INTRODUCTION

Quasi-two-dimensional exciton polaritons are bosons arising due to the strong coupling between excitons and light in a planar microcavity [1–3]. The state of resonantly excited polaritons is macroscopically coherent [4–6] and can be considered by analogy with Bose–Einstein condensates. Nonlinearity caused by the polariton–polariton interaction leads to optical multistability: the cavity response to a plane light wave is characterized by several alternative stable states, between which nonequilibrium transitions are possible [7–11]. It is known that multistability in a laterally homogeneous system can affect the character of parametric scattering [12, 13] or lead to the appearance of complex dissipative structures in the spatial distribution of radiation [14–16]. On the other hand, the polariton system in a quantum-confined micropillar behaves like a simple multistable cell with very short characteristic switch times down to several picoseconds [17].

Various mechanisms of transitions between alternative stable states of a polariton system are known. All of them assume that the excitation frequency is slightly higher than the eigenfrequency of polaritons, and this difference is compensated by the blue shift of the resonance with increasing field amplitude [11]. The simplest variant of the transition can occur upon a smooth increase in the pump intensity in a system with an S‑shaped response, where the lower branch of stability is terminated at a certain critical point, and a jump to the upper branch proceeds [7]. In order to make such transitions well controllable, two-beam excitation schemes are used, in which one wave is continuous and creates conditions for multistability as such, whereas the second is a short pulse that acts as a switch trigger [11, 18]. However, for a “zero-dimensional” polariton system in a micropillar, the implementation of this control method is very difficult due to the need to accurately control the phase difference between the two beams [19]. A deformation acoustic pulse, which shortly disturbs the polariton frequency, can be used as a switch here instead of a resonant optical pulse [20, 21]. Experimental confirmation of at least one-way transitions of this kind has recently been obtained [22, 23].

In this work, we propose a mechanism for controlled two-way switch of coherent states of polaritons in a micropillar cavity, which is accompanied by the reversal of the circular polarization of the emitted light. It is known that alternative states with opposite circular polarizations can exist in a static magnetic field; transitions between them were observed in the pulsed excitation regime with a pulse duration of about 0.2 ns [24]. A pulsed pump with such a short duration cannot yet properly ensure multistability and discrete switches. On the other hand, in the case of longer excitation times, a complex pulse shape related to the hysteresis effect would be required to implement controlled transitions. A distinctive feature of the mechanism presented in this work is that it is insensitive to the pulse shape and makes it possible to achieve an almost direct correspondence between the intensity of the light wave exciting the cavity and the sign of the circular polarization of radiation.

The main idea is that polarization switches in the magnetic field can occur due to the transition of the current polarization state of the system to dynamical chaos. The phase trajectory of the system evolving in a chaotic regime covers a wide semi-continuous region of the phase space and eventually falls within the attraction region of an alternative stable state with opposite polarization. Thus, the nonequilibrium transition between stationary states is not direct but is mediated by a chaotic stage with an indefinite (in the general case) duration. Calculations show that for the characteristic parameters of a 2-μm GaAs-based micropillar, spin switching events are quite reliably detected on a scale of 10–8 s at a polariton lifetime of about 10–11 s.

In what follows, we formulate the model describing the dynamics of polaritons in the mean-field approximation, and justify the passage to a purely zero-dimensional case, for which we study stationary states and their stability and discuss possible switching scenarios. Then, we analyze the transitions between states with opposite polarizations in situations of strictly cw and partially stochastic (with random jumps in phase and amplitude) external excitations.

MODEL

The dynamics of a two-dimensional polariton system is considered in the mean-field approximation in terms of a classical spinor amplitude \({{\Psi }_{ \pm }}({\mathbf{r}},t)\), which satisfies the generalized Gross–Pitaevskii equations including dissipation and coherent pumping [5, 8, 11]:

$$\begin{gathered} i\hbar \frac{{\partial {{\Psi }_{ \pm }}}}{{\partial t}} = \left( {{{{\hat {E}}}_{{{\text{LP}}}}} + U(r) - i\gamma \pm \frac{{{{g}_{M}}}}{2} + \tilde {V}{\text{|}}{{\Psi }_{ \pm }}{{{\text{|}}}^{2}}} \right){{\Psi }_{ \pm }} \\ + \frac{{{{g}_{L}}}}{2}{{\Psi }_{ \mp }} + {{{\tilde {f}}}_{ \pm }}{{e}^{{ - i{{E}_{p}}t/\hbar }}}. \\ \end{gathered} $$
(1)

Here, \({{\hat {E}}_{{{\text{LP}}}}} = {{\hat {E}}_{{{\text{LP}}}}}( - i\hbar \nabla )\) is the dispersion relation of the lower polariton branch; \(U(r)\) is the potential; \(\gamma \) is the decay rate (the corresponding polariton lifetime is \(\tau \) = \(\hbar {\text{/}}\gamma \)); \({{g}_{M}}\) is the splitting of sublevels with opposite spins due to the Zeeman effect in a static magnetic field perpendicular to the surface; \(\tilde {V} > 0\) is the pair interaction constant; \({{g}_{L}}\) is the coupling constant of spin components corresponding to the splitting of modes with orthogonal linear polarizations, which can be caused, e.g., by the mechanical stress in the cavity plane [25, 26]; \({{\tilde {f}}_{ \pm }}\) is the effective spinor amplitude; and Ep/\(\hbar \) is the frequency of the pump wave normally incident on the cavity plane. The constants \({{g}_{L}}\) and \({{g}_{M}}\) have the dimension of energy.

The potential \(U(r)\) corresponds to an infinite quantum well with the radius \(R\); i.e., \(U(r) = 0\) at \(r \leqslant R\) and \(U(r) = \infty \) at \(r > R\). The splitting of the quantum confined levels is proportional to \(1{\text{/}}{{R}^{2}}\). Consequently, if \(R\) is sufficiently small, and the pump frequency Ep/\(\hbar \) is fixed near the lowest level, only one spatial mode is excited in the polariton system, the profile of which is described by the zero-order Bessel function \({{J}_{0}}\) due to the circular symmetry. Under this assumption, we seek solutions of Eqs. (1) in the form Ψ±(r, t) = \(C{{J}_{0}}(\alpha r{\text{/}}R){{\psi }_{ \pm }}(t)\), where \(\alpha \approx 2.4\) is the smallest (in absolute value) zero of the function \({{J}_{0}}\). The choice of the normalization constant C in the form \(C = 1{\text{/}}{{J}_{1}}(\alpha )\), where \({{J}_{1}}\) is the first-order Bessel function, ensures the coincidence of \({\text{|}}\Psi (r,t){{{\text{|}}}^{2}}\) averaged over the micropillar with \({\text{|}}\psi (t){{{\text{|}}}^{2}}\). As a result, the equations for \({{\psi }_{ \pm }}(t)\) have the form

$$\begin{gathered} i\hbar \frac{{d{{\psi }_{ \pm }}}}{{dt}} = \left( {{{E}_{0}} - i\gamma \pm \frac{{{{g}_{M}}}}{2} + V{\text{|}}{{\psi }_{ \pm }}{{{\text{|}}}^{2}}} \right){{\psi }_{ \pm }} \\ + \frac{{{{g}_{L}}}}{2}{{\psi }_{ \mp }} + {{f}_{ \pm }}{{e}^{{ - i{{E}_{p}}t/\hbar }}}, \\ \end{gathered} $$
(2)

where

$${{f}_{ \pm }} = \frac{{2{{{\tilde {f}}}_{ \pm }}}}{\alpha } \approx 0.83{\kern 1pt} {{\tilde {f}}_{ \pm }},$$
(3)
$$V = \frac{{2\tilde {V}}}{{{{\alpha }^{2}}{{{[{{J}_{1}}(\alpha )]}}^{4}}}}\int\limits_0^\alpha {\xi {{{[{{J}_{0}}(\xi )]}}^{4}}d\xi \approx 2.10{\kern 1pt} \tilde {V}} ,$$
(4)
$${{E}_{0}} \approx {{E}_{{{\text{LP}}}}}(k = 0) + \frac{{{{\hbar }^{2}}{{\alpha }^{2}}}}{{2m{{R}^{2}}}}$$
(5)

and m is the effective polariton mass near the mode with zero two-dimensional wavenumber \(k\). In further examples, we consider a GaAs-based cavity with the effective dielectric constant \(\varepsilon = 12\) and identical exciton and photon mode energies \(E = 1.5\) eV for k = 0; in this case, \(m\) = \(2E{\text{/}}\varepsilon {{c}^{2}}\). The Rabi splitting (exciton–photon coupling strength) is 10 meV, the decay rate is \(\gamma = 0.075\) meV, the characteristic parameters of spin splitting and spin coupling are \({{g}_{M}} = 0.2\) meV and \({{g}_{L}} = 0.5\) meV. In practical calculations, the units for f and \(\psi \) are conveniently fixed by the condition V = 1; in this case, in particular, \({\text{|}}{{\psi }_{ \pm }}{{{\text{|}}}^{2}}\) has the dimension of energy and means the blue shift of the resonance.

The solutions of Eqs. (2) coincide quantitatively with the solutions of Eqs. (1) averaged over the micropillar area for \(R \lesssim 1\) µm if the pump energy detuning \({{E}_{{\text{p}}}} - {{E}_{{{\text{LP}}}}}(k = 0)\) in Eq. (1) corresponds to \({{E}_{{\text{p}}}} - {{E}_{0}}\) in Eq. (2) and does not exceed 1 meV. The quantity R = 1 µm is much larger than the exciton Bohr radius; therefore, the classical approximation, within which the original Eq. (1) was formulated, can be assumed valid. The multistability effect was experimentally observed and reproduced by the equations similar to Eq. (1) at least for \(R \approx 1.5\) µm [10, 17, 27].

STATIONARY SOLUTIONS

The substitution of \({{\psi }_{ \pm }}(t) = {{\bar {\psi }}_{ \pm }}{{e}^{{ - i{{E}_{p}}t/\hbar }}}\) into Eq. (2) leads to the time-independent equations

$$\left[ { \pm \frac{{{{g}_{M}}}}{2} - D - i\gamma + V{\text{|}}{{{\bar {\psi }}}_{ \pm }}{{{\text{|}}}^{2}}} \right]{{\bar {\psi }}_{ \pm }} + \frac{{{{g}_{L}}}}{2}{{\bar {\psi }}_{ \mp }} + {{f}_{ \pm }} = 0,$$
(6)

where D = Ep – E0. Having calculated the stationary solution \({{\bar {\psi }}_{ \pm }}\) for any parameters, one should then study its stability with respect to small field fluctuations. To this end, the energies E of elementary excitations are found. In the limit of \({\text{|}}\delta {{\psi }_{ \pm }}{\text{|}} \ll {\text{|}}{{\bar {\psi }}_{ \pm }}{\text{|}}\), the substitution of

$${{\psi }_{ \pm }}(t) = {{\bar {\psi }}_{ \pm }}{{e}^{{ - i{{E}_{{\text{p}}}}t/\hbar }}} + \delta {{\psi }_{ \pm }}{{e}^{{ - iEt/\hbar }}}$$
(7)

into Eq. (2) leads to a homogeneous linear system of equations for \(\delta {{\psi }_{ + }}\), \(\delta {{\psi }_{ - }}\), \(\delta \psi _{ + }^{*}\), \(\delta \psi _{ - }^{*}\), which gives four eigenvalues E. If they all have a negative imaginary part, the solution under consideration \({{\bar {\psi }}_{ \pm }}\) is asymptotically stable (attractive). The fundamental details of the calculations are discussed in [11].

Figure 1 shows characteristic solutions of Eqs. (6) as functions of the total pump intensity \(I = {\text{|}}{{f}_{ + }}{{{\text{|}}}^{2}}\, + \,{\text{|}}{{f}_{ - }}{{{\text{|}}}^{2}}\). Figures 1a and 1b present an important special case where \({{f}_{ + }} = {{f}_{ - }}\) and \({{g}_{M}} = 0\) (the light wave is linearly polarized, the magnetic field is zero) and thus, the equations for \({{\bar {\psi }}_{ + }}\) and \({{\bar {\psi }}_{ - }}\) are completely the same. This problem is exactly solvable [28]. For any I value, there are symmetrical solutions (\({{\bar {\psi }}_{ + }} = {{\bar {\psi }}_{ - }}\)) forming an S‑shaped curve with two stable branches \(({{\pi }_{x}})\). Solutions of the second type (branches σ+ and σ–) are characterized by a high circular polarization and lie in the intermediate region of I and \(u = V({\text{|}}{{\bar {\psi }}_{ + }}{{{\text{|}}}^{2}} + \,{\text{|}}{{\bar {\psi }}_{ - }}{{{\text{|}}}^{2}})\). Only asymmetric solutions can be stable sometimes at certain I values, so the transition from linear to elliptical polarization of radiation is strictly predetermined [25, 26, 29]. It is noteworthy that with increasing \({{g}_{L}}{\text{/}}\gamma \), the solutions of the second type can also become unstable in a certain interval of I, where no allowed stationary solutions remain [30]. In this case (see Fig. 1b, \(I \sim 0.1\)), the system switches to the regime of regular or chaotic oscillations of the intensity and circular-polarization degree \(\rho \) [28].

Fig. 1.
figure 1

(Color online) (Solid lines) Stable and (dashed lines) unstable stationary solutions of Eq. (2). (a) Spin-symmetric model (\({{f}_{ + }} = {{f}_{ - }}\), \(D \approx 12\gamma \), \({{g}_{M}} = 0\), \({{g}_{L}} \approx 1.7\gamma \)). Branches \({{\pi }_{x}}\) are symmetric solutions with the linear polarization (\({{\psi }_{ + }} = {{\psi }_{ - }}\)), branches \({{\sigma }^{ + }}\) and \({{\sigma }^{ - }}\) are the intensity-degenerate solutions with a high circular polarization. (b) Symmetric model with the higher coupling constant \({{g}_{L}} \approx 5.5\gamma \) at \(D = 8\gamma \). (c) Solutions in a magnetic field at \({{g}_{L}} < {{g}_{M}}\) (\({{\rho }_{p}} = ({\text{|}}f_{ + }^{2}{\text{|}} - \,{\text{|}}{{f}_{ - }}{{{\text{|}}}^{2}}){\text{/}}({\text{|}}f_{ + }^{2}{\text{|}} + \,{\text{|}}{{f}_{ - }}{{{\text{|}}}^{2}}) = - 0.5\), \(D = 10\gamma \), \({{g}_{M}} = 3\gamma \), \({{g}_{L}} \approx 1.5\gamma \)). The circular-polarization degree \(\rho \) is indicated by color. (d) Solutions in a magnetic field at \({{g}_{L}} > {{g}_{M}}\) (\({{\rho }_{p}} = - 1{\text{/}}3\), \(\arg f_{ + }^{*}{{f}_{ - }} = - \pi {\text{/}}12\), \(D \approx 6.8\gamma \), \({{g}_{M}} \approx 2.7\gamma \), \({{g}_{L}} \approx 6.7\gamma \)). In all cases, \(\gamma = \) 0.075 meV.

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A magnetic field perpendicular to the cavity plane leads to the splitting \({{E}_{0}} \to {{E}_{0}} \pm {{g}_{M}}{\text{/}}2\) of the energy levels of polaritons with \(\rho = \pm 1\). Consequently, the effective detuning of the pump frequency from resonance is different for the two spin components. Stationary solutions with opposite polarizations now have different intensities and, in general, different critical pump intensities I at which the corresponding branches of solutions begin or end. As a result, controlled transitions between the branches \({{\sigma }^{ + }}\) and \({{\sigma }^{ - }}\) induced solely by smooth changes in I become possible. As before, the solutions with high polarizations can be either stable or unstable. The first case is implemented for all I values regardless of \({{g}_{M}}\), if the ratio \({{g}_{L}}{\text{/}}\gamma \) is relatively small; Fig. 1c shows a typical example for \({{g}_{L}} \approx 1.5\gamma \) and \({{g}_{M}} = 3\gamma \). It is seen that the stationary solutions have a wide hysteresis loop; transitions between the branches σ+ and σ– owing to a smooth change in I can only go through intermediate states with small |ρ| and either a very high (for the transition \({{\sigma }^{ + }} \to {{\sigma }^{ - }}\)) or very low (\({{\sigma }^{ - }} \to {{\sigma }^{ + }}\)) field intensity [24]. It is difficult to make such transitions both fast and well controlled: this would have required precise control of the complex shape and phase of short switching pulses.

A completely new spin switch scenario appears for \({{g}_{M}} > \gamma \) and \({{g}_{L}} > 4\gamma \) (Fig. 1d), when the solutions with high |ρ| become unstable, as in the example in Fig. 1b, however, owing to a lifted degeneracy, only one of two alternative solutions becomes unstable. One may suppose that an increase in I near the corresponding critical point should involve a transition from the branch \({{\sigma }^{ + }}\) to the branch \({{\sigma }^{ - }}\), because, as seen in Fig. 1d, no other stable solutions for this I value remain. Figure 2 shows the intensity, polarization, and stability measure (Γ = max Im(E)) of stationary solutions near the transition point. In addition to a simple transition with reversal of the polarization, a change in the sign of \(\Gamma \) implies the possibility of self-oscillations or dynamical chaos. The chaotic regime and spontaneous recovery from it to the alternative stable state are discussed below.

Fig. 2.
figure 2

(a) Field intensity, (b) circular-polarization degree, and (c) the largest imaginary part of the energy of excitations near the transition point between the branches σ+ and σ– of stationary solutions. The arrows indicate two-way spin switches that occur when the pump intensity increases or decreases in the same critical region. The parameters correspond to Fig. 1d.

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In order to lift the degeneracy in intensity between the solutions with opposite polarizations, generally speaking, only optical pumping with a nonzero circular-polarization degree \({{\rho }_{p}}\) would suffice. However, in the presence of the magnetic field, one branch of solutions can become stable or unstable, and the other branch near the same I value appears or disappears as such, similar to our example in Figs. 1d and 2. Such coincidences, which enable one to implement two-way spin transitions, are always possible in the case of sufficiently large \({{g}_{L}}{\text{/}}\gamma \) under the following conditions. First, the pump frequency should exceed the eigenfrequencies of both spin components of the field; the relation \(D \sim {{g}_{L}}\) is usually close to optimal if \({{g}_{M}} \sim {{g}_{L}}{\text{/}}2\). Second, the sign of the pump circular polarization should be opposite to the sign of \({{g}_{M}}\). In other words, the spin component that is more redshifted from Ep due to the magnetic splitting (\({{\psi }_{ - }}\) at \({{g}_{M}} > 0\) or \({{\psi }_{ + }}\) at \({{g}_{M}} < 0\)) should be excited more strongly by the light wave so that the solutions with opposite polarizations could ultimately have a comparable total intensity and be located in the same I region. The direction of the principal polarization axis, which in our model is determined by the phase difference \(\arg f_{ + }^{*}{{f}_{ - }}\), can be used as a fine-tuning parameter.

DYNAMICS OF SPIN TRANSITIONS

Figure 3 shows the solutions of Eqs. (2) obtained at the slow (for more than 100 ns) variation in the pump intensity I. The system is the same as in Fig. 2. It is seen that oscillations in the polarization degree in the range from –1 to +1 arise at a critical point where the solution with the right-hand circular polarization becomes unstable. Oscillations are chaotic; a small fragment of the explicit time dependence of the polarization degree is shown in Fig. 4. The polarization degree varies noticeably in a time of a few picoseconds (\( \lesssim {\kern 1pt} \hbar {\text{/}}\gamma \)). Despite the lack of periodicity, the signal shape demonstrates certain self-reproducing features, which indicate the presence of an attractor of the phase trajectory. We also note that no intermediate region with a gradual increase in oscillations of the polarization degree upon increasing I is seen in Fig. 3a. This fact independently indicates a transition to chaos, in contrast to the Hopf bifurcation, when stable limit cycles (regular self-oscillations) continuously arise from fixed points [28].

Fig. 3.
figure 3

(Color online) Transitions between stable stationary states under conditions of a slow (a) increase or (b) decrease in the coherent pump intensity. The parameters correspond to Figs. 1d and 2.

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Fig. 4.
figure 4

(Color online) Characteristic dynamics of \(\rho (t)\) in a small part of the chaotic region of Fig. 3a. During the presented time interval (0.3 ns), the intensity of the external field I changes by less than 0.1%.

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Chaos appears because the exponential growth of fluctuations takes the system away from the stationary solution that has become unstable, but cannot deliver it to a new stable branch. The trajectory of the chaotic system covers a wide region of the phase space, only randomly approaches the single asymptotically stable state that exists for a given I value, and eventually gets attracted to it. Thus, the entire transition occurs spontaneously under conditions of intense and irregular fluctuations of the polarization degree. It does not have a precisely defined duration; in calculations, the transition time varied in the range from \( \sim 0.1\) to \( \sim 10\) ns even for small changes in the parameters.

The reverse switching (Fig. 3b) occurs as expected after the disappearance of stationary solutions with left-hand circular polarization with decreasing I. In this case, chaotic oscillations are initially observed similar to those in Fig. 4. Despite the chaos, the regions of \({{\sigma }^{ + }}\, \leftrightarrow \,{{\sigma }^{ - }}\) transitions are located very close, unlike, e.g., Fig. 1c, where the sequential \({{\sigma }^{ + }}\, \to \,{{\sigma }^{ - }}\, \to \,{{\sigma }^{ + }}\) transition would require a change in I by more than an order of magnitude. In the case of conventional optical bistability, a decrease in the hysteresis region I would be accompanied by a convergence of alternative states in the phase space.

Because of the absence of the usual hysteresis effect, the final state of the system ceases to depend on the initial conditions and, in general, on the “history” of the excitation process. Figure 5 shows the phase diagram in the region of two-way polarization switching. Each point was obtained by averaging over thousand independent calculations with random initial conditions for the real and imaginary parts of \({{\psi }_{ + }}\) and \({{\psi }_{ - }}\), which are uniformly distributed on the interval from ‒1 to 1 meV1/2/V 1/2. In addition, white noise was added to the coherent excitation source, providing permanent fluctuations of \({\text{|}}{{\psi }_{ \pm }}(t){{{\text{|}}}^{2}}\) with an amplitude of about 10% of the equilibrium intensity. The evolution time of the system in each calculation was 10 ns, but the results that were eventually averaged were recorded only for the last 0.1 ns in order to reduce the effect of transient phenomena as much as possible.

Fig. 5.
figure 5

(Color online) Transitions between stable states under partially stochastic excitation. Each point corresponds to an averaged series of independent calculations carried out with random initial conditions; the quantities \({{f}_{ \pm }}\) in each calculation experienced time-dependent random perturbations. Colors distinguish the I intervals, where only one of the stationary solutions with \(\rho > 0\) and \(\rho < 0\) is stable and both solutions are stable. The system parameters correspond to Figs. 1d, 2, and 3.

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As seen in Fig. 5, the average polarization is close to zero in a very narrow I region where stable branches \({{\sigma }^{ + }}\) and \({{\sigma }^{ - }}\) exist simultaneously, and, therefore, the choice between them is random. (The width of this region can be easily controlled by varying the parameters.) The shift of I in any direction quickly leads to single-valued solutions corresponding to the stationary model described by Eq. (6). Random perturbations of \({{f}_{ \pm }}(t)\) do not distort the result, but rather play a constructive role since purely periodic (self-oscillatory) states with a low average polarization can occur in their absence. Finally, certain points in the transition region deviate even from the single (for the corresponding I) stable stationary state because the region of its attraction in the phase space is still very small, and a significant amount of time is required to fall within it accidentally, whereas, even an established solution can be destroyed by finite perturbations of \({{f}_{ \pm }}(t)\).

Despite the considered fine effects, in general, a nearly direct correspondence is achieved between the pump intensity and the sign of the circular polarization of radiation. The described mechanism of spin transitions is universal, does not require precise phase control, and can be implemented in both continuous and pulsed (at least on a scale of several nanoseconds) regimes of optical excitation of polaritons. Dynamical chaos is feasible under the condition of the linear coupling between spin components \(({{g}_{L}} > 0)\), which implies a lateral anisotropy of the system. In fact, the only thing that matters is the ratios of \({{g}_{L}}\) and \({{g}_{M}}\) to the linewidth \(\gamma \) that is determined by the quality factor of the cavity. Therefore, all technical restrictions regarding the anisotropy of the system or the required intensity of the magnetic field are relaxed with increasing quality factor.