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Abstract
The generation of squeezed light in semiconductor materials opens opportunities for building on-chip devices that are operated at the quantum level. Here we study theoretically a squeezed light source of polariton dark solitons confined in a geometric potential well of semiconductor microcavities in the strong coupling regime. We show that polariton dark solitons of odd and even parities can be created by tuning the potential depth. When driving the potential depth linearly, a bistability of solitons with the two different parities can be induced. Strong intensity squeezing is obtained near the turning point of the bistability due to the large nonlinear interaction, which can be controlled by the cavity detuning. The phase diagram of the bistability and squeezing of the dark solitons is obtained through large scale numerical calculations. Our study contributes to the current efforts in realizing topological excitations and squeezed light sources with solid-state devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Polaritons are hybrid light and matter quasi-particles that stem from the strong light-matter coupling of quantum well excitons and cavity photons. The peculiar light-matter composition of polaritons [1,2] allows the formation of Bose-Einstein condensation (BEC) [3–6] at elevated temperatures. The two-body interaction produced by the matter component opens a pathway to explore a broad variety of nonlinear phenomena in the solid-system setting [7–9], where the nonlinearity is tunable either by modulating the pump light power [10,11], the cavity detuning [12,13], or through the Feshbach resonance technology [14,15]. In particular, topological excitation of dark solitons attracts great interest rooted from the fundamental importance and all-optical signal processing applications [16]. Due to the giant nonlinearity available in polariton condensates dark solitons can form at the sub-millimeter length scales [17] and on picoseconds time scales [18]. Inspired by the high controllability and on-chip integration of the semiconductor microcavity, enormous efforts have been spent on the study of nonlinear phenomena in the semi-classical regime, including dark [19,20] and half-dark [21,22] solitons, dark soliton trains [23] and their bistability [24], Cherenkov radiation of solitons [25], dark-soliton molecules [26,27], and so on.
On the other hand, the Kerr nonlinearity has been used to generate intensity squeezing that can surpass the quantum noise limit [28,29]. As an effective Kerr medium, the refractive index of the polariton condensate can be controlled by the light intensity [30–32], holding the promise to realize a squeezed light source. As such, it has been shown recently the intensity squeezing of polariton condensate at the turning point of the bistability [33], the periodic squeezing in a Josephson junction [34], and polariton blockade [35–37] in a fiber cavity. However, the squeezing of polariton dark solitons is mainly limited by two obstacles, i.e., the limited stability [38] and the excess noise characterized by a continuum of microcavity polariton modes. The latter is prone to couple with lattice phonons and to reduce the quantum effect, which could be surpassed via the geometric confinement (in the strong localization limit) [39]. The challenge is to realize polariton dark soliton squeezing by overcoming the two limitations simultaneously.
In this work, we study the intensity squeezing of polariton dark solitons near the turning point of bistability, which are stabilized at quantized energy states of the geometric confining potential. Our setting is a semiconductor microcavity pumped nonresonantly with a homogeneous cw light beam and patterned with a central potential [see Fig. 1(a)], leading to localized, quantized energy states. The interference of the discrete states can form polariton dark solitons of even and odd parities, which are stabilized by the strong confinement of the potential well. We show that the two types of dark solitons exhibit bistable when adiabatically scanning the potential depth up and down at a constant rate. The phase diagram as a function of the nonlinear interaction is obtained. By modulating the polariton-polariton interaction coefficient, the squeezing of polariton dark solitons is achieved near the turning point of the bistability curve. Our study might pave a new route to develop low-power squeezed light sources in solid-state devices with the controllable microcavity polaritons.
Fig. 1. (a) Sketch of the quasi-1D semiconductor microcavity. The metallic mesa in the central area creates the centrosymmetric finite square potential well with tunable amplitude $V_0$, which can be tuned by applying an electric field. A homogeneous cw light pumps the semiconductor microcavity to generate polariton BEC. (b) Quantized energy states in the potential well with potential depth $V_0$.
2. Model
In our setting, a metallic mesa is embedded in the central area of a quasi-1D planar semiconductor microcavity [see Fig. 1(a)] to fabricate a centrosymmetric finite potential well,
where the depth $V_0$ is tunable by modulating the electric field applied on the mesa [40,41]. States in the potential well are quantized, while states out of the potential remain continuous. The deeper the depth $V_0$, the more the quantized states are, as depicted in Fig. 1(b). A homogeneous cw optical field excites non-resonantly polariton particles to form two domains of BEC in real space, separated by the potential well.Dynamics of the polariton field is modelled by the non-Hermitian Hamiltonian of polariton condensate in the vicinity of the ground state, as can be described as:
Governed by the Heisenberg equation $i\hbar \frac {\partial {\hat {\Psi }(x)}}{\partial {t}}=-[\hat {H},\hat {\Psi }(x)]$, the Langevin equation for the polariton field is derived as follows:
To allow for the large populations involved in polariton condensate, we expand the polariton field operator as $\hat {\Psi }(x)=(\psi +\delta \hat {\psi })$, where $\psi =\langle \hat {\Psi }(x) \rangle$ is the coherent mean field component, $\delta \hat {\psi }$ represents the fluctuation (noise) operator fulfilling $\langle \delta \hat {\psi } \rangle =0$. Substituting $\langle \hat {\Psi }(x) \rangle =\psi$ into the Langevin equation of Eq. (4), the mean field component of polariton condensate $\psi$ is governed by an open Gross-Pitaevskii equation, coupled to a rate equation for the density of excitation reservoir $n_{\rm R}$ [43]:
Next, by substituting the fluctuation (noise) operator $\delta \hat {\psi }=\hat {\Psi (x)}-\psi$ into the Eq. (4), Let’s characterize the quantum noise behaviour through quantum Langevin equation with the assumption of linear approximation as following:
The correlation of fluctuation operators can be derived from Eq. (7) to satisfy the following relation:
Then the squeezing of intensity noise is characterized through parameter $\xi$ [33,34]:
The squeezing parameter determines statistical properties of fluctuations, which is a Poissonian when $\xi =1$, super-Poissonian (bunched) when $\xi >1$, and squeezed (antibunched) when $\xi <1$, correspondingly.
3. Polariton dark solitons
The quasi-1D polariton condensate is generated when the pump field is above the threshold $P_{0}>P_{th}=\gamma _{\rm C} \gamma _{\rm R}/R^{1D}$ [46]. Without the potential well ($V_0=0$), polariton density distributes evenly in space with $n_{C}^{0}=(P_{0}-P_{th})/\gamma _{\rm C}$ in the steady state. With the centrosymmetric potential well (Fig. 1), two domains of polariton BEC appear in the extended space across the potential well [see Fig. 2]. Away from the potential well, the polariton density ($\approx n_{C}^{0}$) is nearly homogeneous. In the potential region, both the polariton density and phase $\phi$ vary dramatically. As the velocity $v\propto \nabla _{x} \phi$, one notes that the polariton particles in these two domains accelerate towards the center of the potential, leading to counter-propagating waves. Due jointly to the high-velocity propagation and phonon-assisted relaxation, polaritons with opposite wave vectors collide in the potential center, and induce a sink-type interference. Depending on whether the interference is constructive or destructive, $\psi$ exhibits odd or even parity [see Fig. 2(a2), Fig. 2(b2), and discussion below] [47].
Fig. 2. (a) Single dark soliton and (b) a pair of sink-type dark solitons. Both solitons are localized in the potential well region. Top row shows the dynamical evolution of polariton density, and the formation of the polariton dark solitons. Middle row shows polariton amplitude $|\psi (x)|$ and phase $\phi (x)$ at t=10ns, which clearly shows the formation of the dark soliton. Bottom row gives the corresponding spectra energy of these two states by making a Fourier transformation of $\psi (x,t)$ to time t. Other parameters are $V_0=-0.6$meV in (a) and $V_0=-1.0$meV in (b), while the pumping power $P_0=2.3P_{th}$ and $gc=0.4787 \mu$eV $\mu$m.
Dark solitons are stabilized by the nonlinear interaction [see Fig. 2(a1) and 2(b1)]. As low-energy excitations, the dark soliton is additionally affected by the pumping field power and potential depth. To illustrate this dependence, we first consider a scenario with fixed $P_{0}=2.3P_{th}$ and $gc=0.4787 \mu$eV $\mu$m. When −0.8meV$\le V_{0} \le$ −0.5meV, a dark soliton with odd parity forms, depicted in Figs. 2(a). Deepening the potential depth down to −1.6mev$\le V_{0} \le -1.0$meV, a dark soliton of even parity is found, originated from the constructive interference between the counter-propagating waves. Note that the two groups of dark solitons might be regarded as complex modes of the bound states $|1\rangle |$ and $|2\rangle$ as shown in the bottom line of Figs. 2, which could construct a two-level system induced by the quantum effect in confined space. When projected to the bound states, the odd (even) parity soliton maintains the maximum population in state $|1\rangle |$ ($|2\rangle$). This is in sharp contrast to the typically equilibrium case where polariton particles condense on the lowest energy state. As shown in Fig. 2, the destructive and constructive interference result in a neat separation of polariton condensate into two domains by a wavy junction in the potential well. The steep jump of the phase across the junction, together with the corresponding density depletion, demonstrates the spontaneous creation of these two types (odd or even) of dark solitons acting as an insulating barrier [48].
4. Emergent bistabilities
In the odd parity case we obtain a single polariton dark soliton, while the even parity could be regarded as two dark solitons, whose phase changes by $-\pi$ and $\pi$, respectively [from left to right in Fig. 2(b2)]. These two types of dark solitons can transit from one to the other when the potential depth $V_0$ is adiabatically scanned up and down, as shown in Fig. 3(a) and 3(b). Uniquely the bistability is controlled entirely by the potential depth $V_{0}$, shown in Fig. 3(c). When $V_0$ is scanned down to the value between the two phases [Fig. 3(a) and 3(c)], the polariton condensate is not in a steady state, whose density oscillates periodically [see Fig. 3(d)]. With a Fourier transformation for this periodical oscillation, two energy levels appear where spectrums are the same as those respectively in Fig. 2(a3) for the bound state |1> and in Fig. 2(b3) for the bound state |2>. This shows that the snake instability of periodically oscillation stems largely from the linear superposition of the odd parity soliton on bound state |1> and the even parity soliton on bound state |2>, which could be further demonstrated by a numerical simulation $\psi _{snake}=c1*\psi _{odd}+c2*\psi _{even}$ with a normalized condition $c1^2+c2^2=1$. Additionally, because the energy splitting between two bound states |1> and |2> increases with the deepening of potential well (decrease of $V_0$), the oscillating period together with the oscillating amplitude decrease as $V_0$. Note that the appearance of the snake instability is directional, i.e. only found in the odd-even transition (decreasing $V_0$). This is largely due to the fact that the sudden birth of the even parity soliton corresponds to a high energy state. The relaxation of this excitation is very slow as a significant fraction of the population in the state $|1\rangle$ and $|2\rangle$.
Fig. 3. (a),(b) Transition between the even and odd parity polariton dark solitons. The arrow shows the direction when changing $V_0$. (c) Bistability phase diagram of the even and odd parity polariton dark solitons by scanning $V_{0}$ with rate $5\times 10^{-5}$meV/ps. One dark soliton is found in the odd parity and two dark solitons are found in the even parity. The arrows indicate the direction of bistability transitions. (d) Snake instability between the two phases at the parameters denoted by O in (a) and (c).
To further illustrate the directional excitation in the bistable transition, we examine the energy of the polariton dark soliton. In the perturbative regime, we assume that the dark soliton wave function $\psi _{d}$ propagates on a background of spatially homogeneous condensate, given by $\psi =\sqrt {n_{C}^{0}} \times \psi _{d}$ and $n_{\rm R}=n_{\rm R}^{0} + m_R$. Here $n_{C}^{0}$ and $n_{\rm R}^{0}$ are spatially homogeneous densities respectively for polaritons and exciton reservoir. $m_R$ is the perturbation for $n_{\rm R}^{0}$. Using Eq. (5), we obtain the energy of the dark soliton [38]:
As shown in Fig. 4(a) and 4(b), the perturbative energy exhibits different profiles, depending on how the potential depth is varied. It has a similar shape to the bistability transition found in Fig. 3(c). The oscillation of the energy signifies the snake instability, as seen in the density pattern shown in Fig. 3(d). Note that it is found only when $V_0$ is changed across the bistable transition.
Fig. 4. (a)-(b) The perturbative energy and (c)-(d) intensity noise (squeezing parameter) when scanning $V_0$. In (a) the density dependent nonlinear effect dominates, such that squeezing is absent in (c). Increasing of $g_{\rm C}^{\rm 1D}$, the nonlinear interaction suppresses the quantum fluctuations, leading to squeezing, shown in (d). The directional dependent phase diagram for both the bistability and squeezing is shown in (e) by decreasing $V_0$, and (f) by increasing $V_0$ across the bistable region, respectively. In (a) and (c) only stability is found [corresponding to point A in (e) and (f) with BT phase], where the parameter $P_0=2.3P_{th}$, $g_{\rm C}^{\rm 1D}=0.4787 \mu$eV$\cdot \mu$m [the same as in Fig. 3(c)]. In (b) and (d) both bistability and squeezing are found [corresponding to point B in (e) and (f) with BTS phase]. The related parameters are $P_0=1.6P_{th}$ and $g_{\rm C}^{\rm 1D}=1.5958 \mu$eV$\cdot \mu$m. In all panels, the arrows indicate the scanning direction of $V_{0}$.
We want to emphasize that the bistability discussed in this work is realized in a two-level system confined in an effective potential of an off-resonantly pumped polariton condensate. Two kinds of dark solitons involved in the bistability are complex eigenmodes of two energy levels and are originated from the sink-type interference in the center of potential by polariton particles distributed outside of the potential. The bistability is driven by modulating potential depth $V_0$. Such bistability is essentially different from the one found in counter-propagating interacting polariton fluids resonantly excited by two spatially separated pump beams [49]. In the latter case, a train of dark solitons are self-organized due to the linear interference of two counterpropagating polariton fluids and the number of dark solitons are controlled by the phase between two resonant pump beams to realize the bistability.
5. Squeezing of dark solitons
The Kerr-like nonlinear interaction allows to achieve quadrature-squeezed light. Typically the strongest squeezing is found around the turning points of the bistability [45]. Both the nonlinear interaction coefficient $g_{\rm C}^{\rm 1D}$ and pump power plays crucial roles in realizing squeezing ($\xi <1$) in our work. To demonstrate this, we first examine the squeezing parameter corresponding to the situation discussed in Fig. 4(a). As shown in Fig. 4(c), the intensity noise becomes weaker at the turning points of bistability, where $\xi >1$ is relatively smaller (larger) along the odd-even (even-odd) transition. However irrelevant to the direction, squeezing is not found in either case. Both the topological excitation of polariton dark solitons and the intensity squeezing require the strong nonlinear interaction. We note that the polariton dark solitons are classical excitations that depend strongly on the pump field. The stronger the pump power, the higher the polariton density. Though this could generate stronger density-dependent nonlinear effects (e.g., stabilizing the dark soliton), quantum fluctuations also increases significantly at the same time [Fig. 4(c)], such that squeezing of the noise is not seen.
Thanks to the technique for adjusting the polariton-polariton interaction coefficient [12,13], we can maintain strong nonlinearities (via tuning $g_{\rm C}^{\rm 1D}$) even at low polariton densities. In this case, bistability is still clearly visible, as shown in Fig. 4(b). Squeezing, however, is drastically improved, as can be seen in Fig. 4(d). It is found that $\xi$ becomes smaller round the bistable transition region. More importantly, squeezing, i.e. $\xi <1$, is achieved around the turning point of the bistability, which is independent of the direction when changing $V_0$. The squeezing strength, however, depending on the direction. Our numerical simulations show that $\xi =0.667$ at the turning point of the bistability when decreasing $V_0$. In contrast, a slightly weaker squeezing with $\xi =0.768$ is obtained when increasing $V_0$. It is noted that the static potential disorder around 0.1mev in the samples caused by imperfect fabrication techniques doesn’t destroy the bistability and squeezing at the turning point but decreases the value of the strongest squeezing and the potential range to observe the phenomena.
Through large scale numerical simulations, we obtain phase diagrams to capture both bistability and squeezing of polariton dark solitons, shown in Fig. 4(e) and 4(f). To obtain the phase diagram, dynamics of the system is solved by scanning $V_0$ downwards [Fig. 4(e)] and upwards [Fig. 4(f)] with a given set of $g_{\rm C}^{\rm 1D}$ and $P_0$. By examining stability behaviors and squeezing parameters, three distinctive phases, one without bistability (blue region), one with only bistability (BT phase, grey region), and one with both bistability and squeezing (BTS phase, colors other than blue or grey), are identified. For a given $g_{\rm C}^{\rm 1D}$, the BT phase forms when the pump field is strong enough, such that the density (Kerr) nonlinearity can stabilize the two types of solitons. Squeezing, on the other hand, occurs when the nonlinear interaction coefficient $g_{\rm C}^{\rm 1D}$ is large enough while the polariton density is relatively low. Hence the BT phase occupies larger areas than the BTS phase. Such feature is independent of the changing direction of $V_0$, which is consistent with the example shown in Fig. 4(c) and 4(d). In the phase diagram, the strongest squeezing is $\xi =0.516$, which is obtained at $g_{\rm C}^{\rm 1D}=2.473 \mu$eV$\cdot \mu$m and $P_0=1.56P_{th}$ when $V_0$ is decreasing.
6. Conclusions
We have shown that the odd and even parity dark solitons can be generated in a controlled fashion in a nonresonantly pumped semiconductor microcavity containing a central potential well. The solitons are formed due to the strong geometric confinement and nonlinear Kerr like interaction. By varying the potential depth, a bistable phase of the two dark solitons is found. We have revealed that strong intensity squeezing can be achieved at the turning points of the bistability. Through large scale numerical simulations, we have demonstrated that the level of the intensity squeezing can be controlled by changing either $P_0$ or $g_{\rm C}^{\rm 1D}$, where the latter is achieved via the cavity detuning. Our study might open a way to create ultra-low threshold, on-chip integrated topological excitations and squeezed light sources for quantum information-processing applications.
Funding
National Natural Science Foundation of China (12074147); UKIERI-UGC Thematic Partnership (IND/CONT/G/16-17/73); Royal Society through the International Exchanges Cost Share (IEC/NSFC/181078).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
References
1. V. Timofeev and D. Sanvitto, Exciton Polaritons in Microcavities (Springer, 2014).
2. A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy, Microcavities (Oxford University Press, 2007).
3. J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymańska, R. André, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and L. S. Dang, “Bose-Einstein condensation of exciton polaritons,” Nature 443(7110), 409–414 (2006). [CrossRef]
4. A. Amo, J. Lefrère, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdré, E. Giacobino, and A. Bramati, “Superfluidity of polaritons in semiconductor microcavities,” Nat. Phys. 5(11), 805–810 (2009). [CrossRef]
5. R. Su, S. Ghosh, J. Wang, S. Liu, C. Diederichs, T. C. H. Liew, and Q. Xiong, “Observation of exciton polariton condensation in a perovskite lattice at room temperature,” Nat. Phys. 16(3), 301–306 (2020). [CrossRef]
6. M. Dusel, S. Betzold, O. A. Egorov, S. Klembt, J. Ohmer, U. Fischer, S. Hófling, and C. Schneider, “Room temperature organic exciton-polariton condensate in a lattice,” Nat. Commun. 11(1), 2863 (2020). [CrossRef]
7. A. V. Zasedatelev, A. V. Baranikov, D. Urbonas, F. Scafirimuto, U. Scherf, T. Stóferle, R. F. Mahrt, and P. G. Lagoudakis, “A room-temperature organic polariton transistor,” Nat. Photonics 13(6), 378–383 (2019). [CrossRef]
8. I. A. Shelykh, G. Pavlovic, D. D. Solnyshkov, and G. Malpuech, “Proposal for a mesoscopic optical Berry-phase interferometer,” Phys. Rev. Lett. 102(4), 046407 (2009). [CrossRef]
9. K. Rayanov, B. L. Altshuler, Y. G. Rubo, and S. Flach, “Frequency combs with weakly lasing exciton-polariton condensates,” Phys. Rev. Lett. 114(19), 193901 (2015). [CrossRef]
10. I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85(1), 299–366 (2013). [CrossRef]
11. E. Z. Casalengua, J. C. L. Carreño, F. P. Laussy, and E. d. Valle, “Conventional and unconventional photon statistics,” Laser Photonics Rev. 14(6), 1900279 (2020). [CrossRef]
12. Y. Sun, Y. Yoon, M. Steger, G. Liu, L. N. Pfeiffer, K. West, D. W. Snoke, and K. A. Nelson, “Direct measurement of polariton-polariton interaction strength,” Nat. Phys. 13(9), 870–875 (2017). [CrossRef]
13. E. Estrecho, T. Gao, N. Bobrovska, D. Comber-Todd, M. D. Fraser, M. Steger, K. West, L. N. Pfeiffer, J. Levinsen, M. M. Parish, T. C. H. Liew, M. Matuszewski, D. W. Snoke, A. G. Truscott, and E. A. Ostrovskaya, “Direct measurement of polariton-polariton interaction strength in the Thomas-Fermi regime of exciton-polariton condensation,” Phys. Rev. B 100(3), 035306 (2019). [CrossRef]
14. M. Navadeh-Toupchi, N. Takemura, M. D. Anderson, D. Y. Oberli, and M. T. Portella-Oberli, “Polaritonic cross Feshbach resonance,” Phys. Rev. Lett. 122(4), 047402 (2019). [CrossRef]
15. N. Takemura, S. Trebaol, M. Wouters, M. T. Portella-Oberli, and B. Deveaud, “Polaritonic Feshbach resonance,” Nat. Phys. 10(7), 500–504 (2014). [CrossRef]
16. S. Blair and K. Wagner, “Spatial soliton angular deflection logic gates,” Appl. Opt. 38(32), 6749–6772 (1999). [CrossRef]
17. A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet, I. Carusotto, F. Pisanello, G. Leménager, R. Houdré, E. Giacobino, C. Ciuti, and A. Bramati, “Polariton superfluids reveal quantum hydrodynamic solitons,” Science 332(6034), 1167–1170 (2011). [CrossRef]
18. X. Ma, O. A. Egorov, and S. Schumacher, “Creation and manipulation of stable dark solitons and vortices in microcavity polariton condensates,” Phys. Rev. Lett. 118(15), 157401 (2017). [CrossRef]
19. Y. Xue and M. Matuszewski, “Creation and abrupt decay of a quasistationary dark soliton in a polariton condensate,” Phys. Rev. Lett. 112(21), 216401 (2014). [CrossRef]
20. P. M. Walker, L. Tinkler, B. Royall, D. V. Skryabin, I. Farrer, D. A. Ritchie, M. S. Skolnick, and D. N. Krizhanovskii, “Dark solitons in high velocity waveguide polariton fluids,” Phys. Rev. Lett. 119(9), 097403 (2017). [CrossRef]
21. R. Hivet, H. Flayac, D. D. Solnyshkov, D. Tanese, T. Boulier, D. Andreoli, E. Giacobino, J. Bloch, A. Bramati, G. Malpuech, and A. Amo, “Half-solitons in a polariton quantum fluid behave like magnetic monopoles,” Nat. Phys. 8(10), 724–728 (2012). [CrossRef]
22. L. Dominici, G. Dagvadorj, J. M. Fellows, D. Ballarini, M. D. Giorgi, F. M. Marchetti, B. Piccirillo, L. Marrucci, A. Bramati, G. Gigli, M. H. Szymańska, and D. Sanvitto, “Vortex and half-vortex dynamics in a nonlinear spinor quantum fluid,” Sci. Adv. 1(11), 1500807 (2015). [CrossRef]
23. F. Pinsker and H. Flayac, “On-demand dark soliton train manipulation in a spinor polariton condensate,” Phys. Rev. Lett. 112(14), 140405 (2014). [CrossRef]
24. V. Goblot, H. S. Nguyen, I. Carusotto, E. Galopin, A. Lemaître, I. Sagnes, A. Amo, and J. Bloch, “Phase-controlled bistability of a dark soliton train in a polariton fluid,” Phys. Rev. Lett. 117(21), 217401 (2016). [CrossRef]
25. D. V. Skryabin, Y. V. Kartashov, O. A. Egorov, M. Sich, J. K. Chana, L. E. Tapia Rodriguez, P. M. Walker, E. Clarke, B. Royall, M. S. Skolnick, and D. N. Krizhanovskii, “Backward Cherenkov radiation emitted by polariton solitons in a microcavity wire,” Nat. Commun. 8(1), 1554 (2017). [CrossRef]
26. A. Maître, G. Lerario, A. Medeiros, F. Claude, Q. Glorieux, E. Giacobino, S. Pigeon, and A. Bramati, “Dark-soliton molecules in an exciton-polariton superfluid,” Phys. Rev. X 10(4), 041028 (2020). [CrossRef]
27. G. Lerario, S. V. Koniakhin, A. Maître, D. Solnyshkov, A. Zilio, Q. Glorieux, G. Malpuech, E. Giacobino, S. Pigeon, and A. Bramati, “Parallel dark-soliton pair in a bistable two-dimensional exciton-polariton superfluid,” Phys. Rev. Res. 2(4), 042041 (2020). [CrossRef]
28. H. Wang, J. Qin, S. Chen, M.-C. Chen, X. You, X. Ding, Y.-H. Huo, Y. Yu, C. Schneider, S. Höfling, M. Scully, C.-Y. Lu, and J.-W. Pan, “Observation of intensity squeezing in resonance fluorescence from a solid-state device,” Phys. Rev. Lett. 125(15), 153601 (2020). [CrossRef]
29. R. Dassonneville, T. Ramos, V. Milchakov, L. Planat, É. Dumur, F. Foroughi, J. Puertas, S. Leger, K. Bharadwaj, J. Delaforce, C. Naud, W. Hasch-Guichard, J. J. García-Ripoll, N. Roch, and O. Buisson, “Fast high-fidelity quantum nondemolition qubit readout via a nonperturbative cross-Kerr coupling,” Phys. Rev. X 10(1), 011045 (2020). [CrossRef]
30. M. Romanelli, J. Ph. Karr, C. Leyder, E. Giacobino, and A. Bramati, “Two-mode squeezing in polariton four-wave mixing,” Phys. Rev. B 82(15), 155313 (2010). [CrossRef]
31. S. V. Andreev, “Dipolar polaritons squeezed at unitarity,” Phys. Rev. B 101(12), 125129 (2020). [CrossRef]
32. F. Tassone and Y. Yamamoto, “Lasing and squeezing of composite Bosons in a semiconductor microcavity,” Phys. Rev. A 62(6), 063809 (2000). [CrossRef]
33. T. Boulier, M. Bamba, A. Amo, C. Adrados, A. Lemaitre, E. Galopin, J. Bloch, C. Ciuti, E. Giacobino, and A. Bramati, “Polariton-generated intensity squeezing in semiconductor micropillars,” Nat. Commun. 5(1), 3260 (2014). [CrossRef]
34. A. F. Adiyatullin, M. D. Anderson, H. Flayac, M. T. Portella-Oberli, F. Jabeen, C. Ouellet-Plamondon, G. C. Sallen, and B. Deveaud, “Periodic squeezing in a polariton Josephson junction,” Nat. Commun. 8(1), 1329 (2017). [CrossRef]
35. A. Verger, C. Ciuti, and I. Carusotto, “Polariton quantum blockade in a photonic dot,” Phys. Rev. B 73(19), 193306 (2006). [CrossRef]
36. G. Muñoz-Matutano, A. Wood, M. Johnsson, X. Vidal, B. Q. Baragiola, A. Reinhard, A. Lemaître, J. Bloch, A. Amo, G. Nogues, B. Besga, M. Richard, and T. Volz, “Emergence of quantum correlations from interacting fibre-cavity polaritons,” Nat. Mater. 18(3), 213–218 (2019). [CrossRef]
37. A. Delteil, T. Fink, A. Schade, S. Höfling, C. Schneider, and A. imamoĝlu, “Towards polariton blockade of confined exciton-polaritons,” Nat. Mater. 18(3), 219–222 (2019). [CrossRef]
38. L. A. Smirnov, D. A. Smirnova, E. A. Ostrovskaya, and Y. S. Kivshar, “Dynamics and stability of dark solitons in exciton-polariton condensates,” Phys. Rev. B 89(23), 235310 (2014). [CrossRef]
39. M. Bamba, S. Pigeon, and C. Ciuti, “Quantum squeezing generation versus photon localization in a disordered planar microcavity,” Phys. Rev. Lett. 104(21), 213604 (2010). [CrossRef]
40. C. Schneider, A. Rahimi-Iman, N. Kim, J. Fischer, I. G. Savenko, M. Amthor, M. Lermer, A. Wolf, L. Worschech, V. D. Kulakovskii, I. A. Shelykh, M. Kamp, S. Reitzenstein, A. Forchel, Y. Yamamoto, and S. Hófling, “An electrically pumped polariton laser,” Nature 497(7449), 348–352 (2013). [CrossRef]
41. D. K. Loginov, P. A. Belov, V. G. Davydov, I. Ya. Gerlovin, I. V. Ignatiev, A. V. Kavokin, and Y. Masumoto, “Exciton-polariton interference controlled by electric field,” Phys. Rev. Res. 2(3), 033510 (2020). [CrossRef]
42. N. A. Gippius, I. A. Shelykh, D. D. Solnyshkov, S. S. Gavrilov, Y. G. Rubo, A. V. Kavokin, S. G. Tikhodeev, and G. Malpuech, “Polarization multistability of cavity polaritons,” Phys. Rev. Lett. 98(23), 236401 (2007). [CrossRef]
43. T. Byrnes, T. Horikiri, N. Ishida, M. Fraser, and Y. Yamamoto, “Negative Bogoliubov dispersion in exciton-polariton condensates,” Phys. Rev. B 85(7), 075130 (2012). [CrossRef]
44. M. Wouters and V. Savona, “Stochastic classical field model for polariton condensates,” Phys. Rev. B 79(16), 165302 (2009). [CrossRef]
45. L. Hilico, C. Fabre, S. Reynaud, and E. Giacobino, “Linear input-output method for quantum fluctuation in optical bistability with two-level atoms,” Phys. Rev. A 46(7), 4397–4405 (1992). [CrossRef]
46. E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, G. Malpuech, and J. Bloch, “Spontaneous formation and optical manipulation of extended polariton condensates,” Nat. Phys. 6(11), 860–864 (2010). [CrossRef]
47. H. Sigurdsson, T. C. H. Liew, and I. A. Shelykh, “Parity solitons in nonresonantly driven-dissipative condensate channels,” Phys. Rev. B 96(20), 205406 (2017). [CrossRef]
48. D. Caputo, N. Bobrovska, D. Ballarini, M. Matuszewski, M. D. Giorgi, L. Dominici, K. West, L. N. Pfeiffer, G. Gigli, and D. Sanvitto, “Josephson vortices induced by phase twisting a polariton superfluid,” Nat. Photonics 13(7), 488–493 (2019). [CrossRef]
49. H. Ohadi, R. L. Gregory, T. Freegarde, Y. G. Rubo, A. V. Kavokin, N. G. Berloff, and P. G. Lagoudakis, “Nontrivial phase coupling in polariton multiplets,” Phys. Rev. X 6(3), 031032 (2016). [CrossRef]
