Abstract

Two coupled exciton-polariton condensates (EPCs) in a double-well photonic potential are suggested to form the optical Josephson oscillation (JO) whose dependences on the pump arrangement, the potential geometry, and the exciton-photon detuning are studied through the Gross-Pitaevskii equations. When the pump detuning is slightly positive with respect to the polariton energy and the phase difference between the two EPCs is near π/2 (both are controlled by the pump beams), the system demonstrates the optical JO. The optical JO tunneling strength (J) depends on the distance (d) and barrier ($\varLambda$) between the two wells, for which an empirical formula is fitted, i.e., $J\approx 0.42\exp (-d\varLambda /18.4)$ with the energy and length units in meV and μm. Since the double-well potential adopted is general, this fitting relation can show a guidance in practice for designing the optical devices based on the optical JO.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Josephson oscillation (JO) was first proposed in a superconductor/insulator/superconductor junction with a key character of a tunneling current [1]. People have experimentally demonstrated that the quantum oscillation could take place among the exciton-polariton condensates (EPCs) [2], accompanied by many theoretical works that show influences of magnetic fields [35], spin degree [6], phonons [7,8], and instability [9,10]. Much attention was also attracted by related phenomena, such as nontrivial phase coupling [11], spin order [12], self-trapping [13], and quantum entanglement [14]. Exciton polaritons (EPs) are hybrid Bosonic quasiparticles resulting from the strong coupling between quantum-well excitons and cavity photons in semiconductor microcavities [15,16]. The photon part makes the EPs hold ultra-light effective mass to undergo Bose-Einstein condensation [1719] with high condensate temperature, for example, the exciton-polariton condensate temperature is ${\sim }10$ K in GaAs and CdTe cavities and is ${\sim }300$ K in GaN cavities [18,20]. Since the EPs can decay into external photons, the EPCs are a non-equilibrium steady state that keeps a balance between radiative decay and external pumping, being different from cold atom condensates. The thermalization time for cold atoms is shorter with respect to their decay time, allowing the equilibrium approximation to be valid [21]. The polaritons’ lifetime is typically comparable to or shorter than thermalization time, giving them an inherently non-equilibrium nature, while some polariton systems could also show an equilibrium polariton condensate [2224]. This non-equilibrium character is responsible for that the EPCs can simultaneously occur in several states, which provides a way to study the interaction among multi macroscopic quantum states, such as quantum vortices [2527], Bloch oscillation [28,29], spinor [30], quantum phase transition [31], and so on [3237]. These studies inspire the potential EP application [38] to optical amplifiers [39,40], switches [4143], gates [4446], diodes [47], single photon sources [48], and lasers [49,50].

Since the EPs contain the photonic part, the two coupled EPCs can induce an optical analogue to the Josephson oscillation (JO) in electronics, that is, the optical JO. The required double-well potential can be designed by engineering the semiconductor microcavity with the methods of external stress [5153], metal-film patterning [54], photolithographic structures [55,56], interaction effects [22,31,5759], and growth-induced disorder [2,17,60]. These techniques can provide high-quality samples for researchers to study the optical JO based on the two coupled EPCs. The JO shows the dependence on the potential, pump arrangement, and exciton-photon detuning. As a key parameter, the JO strength is mainly determined by the photonic potential. The common treatment is to take the JO strength as a model parameter [4,9,37,61]. Simultaneously, these works usually adopted the two-mode model to analyze the JO and related oscillation dynamics. The relation between the JO strength and the potential geometry is not analyzed quantitively. The aim of the present work is to make this question clear through numerical calculation. As is known, the Gross-Pitaevskii equations (GPEs) are valid for describing the EPCs and so do for the optical JO. To make the optical JO be manifest, the two degenerate EPCs are considered, produced in a photonic double-well potential, see Fig. 1(a). Generally speaking, it is easier to design a photonic potential than an excitonic one. The photonic potential has been achieved in several experiments [56,62]. Two normal incident Gaussian laser beams are used to pump and control the phases of the EPCs in the two wells. The Josephson effect is reflected by the particle number oscillation between them. The tunneling strength is extracted from the numerical results based on the GPEs, which is significant for designing the quantum devices that use the Josephson effect.

 

Fig. 1. (a) Schematic draft indicated the model for a photonic double-well potential in a semiconductor microcavity, pumped by two normal incident Gaussian beams. (b) Photonic potential $V^C(\boldsymbol r)|_{y=0}$ in Eq. (1) with three different distances between the two wells where $R = 6\ \mu$m and $V_0=20$ meV.

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The remainder of this work is organized as follows. In Sec. 2, we introduce the GPEs that describe the EPCs in the double-well photonic potential. In Sec. 3, numerical results are displayed and discussed for the pulsed and continuous-wave Gaussian pumps. Finally, a conclusion is summarized in Sec. 4.

2. Model and formulas

The Fig. 1(b) gives an example for the photonic potential adopted in this work, i.e.,

$$V^C(\boldsymbol r)=V_0 \left\{1-e^{-\left[\left(x+\frac{d}{2}\right)^2+y^2\right]/R^2}\right\} \left\{1-e^{-\left[\left(x-\frac{d}{2}\right)^2+y^2\right]/R^2}\right\},$$
where the in-plane spatial coordinate is $\boldsymbol r=(x,\ y)$. $V_0$ and $R$, respectively, are the strength and width of each potential, and $d$ gives the distance between the two wells. The two wells centered at the points of $(\pm {d\over 2},\ 0)$ hold the barrier, $\varLambda$, given by
$$\varLambda =V_0\left(1-e^{{-}d^2/4R^2}\right)^2.$$
The reason why we use this type potential function is that it is parabolic near the centers of the two wells and very general for covering the main information of the double-well potential without loss of generality.

As a superposition of the quantum well exciton field, $\hat \Psi _X(\boldsymbol r,t)$, and the cavity photon field, $\hat \Psi _C(\boldsymbol r,t)$, the EP field Hamiltonian reads [29,63]

$$\begin{aligned} {\cal H}&=\int d^2{\boldsymbol r}\sum_{i,j}^{\{X,C\}} \hat{\Psi}^{i\dagger}({\boldsymbol r})\left[\mathbf{h}^0_{ij} +V^i ({\boldsymbol r})\delta_{ij}\right]\hat{\Psi}^j({\boldsymbol r})+{g^X\over 2}\int d^2{\boldsymbol r}\hat{\Psi}^{X\dagger}({\boldsymbol r}) \hat{\Psi}^{X\dagger}({\boldsymbol r})\hat{\Psi}^X({\boldsymbol r})\hat{\Psi} ^X({\boldsymbol r})\\ &+\int d^2{\boldsymbol r} \mathcal{F}^C({\boldsymbol r,t}) \hat{\Psi}^{C\dagger}({\boldsymbol r})+\textrm{h.c.}, \end{aligned}$$
where the indices $i,j\in \{X,C\}$ denote the exciton and cavity photon fields, respectively. The Bose commutation rules, $[\hat {\Psi }^i({\boldsymbol r}),\hat {\Psi }^{j\dagger }({\boldsymbol r'})]=\delta ^2({\boldsymbol r}-{\boldsymbol r'})\delta _{ij}$, are maintained by the exciton and photon field operators. $V^X(\boldsymbol r)$ and $V^C(\boldsymbol r)$ represent the potentials of the exciton and photon fields, respectively. Since it is easier to design a photonic potential than an excitonic one, we set $V^X(\boldsymbol r)=0$ and only consider the photonic potential through this work, see Eq. (1) and Fig. 1(b) where the harmonic-like traps are adopted for convenience. Note that most photonic potentials are more square-well-like (e.g. etched pillars), but the polariton JO is more sensitive to the width of the barrier and the distance between the two wells rather than the shape of the potential. Thus it is available to use the harmonic-like traps, as shown in Fig. 1(b). $g_X$ and $\mathcal {F}^C({\boldsymbol r,t})$ give the nonlinear exciton-exciton interaction and external pump field, respectively. The linear single-particle Hamiltonian, $\mathbf {h}^0$, reads [63]
$$\mathbf{ h}^0({-}i\nabla)=\left( \begin{array}{cc} E^X({-}i\nabla) & \Omega_R\\ \Omega_R & E^C({-}i\nabla) \end{array}\right).$$
where the cavity photon dispersion is assumed to be quadratic, $E^C(\boldsymbol k)=E^C(0)+{\hbar ^2\boldsymbol k}^2/(2m_C)$ with $\boldsymbol k$ and $m_C$ being the in-plane wave vector and effective mass of the cavity photons. Since quantum well excitons are far heavier than cavity photons, we neglect the exciton dispersion and assume it is a constant, i.e., $E^X(\boldsymbol k)=E^X(0)$. $\Omega _R$ is the Rabi coupling between the excitons and photons. Equation (4) gives the two-branch (upper and lower) EPs with eigenfrequencies being $E^{UP/LP}(\boldsymbol k)=\frac {1}{2}\left \{E^X+E^C(\boldsymbol k)\pm \sqrt {\left [E^X-E^C(\boldsymbol k)\right ]^2+4\Omega ^{2}_{R}}\right \}.$ Researchers are commonly interested in the lower branch EPs, because they can be in Bose-Einstein condensates.

In order to excite the EPs in the two wells, we use the following two Gaussian pumps with zero in-plane wave vector, namely,

$$\mathcal{F}^C(\boldsymbol r,t) = \mathcal{F}_L^C(\boldsymbol r,t) +\mathcal{F}_R^C(\boldsymbol r,t),$$
where
$$\mathcal{F}_{L/R}^C(\boldsymbol r,t) = f_Ce^{i\phi_{L/R}}\cdot e^{-\left[\left(x\pm\frac{d}{2}\right)^2+y^2\right]/w^2}\cdot f(t)e^{i\omega_pt}.$$
$f_C$, $w$, and $\omega _p$ denote the amplitude, width, and frequency of the two Gaussian beams, centered at the points of $(\pm {d\over 2},\ 0)$. $\phi _L$ and $\phi _R$ represent the pump phases and their difference is denoted as $\Delta \phi =\phi _R-\phi _L$. The time function, $f(t)$, takes 1 for the continuous-wave pump and $e^{-(t-t_c)^2/t_w^2}$ for the pulsed-wave pump, after the initial time of $t=0$. $t_c$ and $t_w$ give the time center and duration.

At the mean-field level, the dynamics of the EP field is described by the well-known GPEs [64]

$$\begin{aligned} i\hbar\frac{d}{d t} \left(\begin{array}{c} \psi^X({\boldsymbol r})\\ \psi^C({\boldsymbol r}) \end{array}\right) = \left[\left(\begin{array}{cc} \begin{array}{l}E^X-\frac{i}{2}\gamma^X\\+g^X|\psi^{X}({\boldsymbol r})|^{2}\end{array} & \Omega_R\\ \Omega_R & \begin{array}{r} E^C(-i\nabla)-\frac{i}{2}\gamma^C\\+V^{C}({\boldsymbol r})\end{array} \end{array}\right)\right] \left(\begin{array}{c} \psi^X({\boldsymbol r})\\ \psi^C({\boldsymbol r}) \end{array}\right) + \left(\begin{array}{c} 0\\ \mathcal{F}^C(\boldsymbol r,t) \end{array}\right), \end{aligned} $$
where $\psi ^{X(C)}(\boldsymbol r)=\langle \hat \Psi ^{X(C)}(\boldsymbol r)\rangle$ is the mean-field wave functions for the excitons (photons) with the decay rate $\gamma ^{X(C)}$. They are derived from the Heisenberg motion equation, i.e.,
$$i\hbar {d\over dt} \hat{\boldsymbol \Psi}(\boldsymbol r) = \left[\hat{\boldsymbol \Psi}(\boldsymbol r),~ ~ {\cal H}\right],$$
with $\hat{\boldsymbol \Psi}(\boldsymbol r)=\left [{\hat \Psi }^X(\boldsymbol r), {\hat \Psi }^C(\boldsymbol r)\right ]^T$. The Eq. (7) is the start point of the numerical calculation in this work. With the exciton and photon fields their particle numbers can be obtained as follows,
$$N_L^{X/C}=\int_0^{+\infty}dx\int_{-\infty}^{+\infty} dy\ |\psi_L^{X/C}(x,y,t)|^2, $$
$$N_R^{X/C}=\int_{-\infty}^{0}dx\int_{-\infty}^{+\infty} dy\ |\psi_R^{X/C}(x,y,t)|^2, $$
in the left and right wells, respectively. The total number for the EPs is
$$N_T^P=N_L^P+N_R^P,$$
where the EP numbers in the left/right wells $N_{L/R}^P$ are
$$N_{L/R}^P=N_{L/R}^X+N_{L/R}^C.$$
The optical JO is just reflected by the photon number oscillation between the two wells, whose oscillation period $T$ depends on the pump and potential geometry and provides the JO tunneling strength of $J\equiv {2\pi \hbar \over T}$.

In calculation, we take the typical GaAs-based microcavity as an example. The corresponding parameters are as follows. The energy of the excitons is set to be the reference point, i.e., $E^X=0$, and the energy of the cavity photons is set to be $E^C(0)=\delta _c$, being the exciton-photon detuning. Other unchanged parameters [42,63,65,66] are $m_C=1\times 10^{-5}m_e$ where $m_e$ is the free electron mass, $g^X=0.015\ \textrm {meV}\cdot \mu \textrm {m}^2$, $\Omega _R=2.5$ meV, and $V_0=20.0$ meV (corresponding to $\varLambda \gtrsim 8$ meV as $d\geq 2R$). Besides, the pump beam is always normal to the microcavity and the pump detuning is defined as $\delta _p=\hbar \omega _p-E^{LP}(0)$. The Gaussian beam size is taken to be $w=4$ $\mu$m. Since the strong interaction can lead to the so-called macroscopic quantum self-trapping [13], we take a weak pump to avoid the nonlinear interaction, namely, $f_C=1\ \textrm {meV}\cdot \mu \textrm {m}^{-1}$, resulting in that $g^X|\psi ^X(\boldsymbol r)|^2$ is much less than the Rabi coupling between the excitons and photons. In addition, two pump cases of continuous and pulsed waves are considered to show the optical JO.

3. Numerical results and discussion

This section is separated into three parts. In subsection 3.1, the pulsed Gaussian pump is directly used to show the optical JO in the time domain. In subsection 3.2, we study the effects of the pump detuning and exciton-photon detuning. For them there is the best parameter space for achieving the JO. In subsection 3.3, we further study the dependence of the optical JO on the potential geometry. The relation of the optical JO tunneling strength ($J$) with the distance ($d$) and barrier ($\varLambda$) between the two wells is fitted as an empirical formula, i.e., $J\approx 0.42\exp (-d\varLambda /18.4)$ with the energy and length units in meV and $\mu$m.

3.1 Optical JO under the pulsed Gaussian pump

The pulsed-wave pump can give a direct observation for the JO, because it permits a free evolution of the excited EPs in the time domain. Figure 2(a) shows the time evolution of the EP number under the two pulsed Gaussian pumps with $\Delta \phi =\pi /2$. During the initial time of $20$ ps, the EPs are excited and accumulated up to the maximum number. After the pump process the EPs move freely and decay exponentially due to the exciton and photon dissipations [see the solid black curve in Fig. 2(a)]. The decay rates of the EPs in both wells are strictly equal to $0.01$ meV because of $\gamma ^X=\gamma ^C=0.01$ meV. The initial EP state created by the two pulsed Gaussian beams is a superposition of the binding and anti-binding eigenmodes. Because of their energy splitting the system shows the JO. The JO is reflected by the clean envelope oscillations of $N_L^P$ and $N_R^P$, see the dashed red and dotted blue curves in Fig. 2(a). The five moments of $t_1$ to $t_5$ alternately denote the positions of the maximum and minimum of $N_L^P$. Their values are shown in Figs. 2(b)–2(k). The exciton and photon dissipations are responsible for the inequality of $2t_2\neq t_1+t_3$. The period of the JO is given by $T={1\over 2}(t_5-t_1)\approx 93.8$ ps and the corresponding JO tunneling strength of $J\approx 0.044$ meV.

 

Fig. 2. (a) Time evolution of the EP numbers, (b-f) density distributions of the photon fields, and (g-k) phase distributions that correspond to (b-f), respectively. The five moments of $t_1$ to $t_5$ for each density or phase map are denoted in (a), alternately corresponding to the positions of the local maximum and minimum of $N_L^P$. For easy observation the photon densities have been normalized to their maximums denoted by “Max”, and for the phase the maximum and minimum are both shown. All panels of (b-k) have the areas of $16\ \mu \textrm {m}\times 8\ \mu {}{\rm m}$. The two pulsed pump beams are adopted. Parameters: $\Delta \phi =\pi /2$, $\delta _p=0.0$, $\delta _c=0.0$, $\gamma ^X=\gamma ^C=0.01$ meV, $V_0=20.0$ meV, $R = 6.0\ \mu$m, $d = 8.0\ \mu$m, $f_C=1.0\ \textrm {meV}/\mu \textrm {m}$, $t_c=6.6$ ps, $t_w=3.3$ ps.

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For confirming the JO we plot the density and corresponding phase distributions for the photon field as time functions in Figs. 2(b)–2(f) and 2(g)–2(k), respectively. The photon field is mainly in the left well at the moments of $t_1$, $t_3$, and $t_5$, while in the right well at the moments of $t_2$ and $t_4$, which demonstrates the JO unequivocally. Since the pump beams are two pulsed waves, the field phase shows a time dependence [see Figs. 2(g)–2(k)] and simultaneously, has a difference between two wells. In view of symmetry, the Josephson effect is just due to the spontaneous gauge symmetry breaking. Its simplest manifestation is that the probability to find the EPs in one trap of the two wells is larger than in the other [see Figs. 2(b)–2(f)].

3.2 Effects of the pump arrangement, exciton-photon detuning, and photon loss

Since the particle number decreases sharply with time, it may be not convenient to observe the optical JO. On the other hand, the EP system would ultimately reach the steady state under a continuous-wave pump, that is, no difference between the particle densities of the two wells. The main effect of the phase difference of $\Delta \phi$ is to influence the total photon number, that is, the particle number strongly relates with the phase difference between the two continuous-wave pumps, while $\Delta \phi$ determines the appearance of the JO, see Fig. 3 where $d=8\ \mu$m and $\delta _p=0.5$ meV. Two resonant continuous pump waves are used to excite the EPs in the two wells after the initial time zero, that is, the continuous pumps are suddenly switched on at $t=0$ and then no longer change. When $\Delta \phi =0$, the two EPCs have the same phase [see Fig. 3(f)], resulting in a big overlap between them [see Fig. 3(a)]. The fast oscillation of the total photon number of $N_T^P$ in Fig. 3(k) dates from the interference between the EPs excited earlier and later by the Gaussian pump beams, whose envelope curves show a monotonic variation with time. When $\Delta \phi$ increases to $\pi /2$, the envelope curves also oscillate before the system reaches its steady state [see Fig. 3(m)], reflecting the EP jumping between the two wells, i.e., the optical JO. When $\Delta \phi$ further increases to $\pi$, the envelope curves monotonically decrease with time again and thus the EP jumping between the two wells disappears simultaneously [see Fig. 3(o)]. As $\Delta \phi$ increases from 0 to $\pi$, the EP phases in both wells become more inconsistent [see Figs. 3(f)–3(j)], resulting in a decrease of the total EP number in the steady state. For example, the steady-state photon number decreases from $29.6$ in Fig. 3(k) to $18.0$ in Fig. 3(o), which provides an indirect way for observing the JO under the continuous-wave pump, i.e., observe the oscillation of the emitted photon number. Since the oscillation of the photon number with $\Delta \phi$ can be up to ${29.6 - 18.0 \over 29.6+18.0}\approx 24\%$, this method is feasible for observing the optical JO as the pump is a continuous wave. The main effect of $\Delta \phi$, for the continuous-wave pump, is to influence the total photon number [see Figs. 3(k-o)]. Therefore, the gauge symmetry breaking has different presentations for the continuous- and pulsed-wave pumps, but both contain the JO information.

 

Fig. 3. Density distribution (left column), phase distribution (central column), and time variation of the particle number (right column) for the photon fields. The inset in panel (m) shows the detail of the curve. For easy observation the photon densities have been normalized to their maximums denoted by “Max” in the left column. The pump beams used are the two continuous Gaussian waves for all figures henceforth. Parameters: $\delta _p=0.5$ meV, $\delta _c=0.0$, $\gamma ^X=\gamma ^C=0.01$ meV, $V_0=20.0$ meV, $R = 6.0\ \mu$m, $d = 8.0\ \mu$m, and $f_C=1.0\ \textrm {meV}/\mu \textrm {m}$.

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The density distributions in Figs. 3(a) and 3(e) show the binding and anti-binding states that arise from the hybridization of the lowest energy mode of each well, agreeing with the condensates in the photonic molecules [56]. The photon field phase difference between the two wells is 0 for the binding state [see Fig. 3(f), while is $\pi$ for the anti-binding state [see Fig. 3(j)]. Those states between the binding and anti-binding states have the phase difference of $\Delta \phi$ in range of $(0,\ \pi )$, resulting in the JO. One example is shown in Fig. 3(m) where there are three oscillations. The fastest oscillation with period of $\sim 0.69$ ps (referred to the inset) is derived from the Rabi coupling between the excitons and photons. But the confinement of the cavity potential on the photons leads to its energy $\left (\sim {2\pi \hbar \over 0.69\ \textrm {ps}} =6.0\ \textrm {meV}\right )$ being a little larger than $2\Omega _R\ (=5\ \textrm {meV})$. The oscillation with period of $\sim 9.4$ ps is due to the interference between the EPs excited earlier and later by the pump and has the energy of $0.44\ \textrm {meV}$, much smaller than the Rabi coupling. The slowest envelope oscillation gives the JO whose period and energy are $T\approx 90.3$ ps with corresponding $J\approx 0.046$ meV, respectively, in agreement of those extracted from the pulsed-wave case in Fig. 2. Because the pump beams in Fig. 3 are continuous waves, the system will reach the steady states after a long time, see Figs. 3(k)–3(o).

Not only the pump phase plays an important role in the optical JO, but also the pump detuning shows a strong influence, see Fig. 4 where we take $R=8\ \mu$m and $\Delta \phi =\pi /2$. Since the deviation of the pump energy relative to the cavity photon energy decreases with increasing $\delta _p$, the oscillation period of the photon number becomes slower and slower and simultaneously, its envelope form changes. There are several EP levels in each well and therefore, none of them is resonantly excited as $\delta _p$ is negative, which leads to a complex envelope oscillation in Figs. 4(a)–4(c). For the opposite case, i.e., $\delta _p$ approaches the barrier of $\varLambda =2.58$ meV, the envelope oscillation disappears [see Fig. 4(f)], because the barrier cannot isolate the two EPCs well from each other. When $\delta _p$ is a few tenths of meV, the envelope oscillation is very obvious, see Figs. 4(d) and 4(e). This means that for a two-well EP system, the pump phase and detuning both show strong influence on the optical JO and should be carefully chosen for a practical system.

 

Fig. 4. Time variation of the photon number under several different pump detunings. Except $\Delta \phi =\pi /2$ and $\delta _p$ given in each panel, all other parameters and pump arrangement are the same with those in Fig. 3.

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In addition, the exciton-photon detuning (i.e., $\delta _c$) also influences the optical JO, since it can change the EP energy, see Fig. 5. As $\delta _c$ increases, the cavity photon energy increases and approaches the barrier $\varLambda$ and therefore, the optical JO becomes faster and faster. That is, the envelope oscillation strengths are about 0.038 meV, 0.046 meV, 0.054 meV, and 0.068 meV, respectively, from Figs. 5(a) to 5(d). The more important fact is that the influence of the the exciton-photon detuning on the JO is much weaker than that of the pump arrangement. That is, the existence of the JO is almost independent of the value of the exciton-photon detuning, which is beneficial for achieving the JO in experiments. Considering that the exciton potential is roughly equal to change the photon-exciton detuning, one can conjecture that the exciton potential shows a weak influence on the existence of the JO too. The weak dependences of the JO on the exciton-photon detuning and exciton potential are consistent with the fact that the polariton JO is mainly due to the tunneling of the photons between the two wells instead of the excitons (note that the effective mass of the excitons is much heavier than that of the cavity photons).

 

Fig. 5. Time variation of the photon number for four different exciton-photon detunings. Note that the panel (b) is the same with Fig. 3(m). Except $\Delta \phi =\pi /2$ and $\delta _c$ given in each panel, the pump arrangement and all other parameters are the same with those in Fig. 3.

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Figure 6 shows the influence of the system loss on the JO. Physically, the low loss of the system is beneficial to the JO. Typical exciton decay rates are $10^{-2}$ to $10^{-3}$ times the cavity photon decay rates and are about $10^{-1}$ times the photon decay rates in very high-Q cavities [67]. The life time of the excitons is about 1 ns, being larger than the decay rate of the exciton used in Figs. 25, i.e., $\gamma _a=0.01$ meV. Thus, the decay rate of the exciton is set to be $\gamma _a=0.01$ meV for the three cases in Fig. 6, while the loss of the cavity photons is increased from 0.02 meV to 0.1 meV, corresponding to the lifetime decrease from $200$ ps to $40$ ps. The increase of the photon decay rate leads to the decrease of the envelop amplitude, see Fig. 6, but it does not influence the existence of the JO. Furthermore, the period of the envelop oscillation (i.e., the JO period) is almost independent of the cavity loss, see the dashed red lines in Fig. 6. The JO period is about 91.0 ps for the three cases, consistent with that in Fig. 3(m).

 

Fig. 6. Time variation of the photon number for three different cavity losses. Three dashed red lines are used to guide eyes. Except $\Delta \phi =\pi /2$ and $\gamma _c$ given in each panel, the pump arrangement and all other parameters are the same with those in Fig. 3.

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3.3 Effects of potential geometry

This subsection focuses on the influence of the potential geometry. Figure 7 shows the influence of the distance between the two wells, $d$, where the density distribution, phase distribution, and time evolution of the particle number for the photon fields are plotted from the left to right columns, respectively. The separation distance between two wells demonstrates strong influence on the optical JO. For small distance of $d=4.0\ \mu$m the barrier $\varLambda$ is too small ($\sim 0.22$ meV) to separate the two wells, resulting in a photon cloud in the well center, see Fig. 7(a). No envelope oscillation for the photon particle number, see the dashed red curves in Fig. 7(k), reflects no the optical JO in the system. As $d$ increases to $6.0\ \mu$m and $\varLambda$ to 0.98 meV the envelope oscillation appears, i.e., the optical JO, see Fig. 7(l), with the period $T\approx 21.3$ ps and simultaneously accompanying with two cloud centers for the photons, see Fig. 7(b). The Fig. 7(l) also has another faster oscillation which originates from the interference between the EPs excited earlier and later by the pump beams, as discussed in Fig. 3(m). Since their periods approach with each other, it is not convenient to identify one of them from another. This can be overcome by increasing $d$ to $8\ \mu$m [see Figs. 7(c), 7(h), and 7(m)], for which the envelope oscillation is more obvious and has the period of $T\approx 99.4$ ps, larger than that in Fig. 7(l). This is due to that the barrier between the two wells is larger for $d=8\ \mu$m ($\varLambda =2.58$ meV) than for $d=6\ \mu$m ($\varLambda = 0.98$ meV). Since $\varLambda >5.01$ meV as $d>10\ \mu$m (see panels in the last two rows of Fig. 7), the EP jumping between the two wells becomes very harder, so that the envelope of the photon number is monotonically varying with time [see Figs. 7(n) and 7(o)], that is, no the optical JO. Though the barrier should be large enough for forming the two phase-different EPCs, it should also be weak to allow the EP jumping between the wells. As $R=6\ \mu$m, the above discussion indicates that $d=8\ \mu$m (a little larger than $R$) is the best choice for the optical JO, which is the reason why we choose $R=6\ \mu$m and $d=8\ \mu$m from Fig. 2 to Fig. 5. In other words, the system can show an obvious optical JO when the barrier between the two wells takes several meV, estimated by Eq. (2).

 

Fig. 7. Density distribution (left column), phase distribution (central column), and time variation (right column) of the photon number under five different values of $d$ given in the leftmost side of each row. For easy observation the photon densities have been normalized to their maxima denoted by “Max” in the left column. The dashed red curves in the right column guide the envelope variation of the total photon number. The pump beams used are the two continuous-wave Gaussian waves, turned on after time zero. Parameters: $\Delta \phi =\pi /2$, $\delta _p=0.0$, $\delta _c=0.0$, $V_0=20.0$ meV, $R = 6.0\ \mu$m, and $f_C=1.0\ \textrm {meV}/\mu \textrm {m}$.

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Since the JO depends on the potential geometry, we show the variation of the JO strength with $d$ and $R$ as the scatter dots in Fig. 8, defined as $J={2\pi \hbar \over T}$. The condition of $d \geqslant R$ is maintained in Fig. 8 to guarantee the separation of the two wells. The JO strength decreases as $d$ increases, because the distance and potential barrier $\varLambda$ between the two wells both increase as $d$ increases. On the other hand, the JO strength increases with increasing the well radius of $R$, because the barrier $\varLambda$ decreases with increasing $R$. Physically, the JO should exponentially decrease with increasing the barrier and distance between the two wells, that is, $J\propto \exp (-d\varLambda )$. This relation is reflected by the consistence between the variations of the $\log _{10} J$ and $-d\varLambda$ on $d$ and $R$, comparing the solid curves with the scatter dots in Fig. 8. By fitting, we can find an empirical formula for $J$, i.e., $J\approx 0.42\exp ({-d\varLambda /18.4})$ with the energy and length units in meV and $\mu$m. Though this formula is a fitting one, it can also show a guidance for more general cases, since the potential used in Eq. (1) can well describe the two-well system in the semiconductor microcavities.

 

Fig. 8. Variations of the JO strength in energy (scatter dots, left axis) and $-d\varLambda$ (solid curves, right axis) vs $d$ and $R$. Except $R$ and $d$, other parameters are the same to Fig. 3.

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4. Conclusion

We studied the optical Josephson oscillation based on the two coupled exciton-polariton condensates in a double-well potential and found that it is strongly influenced by the potential geometry and the pump arrangement. Two normal incident Gaussian beams are used to pump and control the phases of the exciton polaritons. The pulsed-wave pump beams were firstly used to confirm the optical Josephson oscillation. The polariton number oscillates with time, directly reflecting the optical Josephson oscillation. On the other hand, the polariton system under the continuous-wave pumps shows three different oscillations before it reaches a steady state, in which the envelope oscillation of the photon number evolution curves corresponds to the optical Josephson oscillation. When the phases of the exciton polariton condensates in the two wells are fully in (out of) step with each other, the system presents the binding (anti-binding) state. For observing the polariton Josephson oscillation, the pump detuning with respect to the polariton energy should be a slightly positive. At last, we fitted an empirical formula for the Josephson oscillation tunneling strength ($J$) with the distance ($d$) and barrier ($\varLambda$) between the two wells, i.e., $J\approx 0.42\exp ({-d\varLambda /18.4})$ where the energy and length units are in meV and $\mu$m. Since the double-well potential adopted is general in the present work, this fitting relation can show a guidance in practice for designing the optical devices that are based on the optical Josephson oscillation.

Funding

National Natural Science Foundation of China (11304015).

Acknowledgments

Formal funding declarations should not be included in the acknowledgments but in a Funding Information section as shown above.

Disclosures

The authors declare no conflicts of interest.

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References

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    [Crossref]
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2018 (2)

2017 (6)

Y. Sun, P. Wen, Y. Yoon, G. Liu, M. Steger, L. N. Pfeiffer, K. West, D. W. Snoke, and K. A. Nelson, “Bose-einstein condensation of long-lifetime polaritons in thermal equilibrium,” Phys. Rev. Lett. 118(1), 016602 (2017).
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X.-W. Xu, A.-X. Chen, and Y.-X. Liu, “Phononic josephson oscillation and self-trapping with two-phonon exchange interaction,” Phys. Rev. A 96(2), 023832 (2017).
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H. Ohadi, A. J. Ramsay, H. Sigurdsson, Y. del Valle-Inclan Redondo, S. I. Tsintzos, Z. Hatzopoulos, T. C. H. Liew, I. A. Shelykh, Y. G. Rubo, P. G. Savvidis, and J. J. Baumberg, “Spin order and phase transitions in chains of polariton condensates,” Phys. Rev. Lett. 119(6), 067401 (2017).
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W. Casteels and C. Ciuti, “Quantum entanglement in the spatial-symmetry-breaking phase transition of a driven-dissipative bose-hubbard dimer,” Phys. Rev. A 95(1), 013812 (2017).
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S. S. Demirchyan, T. A. Khudaĭberganov, I. Y. Chestnov, and A. P. Alodzhants, “Quantum fluctuations in a system of exciton polaritons in a semiconductor microcavity,” J. Opt. Technol. 84(2), 75–81 (2017).
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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]

2016 (6)

D. Sanvitto and S. Kéna-Cohen, “The road towards polaritonic devices,” Nat. Mater. 15(10), 1061–1073 (2016).
[Crossref]

O. Kyriienko and T. C. H. Liew, “Exciton-polariton quantum gates based on continuous variables,” Phys. Rev. B 93(3), 035301 (2016).
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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).
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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).
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H. Ohadi, Y. del Valle-Inclan Redondo, A. Dreismann, Y. G. Rubo, F. Pinsker, S. I. Tsintzos, Z. Hatzopoulos, P. G. Savvidis, and J. J. Baumberg, “Tunable magnetic alignment between trapped exciton-polariton condensates,” Phys. Rev. Lett. 116(10), 106403 (2016).
[Crossref]

T. Ostatnický, “Oscillations between microcavity polariton spin states coupled by two-particle tunneling,” J. Opt. Soc. Am. B 33(7), C144–C152 (2016).
[Crossref]

2015 (5)

X. Duan, B. Zou, and Y. Zhang, “Suppression of space broadening of exciton polariton transport by bloch oscillation effect,” J. Opt. 17(12), 125401 (2015).
[Crossref]

G. Liu, D. W. Snoke, A. Daley, L. N. Pfeiffer, and K. West, “A new type of half-quantum circulation in a macroscopic polariton spinor ring condensate,” Proc. Natl. Acad. Sci. 112(9), 2676–2681 (2015).
[Crossref]

D. Solnyshkov, O. Bleu, and G. Malpuech, “All optical controlled-not gate based on an exciton–polariton circuit,” Superlattices Microstruct. 83, 466–475 (2015).
[Crossref]

G. Li, T. C. H. Liew, O. A. Egorov, and E. A. Ostrovskaya, “Incoherent excitation and switching of spin states in exciton-polariton condensates,” Phys. Rev. B 92(6), 064304 (2015).
[Crossref]

M. Steger, C. Gautham, D. W. Snoke, L. Pfeiffer, and K. West, “Slow reflection and two-photon generation of microcavity exciton–polaritons,” Optica 2(1), 1–5 (2015).
[Crossref]

2014 (5)

D. V. Vishnevsky and F. Laussy, “Effective attractive polariton-polariton interaction mediated by an exciton reservoir,” Phys. Rev. B 90(3), 035413 (2014).
[Crossref]

D. D. Solnyshkov, H. Terças, and G. Malpuech, “Optical amplifier based on guided polaritons in gan and zno,” Appl. Phys. Lett. 105(23), 231102 (2014).
[Crossref]

S. S. Gavrilov, A. S. Brichkin, S. I. Novikov, S. Höfling, C. Schneider, M. Kamp, A. Forchel, and V. D. Kulakovskii, “Nonlinear route to intrinsic josephson oscillations in spinor cavity-polariton condensates,” Phys. Rev. B 90(23), 235309 (2014).
[Crossref]

T. Byrnes, N. Y. Kim, and Y. Yamamoto, “Exciton-polariton condensates,” Nat. Phys. 10(11), 803–813 (2014).
[Crossref]

R. Hivet, E. Cancellieri, T. Boulier, D. Ballarini, D. Sanvitto, F. M. Marchetti, M. H. Szymanska, C. Ciuti, E. Giacobino, and A. Bramati, “Interaction-shaped vortex-antivortex lattices in polariton fluids,” Phys. Rev. B 89(13), 134501 (2014).
[Crossref]

2013 (6)

P. Cristofolini, A. Dreismann, G. Christmann, G. Franchetti, N. G. Berloff, P. Tsotsis, Z. Hatzopoulos, P. G. Savvidis, and J. J. Baumberg, “Optical superfluid phase transitions and trapping of polariton condensates,” Phys. Rev. Lett. 110(18), 186403 (2013).
[Crossref]

G. Pavlovic, G. Malpuech, and I. A. Shelykh, “Pseudospin dynamics in multimode polaritonic josephson junctions,” Phys. Rev. B 87(12), 125307 (2013).
[Crossref]

M. Abbarchi, A. Amo, V. G. Sala, D. D. Solnyshkov, H. Flayac, L. Ferrier, I. Sagnes, E. Galopin, A. Lemaître, G. Malpuech, and J. Bloch, “Macroscopic quantum self-trapping and josephson oscillations of exciton polaritons,” Nat. Phys. 9(5), 275–279 (2013).
[Crossref]

D. Ballarini, M. De Giorgi, E. Cancellieri, R. Houdré, E. Giacobino, R. Cingolani, A. Bramati, G. Gigli, and D. Sanvitto, “All-optical polariton transistor,” Nat. Commun. 4(1), 1778 (2013).
[Crossref]

T. Espinosa-Ortega, T. C. H. Liew, and I. A. Shelykh, “Optical diode based on exciton-polaritons,” Appl. Phys. Lett. 103(19), 191110 (2013).
[Crossref]

A. Askitopoulos, H. Ohadi, A. V. Kavokin, Z. Hatzopoulos, P. G. Savvidis, and P. G. Lagoudakis, “Polariton condensation in an optically induced two-dimensional potential,” Phys. Rev. B 88(4), 041308 (2013).
[Crossref]

2012 (4)

M. Galbiati, L. Ferrier, D. D. Solnyshkov, D. Tanese, E. Wertz, A. Amo, M. Abbarchi, P. Senellart, I. Sagnes, A. Lemaître, E. Galopin, G. Malpuech, and J. Bloch, “Polariton condensation in photonic molecules,” Phys. Rev. Lett. 108(12), 126403 (2012).
[Crossref]

G. Tosi, G. Christmann, N. G. Berloff, P. Tsotsis, T. Gao, Z. Hatzopoulos, P. G. Savvidis, and J. J. Baumberg, “Sculpting oscillators with light within a nonlinear quantum fluid,” Nat. Phys. 8(3), 190–194 (2012).
[Crossref]

M. Steger, C. Gautham, B. Nelsen, D. Snoke, L. Pfeiffer, and K. West, “Single-wavelength, all-optical switching based on exciton-polaritons,” Appl. Phys. Lett. 101(13), 131104 (2012).
[Crossref]

B. Deveaud-Plédran, “On the condensation of polaritons,” J. Opt. Soc. Am. B 29(2), A138–A145 (2012).
[Crossref]

2011 (4)

C. Zhang and G. Jin, “Magnetic field modulated josephson oscillations in a semiconductor microcavity,” Phys. Rev. B 84(11), 115324 (2011).
[Crossref]

S. Peotta, M. Gibertini, F. Dolcini, F. Taddei, M. Polini, L. B. Ioffe, R. Fazio, and A. H. MacDonald, “Josephson current in a four-terminal superconductor/exciton-condensate/superconductor system,” Phys. Rev. B 84(18), 184528 (2011).
[Crossref]

H. Flayac, D. D. Solnyshkov, and G. Malpuech, “Bloch oscillations of an exciton-polariton bose-einstein condensate,” Phys. Rev. B 83(4), 045412 (2011).
[Crossref]

H. M. Gibbs, G. Khitrova, and S. W. Koch, “Exciton-polariton light-semiconductor coupling effects,” Nat. Photonics 5(5), 273 (2011).
[Crossref]

2010 (4)

A. Amo, T. C. H. Liew, C. Adrados, R. Houdré, E. Giacobino, A. V. Kavokin, and A. Bramati, “Exciton-polariton spin switches,” Nat. Photonics 4(6), 361–366 (2010).
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H. Deng, H. Haug, and Y. Yamamoto, “Exciton-polariton bose-einstein condensation,” Rev. Mod. Phys. 82(2), 1489–1537 (2010).
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K. G. Lagoudakis, B. Pietka, M. Wouters, R. André, and B. Deveaud-Plédran, “Coherent oscillations in an exciton-polariton josephson junction,” Phys. Rev. Lett. 105(12), 120403 (2010).
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E. B. Magnusson, H. Flayac, G. Malpuech, and I. A. Shelykh, “Role of phonons in josephson oscillations of excitonic and polaritonic condensates,” Phys. Rev. B 82(19), 195312 (2010).
[Crossref]

2009 (1)

Y. Zhang and G. Jin, “Polariton bunching and anti-bunching in a photonic dot controlled by a magnetic field,” Phys. Rev. B 79(19), 195304 (2009).
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2008 (6)

D. Sarchi, I. Carusotto, M. Wouters, and V. Savona, “Coherent dynamics and parametric instabilities of microcavity polaritons in double-well systems,” Phys. Rev. B 77(12), 125324 (2008).
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K. G. Lagoudakis, M. Wouters, M. Richard, A. Baas, I. Carusotto, R. André, L. S. Dang, and B. Deveaud-Plédran, “Quantized vortices in an exciton-polariton condensate,” Nat. Phys. 4(9), 706–710 (2008).
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I. A. Shelykh, D. D. Solnyshkov, G. Pavlovic, and G. Malpuech, “Josephson effects in condensates of excitons and exciton polaritons,” Phys. Rev. B 78(4), 041302 (2008).
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J. J. Baumberg, A. V. Kavokin, S. Christopoulos, A. J. D. Grundy, R. Butté, G. Christmann, D. D. Solnyshkov, G. Malpuech, G. Baldassarri Höger von Högersthal, E. Feltin, J.-F. Carlin, and N. Grandjean, “Spontaneous polarization buildup in a room-temperature polariton laser,” Phys. Rev. Lett. 101(13), 136409 (2008).
[Crossref]

D. Bajoni, P. Senellart, E. Wertz, I. Sagnes, A. Miard, A. Lemaître, and J. Bloch, “Polariton laser using single micropillar GaAs–GaAlAs semiconductor cavities,” Phys. Rev. Lett. 100(4), 047401 (2008).
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A. P. D. Love, D. N. Krizhanovskii, D. M. Whittaker, R. Bouchekioua, D. Sanvitto, S. A. Rizeiqi, R. Bradley, M. S. Skolnick, P. R. Eastham, R. André, and L. S. Dang, “Intrinsic decoherence mechanisms in the microcavity polariton condensate,” Phys. Rev. Lett. 101(6), 067404 (2008).
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2007 (6)

R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, and K. West, “Bose-einstein condensation of microcavity polaritons in a trap,” Science 316(5827), 1007–1010 (2007).
[Crossref]

C. Leyder, T. C. H. Liew, A. V. Kavokin, I. A. Shelykh, M. Romanelli, J. P. Karr, E. Giacobino, and A. Bramati, “Interference of coherent polariton beams in microcavities: Polarization-controlled optical gates,” Phys. Rev. Lett. 99(19), 196402 (2007).
[Crossref]

C. W. Lai, N. Y. Kim, S. Utsunomiya, G. Roumpos, H. Deng, M. D. Fraser, T. Byrnes, P. Recher, N. Kumada, T. Fujisawa, and Y. Yamamoto, “Coherent zero-state and π-state in an exciton-polariton condensate array,” Nature 450(7169), 529–532 (2007).
[Crossref]

Y. Zhang, G. Jin, and Y.-Q. Ma, “Phase effects on the exciton polariton amplifier,” Appl. Phys. Lett. 91(19), 191112 (2007).
[Crossref]

M. Wouters and I. Carusotto, “Excitations in a nonequilibrium bose-einstein condensate of exciton polaritons,” Phys. Rev. Lett. 99(14), 140402 (2007).
[Crossref]

S. Christopoulos, G. B. H. von Högersthal, A. J. D. Grundy, P. G. Lagoudakis, A. V. Kavokin, J. J. Baumberg, G. Christmann, R. Butté, E. Feltin, J.-F. Carlin, and N. Grandjean, “Room-temperature polariton lasing in semiconductor microcavities,” Phys. Rev. Lett. 98(12), 126405 (2007).
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2006 (5)

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).
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O. El Daïf, A. Baas, T. Guillet, J.-P. Brantut, R. I. Kaitouni, J. L. Staehli, F. Morier-Genoud, and B. Deveaud, “Polariton quantum boxes in semiconductor microcavities,” Appl. Phys. Lett. 88(6), 061105 (2006).
[Crossref]

R. B. Balili, D. W. Snoke, L. Pfeiffer, and K. West, “Actively tuned and spatially trapped polaritons,” Appl. Phys. Lett. 88(3), 031110 (2006).
[Crossref]

R. I. Kaitouni, O. El Daïf, A. Baas, M. Richard, T. Paraiso, P. Lugan, T. Guillet, F. Morier-Genoud, J. D. Ganière, J. L. Staehli, V. Savona, and B. Deveaud, “Engineering the spatial confinement of exciton polaritons in semiconductors,” Phys. Rev. B 74(15), 155311 (2006).
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Other (4)

A. Kavokin and G. Malpuech, Cavity Polaritons, vol. 32 of Thin Films and Nanostructures (Elsevier, Amsterdam, 2003).

N. P. Proukakis, D. W. Snoke, and P. B. Littlewood, eds., Universal Themes of Bose-Einstein Condensation (Cambridge University Press, 2017).

N. Y. Kim, W. H. Nitsche, and Y. Yamamoto, Berezinskii-Kosterlitz-Thouless Phase of a Driven-Dissipative Condensate (Cambridge University Press, 2017187–204

J. Keeling, L. M. Sieberer, E. Altman, L. Chen, S. Diehl, and J. Toner, Superfluidity and Phase Correlations of Driven Dissipative Condensates (Cambridge University Press, 2017205–230

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Figures (8)

Fig. 1.
Fig. 1. (a) Schematic draft indicated the model for a photonic double-well potential in a semiconductor microcavity, pumped by two normal incident Gaussian beams. (b) Photonic potential $V^C(\boldsymbol r)|_{y=0}$ in Eq. (1) with three different distances between the two wells where $R = 6\ \mu$ m and $V_0=20$ meV.
Fig. 2.
Fig. 2. (a) Time evolution of the EP numbers, (b-f) density distributions of the photon fields, and (g-k) phase distributions that correspond to (b-f), respectively. The five moments of $t_1$ to $t_5$ for each density or phase map are denoted in (a), alternately corresponding to the positions of the local maximum and minimum of $N_L^P$ . For easy observation the photon densities have been normalized to their maximums denoted by “Max”, and for the phase the maximum and minimum are both shown. All panels of (b-k) have the areas of $16\ \mu \textrm {m}\times 8\ \mu {}{\rm m}$ . The two pulsed pump beams are adopted. Parameters: $\Delta \phi =\pi /2$ , $\delta _p=0.0$ , $\delta _c=0.0$ , $\gamma ^X=\gamma ^C=0.01$ meV, $V_0=20.0$ meV, $R = 6.0\ \mu$ m, $d = 8.0\ \mu$ m, $f_C=1.0\ \textrm {meV}/\mu \textrm {m}$ , $t_c=6.6$ ps, $t_w=3.3$ ps.
Fig. 3.
Fig. 3. Density distribution (left column), phase distribution (central column), and time variation of the particle number (right column) for the photon fields. The inset in panel (m) shows the detail of the curve. For easy observation the photon densities have been normalized to their maximums denoted by “Max” in the left column. The pump beams used are the two continuous Gaussian waves for all figures henceforth. Parameters: $\delta _p=0.5$ meV, $\delta _c=0.0$ , $\gamma ^X=\gamma ^C=0.01$ meV, $V_0=20.0$ meV, $R = 6.0\ \mu$ m, $d = 8.0\ \mu$ m, and $f_C=1.0\ \textrm {meV}/\mu \textrm {m}$ .
Fig. 4.
Fig. 4. Time variation of the photon number under several different pump detunings. Except $\Delta \phi =\pi /2$ and $\delta _p$ given in each panel, all other parameters and pump arrangement are the same with those in Fig. 3.
Fig. 5.
Fig. 5. Time variation of the photon number for four different exciton-photon detunings. Note that the panel (b) is the same with Fig. 3(m). Except $\Delta \phi =\pi /2$ and $\delta _c$ given in each panel, the pump arrangement and all other parameters are the same with those in Fig. 3.
Fig. 6.
Fig. 6. Time variation of the photon number for three different cavity losses. Three dashed red lines are used to guide eyes. Except $\Delta \phi =\pi /2$ and $\gamma _c$ given in each panel, the pump arrangement and all other parameters are the same with those in Fig. 3.
Fig. 7.
Fig. 7. Density distribution (left column), phase distribution (central column), and time variation (right column) of the photon number under five different values of $d$ given in the leftmost side of each row. For easy observation the photon densities have been normalized to their maxima denoted by “Max” in the left column. The dashed red curves in the right column guide the envelope variation of the total photon number. The pump beams used are the two continuous-wave Gaussian waves, turned on after time zero. Parameters: $\Delta \phi =\pi /2$ , $\delta _p=0.0$ , $\delta _c=0.0$ , $V_0=20.0$ meV, $R = 6.0\ \mu$ m, and $f_C=1.0\ \textrm {meV}/\mu \textrm {m}$ .
Fig. 8.
Fig. 8. Variations of the JO strength in energy (scatter dots, left axis) and $-d\varLambda$ (solid curves, right axis) vs $d$ and $R$ . Except $R$ and $d$ , other parameters are the same to Fig. 3.

Equations (12)

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V C ( r ) = V 0 { 1 e [ ( x + d 2 ) 2 + y 2 ] / R 2 } { 1 e [ ( x d 2 ) 2 + y 2 ] / R 2 } ,
Λ = V 0 ( 1 e d 2 / 4 R 2 ) 2 .
H = d 2 r i , j { X , C } Ψ ^ i ( r ) [ h i j 0 + V i ( r ) δ i j ] Ψ ^ j ( r ) + g X 2 d 2 r Ψ ^ X ( r ) Ψ ^ X ( r ) Ψ ^ X ( r ) Ψ ^ X ( r ) + d 2 r F C ( r , t ) Ψ ^ C ( r ) + h.c. ,
h 0 ( i ) = ( E X ( i ) Ω R Ω R E C ( i ) ) .
F C ( r , t ) = F L C ( r , t ) + F R C ( r , t ) ,
F L / R C ( r , t ) = f C e i ϕ L / R e [ ( x ± d 2 ) 2 + y 2 ] / w 2 f ( t ) e i ω p t .
i d d t ( ψ X ( r ) ψ C ( r ) ) = [ ( E X i 2 γ X + g X | ψ X ( r ) | 2 Ω R Ω R E C ( i ) i 2 γ C + V C ( r ) ) ] ( ψ X ( r ) ψ C ( r ) ) + ( 0 F C ( r , t ) ) ,
i d d t Ψ ^ ( r ) = [ Ψ ^ ( r ) ,     H ] ,
N L X / C = 0 + d x + d y   | ψ L X / C ( x , y , t ) | 2 ,
N R X / C = 0 d x + d y   | ψ R X / C ( x , y , t ) | 2 ,
N T P = N L P + N R P ,
N L / R P = N L / R X + N L / R C .

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