ii) both correct and compatible, so that many different views and practices about QM can stay essentially unchanged. As usual, there is a price to pay, but the currency will be ontological: it will be a subtle but deep change in what is meant by physical properties, which should not any more be considered as properties of the system itself, but jointly attributed to the system, and to the context in which it is embedded (precise definitions will be given below). We will show also that this ontological change has strong links with quantization as a basic physical phenomenon, and that this combination can explain why QM must be a probabilistic theory.

Let us start with the simple question: what is a quantum state ? Since current answers do not meet a large agreement between physicists, it may be useful to remember that before QM, "classical" physicists used to work with physical states, that pertained to physical systems. Physical systems are entities of the natural world that can be isolated well enough to study them, measure their properties, i.e. "ask them questions" (what is your position, mass, velocity...). The set of answers resulting from 1 Some philosophical background is given in the Annex.

a given set of questions defines a physical state, that is, the ID card of the physical system. Once this ID card is known, the behavior of the system becomes perfectly predictable from the equations of motion, and answers to new questions are predictable as well 2 . In the particular case where the measured properties are constant of the motion, the same questions will invariably give the same answers, whatever their total number, or the ordering of their sequence. This is consistent with attributing a physical state to the system itself, so that the system "is" in that state, even if nobody is there to look at it [START_REF] Mermin | Is the moon here when nobody looks? Reality and the quantum theory[END_REF]. This property is often thought as the core of objectivity, and more generally, corresponds to ordinary ontology, i.e., our usual way of seeing (classical) reality.

With the rise of QM, the notion of physical state was seriously shaken. In particular, it appeared impossible to obtain as many certain answers as possible questions, i.e. to get the full ID card of the system. At early times of the quantum theory, this resulted in the famous Einstein-Bohr debate [START_REF] Einstein | Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?[END_REF][START_REF] Bohr | Can Quantum-Mechanical Description of Physical Reality be Considered Complete?[END_REF][START_REF] Bell | On the Einstein-Podolski-Rosen paradox[END_REF][START_REF] Aspect | Bell's inequality test: more ideal than ever[END_REF], with opposite claims that either the physical state exists like in classical physics, but then QM would be incomplete [START_REF] Einstein | Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?[END_REF], or that QM would be complete, but then the full physical state does not exist, at least not in a classical sense [START_REF] Bohr | Can Quantum-Mechanical Description of Physical Reality be Considered Complete?[END_REF]. Then many different interpretations of QM have been proposed, and our purpose here is not to discuss all of them [START_REF] Laloë | Do we really understand quantum mechanics[END_REF], but rather to examine another, maybe more philosophical, perspective: assuming physical realism, can we spell out a physical reality suitable for QM, and what are the practical consequences of such an ontological approach?

In Sec. II, we will present and discuss the current -often implicit -quantum ontology. Then in Sec. III and IV, we will introduce our approach, closely related to "contextual objectivity" [START_REF] Grangier | Contextual objectivity: a realistic interpretation of quantum mechanics[END_REF][START_REF] Grangier | Contextual objectivity and the quantum formalism[END_REF]. Important consequences will be presented in Sec. V to VIII, and some related philosophical considerations will be discussed in the Annex.

II. USUAL QUANTUM ONTOLOGY

Nowadays in QM, one is accustomed to the idea that when studying a physical system, the full ID card, corresponding to the ensemble of answers to all the possible questions one can ask, is not accessible, and that the description must be restricted to the "quantum state", quite different from what the "classical state" used to be. A paradigmatic example of such quantum state is the photon polarization. It is usually measured with a polarizing beam-splitter, that can be rotated by a continuous angle θ. By definition, the polarization is |H θ (resp. |V θ ) if the photon is transmitted (resp. reflected) with certainty by a θ oriented beam-splitter. In agreement with the Malus law that applies for classical light, a photon of polarization |H θ=0 sent in a θ = 0 oriented beam-splitter will be either transmitted with probability cos 2 (θ), or reflected with probability sin 2 (θ). After the measurement, the photon is projected on the state |H θ (resp. |V θ ).

This simple picture (Fig. 1) gathers important characteristics of quantum mechanics. First, there are indeed less certain answers than possible questions, meaning in that specific example that the photon polarization cannot be known with certainty for all θ angles. Second, results of successive measurements depend on the ordering of the sequence; for instance, it is not equivalent to measure the polarization successively in the basis {0 • , 45 • , 90 • }, or {0 • , 90 • , 45 • } -in other words, measurements do not commute 3 . These characteristics derive from the fact that 3 Classically one could do the same experiment with a polarized light wave, and also get different results according to the ordering of the polarizers. However the interpretation is completely different : the classical polarization "has" an orientation in space, which can be determined by carrying out more measurements, whereas the photon can only give a probabilistic yes/no answer.

measurement is a projection, from an initial state onto an eigenstate associated with a "measurement context". Even if this conception is commonly spread, it holds many unsolved questions that makes the understanding of quantum mechanics a vivid field of debates and interpretations, more than a hundred years after it was born [START_REF] Peres | Is quantum theory universally valid?[END_REF][START_REF] Bohm | A Proposed Solution to the Measurement Problem in Quantum Mechanics by a Hidden Variable Theory[END_REF][START_REF] Ghirardi | Unified dynamics for microscopic and macroscopic systems[END_REF][START_REF] Rovelli | Relational Quantum Mechanics[END_REF][START_REF] Griffiths | Consistent Quantum Mechanics[END_REF][START_REF] Schlosshauer | Decoherence, the measurement problem, and interpretations of quantum mechanics[END_REF][START_REF] Rèdei | Quantum probability theory[END_REF][START_REF] Brukner | Information Invariance and Quantum Probabilities[END_REF][START_REF] Hardy | Reformulating and Reconstructing Quantum Theory[END_REF][START_REF] Zurek | Quantum darwinism, classical reality, and the randomness of quantum jumps[END_REF][START_REF] Schlosshauer | On Zurek's derivation of the Born rule[END_REF]. For instance, on the ontological side, what does the quantum state represent: our knowledge of reality, or reality itself -in other words, is the wave function epistemic, or ontic, or a bit of both [START_REF] Pusey | On the reality of the quantum state[END_REF] ? What happens during a measurement, at the microscopic scale ? Where do probabilities come from, and how to justify them ? What happens during an EPR / Bell experiment carried out on entangled particles, and what is this "spooky action at a distance" that affects Bob's photon, when Alice measures hers ? Last but not least, measurement is at the heart of the usual quantum description -but is it possible to describe the world out of any measurement process, and thus to get rid of the so-called quantumclassical boundary [START_REF] Bell | Against measurement[END_REF][START_REF] Mermin | Quantum mechanics: Fixing the shifty split[END_REF] ?

In the following we propose an alternative ontology for quantum mechanics. Let us emphasize that an ontology, that aims to define "what reality is", can never be demonstrated, but is rather a "thinking framework", which is chosen for both intuitive and practical reasons. Our strategy will therefore consist in going back to the roots of ontology, unveil and question our intuitions on what reality is, and eventually, take the freedom to choose a new direction. In particular, in the same way that quantum physicists progressively gave up with the idea that the quantum state should consist of the full ID card of the system, here we will question the fact that the quantum state should pertain to the system alone. As detailed below, our perspective sheds new light on the set of conundrums reminded above, but it is unavoidable that other ontologies may also fit with the same mathematical theory -which one gives the "best fit" is ultimately the choice to be made.

III. SYSTEM, CONTEXT, AND MODALITIES

To define an ontology, our main guideline is to start from the basic question: what can we be certain of ? More precisely, within the physical framework we are interested in, which phenomena can we predict with certainty, and obtain repeatedly ? Though a full "ID card" in the classical sense is no more available as said above, certainty and repeatability of phenomena will allow us to identify some items of an ID card, by providing necessary conditions to be able to define a "state". Such an approach, supported by quantum experiments, has a clear relationship with the criteria for physical reality given by EPR [START_REF] Einstein | Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?[END_REF] -but the following will be different.

Our quantum ontology will thus involve three entities of different natures. First comes the system, that is -as stated above -a subpart of the world that is isolated well enough to be studied. The system is in contact with other systems, that can be a measuring device, an environment -no need to be more specific at this point. The ensemble of these other systems will be called a context. A given context corresponds to a given set of questions, that can be asked together to the system. A set of answers that can be predicted with certainty and obtained repeatedly within such a context will be called a modality. Given these definitions, let us bind them together by the following rule: In QM, modalities are attributed jointly to the system and the context. This principle will be called "CSM", referring to the combination of Context, System, and Modality4 . As a set of certain and repeatable phenomena, a modality fulfills the above conditions for the objective definition of a quantum state 5 .

At this point we can emphasize that the context is classical, in the sense that no other context has to be specified to define its state. Note that here neither size, nor a quantitative criterium has been made to draw the quantum-classical boundary: the quantum vs classical behavior is only related to the CSM principle itself, i.e., to the very definition of a modality.

Obviously, CSM applies to the photon example quoted above: the system is the photon, the θ-oriented polarizer is the context, and the two possible exclusive answers/modalities in this context are either "transmitted", or "reflected". To predict with certainty if a photon will be transmitted or reflected, one has to know the modality, which includes the context it corresponds to -the angle of the polarizer in that case 6 . In the CSM perspective, a photon does not "own" a polarization, but the ensemble photon-polarizer does. If the context is known, and if the system is available, a modality defined in this same context can be recovered without error. This property has been exploited for years by quantum communication technologies, and provides the core of quantum cryptography protocols [START_REF] Bennett | Quantum cryptography: Public key distribution and coin tossing[END_REF]. Here, we have drawn the consequences of this behavior in ontological terms.

The resulting ontology is clearly different from the classical one, where it is expected that a state should "exist" independently of any context to guarantee objectivity. But even if CSM is fundamentally non-classical, physical realism is not lost: it still pertains to the ensemble made of context, system, and modality. Objectivity, defined as the independence from any particular observer's perception, is still guaranteed, but the "object" is the system and the context, and its "properties" are modalities [START_REF] Grangier | Contextual objectivity: a realistic interpretation of quantum mechanics[END_REF][START_REF] Grangier | Contextual objectivity and the quantum formalism[END_REF].

It might even be argued that the CSM principle should apply to classical ontology as well, since physical states always show up in a given context. As mentioned in the introduction however, in classical physics the ordering of the questions, i.e. of the contexts that are successively in contact with the system, does not have any influence on the results, and all the questions can get a definite answer. As a consequence, the context can be forgotten, and the modality can be fully attributed to the system alone -giving birth to the ordinary classical ontology. On the contrary, in QM, the ordering in the succession of the questions has a strong influence on the answers obtained (see Fig. 1), and therefore the context cannot be forgotten. Hence the non-commutation of measurements mentioned above is intimately related to the CSM principle; we will come back to this in the discussion below.

Finally, after quoting EPR at the beginning of this section, let us emphasize that the CSM principle is not foreign to Bohr's view, as expressed in his answer to the EPR argument [START_REF] Einstein | Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?[END_REF][START_REF] Bohr | Can Quantum-Mechanical Description of Physical Reality be Considered Complete?[END_REF]: "The very conditions which define the possible types of predictions regarding the future behavior of the system constitute an inherent element of the description of any phenomenon to which the term physical reality can be properly attached".

In this sentence, Bohr explicitly states that despite being classical, the "very conditions" (i.e., the context) must appear together with the system in the description of quantum phenomena. In the following, we show that the CSM principle does not come as a bolt from the blue, but is actually tightly bound to a quantization principle.

IV. QUANTIZATION PRINCIPLE

Let us consider the situation pictured in Fig. 2: a quantum system is described within a set of contexts, whose parameters can be changed continuously, whereas the measurement results (the modalities) are discrete and in a finite number 7 . This is a quite general and canonical situation of quantum mechanics, which describes for instance spins interacting with a Stern-Gerlach apparatus, photons analyzed with polarizing beam-splitters, or quantum measurements on an ensemble of qubits.

In a given context, the modalities are "mutually exclusive", meaning that if one modality is true, the others are wrong. On the other hand, modalities obtained in different contexts are generally not mutually exclusive: FIG. 2: Illustration of the quantization principle for a spin 5/2 measured by a Stern-Gerlach apparatus. There is a continuous infinity of possible contexts, labelled by the angle θ of the magnetic field gradient, but only N=6 mutually exclusive modalities. The number of spots, corresponding to the number of modalities for a given orientation θ, does not change while rotating the apparatus.

they are said to be "incompatible", meaning that if one modality is true, one cannot tell whether the others are true or wrong. This terminology applies to modalities, not to contexts, that are classically defined: changing the context results from changing the measurement apparatus at the macroscopic level, that is, "turning the knobs". These context transformations have the mathematical structure of a continuous group, that will be denoted G: the combination of several transformations is associative and gives a new transformation, there is a neutral element (corresponding to no change of context), and each transformation has an inverse. In the following, we will thus consider infinitely many contexts, all related by the continuous group G, and belonging to a set C.

These definitions allow us to state the following quantization principle:

(i) For each well-defined system and context, there is a discrete number N of mutually exclusive modalities. This number N does not depend on any particular context within the set C.

(ii) Modalities, when defined in different contexts within C, are generally not mutually exclusive, and they are said to be "incompatible".

Otherwise stated, whereas infinitely many questions can be asked, corresponding to all possible contexts, only a finite number N of mutually exclusive modalities can be obtained8 (Fig. 2). An essential consequence is that it is impossible to get more details on a given system by combining several contexts, because this would create a new context with more than N mutually exclusive modalities, contradicting the above quantization principle. The resulting impossibility to define a unique context where all modalities are mutually exclusive also makes that the succession of observed modalities depends on the sequence of contexts applied to the system (Fig. 1). As mentioned above, it is therefore forbidden to attribute definite physical properties to the system alone, in agreement with the CSM principle. Note that in our perspective, CSM does not show up because of some practical reasons pertaining to the measurement protocol, but is intimately linked to the quantization principle. In the following, we revisit some of the main features of quantum theory within this new framework.

V. PROBABILITIES

As a first important conceptual consequence, we argue that quantum mechanics must be a probabilistic theory, not due to any "hidden variables", but due to the ontology of the theory. The argument runs as follows: let us consider a single system, two different contexts C 1 and C 2 , and the associated modalities M (C 1 , n) and M (C 2 , m), where n and m go from 1 to N. As said above, the quantization principle forbids to gather all the modalities M (C 1 , n) and M (C 2 , m) in a single set of 2N mutually exclusive modalities. Looking for instance at photon polarization (N=2), it impossible to know with certainty one the four possible results for the photon going through a polarizer oriented at 0 • , and through a polarizer oriented at 45 • . Therefore the only relevant question to be answered by the theory is: if the initial modality is

M (C 1 , n) in context C 1 ,
what is the conditional probability for obtaining modality M (C 2 , m) when the context is changed from C 1 to C 2 ? Again, such a probabilistic description is the unavoidable consequence of the impossibility to define a unique context making all modalities mutually exclusive, as it would be done in classical physics. It appears therefore as a joint consequence of the quantization and CSM principles, i.e. that modalities are quantized, and require a context to be defined.

Note that in the CSM picture as described so far, a "measurement" is nothing but a change of context. As we will see in section VII, this does not forbids to look for a more microscopic description of a measurement, e.g. by coupling the initial system to an "ancilla". But this will simply lead to define some new and larger system and context, in which the previous scheme will apply again.

VI. ABOUT THE EPR ARGUMENT

Second, CSM sheds new light on the EPR argument [START_REF] Einstein | Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?[END_REF]. To show this, let us consider two spin 1/2 particles in the singlet state, shared between Alice and Bob. The singlet state is a modality among four mutually exclusive modalities defined in a context relevant for the two spins, where measurements of the total spin (and any component of this spin) will certainly and repeatedly give a zero value. On the other hand, the singlet state is incompatible with any modality attributing definite values to the spin components of the separate particles in their own (spatially separated) contexts. Now, let us assume that Alice performs a measurement on her particle, far from Bob's particle. Alice's result is random as expected, but what happens on Bob's side? Since Bob's particle is far away, the answer is simply that nothing happens. How to explain the strong correlation between measurements on the two particles? By the fact that after her measurement, Alice can predict with certainty the state of Bob's particle; however, this certainty applies jointly to the new context (owned by Alice) and to the new system (owned by Bob). The socalled "quantum non-locality" arises from this separation, and the hidden variables from the impossible attempt to attribute properties to Bob's particle only, whereas properties must be attributed jointly to Alice's context and Bob's system. Getting them together is required for any further step, hence the irrelevance of any influence on Bob's system following Alice's measurement. Here the separation between context and system is particularly obvious and crucial, since they are in different places 9 .

According to the above reasoning, after Alice's measurement on one particle from a pair of particles in a singlet state, the "reality" is a modality for Bob's particle, within Alice's context. But Bob may also do a measurement, independently from Alice, and then the "reality" will be a modality for Alice's particle, within Bob's context. Does that mean that we have two "contradictories" realities ? Actually no, because these realities are contextual : for instance Alice's modality tells that if Bob does a measurement in the same context as Alice, he will find with certainty a result opposite to Alice's one (given the initial singlet state). This statement is obviously true, as well as the one obtained by exchanging Alice and Bob. But if Bob does a measurement in another context (different from Alice's), then one gets a probabilistic change of context for a N=2 system, as described before.

If Alice and Bob both do measurements with different orientations of their analyzers, the simplest reasoning is to consider the complete context for both particles, which is initially a joint context (with a modality which is the singlet state) and finally two separated contexts, again with 4 possible modalities due to the quantization postulate. Then this is now a probabilistic change of context for a N=4 system, again with the same result.

It is interesting to write a few equations about these 9 In order to define a modality, i.e. a set of values that can be predicted with certainty, there is no need for a physical contact between the context and the system. Moreover, the context is not the whole physical environment, but only the classical data required to define the modality, e.g., the polarizer's orientation, which can be classically broadcasted. But checking the predicted results does require the actual measurement to be done, i.e. the context and the system "meeting" at the same place.

initial, "intermediate" and final modalities, because this allows us to see more explicitly where CSM differs from Bell's hypothesis, even before the quantum formalism is introduced. So let us denote a i , b j the modalities with results i, j = ±1 for some orientation (context) a for Alice, and b for Bob. Given some "hidden variables" λ, and using the vertical bar "|" as the usual notation for conditional probabilities p(X|Y ), Bell's hypothesis are :

p(a i , b j |λ) = p(a i |λ) p(b j |λ)
The equivalent CSM equations, given the initial joint modality µ, are for Alice, who knows µ and a i So these equations clearly differ from Bell's hypothesis, though there is no action at a distance, and no faster than light signalling. However, there is some non-locality, in the sense that the result on one side depends on the result on the other side; but this is only through a (local) redefinition of the context, not through any influence at a distance onto the remote particle. Again, it is essential here to consider that the modality belongs jointly to the particle(s) and to the context, and not to the particle(s) only, otherwise one would be lead to Bell's hypothesis. Another important consequence is that if Alice and Bob both do measurements, their realities must ultimately agree together, since there will be a unique final modality (a i , b j ). Therefore their predictions must also agree together, and one must have

p(a i , b j |µ) = p(a i |µ) p(b j |µ, a i ) = p(a i |µ, b j ) p(b j |µ)
These equations are just the same as the ones we would obtain by the usual "instantaneous reduction of the wave packet", though in our reasoning there is no wave packet, and no reduction, but only a measurement performed by either Alice or Bob on the known initial modality µ. Even more, if we admit that (µ, a i ) is a new modality for Bob, and (µ, b j ) is a new modality for Alice, then p(b j |µ, a i ) or p(a i |µ, b j ) cannot be anything else than the one-particle conditional probabilities, i.e. Malus law for polarized photons (see section II).

So we get a simple explanation about the famous "peaceful coexistence" between QM and relativity, i.e. why quantum correlations are non-local, but also "no signalling" (they don't allow one to transmit any faster than light signal): this is because a modality requires both a system and a context, and for instance (µ, a i ) defines a modality for Bob's particle, by using only Alice's data. Such a situation, though strongly non-classical, does not conflict with physical realism or causality: in the CSM perspective, quantum non-locality is a direct consequence of the bipartite nature of (quantum) reality.

VII. EVOLUTION OF MODALITIES

As said above, a modality is fully predictable and reproducible as long as the context is not changed, and the system does not evolve. More generally, the system will evolve, according to some physical laws, and it may also interact with other systems. As usual in QM, interactions can be treated by combining systems, within a larger context suitable to define appropriate modalities. Modalities for the combined systems may be obtained by defining separate modalities for each sub-system, or by defining "joint" modalities, which are predictable and reproducible only in the combined context. Such modalities correspond to entanglement, as discussed in the above section about the EPR argument. For two systems with respectively N 1 and N 2 mutually exclusive modalities, it is natural to admit (or postulate) that the combined system will admit (N 1 × N 2 ) mutually exclusive modalities.

Considering again one system (possibly combined), the consequence of its evolution will be that predictability and reproducibility will be lost in the initial context. However, one expects that the result of the evolution will be a new modality, obtained from the initial one through a deterministic evolution. Therefore there will be a new context, in which the result of a measurement will again be fully predictable and reproducible. As it was assumed before, this new context will be related to the initial one by some element of the group of contexts transformation G, and this element, as well as the modality itself, should be determined by the equations of motion.

At this point, it is essential to note that in an actual measurement, the system typically interacts with other systems (ancillas), and gets entangled with them within a larger context. By naive application of the usual quantum ontology, one does not know how to "stop this process", and in principle the entanglement builds up until the observer himself is absorbed in the wave function. Nothing like that occurs in the CSM ontology, because it is postulated from the beginning that a measurement carried out in a given context on a system with N exclusive modalities can only give one of these modalities, with some probability. Even entanglement, as long as it remains a meaningful concept, makes sense only with respect to an external context, always required for defining (combined) modalities and using the quantum formalism. Let us point out also that measuring a modality actually means performing an "ideal QND measurement [START_REF] Grangier | Quantum non-demolition measurements in optics[END_REF]", or a "full projective Von Neumann 10 measurement". 10 This calculation was initially done in Chapter VI of [START_REF] John Von Neumann | English translation Mathematical Foundations of Quantum Mechanics[END_REF], which considers the ensemble made of I (quantum system) + II (ancilla) + III (observer's device), and shows that for a properly designed system-ancilla interaction (in modern terms, it should be a QND interaction [START_REF] Grangier | Quantum non-demolition measurements in optics[END_REF]), the same result is obtained by applying the measurement between I and II, or between II and III. Such an approach fully agrees with CSM, and is very far from a "many worlds" point of view [START_REF] Schlosshauer | Decoherence, the measurement problem, and interpretations of quantum mechanics[END_REF].

In this limit, which is idealized but fully relevant for our purpose, a measurement is equivalent to a "state preparation": this is consistent with the idea that a modality is defined by the certainty in the initial context, and not by the uncertainty in the (yet unknown) next contexts. Therefore there is no "measurement problem" in CSM, but rather a crucial issue, which is to set up a mathematical formalism able to calculate probabilities connecting different modalities in different contexts. The next question is then: which mathematical object should we associate with the physical concept of a modality ? This will be quickly discussed in the next section, and more thoroughly in a forthcoming article [START_REF] Auffèves | [END_REF].

VIII. BORN'S FORMULA

We understand now that QM must be a probabilistic theory, and the theorems by Kochen -Specker and Bell [START_REF] Bell | Against measurement[END_REF][START_REF] Aspect | Bell's inequality test: more ideal than ever[END_REF] strongly suggest to find an alternative to classical probability theory, unless one accepts that it becomes contextual and non-local. A new probability theory is therefore desirable, and our framework perfectly fits [START_REF] Auffèves | [END_REF] with the usual postulate: Let us associate to each modality a (rank-one) projector in an Hilbert space, so that any set of N mutually exclusive modalities M (C i , n) in a given context C i is associated to a set of N mutually orthogonal projectors Π(C i , n) summing up to identity. Then Gleason's theorem [START_REF] Gleason | Measures on the Closed Subspaces of a Hilbert Space[END_REF] states that the only possible way to write the conditional probability

P [M (C 2 , m)|M (C 1 , n)] is as Trace[Π(C 2 , m).Π(C 1 , n)]:
this is the usual Born formula. Within the framework of separable Hilbert spaces (or von Neumann algebra), it is well known that our initial restriction to a finite dimension N can actually be lifted, recovering the usual (Dirac -von Neumann) formulation of QM 11 [START_REF] Auffèves | [END_REF].

Since we have now reached the starting point of QM textbooks [START_REF] Cohen-Tannoudji | Mécanique Quantique[END_REF], it appears that the usual structure of QM is fully compatible with our approach; in particular rewriting physical quantities as operators and states as rays is straightforward. With respect to standard textbook presentations, the main differences are:

• quantum probabilities do not happen "by chance"

and have nothing to do with ignorance, but they are a consequence of the quantization postulate;

• quantum non-locality has nothing to do with an 11 Though contexts play a central role in our construction, let us emphasize that it is "noncontextual" in the sense associated with Gleason's theorem: this just means that the same modality can be found in different contexts. As an example, consider a system of two spin 1/2 particles S 1 and S 2 , and define S = S 1 + S 2 . Using standard notations for coupled and uncoupled basis, the |m 1 = 1/2, m 2 = 1/2 modality in the context {S z1 , S z2 } is the same as the |S = 1, m S = 1 modality in the context { S 2 , Sz}, though other modalities in the same two contexts are different.

"action at a distance", but appears because a modality belongs to both a system and a context;

• there is no "measurement postulate", since it is already included in the definition of modalities.

More precisely, in our approach there is no "wave function" developing upwards and utimately branching into a "many-world" universe [START_REF] Laloë | Do we really understand quantum mechanics[END_REF], but only (non-classical) probabilities connecting modalities appearing in different contexts. These probabilities are calculated using the standard quantum formalism, and thus they may involve interfering paths, as usual; however, there is no "wave", but only a "wavelike behaviour" due to the quantum way to calculate probabilities, through projections in an Hilbert space. In this "physically realist" point of view, the mathematical state vector (or wave function) should be carefully distinguished from the modality, i.e. from the phenomenon defining the "object" (see Annex).

IX. CONCLUSION

We shall conclude with a few remarks. First, contextual objectivity [START_REF] Grangier | Contextual objectivity: a realistic interpretation of quantum mechanics[END_REF][START_REF] Grangier | Contextual objectivity and the quantum formalism[END_REF] allows for an ontology to QM, this is the joint reality of the context, system, and modalities (CSM). This leads to reinterpret quantum nonlocality as the situation where the context and the system are separated in space: though the certainty (modality) is present, it cannot be "verified" or "actualized" as long as the context and the system are not put together again. Such a situation has no conflict with physical realism, but never happens in classical physics, where the physical properties are carried by the system alone.

Second, let us note that for many physicists, putting the context in the very heart of the theory implied an unacceptable "shifty split" [START_REF] Bell | Against measurement[END_REF][START_REF] Mermin | Quantum mechanics: Fixing the shifty split[END_REF] between the quantum world (attributed to the system) and the classical world (of the context). A lot of efforts have been made to get rid of it, especially to make the classical world emerge from the quantum world, by attempts to describe contexts within the quantum formalism. Such attempts may exploit the fact that there is a considerable flexibility for defining the boundaries of the system, especially when considering that (weak or strong) measurements can be done by entangling the initial system with a "meter" (or ancilla) system, e.g. by doing Quantum Non Demolition (QND) measurements [START_REF] Grangier | Quantum non-demolition measurements in optics[END_REF]. But in our approach, extending such measurements to include the context is self-contradictory: even by adding more and more "meters", the system can never grow up to the point of including the context. This is because without the context, modalities cannot be defined, leaving the system as a fuzzy object including everything, quite unsatisfactory from the perspective of physical realism.

The quantum-classical boundary has therefore a fundamental character, and QM was born from it, both from a physical and from a philosophical point of view, as it is discussed in more details in the Annex below. In a nutshell, the CSM approach presented here, without restricting the generality nor the applicability of QM, acknowledges the fact that, as a scientific discipline, QM "can explain anything, but not everything" [START_REF] Peres | Is quantum theory universally valid?[END_REF].

As a final remark, Bohr's arguments in [START_REF] Bohr | Can Quantum-Mechanical Description of Physical Reality be Considered Complete?[END_REF] were quite right, but perhaps failed to answer a major question asked in essence by EPR in [START_REF] Einstein | Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?[END_REF]: can a physical theory be "complete" if it does not provide an ontology that should be clearly compatible with physical realism ? Unveiling such an ontology is what we propose to do here.

Annex: Some philosophical remarks on the nature of physical reality.

It should be clear that in present approach, the weight of philosophy is larger than in many other interpretations of QM. Since we are still doing physics, and not philosophy, we present in this Annex some elementary philosophical considerations, spelling out the conceptual shifts introduced by the CSM ontology.

From a philosophical point of view, let us first emphasize again that we adopt the point of view of physical realism, telling that the purpose of physics is to study entities of the natural world, existing independently from any particular observer's perception, and obeying universal and intelligible rules. Therefore philosophical issues like the separation between subject vs object are definitely out of our scope: we are interested in defining "objects" consistent with QM, and we claim this can be done -though not in a "naive" (classical) sense.

Here we want to discuss an objection which can be made to CSM, and could also be made to Bohr's answer to EPR: in ordering to keep the above arguments consistent, the context seems to acquire a special status, "evading" the quantum description of reality which is being built. This objection can be answered relatively easily, but this requires a distinction between two notions of reality, that occur frequently in the history of philosophy.

The first kind of reality is the "ultimate material reality", constituted by all the objects in nature which are, from a scientific point of view, made of particles, waves and all their combinations, giving rise to macroscopic objects. There is no need to be very specific about what this reality is made of, but it must have a major property: it does exist -i.e. it is external to our thinking -even if we know very little about it. And in some sense, we cannot know much about it, because it is just too complicated. It also has to be a "global" reality, because no part of it should play a particular role. It could also be called "absolute reality", here we will call it "ultimate reality".

The second kind of reality is "empirical reality", this is the reality of phenomena that are amenable to scientific knowledge. Empirical reality has two main properties: it is real, it does exist independently of the "observer", because it is obviously included in the ultimate reality; and it can be known, which means that it can be perceived and apprehended by observers, as knowledge must (also obviously) pertain to perceiving and thinking agents. Scientific knowledge of empirical reality is thus a synthesis of facts -"what is really going on" (in some subset of the ultimate reality), -and concepts, elaborated through the observation and formalization of what is going on. It is precisely this synthesis that produces "understanding", what we sometimes call the "aha!" effect.

As said before, this distinction between ultimate and empirical realities is very old, probably as old as philosophy, but physicists often ignore it, and think that physics can address directly the ultimate reality, by defining, attributing and measuring properties that belong unconditionally to "real objects". Unfortunately, this way of thinking leads to a dead end as far as QM is concerned.

Actually, physics always deals with empirical reality, not with ultimate reality. Its duty is to describe phenomena with mathematical tools, which will allow one to predict the values of measurable physical quantities. These measurements and their mathematical formalization take place in a framework where phenomena can occur and eventually be described and measured. In practice, this framework is the classical macroscopic world, and though this appears only as a practical requirement, it can hardly be escaped, due to the very nature of empirical reality. Let us emphasize that this statement does not restrict physics only to "what can be perceived". All along its history, physics has elaborated concepts that take an ontological value, such as atoms, and it is perfectly entitled to do so, because empirical reality is grounded on ultimate reality. Atoms are a very good example of such a progress: in the 19th century, they were introduced as abstract hypothetical entities with a strong explanatory power, then they were identified, and now they can easily be "seen" and manipulated at the individual level. More generally, physical concepts such as photon, electron, charge, mass, energy, fields... are also entities required for describing the empirical reality in a synthetic way, referring again to the above mentioned synthesis of "what is going on" -the facts out there -and of its observation and formalization through physical, conceptual and mathematical tools that belong to science. Now we can consider again the difference between classical and quantum mechanics (see Fig. 3). In both, one deals with empirical reality grounded on ultimate reality, but in classical physics, one can easily get the delusion that knowing the state of the system is knowing directly the ultimate reality. In quantum mechanics, this is clearly wrong, because empirical reality must be mediated by a classical context, and the latter cannot be ignored. The context is always part of the ultimate reality, and its own very details in terms of particle, fields etc. certainly "exist". However, they are not relevant as far as the definition of a (CSM) modality is concerned: here the context is only considered as a necessary practical condition allowing the physicist to make measurements on the system, which is a physically-grounded but abstract concept, like the atoms are. The context's role is to reveal a phenomenon, the modality, which must be accessible to the observer, in ordering for knowledge to take place.

Given all that, a modality in CSM is essentially a "phenomenon" -a matter of fact -which involves a context (as a practical requirement) and a system (as a physicallygrounded concept), and which provides measurement results, that can be known and reproduced with certainty. One should notice that most experiments do not give access to the "full modality", because of experimental imperfections: some properties may not be measured properly; experimental devices may add some noise, etc. However, the essential point is that, according to QM, modalities (i.e., pure quantum states) do exist as real phenomena, and the whole theory is based on that. In addition, the modality as a phenomenon is objective, i.e. it can occur anywhere, and requires no role for belief or for any agent's crucial presence. Though observation is part of scientific knowledge as a human endeavor, it is not required for the phenomenal existence of a modality.

It should be clear also that the "cut" or "split" [START_REF] Mermin | Quantum mechanics: Fixing the shifty split[END_REF] is a requirement at the level of empirical reality, in ordering to specify the observed phenomenon, but at the level of ultimate reality, it does not imply that the macroscopic world is different in nature from the microscopic one. It rather means that in QM, macroscopic properties are required to describe phenomena, because the context cannot be ignored, due to the combination of the CSM and quantization postulates introduced above. Therefore for empirical consistency, the quantum system with its either mutually exclusive or incompatible modalities has to connect somewhere to the macroscopic world, where quantization does not show up at first sight.

As a conclusion, a lot of trouble in QM results from a confusion between ultimate reality and empirical reality, associated with the classical delusion of "speaking directly" about ultimate reality. But this is no more possible in quantum mechanics, due to the empirical frontier imposed by the quantization postulate. Again, quantum mechanics can describe anything, but one should be very careful with attempts at using it to describe everything.

FIG. 1 :

 1 FIG. 1: A H-polarized single photon is successively analyzed by three polarizers oriented at either 0 • , 45 • and 90 • (a) or 0 • , 90 • and 45 • (b). The overall probability of being transmitted is respectively either 1/4 (a) or 0 (b).

p

  (a i , b j |µ) = p(a i |µ) p(b j |µ, a i ) whereas they are for Bob, who knows µ and b j p(a i , b j |µ) = p(a i |µ, b j ) p(b j |µ).

FIG. 3 :

 3 FIG.3: Graphical representations of various ontologies discussed in the text. The CSM ontology is a much better basis for physical realism than the usual quantum ontology.

This is clearly an idealized view of classical mechanics, ignoring e.g. chaotic systems, but what matters is that classical mechanics do behave generically in such a way.

This is clearly an idealized view of how QM works, but besides agreeing with the usual formalism, there is abundant experimental evidence that individual quantum objects do behave this way.

[START_REF] Bell | Against measurement[END_REF] With respect to the usual QM formalism, a modality corresponds to a pure state. We adopt here the usual view that statistical mixtures correspond to an extra layer of classical probabilities, added over a truly quantum structure provided by pure states. We note also that the crucial idea of certainty and repeatability is associated with projective measurements. On the other hand, "blurred" measurements (such as POVM's) may be very useful tools, but they don't provide fully predictable and repeatable results, and therefore they are not relevant for our

purpose.[START_REF] Aspect | Bell's inequality test: more ideal than ever[END_REF] In practice our definitions can be restricted to some sub-ensemble of contexts, relevant for some degrees of freedom of the complete physical system. What actually matters is that the transformations within the relevant set of contexts have the mathematical structure of a continuous group, see section IV.

For continuous systems, see remark in section VII. The full consideration of infinite dimension deserves more discussion, and is postponed to another article[START_REF] Auffèves | [END_REF].

This principle is reminiscent of other approaches which bound the information extractable from a quantum system[START_REF] Rovelli | Relational Quantum Mechanics[END_REF][START_REF] Brukner | Information Invariance and Quantum Probabilities[END_REF]. However, in the realist perspective we chose, quantization has not a purely informational character, but characterizes reality itself.

Acknowledgements

The authors thank Nayla Farouki for essential contributions, especially in the Annex, and Francois Dubois, Franck Laloë, Maxime Richard, Augustin Baas, Cyril Branciard for many useful discussions.


A Auffèves

D Gerace

J.-M Gérard

M França Santos

L C Andreani

J.-P Poizat

Controlling the dynamics of a coupled atom-cavity system by pure dephasing: basics and potential applications in nanophotonics

Keywords: numbers: 42.50.Pq, 42.50.Ct, 42.50.Gy, 42.65.Hw

The influence of pure dephasing on the dynamics of the coupling between a two-level atom and a cavity mode is systematically addressed. We have derived an effective atom-cavity coupling rate that is shown to be a key parameter in the physics of the problem, allowing to generalize the known expression for the Purcell factor to the case of broad emitters, and to define strategies to optimize the performances of broad emitters-based single photon sources. Moreover, pure dephasing is shown to be able to restore lasing in presence of detuning, a further demonstration that decoherence can be seen as a fundamental resource in solid-state cavity quantum electrodynamics, offering appealing perspectives in the context of advanced nano-photonic devices. We propose experimental strategies to develop a new type of versatile device that can be operated either as a single photon source or as a laser, based on the control by decoherence of the coupling between a single quantum dot and a solid-state cavity.

I. INTRODUCTION

Cavity Quantum Electrodynamics (CQED) aims at describing light-matter interaction when light and matter reduce to canonical systems, i.e. when light can be modeled by a single mode of the electromagnetic field and matter by a single two-level system. To achieve this situation, one should couple a given transition of the matter field to a monomode cavity. The losses and dephasing times of each system should happen on a time scale much longer than the coupling timescale. Energy can thus be coherently exchanged between the atom and the cavity, allowing to implement fundamental tests of quantum mechanics, and opening the way to quantum information processing. On the way to the quantum regime, Purcell enhancement, which occurs when the emitter lifetime is modified as a consequence of its resonant coupling to the cavity mode, is a well known milestone [1]. Historically, first CQED evidences were demonstrated with atoms coupled to microwave [2] and optical [3] cavities, respectively. These systems are characterized by a very long radiation lifetime of the isolated emitter, which in the spectral domain corresponds to a very narrow dipole resonance coupled to a broad cavity. This picture has been the usual paradigm for CQED so far.

On the other hand, CQED experiments can now be performed with solid-state emitters (so called artificial atoms) and cavities. The strong coupling regime has been reached for the excitonic transition of quantum dots [4] and nanocrystals [5,6] coupled to optical semiconductor cavities, as well as for superconducting qubits coupled to microwave cavities [7]. In all of these systems, the cavity mode quality factor can be very large, while solid state emitters are intrinsically coupled to the matrix they are embedded in. In fact, decoherence and phase relaxation unavoidably broaden any transition between the discrete states of such artificial atoms. These new conditions open an unexplored regime for CQED so far, where the emitter's linewidth can be of the same order of magnitude, or even broader than the cavity mode one. Different mechanisms contribute to the decoherence-induced broadening of artificial atoms, among which phonon-assisted mechanisms [8], or spectral diffusion [9]. If spectral diffusion happens on a timescale much shorter than the typical spontaneous emission timescale, it can safely be modeled by a simple pure dephasing channel in the master equation describing the dynamics of the system. Because of its simplicity, the scheme of a two-level system undergoing pure dephasing can be regarded as an appealing tool to explore this new regime of CQED, as well as a useful effective model to describe solid-state emitters [10,11].

Such a model has mostly been used to study the spectral properties of the light emitted by the atom-cavity system so far, especially with the aim of describing microphotoluminescence experiments performed on quantum dots coupled to semiconductor cavities [12][13][14]. In particular, pure dephasing has been identified as a potential mechanism for the so-called cavity feeding process [10,11], namely the emission of photons at the cavity frequency that shows up even if the emitter is coupled to a detuned cavity mode [14][15][16][17][18][19]. This picture for artificial atoms in the solid state has shed new light on the effects of decoherence, which can be considered as a supplementary degree of freedom as compared to isolated atoms, offering appealing perspectives to achieve advanced nanophotonic devices controlled by pure dephasing [11].

Nevertheless, little attention has been paid to the dynamics of an emitter subject to pure dephasing and coupled to a cavity up to now. In this work, we consider the temporal evolution of the emitter's and cavity mode populations, and show that the system in the incoherent regime can be described by a classical (rate-equation) model with an effective atom-cavity coupling rate. Such a coupling rate is a key parameter: it allows to define a generalized Purcell factor, and beyond, to revisit the notions of good and bad cavity regimes, respectively. We show that pure dephasing can increase the effective atom-cavity coupling, thus enhancing the effective Purcell factor of the system. Switching to the non-linear regime, we also show that the lasing properties of a single emitter can be evidenced by looking at the statistical properties of the emitted light, from which we characterize the conditions for the lasing onset to be achieved. In particular, we define parameters for which pure dephasing can even induce lasing, thus showing that decoherence can turn out to be an extremely positive resource in the framework of prospective solid-state nanophotonic devices. Within these framework, quantum dots appear as promising candidates, as the experimental strategies to tune their environment, thus the pure dephasing rate that controls their homogeneous linewidth, have already started to be developed. As a consequence, we have mostly used parameters consistent with state of the art experiments using quantum dots coupled to optical semi-conducting cavities. On top of it, we propose to implement a versatile device, that can be operated either as a single photon source or as a laser, based on the control of decoherence induced by the environment of the quantum dot. We also propose an innovative strategy to control this environment.

The paper is organized as follows. We first introduce an effective atom-cavity coupling parameter, and use it to define the good and bad cavity regimes, when pure dephasing is properly taken into account. Focusing our attention onto the bad cavity regime, we derive a generalized Purcell factor, and analyze how to optimize the rate of emitted photons by a broad emitter-based single photon source. We then address the good cavity regime by revisiting the properties of the single emitter laser, paying attention to the influence of pure dephasing. We finally analyze the relevance of the developed model to the case of quantum dots coupled to semiconductor photonic cavities.

II. EFFECTIVE ATOM-CAVITY COUPLING

The system under study is represented in Fig. 1a. A two-level atom of frequency ω x is initially pumped in its excited state. It is coupled to a single mode cavity of frequency ω a , with a strength g. The detuning between the atom and the cavity is denoted δ = ω x -ω a , the losses from the isolated atom and from the cavity mode are respectively γ and κ. The atom undergoes pure dephasing with a rate γ * . One first recalls some results related to the relaxation of the atom-cavity system, the atom having initially been excited [11]. The full quantum evolution of this system is described by the master equation

ρ = i[ρ, Ĥ] + L cav damp + L at damp + L deph , (1)
where the total Hamiltonian of the system is ( = 1)

Ĥ = ω x σ+ σ-+ ω a â † â + ig(â † σ--σ+ â) . (2)
Here, â (â † ) is the annihilation (creation) operator for a photon in the cavity mode, while σ-(σ + ) is the lowering (rising) operator for the atom. The damping part for both the atom and the cavity mode can be described by operators in the Lindblad form within the master equation, written as

L (cav) damp = κ 2 (2âρâ † -â † âρ -ρâ † â) L (at) damp = γ 2 (2σ -ρσ + -σ+ σ-ρ -ρσ + σ-) . (3)
We notice that in most of the experimental situations currently accessible, the spontaneous emission rate γ is much lower than the other typical rates, particularly the cavity damping rate κ. In the following, we will keep the spontaneous emission rate in the formulas but shall neglect this parameter in comparison with the cavity decay rate as far as possible. The pure dephasing channel is described by the following Lindblad operator:

L deph = γ * 4 (σ z ρσ z -ρ) . (4)
From the master equation, the evolution of the populations and coherences is described by the following equations of motion:

d â † â dt = -κ â † â + g σ+ â + g â † σ- d σ+ σ- dt = -γ σ+ σ--g σ+ â -g â † σ- d σ+ â dt = iδ σ+ â - γ + γ * + κ 2 σ+ â +g( σ+ σ--â † â ) . (5)
The coherent or quantum regime is characterized by the reversible exchange of a quantum of energy between the atom and the cavity mode. At resonance, this so-called vacuum

Rabi oscillation shows up if the vacuum Rabi splitting 2g essentially overcomes the effective dephasing rate γ + γ * + κ. The opposite case corresponds to a regime where the energy is irreversibly spread between the atom, the cavity and the environment. In this incoherent, classical regime, the crossed terms â † σ and σ+ â , which are responsible for the Rabi oscillation, can be adiabatically eliminated. Out-of-resonance, the adiabatic elimination is valid for δ > g.

At this step it is worth pointing out the difference between the coherent regime and the so-called strong coupling regime. Strong coupling is reached when the spontaneous emission spectrum of the system consists in two peaks of distinct frequencies [20], which corresponds, taking into account pure dephasing, to 2g > |γ + γ * -κ|. If this condition is fulfilled, the width of each peak equals (κ + γ + γ * )/2. Coherent coupling is reached when the doublet is resolved, which is the spectral counterpart of the vacuum Rabi oscillation. In the standard CQED picture where the atomic width γ * + γ is negligible, the strong coupling regime perfectly matches the coherent regime. On the contrary, when γ * is of the same order of magnitude as κ, strong coupling is a necessary but not sufficient condition to observe a coherent exchange of energy between the atom and the cavity mode. Coherent regime thus appears as more demanding than strong coupling.

In the incoherent regime, the adiabatic elimination leads to the set of coupled dynamical equations:

d â † â dt = -(κ + R) â † â + R σ+ σ- d σ+ σ- dt = -(γ + R) σ+ σ-+ R â † â , (6)
where we have introduced the quantity

R = 4g 2 κ + γ + γ * 1 1 + 2δ κ+γ+γ * 2 . ( 7
)
As it was underlined in Ref. [11], the quantity R can be seen as an effective coupling rate between the atom and the cavity mode, i.e. the system is formally equivalent to two coupled boxes (as represented in Fig. 1b). The "atomic" box is initially charged with a quantum of energy that can escape in the environment at rate γ, or in the "cavity" box at rate R. In the same way, the cavity box can lose its excitation with a rate κ, or give it back to the atomic box with a probability per unit time R. The parameter R is also involved in the efficiency of the corresponding single photon source, which reads

β = Rκ/(R + κ) γ + Rκ/(R + κ) . (8)
Seen from the atom point of view, the cavity mode appears as a further loss channel, whose effective rate is Rκ/(R+κ) (see Fig. 1c). This result could have been straightforwardly derived from the classical picture. Such an expression for β is valid in any regime, even out of the incoherent regime, the only one in which the adiabatic elimination is supposed to be valid. Thus, the effective coupling rate R appears as a key parameter, allowing us to revisit the notions of good and bad cavity regimes, respectively. The bad cavity regime is achieved when κ > R, namely when the cavity damping time is shorter than the typical atom-cavity coupling time. In this case, the quantum exits the cavity mode as soon as it is released from the emitter: the cavity behaves as a supplementary loss channel. This is the usual regime for single photon sources and it is studied in the next two sections. In the good cavity regime, which is achieved when R > κ, the quantum of energy is emitted by the atom and can stay in the cavity mode before being lost in the environment. We stress here that the good cavity regime, which at resonance is achieved when 2g > κ(κ + γ + γ * ), is more demanding than the strong coupling regime if γ * becomes non negligible (whereas again, the two regimes coïncide in the standard CQED picture). On the contrary, the good cavity regime does not necessarily imply a coherent energy exchange between the atom and the cavity (which on resonance requires 2g > κ + γ + γ * ), as it would be the case in usual CQED experiments performed with atoms. On the contrary, pure dephasing opens a new regime where the quantum of energy can stay in the cavity mode without being reabsorbed by the atom. As it will be studied in section V, the good cavity regime is a necessary condition to implement single emitter lasers.

III. GENERALIZED PURCELL FACTOR

In this section, we focus on the bad cavity regime and show that the effective coupling R allows to define a generalized Purcell factor. By definition, in this regime the cavity behaves like a source of losses, and the atom-cavity coupling is incoherent. As a consequence, the parameter R has the dynamical meaning of an effective spontaneous emission rate. If R κ (which corresponds to the so called Purcell regime), one can easily extract the atomic relaxation rate from the set of Eqs. ( 6), which is γ + R. As expected, "switching on" the cavity mode corresponds to creating an additional relaxation channel for the atom, whose typical rate is R. One can thus define a generalized Purcell factor F * = R/γ, quantifying the enhancement of spontaneous emission rate that simultaneously takes into account the influence of pure dephasing. This factor can be expressed as

F * = 4g 2 γ(κ + γ + γ * ) 1 1 + 2δ κ+γ+γ * 2 . (9)
We can notice that one recovers the usual expression for the Purcell factor, F = 4g 2 /κγ [21][22][START_REF] Haroche | Advances in Atomic and Molecular Physics[END_REF], for γ * = 0. With respect to the standard expression, F * is obtained by replacing the cavity mode linewidth, κ, with the sum of κ and the total emitter's linewidth, γ + γ * .

This essentially reduces to replacing the bare cavity mode Q-factor, Q cav = ω cav /κ, which usually appears in the standard Purcell expression, with an effective quality factor, Q ef f , depending also on the emitter's quality factor,

Q em = ω 0 /(γ + γ * ), as 1 Q ef f = 1 Q cav + 1 Q em . ( 10
)
The existence of a generalized Purcell factor had already been heuristically derived in [21,[START_REF] Yokoyama | Confined Electrons and Photons: New Physics and Applications[END_REF] and finds here a demonstration in the case of a single emitter homogeneously broadened.

Note that this effective quality factor gives a symmetrical role to the emitter and to the cavity mode as long as one does not exit the bad cavity regime. In experiments exploiting CQED effects, increasing the Purcell factor is an important goal, giving rise to a quest to increase the quality factor of the cavity mode. According to the generalized expression we have derived, it makes sense to search for the highest possible Q cav (even when it overcomes Q em ) compatible with the bad cavity regime. In particular, by making Q cav bigger than Q em , one can double the effective Purcell factor of the system in the case where the atom is resonant with the cavity mode. On the contrary, increasing pure dephasing γ * leads to a decrease of the emitter's quality factor Q em , thus to a decrease of the effective quality factor Q ef f . Consequently, the spontaneous emission rate is reduced as it appears in Fig. 2a, where we have represented the relaxation of an initially excited atom in the resonant case for different values of the pure dephasing rate.

The influence of pure dephasing is dramatically different if the atom and the cavity are detuned, as already pointed out in Ref. [28]. In this case indeed, it makes sense that a Tuning of the pure dephasing rate can be achieved using temperature or pump power as it clearly appears in the studies described in [25]. The observed behavior was attributed to the fluctuation in the occupancy of electron traps in the QD neighbourhood [9]. In particular, these experiments show that the linewidth of the QD exciton can be tailored almost at will over a large range of experimental values (1-500 µeV). Nevertheless, non-resonant pumping was used in these studies, thus allowing the trapping of more than a single exciton in the quantum dot and effectively hindering the validity of the two-level atom model used in the present paper. To preserve the validity of our approach, quasi-resonant pumping (either in the p-shell, or using a resonant mechanism assisted by the creation of a phonon) or direct resonant pumping should be used [26,27], where the pump power is expected to have no influence on the density of carriers around the quantum dot and thus has no influence on the pure dephasing rate. Moreover, as the temperature also acts on the QD-cavity detuning, it is necessary to use another parameter to adjust it, for instance the electric-field [14] or a tunable microcavity [17]. Another promising approach could consist in optically pumping a quantum well positioned in the vicinity of the quantum dot: by controlling the carrier density in the quantum well, one could adjust the amount of decoherence induced by Coulomb interaction between the trapped exciton and the nearby quantum dot under study. The development of such a device would allow to explore the new regimes for CQED studied in the present paper and directly check the influence of the pure dephasing rate on the dynamics of the QD cavity coupling. Moreover it would realize a novel type of nano-photonic device based on the exploitation of decoherence, showing that pure dephasing is a resource specific to solid-state emitters, as it will be shown in the following examples. In particular, we propose strategies based on this control to optimize the figures of merit of single photon sources and nanolasers.

decrease of Q em , thus of Q ef f ,

IV. BROAD EMITTER-BASED SINGLE PHOTON SOURCES

A figure of merit usually considered for single photon sources is the efficiency of the device. Assuming that the detector is perfectly geometrically coupled to the cavity channel of losses, this corresponds to the probability for the photons to be spontaneously emitted in the cavity mode, which is usually denoted β. The behavior of this parameter, also considering the effects of pure dephasing and atom-cavity detuning, is studied in Ref. [11], However, the system was studied in the spontaneous emission picture, which does not model a usual photoluminescence experiment where the atom is typically pumped in continuous wave. In this context, an interesting figure of merit for a single photon source is the rate of photons that is emitted in the cavity loss channel, N . To describe the pumping mechanism on the atom, one has to add to the master equation a Lindblad operator that is formally expressed as

L at pump = P x 2 (2σ + ρσ --σ-σ+ ρ -ρσ -σ+ ) . (11)
If we restrict our analysis to the bad cavity regime, we can safely assume that the dynamics is restricted to the subspace spanned by the three states {|g, 0 , |g, 1 , |e, 0 }, irrespective of the pumping rate P x . This allows us to obtain steady state solutions for the atomic (n x = σ+ σss ) and the cavity mode (n a = â † â ss ) populations, respectively. We obtain

n x = P x P x + (γ + κ R κ+ R ) (12) n a = R κ + R n x , (13)
where we have introduced the effective rates

Γ 2 = P x + γ + γ * + κ 2 (14) R = 4g 2 Γ 1 1 + (2δ/Γ) 2 . ( 15
)
In the case where P x γ + γ * + κ, one obtains for R the expression of the effective coupling R defined in Eq. ( 7), making transparent the expression of the atomic population n x . It corresponds indeed to the incoherent pumping of a two-level system connected to two different loss channels, the first being due to the continuum of leaky photonic modes with a rate γ, and the second to the coupling to the cavity mode as it appears schematically in Fig. 1d. The effective loss rate of this second channel is, again, κ R/(κ+ R), as already showed in the spontaneous emission picture. The latter considerations reinforce the generality of the physical interpretation for the parameter R. In the following, we identify the two coupling strengths and use a unified notation for it, R.

Finally, the rate of photons emitted in the bad cavity regime can be explicitly given as

N = κn a = κR κ + R P x P x + γ + κR κ+R . ( 16
)
In the limiting case of very low pumping rate, i.e. for P x γ + κR/(κ + R), the rate of photons exiting the cavity can be expressed as N = βP x , where β is the efficiency of the single photon source given in eq.( 8), showing that the low incoherent pumping scheme can safely be modeled as a series of spontaneous emission events with a rate P x . This conclusion completely restores the continuity between the spontaneous emission picture and the incoherent pumping one, and it justifies to optimize the efficiency β when the device is operated below saturation. On increasing pump power, the rate of emitted photons saturates to the value N sat = κR/(κ + R), which in the bad cavity regime reduces to N sat = R. In order to maximize the rate of emitted photons, one has to maximize the parameter R, or equivalently F * , either by increasing Q ef f in the resonant case, or by lowering it, by playing on pure dephasing, in the detuned case. Note that the behavior of the parameter N with respect to the atom-cavity detuning, δ, is dramatically different depending on the atom being saturated or not. If the pump power is low, then N evolves as β(δ), whereas if the pump power is high, N evolves like the atom-cavity coupling R(δ). Such a change of behavior turns out to be a fruitful method, e.g., to measure the Purcell factor of a single quantum dot coupled to a semiconductor microcavity, as it was evidenced in Ref. [29].

V. SINGLE TWO-LEVEL EMITTER LASER

In the previous Section we have evidenced that in the bad cavity regime, the rate N of photons emitted by the cavity first evolves linearly with respect to pump power (same as the atomic population), before reaching an upper bound imposed by the spontaneous emission rate R (while the atomic population gets totally inverted). This limit is due to the saturation of the two-level emitter. A way to overcome it is to reach the stimulated emission regime, where the atom-cavity coupling scales like the number of photons in the cavity. In this case, the system is operated as a single emitter laser. This ideal device where the gain medium is quantified at the single emitter level, is of tremendous conceptual interest, and has motivated many fundamental studies since [30]. The primary interest of the lasing regime is that photons are mostly funneled into the cavity mode, allowing to achieve the highly efficient conversion of the incoherent power carried by the pump into single mode light. Second, as the emitter's cycling rate is enhanced, it can be pumped at a much higher rate before saturation is reached. In this Section we revisit the single atom laser topics, building on the notion of good cavity regime. We examine to which extent pure dephasing can be a resource in the frame of solid-state lasers, and relate this study to recent experimental demonstrations of single quantum dot lasers.

A. Why the good cavity regime is mandatory

Here we show that the good cavity regime is a necessary condition to reach stimulated emission. A heuristic demonstration has been developed in [21,31] and is based on the search of proper conditions to reach a steady state cavity population n a ∼ 1: in this view, the production rate of photons in the cavity, which is at most βP x , should overcome the cavity mode dissipation rate κ. As the cycling rate P x is limited by the typical spontaneous emission rate R, a necessary condition can be formulated as R > κ, which is the condition for good cavity regime.

This reasoning is confirmed by analyzing the rate equations for the incoherently pumped atom, which can be derived from Eqs. ( 5) and ( 11), respectively. One finds

d σz dt = -(R + γ)( 1 + σz ) + P x (1 -σz ) -2R â † â σz (17) d â † â dt = R 2 (1 + σz ) + R â † â σz -κ â † â , (18)
where σz = |e e| -|g g| stands for the population inversion, while the parameter R is still the effective atom-cavity coupling defined in Eq. (15). As it appears in Eqs. (17)(18), the evolution of the population inversion depends on three terms respectively due to spontaneous emission, pumping, and stimulated emission, whereas the evolution of the cavity mode involves spontaneous emission, stimulated emission and cavity losses. In steady state, the atomic population inversion I = σz ss and the cavity population n a = â † â ss are coupled in the following way:

I = P x -(R + γ) R(1 + 2n a ) + γ + P x ( 19
)
n a = 1 2 R(I + 1) κ -RI . ( 20
)
We have represented in Fig. 3 the behavior of the steady state cavity population, n a , with respect to the atomic inversion, I, by keeping the parameter R constant, both in the good and bad cavity regimes. When the population is not inverted (I → 0), which happens for low pumping rates, n a evolves linearly with respect to I: this is the spontaneous emission regime. On the contrary, if stimulated emission can be reached, cavity population evolves linearly with the pump rate, while the atomic population inversion remains clamped to a value where the optical gain compensates for the losses. This behavior is characteristics of lasing, whatever the type of device (conventional lasers, high β lasers, nanolasers). Namely, in this highly non-linear regime, the cavity population diverges with respect to the atomic one. As it appears in the figure, this can only happen in the good cavity regime, confirming the prediction above.

Two strategies can be adopted to enter the good cavity regime: decreasing the cavity losses, i.e. κ, or increasing the effective atom-cavity coupling, R. The first approach has been explored to study the potential of a quantum dot coupled to a high Q microsphere to show lasing [31]. The other approach is more promising, as it allows to simultaneously increase the fraction β of photons spontaneously emitted in the cavity mode. In the limit where β → 1, the device has a perfect quantum efficiency, even before stimulated emission is reached. In such kind of devices, no kink can be observed in the input-output curve (namely, the rate of emitted photons with respect to pump power), thus justifying the denomination of "thresholdless laser" [32,33]. In the following we restrict the study to the case of a high β single atom laser, and define signatures of the lasing regime.

B. Single two-level emitter lasing criteria

We have studied the evolution of three main properties of the single emitter device with respect to pump power: the cavity population, n a , the atomic population, n x , and the autocorrelation function of the field at zero time delay, defined as g 2 (0) = â † â † ââ /n 2 a . To this end, we have numerically solved the master equation, Eq. ( 1), for the model hamiltonian in Eq. ( 2) and the Lindblad operators in Eqs. (3), (4), and (11). The operators are explicitly built in matrix form on a Fock basis of occupation numbers for the atom and the cavity mode, respectively. For any given set of model parameters, the steady state density matrix can be obtained by numerically searching for the eigenvector |ρ ss corresponding to the eigenvalue λ ss = 0 of the linear operator equation L|ρ = λ|ρ . In the latter, |ρ is essentially the density operator mapped into vectorial form, and L is the linear matrix corresponding to the Liouvillian operator in the right-hand side of the master equation. If it exists, as it is always the case for the parameters considered, the steady state solution is unique [34].

After recasting the vector |ρ ss in matrix form, the relevant observable quantities can be calculated as O ss = T r{ Ôρ ss }. In the following simulations, we kept up to 30 photons in the basis, which is largely sufficient for convergence. We show the results in Figs. 4a,b,c respectively, for δ = 0. First we focus on the case where pure dephasing is negligible (blue solid line). With the set of parameters used, this corresponds to the good cavity regime.

Note again that the chosen parameters are within reach of current technology regarding quantum dots coupled to optical semi-conducting microcavities. Typical coupling strengths of g = 50 -100 µeV can be reached [15,35], whereas quality factors exceeding Q cav = 10 5 (i.e. cavity linewidths smaller than κ = 10 µeV) have separately been demonstrated [36].

However, the physics of single two-level atom lasers and single quantum dot lasers are drastically different from one another. We will come back to discuss this point in Sec D.

As it can be seen from the plotted quantities, the device perfectly converts the pump energy into cavity photons, whatever the pump power (log-log scale), which was expected since the device shows a high β [32,33]. Very intuitively, the critical value n a = 1 is reached as soon as the pump rate is of the same order of magnitude as the cavity damping rate. At this point, the atomic population remains clamped at a value nearly equal to n x ∼ 0.5, which is already a signature of lasing. This is confirmed studying the statistics of the emitted field, that clearly shows a transition from antibunched (g 2 (0) < 1) to Poissonian (g 2 (0) = 1), for nearly the same value of the pump power. Qualitatively similar results were recently shown in [13,37]. Indeed, in the spontaneous emission regime the device produces streams of single photons, and the emitted field is antibunched [38]. When stimulated emission is reached, more than one photon can be stored in the cavity mode before the intra-cavity field is dissipated, leading to the buildup of a Poissonian field that reflects the statistics of the single atom excitation events during a typical cavity lifetime. Thus, in the single atom device, in addition to be an efficient relaxation channel (just as in the conventional laser case), the cavity plays the role of a photon delayer, which keeps the photons emitted by a single atom for a sufficiently long time so that a Poissonian field can build up in the mode. This crossover in the statistics of the emitted field is a signature of the transition from the "single photon source" to the "single two-level emitter laser" operating regime.

We mention here that this behavior is quite different from what happens for "conventional" high-β lasers involving several emitters. In the latter case indeed, the statistics of the field maps the statistics of the pump, whatever its power is [32]. The single photon source regime has been observed, e.g., for a Caesium atom strongly coupled to an optical cavity [39].

A crossover to Poissonian statistics has been observed -to a certain extent -for a single quantum dot coupled to a micropillar cavity [40], and a photonic crystal cavity [35,41]. The single quantum dot laser case, and its differences and similarities with respect to the two level atom case, are discussed in Subsec. D.

When the pumping rate is too large, the atomic emission becomes incoherent, leading to a decrease of the cavity population, and to the corresponding increase of the atomic population, until total inversion in reached. The emitted field becomes thermal and the parameter g 2 (0) converges to its limiting value, 2. This phenomenon is known as the quenching of the laser, and was first predicted in [30]; it was attributed to the saturation of the two-level emitter. The notion of good cavity regime sheds new light on this feature. As a matter of fact, the atom-cavity coupling R decreases with respect to pump power P x . Once the lasing regime is reached, one can increase P x , and consequently the number of photons in the lasing mode, as long as one remains in the good cavity regime; when this condition is no longer satisfied, the laser gradually switches off. Thus, in this view quenching is due to the transition to the bad cavity regime, induced by power broadening. We have plotted the evolution of the parameter R with respect to pump power P x (inset): as it can be seen in the figure, quenching happens for a typical value of the pump P x ∼ 20g, for which the atom-cavity coupling constant becomes lower than κ, confirming our initial guess.

Before examining the influence of pure dephasing on the lasing signatures, we mention another usual criterium for lasing in conventional devices, namely the narrowing of the spectrum emitted by the cavity mode. In the single atom device, it has been predicted by [13,37,42], and maybe experimentally observed in [41], that at low pump power the spectrum consists in a series of peaks (the so-called "Jaynes-Cummings forks" [13]). The authors of [41] have talked about a "coexistence of the strong coupling regime and the lasing regime". We underline here that the emission of Jaynes-Cummings forks is a natural feature of the single emitter laser [42]. Reaching stimulated emission in the steady state regime simply means that one observes photons coming from radiative transitions between states of the excited manifolds in the Jaynes Cummings ladder, which naturally results in series of peaks centered around the cavity frequency. Increasing the pump power leads to the broadening of the peaks and to their convergence towards a single one at the cavity frequency, as mentioned in [13,37,42]. The transition between multi-peaks and single peak emission takes place after the lasing threshold, as evidenced in [41].

C. Influence of detuning and pure dephasing

In Subsec. A, we have evidenced that a necessary condition for lasing is to achieve the good cavity regime. In the previous Sections we have seen that pure dephasing strongly influences the effective atom-cavity coupling rate. As a consequence, one expects that it should affect the lasing conditions of the system as well. When considering the resonant case of Fig. 4, we see that increasing γ * lowers the effective coupling R(δ, γ * ), up to the point where the lasing criteria are completely lost (e.g., curves for γ * = 40g in Figs. 4a andb). In particular, the clamping of the autocorrelation function to the Poissonian value g 2 (0) = 1 disappears, and the emitted field continuously evolves from antibunched to thermal without showing any coherent character. Loss of the lasing criteria appears for a value of the pure dephasing rate γ * ∼ 20g. As evidenced in Fig. 4c, this corresponds to the transition from the good to the bad cavity regime, confirming our previous intuition. By the very same mechanism, lasing can also be lost by increasing the atom-cavity detuning, as it appears by plotting the same quantities in Fig. 5. Starting from δ = 0, the switching from good to bad cavity regime happens for δ ∼ 2g, which again yields the disappearance of any lasing signature.

On the other hand, we have seen in Sec. II that if the atom and the cavity are detuned, increasing pure dephasing can even increase the effective coupling between the two systems. This induces a transition from the bad cavity to the good cavity regime, and it allows to recover the lasing conditions. This result is shown in Fig. 6c for a typical pure dephasing rate γ * ∼ 2g; in particular, one recovers a clear clamping of the autocorrelation function to g 2 (0) ∼ 1, as in Fig. 4b for δ = 0 and γ * = 0. In other words, under such conditions it turns out that pure dephasing compensates for atom-cavity detuning. This effect, which in the previous Section was responsible for an improvement of the single photon source figure of merit (for system parameters in the bad cavity regime), leads here to a recovery of the lasing signatures. Again, pure dephasing appears as a valuable resource for solid-state nanophotonic devices. Finally, we discuss now the interest of pure dephasing in the frame of conventional lasers.

If one is just interested in efficient energy conversion, lasers involving a high number of emitters are naturally to be preferred over the single emitter device, as they are less subject to saturation and quenching. Nevertheless, the criterium of high β is challenging to realize for each emitter, because of inhomogeneous broadening in the solid-state environment, or the atomic motion in gas lasers. In such cases, pure dephasing could provide an effective tool to overcome this problem. As a matter of fact, increasing the homogenous linewidth of a bunch of detuned emitters would not only help reaching the good cavity regime, which is less critical to fulfill in the N emitter's case, but would also increase their individual β factors so that low-threshold lasing could be favourably achieved. We stress that pure dephasing here is nothing but a very effective model for the broadening of solid-state emitters because of their interaction with the solid-state matrix. Still, such a mechanism may explain some unconventional lasing characteristics of few quantum dots detuned from high-quality photonic crystal cavity modes [43], and again, it looks promising in the context of high β multi-emitters lasers.

D. Single quantum dot lasers

The potential of a single QD coupled to a semiconducting cavity to achieve a solid-state single emitter laser has been explored theoretically [13,31,37,44], and recent experimental demonstrations tend to show that laser gain at the single quantum dot level is within reach [35,40,41]. However, as it was underlined by the authors themselves [35,41], a laser based on single QD emission does not simply maps the physics of the single atom-laser into a solidstate system. One main source of differences is that solid-state cavities are always coupled to a bunch of background emitters that can efficiently feed the mode even if the emitters and the cavity are detuned, as it was experimentally evidenced in [15][16][17][18], and theoretically explored in [10,11,19]. Cavity feeding is generically attributed to the Purcell enhancement of relaxation processes that are resonant with the cavity mode, the very existence of these processes being due to phonon-assisted decay [19] or pure dephasing [10,11] that broaden the emitters' linewidths. Because of this background, cavity field at low pumping rate is not totally antibunched, as it also appears in [35,40,41], giving a quantitative measurement of the contribution of the single QD to the cavity emission [38].

Most importantly, even in the ideal situation where only a single QD is coupled to the cavity mode, the usually employed non-resonant pumping scheme to operate lasers allows to pump more than one exciton in the QD, making the physics of the single QD laser essentially different from a single atom laser. As multiexcitonic transitions happen at different frequencies because of exciton-exciton interaction, it was initially thought that such a device would suffer from blinking [21]. In fact, it can be intuitively argued that if the excitonic transition of the QD is resonant with the cavity mode, the QD decouples from the cavity as soon as it contains two excitons, thus leading to the device switch off.

Although, experimental evidence has shown that single QDs-based lasers do not suffer from blinking, and their lasing transitions display different statistical behaviors ranging from partial antibunching to Poissonian [35,41]. In this context, the evolution of the statistics of the cavity light field with respect to the pump power has not the same physical meaning as in the two-level atom case, which has been the subject of the present work, as it was also underlined in [41]. As a matter of fact, because of pure dephasing, phonon-assisted processes, or power broadening (as it was evidenced above), increasing pump power leads to a broadening of the higher order excitonic transitions, that can also contribute to cavity feeding. In the limiting case where all the transitions are efficiently coupled to the cavity, the statistics of the cavity field will simply map the Poissonian statistics related to the number of excitons in the QD. As a consequence, the role played by the cavity is drastically different from the single atom case: it does not act as a photon trap, delaying the emission of the field until a Poissonian statistics has built in the mode, but rather as a common relaxation channel for all the transitions of the QD. Thus, the crossover from the antibunched to the Poissonian field just reflects the excitation and the broadening of the multiexcitonic transitions. Note that the observation of such a transition does not require the good cavity regime, which explains why lasing was also observed in Refs. [35,40], where the QD and the cavity mode were only weakly coupled.

We propose an alternative strategy to realize a single atom-like laser with a QD coupled to a cavity mode, namely, to use a QD doped with a single electron embedded in a high-Q/low-V microcavity in the good cavity regime (see schematic picture in Fig. 7). Such a device, including tuning parameters to control the charge state of the QD and the QDcavity detuning, is within reach of current technology [45,46]. One can selectively inject an additional exciton in the QD through a quasi-resonant optical pumping, using a transition assisted by the creation of a phonon (either optical or acoustic). If the energy levels in the QD are energetically well separated, the injection of an additional exciton is forbidden because of Pauli blocking. This experimental approach will permit to realize the device proposed in the present manuscript, namely a compact and solid-state based system that can be operated either as a single photon source or as a nanolaser, depending on the pumping and pure dephasing rates. In particular, the dramatic influence of pure dephasing on the lasing threshold could be evidenced on such a device.

VI. CONCLUSION

We have studied the dynamics of a two-level atom undergoing pure dephasing, coupled to a single cavity mode, and derived an effective atom-cavity coupling rate, which has been shown to be a useful and conceptually simple parameter to be used in the description of the physics of the problem. In particular, it allowed us to generalize the notions of good and bad cavity regimes. In the bad cavity regime, we have defined a generalized Purcell factor, and studied the strategies to optimize broad emitter-based single photon sources. We have shown that if the atom and the cavity are detuned, increasing pure dephasing can improve the figures of merit of the device. In the same way, we have shown that in the good cavity regime, lasing can even be induced by increasing pure dephasing. These results enforce the idea that pure dephasing is a promising resource, specific to solid-state emitters, which might be used to develop advanced nano-photonic devices like single photon sources and nanolasers, and could allow to improve their performances. These ideas could be directly checked on an innovative and versatile device based on the control by pure dephasing of the coupling between a single QD and a semi-conducting cavity.

  induced by pure dephasing leads to the enhancement of the spontaneous emission rate as it clearly appears in Fig. 2b. To quantify the maximal enhancement we can get, it is worth noticing that the parameter R is maximized when κ + γ + γ * = 2δ, allowing to reach an optimal value R max = 2g 2 /δ and a maximal effective Purcell factor F * max = 2g 2 /δγ. Higher values of pure dephasing rate lead to a decrease of the effective atom-cavity coupling, and consequently of the spontaneous emission rate. By playing on pure dephasing, one can significantly improve the effective Purcell factor by a factor F * max /F ∼ δ/κ. Note that we have used state of the art parameters of quantum dots coupled to optical semi-conducting cavities, where coupling strengths like g = 50 µeV and cavity linewidths κ = 250 µeV (corresponding to quality factors of Q cav ∼ 5000) are commonly reached.

Figure 1 :

 1 Figure 1: (Color online). (a) System under study: a two-level atom undergoing pure dephasing coupled to a cavity. (b) Equivalent classical system: two connected boxes exchanging a quantum of energy. (c) Equivalent system in the spontaneous emission regime, and (d) in the continuous pumping regime, respectively.

Figure 2 :

 2 Figure 2: (Color online). Evolution of the atomic population as a function of time for resonant (a) and (b) detuned (with δ = 10g) case. Dotted line: the atom and the cavity are not coupled. Solid black curve: γ * = 0. Dashed red curve: γ * = 20g. Corresponding behavior of the effective atom cavity coupling, R, as a function of pure dephasing is shown for both (b) resonant and (d) detuned cases. The black arrow indicates the case where γ * = 0 [corresponding to the red curves in (a) and (c)], while the red dashed arrow is for the case γ * = 20g. [corresponding to the red curves in (a) and (c)]. The other parameters of the model for these calculations are: γ = 0.01g, κ = 5g.

Figure 3 :

 3 Figure 3: Evolution of the steady state cavity population n a with respect to population inversion I We took κ = 0.2g. (a) R = g (good cavity regime); (b) R = 0.1g (bad cavity regime).

Figure 4 :

 4 Figure 4: (Color online). Numerically calculated (a) cavity population, (b) statistics of the emitted field, and (c) atomic population as a function of pump power, for different pure dephasing rates. (d) Effective atom-cavity coupling as a function of the pure dephasing rate (solid line), as compared to the cavity damping rate (dashed line). Parameters are: κ = 0.2g and δ = 0. The inset shows the effective atom-cavity coupling as a function of P x , for δ = 0, γ * = 0.

Figure 5 :Figure 6 :

 56 Figure 5: (Color online). Numerically calculated (a) cavity population, (b) statistics of the emitted field as a function of pump power, and (c) atomic population for different atom-cavity detunings. (d) Effective atom-cavity coupling as a function of the detuning (solid line), as compared to the cavity damping rate (dashed line). Parameters are: κ = 0.2g and γ * = 0.

Figure 7 :

 7 Figure 7: Proposal to achieve a single atom-like laser using a QD doped with a single electron. A quasi-resonant pumping of the QD using a phonon-assisted optical transition is used to excite a trion. Due to this particular pumping protocol, the emitter has only two possible states: (a) no exciton in the QD; (b) a single exciton in the QD.

Acknowledgments

This work was supported by NanoSci-ERA consortium and by the EU under ERANET project LECSIN, by the Nanosciences Foundation of Grenoble, the CNPq, the Fapemig, and the ANR project CAFE. DG acknowledges stimulating discussions with A. Imamoǧlu and T. Volz.