Temperature Gradient Driven Lasing and Stimulated Cooling

A laser can be understood as a thermodynamic engine converting heat to a coherent single mode field close to Carnot efficiency. To achieve lasing, spectral shaping of the excitation light is used to generate a higher effective temperature on the pump than on the gain transition. Here, using a toy model of a quantum well structure with two suitably designed tunnel-coupled wells kept at different temperatures, we predict that lasing can also occur on an actual spatial temperature gradient between the pump and gain regions. Gain and narrow band laser emission require a sufficiently large temperature gradient and resonator quality. Lasing appears concurrent with amplified heat flow between the reservoirs and points to a new form of stimulated solid state cooling. In addition, such a mechanism could reduce intrinsic heating and thus extend the operating regime of quantum cascade lasers by substituting phonon emission driven injection by a phonon absorption step.

Figure 1

Sketch of the two-well structure and the energy levels. The wave functions are assumed to be well localized in each well, while the coupling is described by the tunneling constants $J_1$ and $J_2$. An alternative design including an absorptive step between $|I_4⟩$ and $|u^{\prime }⟩$ is indicated in green (gray).

Figure 2

(a) Numerical solution for the steady-state number of photons $⟨a^{†}a{⟩}_{\mathrm{st}}$ in the cavity as a function of $T_1$ and $T_2$ for $N={10}^7$. The dashed line indicates a minimum $T_{1,\mathrm{th}}$ for cooperative behavior (valid for $T_2$ small enough to set $n_1=n_3=n_4=0$). (b) $⟨a^{†}a{⟩}_{\mathrm{st}}$ as a function of $T_1$ and $N$ ($T_2=0.1\mathrm{K}$). The analytical solution gives $N_{\mathrm{th}}(T_1)$ (dashed line) and reliably predicts the onset of cooperative behavior and a significant increase in $⟨a^{†}a{⟩}_{\mathrm{st}}$.

Figure 3

Occupation ratio in the first quantum well for the coupled and uncoupled case versus $T_2$. The temperature of the hot part $T_1$ is fixed to 400 K. For $T_2<9\mathrm{K}$ we see a reduction of $⟨|I_{3,j}⟩⟨I_{3,j}|⟩/⟨|I_{2,j}⟩⟨I_{2,j}|⟩$ (solid line) compared to the uncoupled case (dashed line).

Figure 4

Linewidth of the cavity output spectrum for varying $T_1$ and $T_2$ with $N={10}^7$ (plotted on a logarithmic scale). The linewidth (FWHM) of the empty cavity $2\frac{\kappa }{2\pi }=1\mathrm{MHz}$ corresponds to the dark red (gray) color (upper boundary of the scale). The linewidth $\Delta f$ falls slightly below that limit with increasing $T_1$, until it finally drops off dramatically as the cooperative effects emerge. For small $T_2$ the onset of cooperative behavior is marked by $T_{1,\mathrm{th}}$.

Figure 5

(a) Number of photons in the mode versus $T_1$ and $T_2$. For $T_2>T_{2,\mathrm{th}}$ we find a sudden increase in $⟨a^{†}a{⟩}_{\mathrm{st}}$ and in (b) a corresponding drop in the linewidth of the cavity emission. This is due to the absorptive step in the revised model where ${\omega }_u^{\prime }={\omega }_u+2\pi \times 0.05\mathrm{THz}$. The remaining parameters are unchanged except for ${\gamma }_3={\gamma }_1=2\pi \times 4\times {10}^2\mathrm{Hz}$.