1.1.

BCS with a Bose–Einstein Condensate as a Binding Agent

BCS is a pillar of superconductivity theory, which relies on three main tenets:

These are the three insights that were mainly contributed by Cooper, Bardeen, and Schrieffer, respectively, and that they could assemble into the BCS theoretical edifice and exploit to reproduce strikingly or predict successfully most of the superconductivity phenomenology.

The first point follows from Cooper’s observation²⁷ that an arbitrarily small attractive interaction between two electrons on top of the Fermi sea leads to a bound state (the Cooper pair), thanks to the truncation of the momentum space for states with wavevector $k>k_{\mathrm{F}}$ (Fermi wavevector). This is a general result, which follows from the Bethe-Goldstone equation for the two-electron problem.

The second point is the identification of an effective attraction between electrons that normally experience bare Coulomb repulsion. This attraction is attributed for conventional superconductors to an interaction through phonons, the Bardeen-Pines potential,²⁸ that consists, vividly, of one electron wobbling the lattice at a first time, which affects another electron at a second time (and at a larger timescale, since the lattice dynamics is much slower than that of electrons).²⁹ If the frequency of the lattice vibration is smaller than that of the propagating electron, the net effect results in an effective attraction [Leggett offers an insightful toy model of coupled oscillators to capture the essence of the interaction character (attractive or repulsive)].³⁰

The last point is the so-called BCS state, which is a coherent superposition of paired bound states and which brings the two-particle Cooper effect to a collective behavior of all electrons in the system.³¹

With these three ingredients put together, the BCS theory is complete.³² For our purposes, points 1 and 3 will be regarded as fundamental and well-established features of (BCS type of) superconductivity. Point 2, which might appear a mere desiderata for point 1 to apply, leaves us room for identifying and designing new types of attractive potentials, optimizing the range of applicability and strength so as to obtain robust superconductivity (e.g., holding at high temperatures or high magnetic fields) or its manifestation in a new class of systems (in microcavities). Bardeen himself, with coworkers,³³ investigated possibilities to engineer a more robust BCS state in a bilayer structure where excitons replace phonons as mediators of the interaction. The idea of substituting phonons by excitons was pioneered by Little³⁴ and developed by Ginzburg,³⁵ who coined the term and theorized the possibility of high-temperature superconductivity, much before it came to fruition with cuprates.³⁶ Cuprates, however, exploit another (still unknown) mechanism different to BCS.³⁷

In the following, we revisit the Ginzburg mechanism, based on BCS, with emphasis on maximizing the strength of interaction between electrons, so as to maintain their binding, and therefore superconductivity, to higher temperatures. The text is organized as follows: in Sec. 2, we give a short overview of the BCS mechanism and the exciton (Ginzburg) mechanism, outlining the points of special interest in our case. In particular, we introduce the gap equation. In Sec. 3, we introduce our hybrid Bose–Fermi system configuration, its Hamiltonian and microscopic interactions, and the effective electron-electron Hamiltonian that results from a mean-field approximation for the condensate and the usual Frölich transformation. We obtain the shape of the effective electron-electron interaction $U$, that we find to be quite different in character to the Cooper (square well) potential. We also consider possible variations of our scheme, namely, a microcavity with a condensate of exciton-polaritons and a condensate of indirect excitons in coupled quantum wells.³⁸ The system of coupled quantum wells explored by several groups^39,40 might be easier to realize and study (it does not need a cavity) and presents some interesting differences as compared to the polariton system. On the other hand, polaritons condense at much higher temperatures.¹⁸ In Sec. 3.2, we study the gap equation for a Bose condensate-mediated effective interaction. Because the potential is not positive-definite, the problem is not well-posed numerically. We propose a simplified potential and an approximate solution of the gap equation, which we motivate by studying its validity on well-established approximations. We obtain the critical temperature in this case. In Sec. 4, we give our conclusions and perspective on this new application of excitons and polaritons and discuss how to measure the effect experimentally.

3.1.

Electron-Electron Interaction

In the original BCS mechanism, electron-electron repulsion is either neglected altogether or overcome by the attractive mechanism and not manifested outside of the attractive window. We take it into account here since it is a detrimental factor for binding and most of our concern for experimental realization is to optimize this value. The full form of the potential is given by the Yukawa potential:

Fig. 2

Average over the Fermi surface, in 3D and 2D.

Our system is 2D, in which case, from Eqs. (5) and (11):

When $\kappa >2k_{\mathrm{F}}$, both the numerator and the denominator become pure imaginary so their quotient remains real. When $\kappa ={2k}_{\mathrm{F}}$, ${\overline{\mathrm{V}}}_{\mathrm{C}}=e^2\backslash {\mathrm{\Phi }}^2\backslash \mathrm{\epsilon }\mathrm{A}\backslash \kappa$. This is plotted in Fig. 3. Since we are trying to maximize attraction, that is, minimize repulsion, systems with small screening length and large wavevectors should be favored (but these parameters play critically on other aspects of the mechanism and the optimum is not compulsorily $k_{\mathrm{F}}/\kappa \gg 1$).

Fig. 3

Average electron-electron repulsion (in natural units) in the QW. This should be made as small as possible, which is obtained for higher Fermi wavevectors (for a given screening length).

3.2.

Electron-Exciton Interaction

The electron-exciton or exciton-polariton interaction is one of the most important ingredients of the mechanism, as it ultimately determines the shape of the effective potential. In the microcavity, an electron (from the 2DEG) interacts with a polariton (from the condensate) through its excitonic component, so this is really the electron-exciton interaction that is to be computed,⁴⁴ weighted by the Hopfield coefficient (the excitonic fraction) $X$. Let us consider, therefore, the scattering of an electron in one of the parallel QW, separated by a distance $L$ from the QW with excitons.⁴⁵ The matrix element of the direct interaction between excitons and electrons reads:

Fig. 4

Electron-exciton interaction in the geometry of Fig. 1, decomposed as the direct interaction (dashed magenta) and dipolar interaction (solid blue) when the exciton is induced with a dipole moment $d$. The latter is both much larger and maximum at zero exchanged momentum.

3.3.

Polariton-Polariton Interaction

We treat the polariton-polariton interaction within the $s$-wave scattering approximation with strength $U=6a_B^2{R}_y{X}^4/A$ (where $a_B$ is the exciton Bohr radius, $R_y$ the exciton binding energy and $A$ the normalization area. $X$ is the exciton Hopfield coeffcient, the square of which quantifies the exciton fraction in the exciton-polariton condensate).⁴⁶ For interaction between bare excitons, $X=1$. Exciton-exciton (polariton-polariton) interactions are repulsive, in general. They result in linearization of the elementary excitation spectra, the Bogoliubov dispersion, but its role is not crucial to the mechanism of superconductivity we discuss, since at the wavevectors of interest, the changes brought by this term are very small compared to the kinetic energy of noninteracting excitons (exciton-polaritons).

3.4.

Effective Interaction

We now proceed to bring our microscopic model toward a form suited to study Cooper-pairing and superconductivity, that is, we apply the canonical Fröhlich transformation that will result in an effective BCS Hamiltonian. Just as in the case of phonons, we start by getting rid of polaritons. We assume a condensate is formed with mean population $N_0$. We do not consider which mechanism, coherent or incoherent, is responsible for creating and maintaining this state. We do assume, however, it is coherent and with a definite phase, so that we can apply the mean-field approximation $a_{{\mathbf{k}}_1+\mathbf{q}}^{†}{a}_{{\mathbf{k}}_1}\approx \langle a_{{\mathbf{k}}_1+\mathbf{q}}^{†}\rangle a_{{\mathbf{k}}_1}+a_{{\mathbf{k}}_1+\mathbf{q}}^{†}\langle a_{{\mathbf{k}}_1}\rangle$ and $\langle a_{\mathbf{k}}\rangle \approx \sqrt{N_{0}A}{\delta }_{\mathbf{k},0}$ with $N_0$ the density of polaritons in the condensate. This allows us to obtain the following expression for the Hamiltonian, after diagonalizing the polariton part by means of a Bogoliubov transformation (that leaves the free propagation of electrons and their direct interaction, $H_{\mathrm{C}}$, invariant):

The last term of Eq. (16) coincides with the Fröhlich electron-phonon interaction Hamiltonian, which allows us to write an effective Hamiltonian for the bogolon-mediated electron-electron interaction. This results in an effective interaction between electrons, of the type $\sum _{{\mathbf{k}}_1,{\mathbf{k}}_2,\mathbf{q}}{V}_{\mathrm{eff}}(\mathbf{q},\omega ){\sigma }_{{\mathbf{k}}_1}^{†}{\sigma }_{{\mathbf{k}}_1+\mathbf{q}}{\sigma }_{{\mathbf{k}}_2+\mathbf{q}}^{†}{\sigma }_{{\mathbf{k}}_2}$. The effective interaction strength reads $V_{\mathrm{eff}}(\mathbf{q},\omega )=V_{\mathrm{C}}(\mathbf{q})+V_{\mathrm{A}}(\mathbf{q},\omega )$, with:

Equation (19) recovers the boson-mediated interaction potential obtained for a Bose–Fermi mixture of cold atomic gases,⁴⁷ in the limit of vanishing exchanged wavevectors. It describes the BEC induced attraction between electrons. Remarkably, it increases linearly with the condensate density $N_0$. This represents an important advantage of this mechanism of superconductivity with respect to the earlier proposals of exciton-mediated superconductivity,^33,34,48 as the strength of Cooper coupling can be directly controlled by optical pumping of the exciton-polariton condensate. The attractive potential is displayed for various exchanged energies in Fig. 5, as a function of $\theta$, defining the exchanged momentum expressed directly through the angle $\theta$ defined in the Fermi circle (cf. Fig. 2). As commented earlier, the negative part corresponds to attraction, and the potential alternates between repulsive and attractive character, obtained at different exchanged momenta. We do not want to keep track of such complicated wavevector dependence, and therefore will average the interaction over the Fermi sea. A notable feature is, for most values of $\omega$, the presence of a pole ${\theta }_0$, where $E_{\mathrm{bog}}[\sqrt{2k_{\mathrm{F}}^2(1+\cos {\theta }_0})]=\hslash \omega$. As is seen in the figure, ${\theta }_0$ separates the attractive part from the repulsive part. The average will bring the additional convenience of canceling such divergencies. We note as well that the spectrum of excitations of the exciton-BEC may be changed in the presence of the electron gas, so that their eventual dispersion may be different to $E_{\mathrm{bog}}$.⁴⁹ This has no effect on the Cooper-pairing of electrons which we discuss here.

Fig. 5

Effective electron-electron interaction as a function of θ, defining the exchanged momentum $q=\sqrt{2}{k}_{\mathrm{F}}\sqrt{1+\cos \theta }$ on the Fermi sea at energies $\hslash \omega =40$, 50, and 60 meV, respectively. The potential is symmetric around $\pi$. In the first case, the whole dependency (over $[0,2\pi ]$) is shown, with a first zoom in the inset showing two poles and another the attractive region that is of small amplitude but extends over a large range. With increasing energies, the poles recede towards smaller values of $\theta$ with a dominating effect of the repulsive (positive) energy. In the central panel, in dashed magenta, the regularized potential is superimposed.

We therefore wish to perform the average

Fig. 6

Exchanged-momentum averaged interaction $U_0(\omega )$ between electrons of the 2DEG, as $N_0$ is increased. It is dimensionless and is attractive (resp. repulsive) when negative (resp. positive). Densities $N_0$ are shown in units of $10^{12}/{\mathrm{cm}}^2$. In insets (a) and (b), a zoom of the regions delineated on the left panel, firstly in the small energies (long time) range, where the attraction is seen to be repulsive for smaller densities, because of the Coulomb repulsion (shown in red), and secondly for the densities $\approx 10^{12}/{\mathrm{cm}}^2$ in the area where the character of the interaction changes abruptly from attractive to repulsive (higher densities are shown in dotted lines).

If we take instead of $E_{\mathrm{pol}}$ a quadratic dispersion for the excitation

Fig. 7

Same as Fig. 6 (also in units of $10^{12}/{\mathrm{cm}}^2$) but for the case of an exciton condensate. A larger dipole moment $d$ has also been assumed. The potential is different in character, much closer to Cooper’s potential with larger attraction at longer times.

From these potentials, one can proceed to solve the gap equation.

4.1.

Cooper Potential

The BCS gap equation (2) is easier to tackle as a continuous equation:

Equation (26) is better known as its approximation when $U_0\ll 1$, in which case it takes the form of the famous BCS gap expression,

Solving exactly the gap equation calls for some numerical method. Equation (24) is a nonlinear integral equation, of the type studied by Hammerstein, i.e.,

Fig. 8

Numerical solution to the gap equation with the BCS square well potential (a). The gap $\mathrm{\Delta }(\xi )$ is not step-wise as approximated in the BCS model, but its value at zero exchanged energy, $\mathrm{\Delta }(0)$, is in close agreement with the analytical expressions, as seen in (b) and (c): $\mathrm{\Delta }(0)$ as obtained from numerically solving the gap equation (dashed blue) and from Eq. (26) (solid magenta), (b) in the weak-coupling limit when $U_0\ll 1$, with small deviations as coupling is increased, and (c) in the strong-coupling where the agreement becomes perfect again. Note that $\mathrm{\Delta }$ increases linearly with the coupling strength out of the weak-coupling limit.

4.2.

Bogoliubov Potential

The potential is not always attractive when, for instance, some overall repulsion, such as direct Coulomb interaction, is superimposed on the attractive Cooper potential, as shown on Fig. 9. The Coulomb interaction is time independent and should therefore extend to all $\omega$ but here also a cutoff ${\omega }_{\mathrm{C}}$ is introduced to avoid divergencies. This results in an attractive, Cooper-like potential, flanked by two repulsive windows. Such a potential is known as the Bogoliubov potential.^56,57

Fig. 9

Adding an overall repulsive Coulomb repulsion $V_{\mathrm{C}}$ (red, left) until a cutoff ${\omega }_{\mathrm{C}}$ to the BCS potential $V_0$ (green, left) results in the Bogoliubov step-wise potential (right). We take the convention $V_0$, $V_{\mathrm{C}}$, $U_0$ positive in the above representation and $U_0=V_0-V_{\mathrm{C}}$, so that, e.g., $V_{\mathrm{C}}<0$ means attractive contribution of the “Coulomb repulsion.”

This approximation has been used to show extremely counterintuitive behavior of the gap equation and justify a-posteriori another heavily criticized approximation of BCS, neglecting Coulomb repulsion: the BCS mechanism indeed assumes only attraction between electrons, which can be dimmed by Coulomb repulsion, but which never explicitly appears as such (like in the Bogoliubov scenario). The great result of Cooper was that binding occurs at arbitrarily small attraction. An important result of the Bogoliubov potential is to show that the detrimental effects of Coulomb repulsion are greatly reduced in the gap.⁵⁷

We now present a linearization of the gap equation that allows one to obtain an approximate solution for the critical temperature.⁵⁶ By assuming the gap equation to be a two-step valued function $\mathrm{\Delta }={({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2)}^T$, the gap equation becomes $(I-1)\mathrm{\Delta }=0$ with

In this form, one can see how Coulomb repulsion indeed introduces a small correction to the original BCS formula. The expression also seems to indicate that $U_0$ could be repulsive ($V_0-V_{\mathrm{C}}<0$) and still lead to a gap as long as the denominator in Eq. (31) remains positive.

In the following, we compare these predictions with numerical solutions of the gap equation, keeping in mind that the iterative procedure is not assured, mathematically, to converge. We have observed that indeed, it sometimes encounters problems and exhibits strong instabilities, with bifurcations of solutions, for example.

In Fig. 10, we show the evolution of the gap function as $V_{\mathrm{C}}$ is increased, from zero (BCS) to a point where the overall potential is essentially repulsive. With onset of the repulsion, the gap acquires two negative sides and becomes a highly distorted function for large $V_{\mathrm{C}}$. Note that the approximation of constant gap over the various regions is at least as good as for the case of BCS.

Fig. 10

Gap (thick magenta and, magnified, thick khaki) of the Bogoliubov potential (filled blue) solved numerically for the parameters: $\hslash {\omega }_{\mathrm{D}}=1$, $\hslash {\omega }_{\mathrm{C}}=2$, $V_0=1$ and $V_{\mathrm{C}}$ taking values from (a) to (d) of: 0 (BCS), 0.1, 0.3 and 0.495. Coulomb repulsion results in a dip in the gap function that, for increasing values, results in oscillations in the gap function. Paradoxically, even when it is large and dominating attraction, repulsion does not prevent a gap, as seen in (d).

In Fig. 11, we now show the case where $U_0=V_0-V_{\mathrm{C}}$ is held constant as the strength of the repulsion is varied independently. In foresight of what is to come later, we also allow the elbows to be negative, that is, to contribute an additional attraction to the conventional mechanism (for now we do not consider physical justification of this). Another unexpected result is obtained: the repulsive potential is in this case favoring a larger gap, as can be seen by comparing (a), where the gap function is highly oscillatory, to (d) where it recovers the BCS bell-shape. In (h), the distorted but overall attractive potential still results in a BCS type gap, but wider and larger. Note how the repulsion, by “squeezing” the gap, allows it to achieve much higher values than for the case of smaller or no repulsion.

Fig. 11

Gap (thick magenta and, magnified, thick khaki) of the Bogoliubov potential (filled blue) solved numerically for the following parameters: $\hslash {\omega }_{\mathrm{D}}=0.5$, $\hslash {\omega }_{\mathrm{C}}=2$, $V_0-V_{\mathrm{C}}=0.5$ (fixed) and $V_{\mathrm{C}}$ (the height of the elbow) taking values from (a) to (h) of: $-5$, $-0.5$, $-0.1$, 0 (BCS), 0.3, 0.5 (BCS), 1.5 and 5. The repulsive nature of the elbows change the character of the gap from bell-shaped to an oscillating function. The oscillating gap in the presence of strong repulsion allows, paradoxically, high values of the gap at zero-exchanged energy.

These unexpected results are confirmed phenomenologically by the Bogoliubov approximation Eq. (31), which reproduces qualitatively the trend of the gap $\mathrm{\Delta }(0)$ as shown in Fig. 12. Here we should emphasize that the two quantities are not meant to be compared quantitatively, since one, $\mathrm{\Delta }(0)$ (computed numerically) is the gap at zero temperature while the other, $k_{\mathrm{B}}{T}_{\mathrm{C}}$ is the temperature at which the gap vanishes. There is a monotonous relationship between the two, that is, increasing $\mathrm{\Delta }(0)$ implies increasing $T_{\mathrm{C}}$, so one can appreciate the consistency of the results by observing similar trends. The obstacle to conducting an extensive numerical comparison is that it is an intensive task numerically to compute $T_{\mathrm{C}}$, since this requires solution of the gap equation for various temperatures until the curve $\mathrm{\Delta }(T)$ is obtained and its intersect with zero is found. A critical slowing down phenomenon makes the iterative process slower as the critical temperature is approached. In addition, numerical instabilities are stronger at nonzero $T$. Therefore, although it is relatively straightforward to compute $\mathrm{\Delta }(0)$ numerically, it is not convenient to use this method to obtain $T_{\mathrm{C}}$. On the other hand, the Bogoliubov approximation gives a fair estimate of $T_{\mathrm{C}}$, but is not able to provide the gap at zero temperature, since at the core of its method, there is an assumption of vanishing $\mathrm{\Delta }$. Therefore, we have two complementary methods, each suited to provide a relevant aspect of the problem. We note that the gap at zero temperature is an important quantity which can be measured independently from $T_{\mathrm{C}}$ by Andreev reflection in conductivity experiments.

Fig. 12

$\mathrm{\Delta }(0)$ for the case of Figs. 10(a) and 11(c), for their respective parameters, and (b), with $\hslash {\omega }_{\mathrm{D}}=.5$, $V_0=1$, ${\omega }_{\mathrm{C}}$ changing as indicated and $V_{\mathrm{C}}$ changing such that the area of the repulsive elbow is conserved.

4.3.

Polariton Potential

In the case of the polariton problem, we have seen that, even when neglecting Coulomb repulsion (as in the original BCS formulation), the potential $U(\omega )$ departs strongly from the Cooper potential and features two large attractive regions far from small energies, immediately followed by two strong repulsive windows. We extend the Bogoliubov method to a three-step approximation of this potential, such as displayed in Fig. 13, with, in reference to previous potentials, notations ${\omega }_{\mathrm{D}}$, ${\omega }_{\mathrm{C}}$ and ${\omega }_{\mathrm{B}}$ for the boundaries of the central, shallow attractive region, narrow, deep attractive region and repulsive region, respectively. This is a notation only and is not mean to be understood as referring to Debye, Coulomb, or Bogoliubov in any strict sense. Following the same premises, we approximate the gap equation by a three-step valued function $\mathrm{\Delta }={({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)}^T$. This approximation turns out to be an exceedingly good one in certain cases, such as the one displayed in Fig. 13, where the gap itself is also a three-steps function in good approximation. Here there is even more room to choose a parametrization of the $I$ matrix. We now give general guidelines on how to build this matrix. The simplest method is to fix $\xi$ on the lhs of Eq. (25) at the center of each region and, in the corresponding row, take for each column the potential that is sampled more by the difference $|\xi -{\xi }^{\prime }|$. Refinements are possible, such as weighting elements of $I$ by coefficients which reflect how much time the variables $\xi$ and ${\xi }^{\prime }$ spend in the regions that determine the matrix equation. This problem has the following mathematical expression: how is the random variable $X-Y$ distributed when $X$ (resp. $Y$) is uniformly distributed in an interval $[g_i,g_{i+1}]$ (resp. $[h_i,h_{i+1}]$). The solution is easily obtained as proportional (normalize to unity) to:

Fig. 13

(solid blue) Three-step potential that models the form of effective electron-electron interaction when polariton-mediated with the three parameters ${\omega }_{\mathrm{B},\mathrm{C},\mathrm{D}}$ and, (thick magenta) the gap $\mathrm{\Delta }(\xi )$ as computed numerically for this potential. In good approximation, the gap is also itself a three-steps function.

As before, we estimate the critical temperature by the condition that gives vanishing values of the gap on the whole interval. The first integral ${\mathcal{I}}_1$ is evaluted in the same approximation as for the usual BCS (or Bogoliubov) potential:

${\mathcal{I}}_2$ and ${\mathcal{I}}_3$ are also essentially logarithmic in their energy range and we take:

We solve the gap equation by setting its determinant to zero and solving for $T_{\mathrm{C}}$, giving

We see that the impact of the polariton potential shape on the gap is rather intricate, here as well, with some unexpected behavior, produced both numerically and from this formula. In Fig. 14, we show the effect of widening the central, attractive region of the potential. Naively one would expect this to increase the gap (or critical temperature), since $\mathrm{\Delta }(0)$ increases linearly with $\hslash {\omega }_{\mathrm{D}}$. However, Eq. (37) predicts a decrease of the critical temperature, as shown in Fig. 15. The reason why is understood by computing the gap at zero temperature, as seen in Fig. 14, where the repulsive barrier is shown to behave as a trap for the gap function, which results, as the width is increased, in a loosening of the gap strength, akin to the quantum mechanical situation of a bound state in a square potential. The numerical and linearized models display a good qualitative agreement, as seen in Fig. 15. Let us repeat that we are not considering here quantities that can be directly compared, since $\mathrm{\Delta }(T_{\mathrm{C}})$ should be used for this purpose.

Fig. 14

Potential (filled blue) and its corresponding gap equation, solved numerically, as the size of the central attractive plateau is increased. Somewhat unexpectedly, the gap decreases as a result. The repulsive barriers have the effect of squeezing the gap up.

Fig. 15

$\mathrm{\Delta }(0)$ (joined points, blue) and $k_{\mathrm{B}}{T}_{\mathrm{C}}$ (magenta) from the three-step Bogoliubov approximation, in the case where the width of the central attraction is increased. Unexpectedly, this results in a decrease of the critical temperature and of the gap.

Another example is shown in Fig. 16, where both the attractive and repulsive parts of the potential are increased together. This results in a rapid increase of the gap. The result is also confirmed by numerical simulations, as shown in Fig. 17, and a good agreement is obtained. Unexpectedly, if the repulsive part grows twice as fast as the attractive one, not only does this also result in an increase of the gap, but even a faster one. This, too, is confirmed by the numerics. Note that in this case, we reach a region where our numerical procedure jumps to other solutions (usually of a highly oscillatory character, such as those reported in⁵⁷), but behaves as expected until then. In this regard, the value of having an approximate analytical solution is obvious.

Fig. 16

Numerical solution of the gap equation when both the attractive and repulsive parts of the potential are increased together (here in the same proportions), showing a rapid increase of the gap.

Fig. 17

$\mathrm{\Delta }(0)$ (joined points, blue) and $k_{\mathrm{B}}{T}_{\mathrm{C}}$ (magenta) from the three-step Bogoliubov approximation.

This analysis is finally applied to the case of the gap equation with the numerical results of the critical temperature obtained in both the polariton and exciton cases for the cases of Figs. 6 and 7. The parameters are gathered in Table  1 and the results plotted in Fig. 18. Both cases show a strong variation of the critical temperature with moderate variations of the condensate density (one order of magnitude). The polariton case is steeper and roughly linear while the exciton case increases less quickly but begins earlier. In both cases, temperatures are very high, and other effects will surely break the mechanism, for instance the loss of the condensate. This shows, however, the robustness of the mechanism in the conditions where it should apply.

Fig. 18

Critical temperature as a function of the condensate density $N_0$ for the case of a polariton condensate (solid blue) and of an exciton condensate (dotted magenta).