Dissipation engineered directional filter for quantum ratchets

We demonstrate transport rectification in a Hermitian Hamiltonian quantum ratchet by a dissipative, dynamic impurity. While the bulk of the ratchet supports transport in both directions, the properly designed loss function of the local impurity acts as a direction-dependent filter for the moving states. We analyze this scheme theoretically by making use of Floquet $S$-matrix theory. In addition, we provide direct experimental observation of one-way transmittance in periodically modulated plasmonic waveguide arrays containing a local impurity with engineered losses.

Figure 1

Sketch of the dimerized tight-binding Floquet chain with local time-periodically modulated decay rates ${\gamma }_{\mathrm{A}/\mathrm{B}}\left(t\right)$ and hopping amplitudes $J_1\left(t\right)$ and $J_2\left(t\right)$. We partition the chain in sublattices A/B and label the unit cell by $j$. The reflection $r_{k,\alpha }$ and transmission $t_{k,\alpha }$ of the Floquet state with quasimomentum $k$ and band index $\alpha$ are schematically indicated by arrows.

Figure 2

(a) Band structure in the first FBBZ at three different driving frequencies: 1, $\omega =0.597J_0$; 2, $\omega =1.195J_0$; and 3, $\omega =2J_0$. The color code shows the corresponding group velocity $v_{\alpha ,k}^{\mathrm{g}}$. (b) Average absolute value of the group velocity as a function of $\omega$. Gray lines mark values from Eqs. (3) and (4) and numbers highlight the frequencies from panel (a). (c) Quasienergy spectra in dependence on the driving frequency. (d) Squared absolute value of the state amplitude at $\omega =1.195J_0$ with positive ($\alpha =1$) and negative ($\alpha =2$) group velocities along one period at sublattices A and B. Red lines show the time-dependent losses ${\gamma }_{A/B}\left(t\right)$ on sublattices A and B for $\phi =0$ which consist of temporal intervals $L$ of the constant decay rate ${\gamma }_0$. Note that in our model ${\gamma }_{A/B}\left(t\right)$ are applied only to the central unit cell as illustrated in Fig. 1.

Figure 3

Transmission of (a) right and (b) left movers over the impurity parameters ${\gamma }_0$ and $L$ at ${\omega }_0=1.195J_0$. (c) Decadic logarithm of $T_{\alpha =1}/T_{\alpha =2}$ for ${\omega }_0=1.195J_0$; white line marks the function ${\max }_L(T_{\alpha =1}/T_{\alpha =2})\left({\gamma }_0\right)$. (d) Transmission (blue) and reflection (red) of right (solid) of left (dashed) moving states over the shift angle $\phi$ for ${\gamma }_0=1.5J_0$ and $L=0.25T$. (e) Transmission and (f) reflection of right movers for $L=0.25T$ and ${\gamma }_0=1.5J_0$ over driving frequency $\omega$ and quasimomentum $k$.

Figure 4

(a) Sketch of a plasmonic waveguide array featuring unidirectional transmittance. Green and red arrows indicate the low- and high-loss directions, respectively. (b) SEM scan of a typical sample. Inputs A and B are marked by red boxes and blue dotted lines to highlight the region with periodic dissipation. The chromium stripe used to implement losses is magnified in the top right corner.

Figure 5

(a) Measured mean group velocity of a wave packet vs driving frequency $\omega$ for the single-site excitation at the input A. (b) Real-space SPP intensity distributions corresponding to the arrays modulated with frequency ${\mathrm{\Omega }}_1=0.23\mu {\mathrm{m}}^{-1}$ and ${\mathrm{\Omega }}_2=0.39\mu {\mathrm{m}}^{-1}$. (c) Fourier-space SPP intensity distributions for the same arrays as in panel (b).

Figure 6

Real space intensity distributions for the DLSPPW arrays with local modulated dissipation (highlighted by white dashed lines) featuring unidirectional transmittance at ${\mathrm{\Omega }}_1$. The wave packet is excited at $x>0$ (input B, top) or at $x<0$ (input A, bottom). The area plots at the right side from the real-space data show the intensity distribution after the propagation distance $z=5T$. (a) Chromium stripes with $L=0.3T$ were deposited below two waveguides. The red arrows point to reflected wave. (b) The same as in panel (a), but the length of the Cr stripe was reduced to $L=0.15T$.