Adiabatic transfer of light in a double cavity and the optical Landau-Zener problem

We analyze the evolution of an electromagnetic field inside a double cavity when the difference in length between the two cavities is changed, e.g., by translating the common mirror. We find that this allows photons to be moved deterministically from one cavity to the other. We are able to obtain the conditions for adiabatic transfer by first mapping the Maxwell wave equation for the electric field onto a Schrödinger-like wave equation and then using the Landau-Zener result for the transition probability at an avoided crossing. Our analysis reveals that this mapping only rigorously holds when the two cavities are weakly coupled (i.e., in the regime of a highly reflective common mirror) and that, generally speaking, care is required when attempting a Hamiltonian description of cavity electrodynamics with time-dependent boundary conditions.

Figure 1

Double-cavity setup consisting of two perfectly reflecting mirrors separated by a partially transmissive common central mirror. $\Delta L\equiv L_1-L_2$ is the difference in length between the two cavities.

Figure 2

The wave numbers $k$ of the modes in a double cavity plotted as a function of $\Delta L\equiv L_1-L_2$ which is the difference in length between the two cavities. The wave numbers are obtained by solving the transcendental equation [Eq. (7)]. When the reflectivity of the common mirror is infinite ($\alpha \to \infty$), we obtain the localized modes (dashed red lines), which are trapped on one side or the other of the mirror. When the reflectivity is finite we obtain the global modes (solid blue curves) which feature avoided crossings. The total cavity length is set at $L=1\times {10}^{-4}$m, and the global curves have $\alpha =1\times {10}^{-6}$m, which translates to a transmission probability of the common mirror of about $4\%$.

Figure 3

A closeup of one of the avoided crossings shown in Fig. 2 corresponding to $n=128$. Three pairs of curves are shown: The exact numerical solutions of Eq. (7) are plotted as solid (blue) curves, the approximate analytical solutions given by Eqs. (11) and (12) are plotted as short dashed (black) curves, and the localized solutions are plotted as long dashed (red) curves. The difference between the exact numerical solutions and the approximate analytical ones is hard to discern. Note that in contrast to the double-well problem in quantum mechanics, the even mode lies at a higher frequency, $\omega =ck$, than the odd mode.

Figure 4

This series of plots shows how each global wave mode swaps from one side of the double cavity to the other as the common mirror is moved through an avoided crossing. The odd mode $U_{n,\mathrm{odd}}\left(x\right)$ plotted as a function of position $x$ is shown in the left column, and the even mode $U_{n,\mathrm{even}}\left(x\right)$ is shown in the right column. Each row corresponds to a different value of $\Delta L/L$: The top row has $\Delta L/L=-4\times {10}^{-4}$, the second row has $\Delta L/L=-0.6\times {10}^{-4}$, the middle row has $\Delta L/L=0$, the fourth row has $\Delta L/L=0.6\times {10}^{-4}$, and the bottom row has $\Delta L/L=4\times {10}^{-4}$. Strictly speaking, the modes have well-defined parity only at $\Delta L=0$, but it is convenient to retain the labels “even” and “odd” for $\Delta L\ne 0$. The parameters used to make this figure were $L=1\times {10}^{-4}$ m, $\alpha /L=0.3$, and $n=128$ so $k\approx 8\times {10}^6$ m${}^{-1}$.

Figure 5

(Color online) The relative amplitudes $A_n/B_n$ and $B_n/A_n$ of the modes as a function of the displacement $\Delta L$ of the common mirror. The parameters used to make this plot were $\alpha /L=0.3$ and $n=128$.

Figure 6

(Color online) The absolute value of the maximum value the relative amplitude $B_n/A_n$ can take as a function of the common mirror parameter $\alpha$. The exact numerical result is plotted as the solid curve, and the approximate analytical result, as given by Eq. (23), is plotted as the dashed curve. Note that the difference between these two curves is hard to discern. We set $n=128$ to make this plot.

Figure 7

Comparison of second-order-in-time dynamics, as given by the full Maxwell wave equation (48), versus first-order-in-time dynamics, as given by the Schrödinger-like equation (55), as the mirror is swept through an avoided crossing. The initial conditions are the same in every panel: $A_L=1$ and $A_R=0$ at $\tau =-25$, where $\tau \equiv \Delta t/\hslash$ is dimensionless time. The first-order-in-time dynamics are also the same in every panel and are set by $\tilde{\vartheta }\equiv \hslash \vartheta /{\Delta }^2=1$, which is the single dimensionless parameter appearing in the Landau-Zener formula (27) and in Eq. (55). The second-order-in-time dynamics depend additionally on the value of $\Delta /\hslash {\omega }_{\mathrm{av}}$; panel a has $\Delta /\hslash {\omega }_{\mathrm{av}}=1/100$, panel b has $\Delta /\hslash {\omega }_{\mathrm{av}}=1/500$ and panel c has $\Delta /\hslash {\omega }_{\mathrm{av}}=1/1000$. Four curves are shown in each panel: the two curves (black and red) that begin on the upper left at $\tau =-25$ both give $A_L$. The black curve (which climbs higher) is calculated from Eq. (48). The red curve is given by Eq. (55). The other two curves (blue and green) that end on the upper right at $\tau =50$ give $A_R$. The blue curve (which is higher) is calculated from Eq. (48) and the green curve is from Eq. (55).

Figure 8

(Color online) A closeup of Fig. 7 combining the upper right part of all the panels in order to show the time dependence of $A_R$ at the end of the sweep. Like Fig. 7, this figure illustrates how the solutions of the (second-order) Maxwell equation reduce to those of the (first-order) Schrödinger equation as $\Delta /\left(\hslash {\omega }_{\mathrm{av}}\right)\to 0$. The solid black curve is calculated from the first-order Eqs. (55), and the remaining curves from the second-order Eqs. (48); $\Delta /\left(\hslash {\omega }_{\mathrm{av}}\right)=1/100$ (blue short-dashed line), $\Delta /\left(\hslash {\omega }_{\mathrm{av}}\right)=1/500$ (green medium-dashed line), and $\Delta /\left(\hslash {\omega }_{\mathrm{av}}\right)=1/1000$ (red long-dashed line).

Figure 9

(Color online) A plot showing the deviation of $|A_L{|}^2+{\left|A_R\right|}^2$ from unity as a function of time, as calculated from the full two-mode Maxwell equations (48) which include the second-order time derivatives. As shown in Eq. (57), this quantity is proportional to the energy stored in the electromagnetic field. Each curve in this plot has $\tilde{\vartheta }=1$, but the upper curve (short dashed blue line) has $\Delta /\hslash {\omega }_{\mathrm{av}}=1/100$, the middle curve (medium dashed green line) has $\Delta /\hslash {\omega }_{\mathrm{av}}=1/500$, and the lower curve (long dashed red line) has $\Delta /\hslash {\omega }_{\mathrm{av}}=1/1000$.

Figure 10

A plot showing the various regimes discussed in Sec. 8 as a function of the transmission $T$ of the common mirror and ratio of the free spectral range ${\omega }_{\mathrm{FSR}}$ to the average optical frequency ${\omega }_{\mathrm{av}}$. The Schrödinger-like equation (55) is valid below the dashed line, which is given by Eq. (70), the left-hand side of which we have assumed to be equal to ${10}^4$. This is a very conservative estimate but is still a factor of 10 less than the value given by assuming the parameters: $L=100$ $\mu$m, $\alpha =0.3$ L, and $\lambda =780$ nm. If the Schrödinger-like theory holds, then we can apply the Landau-Zener criterion for adiabaticity, given by Eq. (69) and which is plotted as the solid line. To plot this line we assumed adiabaticity sets in when the left-hand side is greater than 10 (the exponential dependence of $P_{\mathrm{LZ}}$ certainly ensures that this is the case). The regime of adiabatic evolution of the electromagnetic field is above the solid line. The shaded region shows the regime where the Schrödinger-like theory applies and the evolution it predicts is adiabatic.

Figure 11

A comparison of the transmission probability $T$ for the finite and $\delta$-function mirrors, as given by Eqs. (B1) and (B2), respectively. The resonances of the finite mirror occur at the peaks of its transmission function. For the parameters used in this plot, which are the same as those given in the caption of Fig. 12, the two curves intersect near a minimum of the finite barrier transmission. This means that the two transmission probabilities remain in good agreement for a larger range of $k$ than they might otherwise do.

Figure 12

Comparison between the wave number structure produced by a mirror of width $2M$ (solid blue curves) and the $\delta$ mirror (dashed red curves) inside a double cavity. This plot is for the case when we are far from a mirror resonance. In order to make the comparison we matched the transmission probabilities $T$ for each mirror over the relevant range of wave numbers as best we could (see Fig. 11). Up to a relative vertical shift, the agreement is very good, including the magnitude of the gap $2\Delta$ at the avoided crossings. In the calculation for the $\delta$-function mirror we set $\alpha /(L-2M)=0.009735$. In the finite mirror calculation we set $M/(L-2M)=0.001334$ and $2n_r^{2}M/(L-2M)=0.1707$. Note that to make this plot clearer we have included only the wave numbers for a single pair of modes in each case.

Figure 13

The wave number structure produced by a mirror of width $2M$ inside a double cavity. This plot includes wave numbers for which there is a mirror resonance. At the bottom we see the familiar avoided crossing structure which couples even and odd modes with the same index $n$, but for higher wave numbers we pass into resonance and the structure changes to one where there are vertical avoided crossings that allow for the possibility of adiabatically coupling localized cavity modes with different values of the index $n$. The parameters used to make this plot were $M/(L-2M)=0.0005005$ and $2n_r^{2}M/(L-2M)=0.01001$.

Figure 14

A plot of the relative amplitudes versus mirror displacement $\Delta L$. It is clear that one only needs small displacements to achieve large amplitude transfers. Here $\alpha /L=0.3$ and $n=128$.

Figure 15

(Color online) By simultaneously moving all three mirrors, it is possible to double the effective speed of the common mirror, i.e., quadruple the rate of change of the relative cavity length $\Delta L\left(t\right)=L_1\left(t\right)-L_2\left(t\right)$.

Figure 16

Time derivatives of the diabatic amplitudes, as obtained by numerical solution of Eq. (55). The green curve (that takes the value one at $\tau \equiv t\Delta /\hslash =-50$) gives the modulus of $d{\tilde{A}}_R/d\tau$, and the red curve gives the modulus of $d{\tilde{A}}_L/d\tau$. The values of the parameters are set to $\tilde{\vartheta }\equiv \hslash \vartheta /{\Delta }^2=1$.

Figure 17

Early time behavior of the real part of ${\tilde{A}}_L$. The solid red curve gives the numerical solution of Eq. (55) and the dashed black curve gives the approximate analytic solution given by Eq. (E6). The values of the parameters are the same as those in Fig. 16.

Figure 18

Early time behavior of the imaginary part of ${\tilde{A}}_L$. Two curves are plotted, one giving the numerical solution of Eq. (55) and the other giving the approximate analytic solution as given by Eq. (E6), but the difference between the two is hardly visible for the times shown. Only for times greater than $\tau =-22$ do differences really become apparent. The values of the parameters are the same as those in Fig. 16.