Multi-threshold second-order phase transition

We present a theory of the multi-threshold second-order phase transition, and experimentally demonstrate the multi-threshold second-order phase transition phenomenon. With carefully selected parameters, in an external cavity diode laser system, we observe second-order phase transition with multiple (three or four) thresholds in the measured power-current-temperature three dimensional phase diagram. Such controlled death and revival of second-order phase transition sheds new insight into the nature of ubiquitous second-order phase transition. Our theory and experiment show that the single threshold second-order phase transition is only a special case of the more general multi-threshold second-order phase transition, which is an even richer phenomenon.

In this Letter, we present a theory of the multithreshold SOPT, and demonstrate experimentally the multi-threshold SOPT phenomenon in an external cavity diode laser system with carefully selected parameters. Our results show that the single threshold SOPT is only a special case of the more general multi-threshold SOPT. Our theory and experiment imply that previous studied fascinating SOPTs, such as spontaneous magnetization [17], superfluidity [18,19] and superconductivity [20][21][22], which are solely limited to single criticalpoint behavior, may be extended to the multi-threshold SOPT.
The Landau expression for the free energy F near the critical point [1,2], a function of an order parameter ψ, is generally applied to all aforementioned SOPTs. Here, the coefficient α = α c (T − T c ), is assumed by Landau to be a simple linear function of temperature T when T is lower than the critical temperature T c . F n is the free energy of the normal phase. However, if the coefficient α consists of periodic function of T, for example, a trigonometric function, the system can evolve to a multi-threshold SOPT phenomenon with suitable experimental parameters, as demonstrated * Electronic address: jbchen@pku.edu.cn in this Letter. In this case, the free energy is expressed as, Each period of the cosine function of T produces a threshold. As we will see later, in this case the order parameter ψ will be a function with a periodic pattern of T. If φ(T ) only weakly depends on the temperature T, Eq.(2) is reduced to Eq.(1).
The lasing action of a laser can be considered as a Bose-Einstein condensation of photons, which also belongs to the SOPT. The analogies with superconductivity and ferromagnet, developed by Haken, Scully and others [10][11][12][13][14][15][16], show that the effective Ginzburg-Landau free energy of laser is, Where the amplitude of electric field u corresponds to the order parameter ψ in Eq.(1) and Eq.(2), G 0 corresponds to F n . A is the gain in the active medium, B is the saturation coefficient, C is the cavity loss rate, and the photon density in laser cavity is n.
In order to experimentally demonstrate the multithreshold SOPT in a laser, the key challenge is to carefully set the gain and cavity loss term A-C in Eq.(3) to be a periodic function of the effective temperature (the population inversion in laser corresponds to negative temperature [23]), which is related to injection current I in a diode laser.
Since the refractive index of semiconductor chip depends on the injection current(see Eq.(12)), it can be shown that for a diode laser system, In Landau's theory, the dependence of the order parameter on the effective temperature near the critical point is determined by the condition that the free energy as a function of the order parameter is minimized [1,2,6]. For a laser near the threshold, from Eq. (3) we obtain Near and above threshold, the exact solution of the photon density obtained from general laser rate equations involving carrier density N and photon density n in a diode laser, should be, Here the gain A and saturation coefficient B are Particularly, the cavity loss rate C is a trigonometric function of I and is given by where, R3−R2 , q is the electron charge, φ 0 is a constant phase, and other symbols are listed in Table 1. Equation (13) describes the effect of injection current through thermal effect and threshold gain through carrier density on the refractive index [24,25].
It should be noted that, Eq.(8) has the same form as the exact solution of laser quantum theory [16]. This expression of Eq.(8) can be reduced to Eq.(5) when laser is near threshold, i.e. A≈C. The measured output power of laser is hence given by P=κCnhνV p , where κ is the output coupling efficiency, hν is the photon energy, V p is the light field mode volume.
In an external cavity diode laser (ECDL) system,with carefully selected parameters for realizing a desired φ(I) in A-C expression, we observed SOPT with multiple (three or four) thresholds in the measured power-current  Figure 1a shows the measured three-dimensional (3D) phase diagrams of multi-threshold SOPT of the ECDL's output power as a function of the diode temperature T d and the injection current I. It can be clearly seen that up to four lasing thresholds can exist at a certain operating diode temperature and that the laser revives at a new threshold after each "death". The multiple threshold currents as a function of the operating diode temperature are shown in Fig. 1b. A linear dependence of threshold current on diode temperature T d can be observed, and the vertical displacement between two adjacent lines are almost 12.3 mA. The thresholds of the injection current I decrease as the diode temperature T d is increased, which is in contrast to that in a usual ECDLs [24,25] where the threshold current increases with the diode temperature. In Fig. 1c, we show the detailed phase diagrams of our laser system (the measured power-current curves at different diode temperatures). In our experiment, GaAs diode laser at a wave-length λ = 830 nm with the output facet coated with a multi-layer antireflection coating was operated with the standard Litterman-Metcalf external cavity configuration. The size of the active region of laser diode is 0.1µm×2µm×300µm (height×width×length). The distance between laser diode and optical grating over a range from 35 mm to 1000 mm does not affect the results. The calculations shown in Fig.3b are based the laser parameters listed in Table 1, and the nonlinear gain effect has been included in calculation.
In the same diode laser, measurements during the last 38 months show exactly the same multi-threshold SOPT behavior. Once the laser is "alive", it operates at a single longitudinal mode, and the laser wavelength is identical at the beginnings of different thresholds. The beam profile of our ECDL is shown in Fig. 2, which clearly shows that it operates at both single longitudinal mode and fundamental Gaussian traverse mode. When the laser is at "death" state, the measured fluorescence spectrum is as wide as 20 nm. We have also measured the P -I curves at different lasing wavelengths within the 20 nm gain range, and the multi-threshold SOPT consistently exists. Fig. 3a shows the dependence of the ECDL's output power as a function of the injection current I at diode temperature T d = 23 o C. The corresponding theoretical result obtained by soloving Eq. (8) is given in Fig. 3b. The blue curves of both figures are the measured laser power as the injection current increased from 100 mA to 160 mA, while the red curves are the measured results when the injection current is decreased. Four thresholds exist in both cases, with the first threshold appearing at I=107 mA (there is no threshold below 100 mA injection current). Each laser threshold is followed with a bi-stable state, which corresponds to the first-order phase transition [26]. In our case, when the injection current is gradually increased, the laser begins to lase at each threshold, and its power drops to zero almost suddenly after reaching a maximum value. However, if the injection current is decreased, the lasing action begins at a point that is lower than the previous maximum, and the output power decreases linearly to zero at the same threshold.
Although Landau's theory of the SOPT [1, 2] and Ginzburg-Landau equations of superconductivity [6,27] have been used to describe laser action [11][12][13][14][15][16]28], all fascinating SOPT explained so far in numerous physical systems such as spontaneous magnetization [17], superfluidity [18,19] and superconductivity [20][21][22], are solely limited to single critical point category. We believe that the SOPT with multiple critical points described by Eq.(2) is more general and even richer phenomenon than the conventional SOPT. The single threshold behavior is only a special case of the multi-threshold SOPT as Eq. (2) can be reduced to Eq.(1) if φ(T ) does not depend on the temperature T. The simple laser rate equations with the gain and cavity loss term A-C depending linearly on population inversion only describes single-threshold SOPT [11][12][13][14][15][16]29] or multi-threshld of first-order phase transition [16,29]. We have also observed the multithreshold SOPT phenomenon in other diode lasers operating at a wavelength of 852 nm. The key is to carefully select the parameters to satisfy the gain and loss condition shown in Eq.(4).
In summary, we proposed and demonstrated that once the effective temperature periodically affects the order parameter, a physical system described by Landau's theory of SOPT can be extended to exhibit multi-threshold SOPT. Our findings lead to a natural question: can multi-threshold SOPT also emerge in other physical systems, such as superconductivity, superfluidity, ferromagnetic system, Bose-Einstein condensates etc., which usually display single SOPT as in a laser?