Parametric amplification of light in a cavity with a moving dielectric membrane: Landau-Zener problem for the Maxwell field

We perform a theoretical investigation into the classical and quantum dynamics of an optical field in a cavity containing a moving membrane (“membrane-in-the-middle” setup). Our approach is based on the Maxwell wave equation and complements previous studies based on an effective Hamiltonian. The analysis shows that for slowly moving and weakly reflective membranes the classical field dynamics can be approximated by first-order-in-time evolution given by an effective Schrödinger-type equation with a Hamiltonian that does not depend on the membrane speed. This approximate theory is the one typically adopted in cavity optomechanics and we develop a criterion for its validity. However, in more general situations, the full second-order wave equation predicts light dynamics which do not conserve energy, giving rise to parametric amplification (or attenuation) that is forbidden under first-order dynamics and can be considered to be the classical counterpart of the dynamical Casimir effect. The case of a membrane moving at constant velocity can be mapped onto the Landau-Zener problem, but with additional terms responsible for field amplification. Furthermore, the nature of the adiabatic regime is rather different from the ordinary Schrödinger case since mode amplitudes need not be constant even when there are no transitions between them. The Landau-Zener problem for a field is therefore richer than in the standard single-particle case. We use the work-energy theorem applied to the radiation pressure on the membrane as a self-consistency check for our solutions of the wave equation and as a tool to gain an intuitive understanding of energy pumped into or out of the light field by the motion of the membrane.

Figure 1

Schematic of the amplitude of light in a double cavity with perfectly reflective end mirrors and a partially transmissive, movable central membrane.

Figure 2

The wave numbers of the normal modes inside a double cavity form a network of avoided crossings when plotted as a function of the difference in length between the two subcavities. The total length of the double cavity is $L=100\mu \mathrm{m}$. The red dashed lines correspond to a perfectly reflective central membrane ($\alpha \to \infty$). The green solid lines correspond to a membrane of reflectivity $98\%$ (i.e., $\alpha =1.7\times {10}^{-6}$ m), the magenta, small dotted lines correspond to a membrane reflectivity of $91\%$ (i.e., $\alpha =8.0\times {10}^{-7}\mathrm{m}$), the blue dashed dotted lines correspond to $61\%$ (i.e., $\alpha =3.1\times {10}^{-7}\mathrm{m}$), and the larger, black dotted lines correspond to a membrane reflectivity of $28\%$ (i.e., $\alpha =1.6\times {10}^{-7}\mathrm{m}$). All curves except the red curve have avoided crossings. The gap at the avoided crossing ($2\mathrm{\Delta }$) goes down as the reflectivity is increased.

Figure 3

Dynamics of an initially empty mode when traversing an avoided crossing at five different speeds. We simulated the field dynamics using the ASOEs given in Eq. (42) in the two-level approximation near an avoided crossing. $c_2$ is the amplitude associated with the upper adiabatic mode, where the initial condition is $c_1=1$ and $c_2=0$. According to Eq. (34), $|c_n{|}^2$ is proportional to the electromagnetic energy of the $n\mathrm{th}$ mode. We see that as the membrane speed goes down, the energy pumped into the initially unpopulated mode tends to zero. Parameters: membrane reflectivity $98\%$ (i.e., $\alpha =1.5\times {10}^{-6}\mathrm{m}$); length of double cavity $100\mu \mathrm{m}$; maximum membrane displacement $\mathrm{\Delta }L/2=\pm 1\times {10}^{-7}\mathrm{m}$. The adiabatic modes shown are those with $n=128$, where we label the modes in terms of the wave numbers for a perfectly reflecting membrane for which $k_n=2\pi n/(L\pm \mathrm{\Delta }L)$. These perfectly localized modes come in pairs that are degenerate at $\mathrm{\Delta }L=0$.

Figure 4

Dynamics of an initially excited mode when traversing an avoided crossing at different speeds as calculated using the ASOEs. This figure is for exactly the same setup as Fig. 3 except here we plot the results for the lower mode. We see that as the membrane speed is decreased, the energy of this mode is not conserved but has a slight upward curve. Combined with Fig. 3, this tells us that while the slowly moving membrane limit is sufficient to avoid nonadiabatic transitions, energy is not conserved.

Figure 5

The fractional change in total energy of the system as it traverses an avoided crossing as calculated using the ASOEs. Here, ${\mathcal{E}}_0$ is the initial energy and we use the same parameters as in Figs. 3 and 4. The plot shows that even at very slow membrane speeds, the energy change as a function of time does not vanish but instead tends to a limiting curve. Hence, even though $v\to 0$, we find ${\sum }_n{\left|c_n\left(\tau \right)\right|}^2\ne \mathrm{constant}$, confirming that adiabaticity does not imply energy conservation.

Figure 6

Dynamics on traversing an avoided crossing as seen in the diabatic basis, with the field initially localized on the right. Membrane speed: $5000{\mathrm{ms}}^{-1}$. All other parameters are the same as in Fig. 3. The results were calculated using both the ASOE and DSOE schemes with the two sets of curves lying right on top of each other.

Figure 7

A comparison of the fractional change in energy calculated using the ASOE and DSOE for the same avoided crossing dynamics as shown in Fig. 6 except that here we also vary the membrane speed. We see that the order of magnitude of difference between ASOE and DSOE is of the order of $1\times {10}^{-5}$. Here, $\mathrm{\Delta }{\mathcal{E}}_{\mathrm{ASOE}}/{\mathcal{E}}_0$ is generated by Eq. (31) and $\mathrm{\Delta }{\mathcal{E}}_{\mathrm{DFOE}}/{\mathcal{E}}_0$ is generated by Eq. (46). Although a speed of $20000{\mathrm{ms}}^{-1}$ seems very high, such effective speeds can be achieved by changing the background index of refractions rather than physically moving the mirror.

Figure 8

A comparison of the fractional change in energy calculated using the ASOE and DSOE for the same avoided crossing dynamics as shown in Fig. 6 except that here we also vary the membrane reflectivity. The results lead us to the same conclusion as in Fig. 7. Here, $\mathrm{\Delta }{\mathcal{E}}_{\mathrm{ASOE}}/{\mathcal{E}}_0$ is generated by Eq. (31) and $\mathrm{\Delta }{\mathcal{E}}_{\mathrm{DFOE}}/{\mathcal{E}}_0$ is generated by Eq. (46).

Figure 9

This figure shows the trend of agreement between DSOE and DFOE as we vary the membrane reflectivity for the same avoided crossing dynamics as shown in Fig. 6. For first-order dynamics, $\mathrm{\Delta }{\mathcal{E}}_{\mathrm{DFOE}}/{\mathcal{E}}_0$ has to be identically zero, while for second-order dynamics it is generally nonzero. Hence, the difference of this quantity from zero can be used to quantify the validity of the first-order model. As reflectivity goes up, the first-order approximation becomes less valid.

Figure 10

This figure shows the trend of agreement between DSOE and DFOE as we vary the membrane speed for the same avoided crossing dynamics as shown in Fig. 6. As our intuition would suggest, the first-order approximation becomes less valid for higher speeds.

Figure 11

A plot of the analytical condition given in Eq. (62) for the validity of the DFOE as a function of membrane reflectivity and speed. When $r$ is large, the DFOE is a good approximation. It can be clearly seen that $r$ becomes large at small speeds. The dependence on reflectivity is much gentler, but there is still a discernible monotonic increase in $r$ as the reflectivity is reduced. These trends are in agreement with the numerical results shown in previous sections, showing that the DFOE becomes a better approximation at low membrane speed and reflectivity.

Figure 12

Comparison of the change in energy $\mathrm{\Delta }\mathcal{E}$ of the optical field (solid blue curves) with the work done $\mathcal{W}$ by radiation pressure on the membrane (red dashed-dotted curves) during passage through an avoided crossing. Both quantities are in units of the initial energy ${\mathcal{E}}_0$. Equation (69) is used to calculate $\mathrm{\Delta }\mathcal{E}$ and the radiation pressure is obtained by summing the contributions given by Eq. (73) for the two modes separately. The agreement is good but breaks down at higher membrane reflectivities. The membrane speed is 5000 ${\mathrm{ms}}^{-1}$. The same mode amplitudes $\{c_1\left(t\right),c_2\left(t\right)\}$ were used for both sets of curves and were calculated using the ASOE. As usual, the light is initially localized on the right-hand side of the membrane.

Figure 13

Same as Fig. 12, except that the radiation pressure is calculated using Eq. (76) which includes interference between the contributions from each mode. As can be seen, the agreement is excellent at all reflectivities.

Figure 14

Evolution of the radiation pressure on the central membrane during passage through an avoided crossing for four different membrane reflectivities. The membrane speed is held constant at 5000 ${\mathrm{ms}}^{-1}$ and the radiation pressure is calculated using (76). The maximum radiation pressure is greater at larger membrane reflectivities, however, at $98\%$ reflectivity the radiation pressure exhibits oscillatory behavior due to the nonadiabatic nature of the optical dynamics.

Figure 15

Same as Fig. 14 except that here we fix the membrane reflectivity at $98\%$ but choose five different membrane speeds. The highest mirror speed leads to nonadiabatic optical dynamics and consequently an oscillatory behavior of the radiation pressure. This figure corresponds to the optical dynamics shown in Figs. 3 and 4.