Dipolar polaritons squeezed at unitarity

TL;DR

This paper explores how tuning dipolar polaritons via shape resonance can produce strongly interacting, squeezed polariton populations, offering new insights for quantum photonics applications.

Contribution

It derives an analytical expression for the polariton coupling constant, revealing conditions for large interactions in experimental settings.

Findings

Large polariton coupling constants are achievable experimentally.

Squeezed polariton populations can persist on the repulsive side of the resonance.

Provides a new theoretical approach for enhancing interactions in quantum photonics.

Abstract

Interaction of dipolar polaritons can be efficiently tuned by means of a shape resonance in their excitonic component. Provided the resonance width is large, a squeezed population of strongly interacting polaritons may persist on the repulsive side of the resonance. We derive an analytical expression for the polariton coupling constant and show that it may have very large values in typical experimental conditions. Our arguments provide a new direction for the quest of interactions in quantum photonics.

Equations

$$\beta \gg \epsilon$$

$$p\ll \sqrt{m_{\mathrm{LT}}\beta }/\hslash ,$$

$$\begin{array}{rl}\hat{H}=& \sum _{\text{p},\sigma }E(\mathbf{p}){\hat{c}}_{\sigma ,\text{p}}^{†}{\hat{c}}_{\sigma ,\text{p}}+\sum _{\text{k}}[2E(\mathbf{k}/2)+\epsilon ]{\hat{C}}_{\text{k}}^{†}{\hat{C}}_{\text{k}}+\frac{g}{2S}\sum _{{\text{p}}_1,{\text{p}}_2,\text{q},\sigma }{\hat{c}}_{\sigma ,{\text{p}}_1+\text{q}}^{†}{\hat{c}}_{\sigma ,{\text{p}}_2-\text{q}}^{†}{\hat{c}}_{\sigma ,{\text{p}}_1}{\hat{c}}_{\sigma ,{\text{p}}_2}\\ & +\sqrt{\frac{{\hslash }^2\beta }{2\pi mS}}\sum _{\mathbf{k},\mathbf{p}}\left({\hat{c}}_{\uparrow ,\mathbf{p}+\frac{\mathbf{k}}{2}}^{†}{\hat{c}}_{\downarrow ,-\mathbf{p}+\frac{\mathbf{k}}{2}}^{†}{\hat{C}}_{\mathbf{k}}+{\hat{c}}_{\uparrow ,-\mathbf{p}+\frac{\mathbf{k}}{2}}{\hat{c}}_{\downarrow ,\mathbf{p}+\frac{\mathbf{k}}{2}}{\hat{C}}_{\mathbf{k}}^{†}\right).\end{array}$$

$$i\hslash \frac{d{\hat{c}}_{\sigma ,\mathbf{p}}}{dt}$$

$$i\hslash \frac{d{\hat{C}}_{\mathbf{k}}}{dt}$$

$${\mu }_{\sigma }=\frac{g}{S}\sum _{\mathbf{q}}|c_{\sigma ,\text{q}}{|}^2=g{n}_{\sigma },$$

$${\hat{c}}_{\sigma ,\mathbf{p}}={\hat{\mathsf{c}}}_{\sigma ,\mathbf{p}}{e}^{-i[E(\mathbf{p})+{\mu }_{\sigma }]t/\hslash },$$

$${\hat{C}}_{\mathbf{k}}=\frac{\sqrt{\frac{{\hslash }^2\beta }{2\pi mS}}}{{\mu }_{\uparrow }+{\mu }_{\downarrow }-{\epsilon }_{\mathbf{k}}^{\prime }}\sum _{\mathbf{p}}{\hat{c}}_{\uparrow ,-\mathbf{p}+\frac{\mathbf{k}}{2}}{\hat{c}}_{\downarrow ,\mathbf{p}+\frac{\mathbf{k}}{2}},$$

$${\epsilon }_{\mathbf{k}}=\frac{{\sum }_{\mathbf{p}}{E}_{\mathrm{rel}}(\mathbf{p},\mathbf{k})<{\hat{\mathsf{c}}}_{\uparrow ,-\mathbf{p}+\frac{\mathbf{k}}{2}}{\hat{\mathsf{c}}}_{\downarrow ,\mathbf{p}+\frac{\mathbf{k}}{2}}>}{{\sum }_{\mathbf{p}}<{\hat{\mathsf{c}}}_{\uparrow ,-\mathbf{p}+\frac{\mathbf{k}}{2}}{\hat{\mathsf{c}}}_{\downarrow ,\mathbf{p}+\frac{\mathbf{k}}{2}}>}$$

$$\begin{array}{rl}{\hat{H}}^{\prime }=& \sum _{\text{p},\sigma }E(\mathbf{p}){\hat{c}}_{\sigma ,\text{p}}^{†}{\hat{c}}_{\sigma ,\text{p}}+\\ & \frac{1}{2S}\sum _{{\text{p}}_1,{\text{p}}_2,\text{q},\sigma ,{\sigma }^{\prime }}{\hat{c}}_{\sigma ,{\text{p}}_1+\text{q}}^{†}{\hat{c}}_{{\sigma }^{\prime },{\text{p}}_2-\text{q}}^{†}{g}_{\sigma {\sigma }^{\prime }}{\hat{c}}_{\sigma ,{\text{p}}_1}{\hat{c}}_{{\sigma }^{\prime },{\text{p}}_2},\end{array}$$

$$g_{\uparrow \downarrow }=\frac{{\hslash }^2}{2\pi m}\frac{\beta }{({\mu }_{\uparrow }+{\mu }_{\downarrow }-{\epsilon }_{{\mathbf{p}}_1+{\mathbf{p}}_2}^{\prime })}.$$

$$\begin{array}{rl}& {\hat{c}}_{\sigma ,p_0}(0)|\psi ⟩=\sqrt{N}|\psi ⟩\\ & {\hat{c}}_{\sigma ,p\ne p_0}(0)|\psi ⟩=0.\end{array}$$

$${\hat{C}}_{2p_0}(0)|\psi ⟩=0.$$

$$\begin{array}{rl}\varphi (t)& ={\varphi }_{\uparrow }(t)+{\varphi }_{\downarrow }(t)\pm \pi /2\\ {\varphi }_{\sigma }(t)& ={\varphi }_{\sigma }(0)\end{array}$$

$$\begin{array}{rl}{\rho }_{\sigma }(t)& =\sqrt{N}{\cosh }^{-1}(t/{\tau }_0)\\ \rho (t)& =\sqrt{N}\tanh (t/{\tau }_0).\end{array}$$

$${\tau }_0=\sqrt{\frac{2\pi m}{\beta n}}$$

$${\mu }_{\sigma }^{\prime }=gn+\frac{{\hslash }^{2}n}{2\pi m}\frac{\beta }{(2ng-\epsilon )}.$$

$$\eta (d,n)=\frac{{\hslash }^2}{2\pi mg}\frac{\mathcal{B}d}{[2ng-\mathcal{E}(d-d_c)]}.$$

$$\begin{array}{rl}{\hat{x}}_{\sigma }& =\frac{1}{2}({\hat{\mathsf{c}}}_{\sigma ,p_0}{e}^{-i{\varphi }_{\sigma }}+{\hat{\mathsf{c}}}_{\sigma ,p_0}^{†}{e}^{i{\varphi }_{\sigma }})\\ {\hat{y}}_{\sigma }& =\frac{1}{2i}({\hat{\mathsf{c}}}_{\sigma ,p_0}{e}^{-i{\varphi }_{\sigma }}-{\hat{\mathsf{c}}}_{\sigma ,p_0}^{†}{e}^{i{\varphi }_{\sigma }})\end{array}$$

$$\begin{array}{rl}\hat{X}& =\frac{1}{2}({\hat{\mathsf{C}}}_{2p_0}{e}^{-i\varphi }+{\hat{\mathsf{C}}}_{2p_0}^{†}{e}^{i\varphi })\\ \hat{Y}& =\frac{1}{2i}({\hat{\mathsf{C}}}_{2p_0}{e}^{-i\varphi }-{\hat{\mathsf{C}}}_{2p_0}^{†}{e}^{i\varphi }).\end{array}$$

$$\begin{array}{rl}\frac{d}{dt}\delta {\hat{x}}_{\uparrow ,\downarrow }& =\pm \sqrt{\frac{\beta }{2\pi mS}}({\rho }_{\downarrow ,\uparrow }\delta \hat{X}+\rho \delta {\hat{x}}_{\downarrow ,\uparrow })\\ \frac{d}{dt}\delta {\hat{y}}_{\uparrow ,\downarrow }& =\pm \sqrt{\frac{\beta }{2\pi mS}}({\rho }_{\downarrow ,\uparrow }\delta \hat{Y}-\rho \delta {\hat{y}}_{\downarrow ,\uparrow })\\ \frac{d}{dt}\delta \hat{X}& =\mp \sqrt{\frac{\beta }{2\pi mS}}({\rho }_{\uparrow }\delta {\hat{x}}_{\downarrow }+{\rho }_{\downarrow }\delta {\hat{x}}_{\uparrow })\\ \frac{d}{dt}\delta \hat{Y}& =\mp \sqrt{\frac{\beta }{2\pi mS}}({\rho }_{\uparrow }\delta {\hat{y}}_{\downarrow }+{\rho }_{\downarrow }\delta {\hat{y}}_{\uparrow }),\end{array}$$

$$\begin{array}{rl}⟨\delta {\hat{x}}_{\sigma }^2(t)⟩& \sim e^{\pm t/{\tau }_0}\\ ⟨\delta {\hat{y}}_{\sigma }^2(t)⟩& \sim e^{\mp t/{\tau }_0},\end{array}$$

$$\hslash {\omega }_{\mathrm{ph}}(\mathbf{p})=\hslash c\sqrt{p^2+{\pi }^2/L_c^2},$$

$$\hslash {\omega }_{\mathrm{ph}}(\mathbf{p})=E_{\mathrm{ph}}+\frac{{\hslash }^2{p}^2}{2m_{\mathrm{ph}}}$$

$$\hslash {\omega }_{\mathrm{X}}(\mathbf{p})=E_{\mathrm{X}}+\frac{{\hslash }^2{p}^2}{2m_{\mathrm{X}}},$$

$$m_{\mathrm{X}}\gg m_{\mathrm{ph}}.$$

$$\begin{array}{rl}{\hat{H}}_{\mathrm{sp}}& =\sum _{\text{p},\sigma }\hslash {\omega }_{\mathrm{ph}}(\mathbf{p}){\hat{a}}_{\sigma ,\text{p}}^{†}{\hat{a}}_{\sigma ,\text{p}}+\sum _{\text{p},\sigma }\hslash {\omega }_{\mathrm{X}}(\mathbf{p}){\hat{b}}_{\sigma ,\text{p}}^{†}{\hat{b}}_{\sigma ,\text{p}}\\ & +\sum _{\text{p},\sigma }i\hslash g(\mathbf{p})({\hat{b}}_{\sigma ,\text{p}}^{†}{\hat{a}}_{\sigma ,\text{p}}-{\hat{b}}_{\sigma ,\text{p}}{\hat{a}}_{\sigma ,\text{p}}^{†})\end{array}$$

$$\hat{V}=\frac{1}{2S}\sum _{{\text{p}}_1,{\text{p}}_2,\text{q},\sigma ,{\sigma }^{\prime }}{\hat{b}}_{\sigma ,{\text{p}}_1+\text{q}}^{†}{\hat{b}}_{{\sigma }^{\prime },{\text{p}}_2-\text{q}}^{†}{V}_{\sigma {\sigma }^{\prime }}(\mathbf{q}){\hat{b}}_{\sigma ,{\text{p}}_1}{\hat{b}}_{{\sigma }^{\prime },{\text{p}}_2}$$

$$V_{\sigma {\sigma }^{\prime }}(\mathbf{q})=\int e^{-i\mathbf{q}\mathbf{r}}{V}_{\sigma {\sigma }^{\prime }}(\mathbf{r})d\mathbf{r}.$$

$$r_{\ast }=\frac{m{e}^2{d}^2}{\kappa {\hslash }^2}$$

$${\hat{p}}_{\sigma ,\mathbf{p}}=u_{\mathbf{p}}{\hat{b}}_{\sigma ,\mathbf{p}}+{\upsilon }_{\mathbf{p}}{\hat{a}}_{\sigma ,\mathbf{p}}$$

Full text

††preprint: APS/123-QED

The commonly adopted strategy to introduce interactions into the quantum optics of semiconductors is tailoring the non-linearity due to excitonic transitions Ciuti . In the regime of strong light-matter coupling the macroscopic population of the cavity mode is efficiently transferred into the exciton field, which can be regarded as a gas of bosonic quasiparticles Keldysh ; Hanamura ; Littlewood ; Amand . In particular, the blueshift of the polariton dispersion is governed by low-energy s-wave collisions between the pairs of excitons Keldysh . This naturally refers to ultra-cold atomic systems, where enhancement of interactions has been demonstrated by working with species having dipole moments Lahaye , Rydberg excitations Low and using the technique of Feshbach resonance Feshbach . The latter provides a possibility to tune the scattering length from positive to negative values through the unitary limit by adjusting the position of the scattering threshold with respect to a bound state.

On the technological side, exceptional excitonic properties are found in atomically thin heterostructures of transition metal dichalcogenides (TMD’s) Colloquium . The atom-like Lennard-Jones interaction between excitons has been demonstrated in these materials LennardJones . Such interaction naturally admits a bound state and, indeed, biexcitons have recently been observed in several types of monolayers XXinTMD . A fundamental difference from the atomic clouds is, however, a purely two-dimensional (2D) character of the exciton translational motion. The interactions in a 2D ultra-cold gas are generically weak due to the properties of 2D kinematics. Thus, in contrast to three dimensions, quantum scattering off a weakly-bound state has a vanishingly small amplitude Landau . At sufficiently low exciton densities these arguments apply also for semiconductor quantum wells (QW’s).

As was proposed by the author RP , a 2D analog of the Feshbach resonance may be realized with dipolar excitons formed of electrons and holes residing in spatially separated layers. The dipolar repulsion introduces a potential barrier between the outer continuum and the bound state (biexciton), which enables a quasi-discrete level with tunable energy and lifetime. Both parameters can be controlled by changing the distance $d$ between the layers. The attractive side of such resonance was theoretically explored in the context of roton-maxon excitations and supersolidity in dipolar Bose-Einstein condensates (BEC’s) Andreev2015 ; Andreev2017 . On the repulsive side, the equilibrium ground state is a condensate of biexcitons, distinguished from the exciton condensate by suppressed coherence of the photoluminescence (PL) and a gapped excitation spectrum RP . These predictions hold for a wide variety of bilayer structures, where the exciton lifetime is sufficiently long to establish a thermodynamic equilibrium. Thus, the numerical calculations of the exciton interaction potential in coupled QW’s Zimmerman suggest that the shape resonance may be responsible for the formation of a fragmented-condensate solid of excitons Andreev2013 ; Andreev2014 ; Andreev2015 ; Andreev2017 .

Several groups have recently reported an increase of the polariton interaction due to the dipolar moment in the excitonic component Savvidis ; Rapaport ; Imamoglu . The results presented in Ref. Rapaport are particularly compelling: a factor of 200 enhancement of the dipolar polariton interaction strength as compared to unpolarized polaritons has been detected. Dipolar repulsion alone cannot explain such tremendous blueshift of the polariton PL. The mystery is deepened by very low values of polariton densities at which the experiment was done.

Motivated by these experimental observations, the paper presents a phenomenological model of resonantly paired dipolar polaritons. In contrast to dipolar excitons, microcavity polaritons are far from the thermodynamic equilibrium, their statistics being closer to lasers rather than to atomic BEC’s PolaritonStatistics . This makes possible existence of a metastable polariton population on the repulsive side of the shape resonance. Coupling to a transient bipolariton mode in this case yields divergent behaviour of the 2D effective interaction, akin to the unitary limit in three-dimensional atomic clouds. We derive an analytical expression for the interaction enhancement factor as a function of the polariton dipole moment and density, and show that it can be very large in typical experimental conditions. Being exact in the dilute regime, this result ultimately holds for two polaritons in vacuum. Another interesting prediction of our theory is that the many-body polariton states become squeezed by the resonance. This could be verified in current experiments by examining the statistics of emitted photons.

Let us discuss the relevant timescales of the problem. First, we shall assume that the polaritons do not condense into the paired state, which is the equilibrium ground state when the discrete level $\varepsilon$ is below the scattering threshold. Second, the width of the resonance $\beta$ must be sufficiently large for polaritons could feel the interior of the barrier during their lifetime $\tau$. We shall assume a broad resonance, which seems to be the most likely for polaritons because of their very low effective mass SI . Hence, we let

[TABLE]

and $\hbar/\beta\ll\tau\ll\tau_{\bm{p}}$, where $\tau_{\bm{p}}$ is the thermalization time footnote .

The system is a mixture of two polariton flavours $\hat{c}_{\sigma}$ with $\sigma=(\uparrow,\downarrow)$ and their bipolaritonic pairs $\hat{C}$. In practice, ”$\uparrow$” and ”$\downarrow$” typically correspond to left- and right-circularly polarized photons Schneider . In planar dielectric microcavities the lower polariton dispersion is split into the linearly polarized TM and TE modes Kaliteevskii . This splitting acts as an effective magnetic field which flips the polariton pseudospin $\sigma$. Provided that

[TABLE]

the decay of a bound state due to the polariton spin-flip is subdominant with respect to the tunneling under the dipolar potential barrier. Here, the parameter $m_{\mathrm{LT}}$ accounts both for the bare photonic and low-$k$ excitonic TE-TM splitting Sham ; Glazov . In confined geometries Lundt ; Bloch one should let $p\sim\pi/L$, where $L$ is the characteristic length of the confinement. Under the condition (2), we may consider a single polariton branch $E(\bm{p})$ characterized by the effective mass $m$ in the vicinity of its minimum. Analysis of a possible departure from this approximation will be given in a separate paper.

The many-body Hamiltonian reads

[TABLE]

Here the first two terms describe the dispersions of single- and bipolaritons, respectively, with $\bm{p}=(p_{x},p_{y})$ and $E(\bm{p})=\hbar^{2}p^{2}/2m$ at the bottom of the band ($\bm{p}\rightarrow 0$). In the limit of zero exciton dipole moment the discrete level $\varepsilon$ can be identified with the binding energy of a loosely bound bipolariton molecule Rocca . The next two terms is the usual background interaction between the polaritons with alike spins (accounting both for the short-range part and the dipolar tail of the bare exciton potential) in the quantization area $S$. The last term models the interaction of polaritons with opposite spins by converting them into the bipolariton mode and vice versa. The square-root prefactor is constructed in such a way as to reproduce the low-energy 2D scattering amplitude for two particles in vacuum RP ; PolaritonicFR .

By using the standard commutation relations for bosons, one obtains the following set of Heisenberg equations of motion

[TABLE]

where we have replaced the Hartree groups of operators by $c$-numbers and defined

[TABLE]

with $n_{\sigma}$ being the polariton densities in each component. By introducing the slowly-varying amplitudes

[TABLE]

we notice existence of a stationary ($d\mathsf{\hat{c}}_{\sigma,\bm{p}}/dt=0$) solution of Eq. (4b) in the form

[TABLE]

where the motion of the bipolariton mode is reduced to that of a pair of polaritons with opposite spins. Here $\varepsilon^{\prime}_{\bm{k}}=\varepsilon-\varepsilon_{\bm{k}}$ with

[TABLE]

being the kinetic energy of the relative motion in the pair, $E_{\mathrm{rel}}(\bm{p},\bm{k})=E(\bm{p}+\bm{k}/2)+E(-\bm{p}+\bm{k}/2)-2E(\bm{k}/2)$.

The condition (1) provides a physical meaning to the solution (7). The objects $\hat{C}_{\bm{k}}$ should be regarded as auxiliary fields describing onset of pair correlations between the polaritons, rather than new (quasi)particles. Indeed, substituting (7) into the last term of the Hamiltonian (3), one obtains an effective model

[TABLE]

with $g_{\uparrow\uparrow}=g_{\downarrow\downarrow}=g$ and

[TABLE]

One can see, that the pair correlations between the polaritons manifest themselves as resonantly strong two-body interactions. For a broad resonance considered in this work, large magnitude of $g_{\uparrow\downarrow}$ is primarily due to the relation (1) (the $\mu_{\sigma}$’s are on the order of $\varepsilon$). However, by reducing the difference $\mu_{\uparrow}+\mu_{\downarrow}-\varepsilon^{\prime}_{\bm{p}_{1}+\bm{p}_{2}}$ (by, e.g., decreasing the polariton density) one may also observe the typical resonant growth of the interaction strength. Though here we have in mind the case $\mu_{\uparrow}+\mu_{\downarrow}>\varepsilon^{\prime}_{\bm{p}_{1}+\bm{p}_{2}}$, the formula (10) can be used on the attractive side $\mu_{\uparrow}+\mu_{\downarrow}<\varepsilon^{\prime}_{\bm{p}_{1}+\bm{p}_{2}}$ as well.

Assume now a resonant excitation of a coherent mixture of ”$\uparrow$” and ”$\downarrow$” polaritons at some point $\bm{p}_{0}$ on the dispersion curve $E(\bm{p})$. The output signal is registered at the distance $l\sim\upsilon_{gr}\tau$ from the excitation spot. Here $\upsilon_{gr}$ is the corresponding group velocity. For simplicity assume equal densities $n_{\uparrow}=n_{\downarrow}\equiv n=N/S$, which in practice may be achieved by using linearly polarized light. We therefore let the following initial ($t=0$) configuration:

[TABLE]

The bipolariton part of the many-body wavefunction $\ket{\psi}$ is initially in the vacuum state:

[TABLE]

By substituting the slowly varying $c$-numbers [see the definition (6)] $\mathsf{c}_{\sigma,p_{0}}=\rho_{\sigma}e^{i\phi_{\sigma}}$ and $\mathsf{C}_{2p_{0}}=\rho e^{i\phi}$ into Eqs. (4), and omitting the terms scaling as $\sqrt{\varepsilon/\beta}$, one can find

[TABLE]

and

[TABLE]

Eq. (14) shows that on the characteristic time scale

[TABLE]

the coherent mixture is entirely converted into the paired state (7). According to (9) and (10), the modified polariton blueshift is given by

[TABLE]

Note, that spreading of the polariton distribution over the $\bm{k}$-space region where the dispersion can be approximated by a linear function will not affect the result (16) since, according to (8), in this region one has $\varepsilon_{\bm{k}}\equiv 0$.

The relation (1) refers to the typical case $d\sim d_{c}$, where $d_{c}$ is the critical value at which the true bound state disappears dc . In general, the width of the resonance changes from [math] to $\infty$ (the latter describing the ultimate case where the level washes out) as the exciton dipole moment is tuned from $d\ll d_{c}$ to $d\gg d_{c}$. For $0\leqslant d\lesssim d_{c}$ one may take $\beta(d)=\mathcal{B}d$. The corresponding dependence for the position of the level has the form RP $\varepsilon(d)=\mathcal{E}(d-d_{c})$.

Following Ref. Rapaport , one may then consider the ”interaction enhancement factor” $\eta(d,n)=\mu_{\sigma}^{\prime}/\mu_{\sigma}-1$ as a function of the dipole moment $d$ and density $n$. Substituting the above relations for $\beta(d)$ and $\varepsilon(d)$ into Eq. (16), we obtain

[TABLE]

Though our analytical methods do not allow us to estimate the typical values of the parameters $\mathcal{B}$ and $\mathcal{E}$, by virtue of (1) one may expect $\mathcal{B}\gg\mathcal{E}$ and, consequently, $\eta\gg 1$. The relation (1), in turn, is guaranteed by very low values of the polariton mass $m$ as compared to bare excitons SI .

Interestingly, the strong correlations in the paired state (7) squeeze the polariton wavefunctions. To illustrate this point, consider again the situation discussed above, where one starts from a coherent state (11) for polaritons and a vacuum state (12) for their pairs. Introduce rotated quadratures

[TABLE]

and

[TABLE]

Write $\hat{x}_{\sigma}=x_{\sigma}+\delta\hat{x}_{\sigma}$ and the same for $\hat{y}_{\sigma}$, $\hat{X}$, $\hat{Y}$. The linearized equations of motion for the quadrature fluctuations read

[TABLE]

where the sign ”$+$” or ”$-$” corresponds to the two possible choices of the phase shift in Eq. (13), and $\rho$, $\rho_{\sigma}$ are given by (14). At $t=0$ one can use Eqs. (18) and (11) to find $\braket{\delta\hat{x}_{\sigma}^{2}(0)}=1/4$ and $\braket{\delta\hat{y}_{\sigma}^{2}(0)}=1/4$, the well-known property of a coherent state Scully . In contrast, at $\tau_{0}\ll t\lesssim\tau$, where $\tau_{0}$ is given by Eq. (15), one can substitute $\rho_{\sigma}=0$ and $\rho=\sqrt{N}$ into the first pair of Eqs. (20) to obtain

[TABLE]

showing that the polaritons exhibit 100 $\%$ squeezing in either of the two quadratures at the output.

The requirement $\tau_{0}\ll\tau$ sets the lower value of the polariton density at which the predicted effects may be observed. For sufficiently large $\beta$ one may operate in the ultra-dilute regime and even in the few-particle limit. In Eqs. (10),(16) and (17) the latter is formally achieved by letting $n=0$. Strong repulsive interactions between just two polaritons may be particularly promising for implementation of the dual-rail polariton logic Lisyansky .

The condition (1) has allowed us to neglect the dissipation and make our arguments particularly transparent. Pair-breaking events due to leakage of the single photons from the cavity result in loss of correlations and, at a first glance, would reduce the degree of squeezing. In practice, however, this reduction may be fully compensated by the noise of the external vacuum (see Ref. Yurke ), which restores the significance of the result (21).

Our last remark concerns the choice of the sign in Eq. (13). Under the condition (1), the Josephson coupling of the polariton states to the resonance stabilizes a definite phase relation during the signal propagation. The initial configuration is, however, chosen stochastically and may vary from one laser pulse to another. This circumstance should be taken into account when verifying the prediction (21) experimentally.

Besides the already mentioned state-of-the-art in QW microcavities Savvidis ; Rapaport ; Imamoglu , in Fig. (1) we sketch a possible implementation of unitary polaritons with TMD’s. A convenient choice may be a homobilayer of MoX2, showing large oscillator strengths and spin-valley selection rules analogous to the monolayer excitons Urbaszek ; Koch . In addition, natural separation between the layers here is close to the threshold $d_{c}$, the latter being on the order of the exciton Bohr radius RP . The electric-field tuning of the resonance position in this case of fixed $d$ may be possible due to suppression of the satellite carrier wavefunction in one of the layers (the residue of the intra-layer exciton) Urbaszek .

To conclude, we predict anomalously large enhancement of repulsive interactions in a system of dipolar polaritons. The proposed model is based on the physics of a bound state separated from the outer continuum by a potential barrier. Our results apply to a wide variety of 2D semiconductor heterostructures, such as atomically thin layers of TMD’s and quantum wells. An intriguing prediction of our theory is that the resonantly paired polaritons represent an efficient source of squeezed radiation. This might be readily verified by examining the statistics of emitted photons with the balanced homodyne detection homodyne . The idea of using the shape resonance to produce strong pair correlations and squeezing at ultra-low polariton densities opens wide perspectives for future research and applications. Thus, an interesting new direction would be application of the physics discussed in this work to the recently established field of topological polaritons Nalitov ; topological .

The author acknowledges the support by Russian Science Foundation (Grant No. 18-72-00013). I also thank Misha Glazov for helpful reading of the manuscript.

Supplemental material

.1 Polariton Fano-Anderson model

Consider 2D excitons coupled to photon modes in a planar microcavity of width $L_{c}$ and area $S$ with perfect mirrors. The photon dispersion reads

[TABLE]

that at low $\bm{p}$ has a massive form

[TABLE]

with $m_{\mathrm{ph}}=\pi^{2}\hbar/cL_{c}$. The exciton dispersion is given by

[TABLE]

and in typical all-dielectric microcavities one has

[TABLE]

Thus, for GaAs-based structures $m_{\mathrm{X}}\sim 10^{4}m_{\mathrm{ph}}$.

The prototype Hamiltonian may be recast as a sum of two contributions $\hat{H}=\hat{H}_{\mathrm{sp}}+\hat{V}$, where

[TABLE]

is the single-particle term describing formation of exciton-polaritons (see below) and

[TABLE]

being the two-body interaction between the excitons,

[TABLE]

We shall be particularly interested in $V_{\uparrow\downarrow}(\bm{r})$, schematically shown by the solid black line in Fig. 2. At large distances this potential is identical to $V_{\uparrow\uparrow}(\bm{r})$ and $V_{\downarrow\downarrow}(\bm{r})$, and is characterized by the dipolar tail $\hbar^{2}r_{\ast}/mr^{3}$, where

[TABLE]

is the dipolar length. At $r\sim r_{\ast}$, however, it features a potential well. At sufficiently small $d$ this potential well admits a true bound state - biexciton. Evolution of this bound state into a resonance as one increases $d$ was described in RP . Here we aim at extending that phenomenology to polaritons.

The standard procedure is to diagonalize the single-particle term (26) by doing the Hoppfield transformation

[TABLE]

with $\lvert u_{p}\rvert^{2}+\lvert\upsilon_{p}\rvert^{2}=1$ in order to preserve the boson commutation relation. One gets two eigenvalues $E_{\pm}(\bm{p})$ corresponding to the upper and lower polariton branches. The Hoppfield coefficients are related by

[TABLE]

In the long-wavelength limit $\bm{p}\rightarrow 0$ we may take $u_{\bm{p}}^{\pm}\approx u_{0}^{\pm}$ and $\upsilon_{\bm{p}}^{\pm}\approx\upsilon_{0}^{\pm}$. Furthermore, assume $E_{\mathrm{ph}}=E_{\mathrm{X}}$ in Eqs. (23) and (24). We get

[TABLE]

and

[TABLE]

One may also estimate $m_{+}=m_{-}\equiv m\approx 2m_{\mathrm{ph}}$. By virtue of (25)

[TABLE]

We define

[TABLE]

for the lower and upper polaritons, respectively. In terms of the new operators the single-particle Hamiltonian (26) reads

[TABLE]

By substituting

[TABLE]

into Eq. (38) and assuming occupation of the lower polariton branch only, one gets

[TABLE]

Thus, one can see that the low-momentum scattering of polaritons occurs in the same potential (rescaled by a numerical factor) as the scattering of excitons, but with a drastically reduced effective mass (34). By changing notations $E_{-}(\bm{p})\equiv E(\bm{p})$ and repeating the arguments of Ref. RP , one may derive the Fano-Anderson model (2) used in the main text. The predictions and physical meaning of this model depend on the ratio $\varepsilon/\beta$. Below we shall argue that the relation (34) implies $\varepsilon/\beta\ll 1$, so that quite generally one may expect the polaritons to interact via a broad resonance.

.2 Dissociation of a dipolar bipolariton in two dimensions

As we have established in the previous section, the polariton-polariton interaction potential apart of a numerical factor coincides with the bare exciton potential. For the $\uparrow\downarrow$ scattering channel of interest here this potential is schematically illustrated in Fig. 2. We first note, that the nominal huge potential barrier becomes effectively suppressed due to the particularities of the 2D kinematics. To see this, consider the radial 2D Schrodinger equation for the relative motion with zero angular momentum ($s$-wave):

[TABLE]

The dissociation rate of $\phi(r)$ may be estimated by considering the tunnelling in the effective one-dimensional model

[TABLE]

which is obtained from (39) by substituting $\phi(r)=\chi(r)/\sqrt{r}$. Here

[TABLE]

where we have taken $V_{\uparrow\downarrow}(r)\approx\hbar^{2}r_{\ast}/mr^{3}$ and defined $x=r/r_{\ast}$. This potential is shown by the dashed line in Fig. 2. One can see that the tunnelling at $E=0$ occurs in a narrow region from $r_{a}\approx r_{\ast}$ to $r_{b}=4r_{\ast}$. For a crude estimate we may use the quasiclassical approximation Landau to calculate the tunnelling probability as a product of the number of the particle beatings per unit of time $\sim p/mr_{\ast}\sim\hbar/mr_{\ast}^{2}$ and the barrier transmission

[TABLE]

By evaluating the integral one gets $D(0)\sim 0.25$. Thus, we may write

[TABLE]

Hence, we see that, by virtue of (34), the bipolariton resonance width should greatly exceed the analogous quantity for bare excitons $\beta_{\mathrm{X}\mathrm{X}}$,

[TABLE]

On the other hand, by using the formula

[TABLE]

where

[TABLE]

and taking into account that $\Delta d\sim\beta$ RP , we may write

[TABLE]

Thus, even if one has $\Delta d_{\mathrm{X}\mathrm{X}}\sim(d_{c}-d)$ and, consequently, $\varepsilon_{\mathrm{X}\mathrm{X}}\sim\beta_{\mathrm{X}\mathrm{X}}$ for the bare excitons, for the polaritons one would obtain

[TABLE]

We conclude that a dipolar ”bipolariton” may exist only as a broad resonance. This result generalizes the earlier estimate of the bipolariton binding energy at $d=0$ (non-dipolar excitons) Rocca .

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