Scattering theory for Floquet-Bloch states

Motivated by recent experimental implementations of artificial gauge fields for gases of cold atoms, we study the scattering properties of particles that are subjected to time-periodic Hamiltonians. Making use of Floquet theory, we focus on translationally invariant situations in which the single-particle dynamics can be described in terms of spatially extended Floquet-Bloch waves. We develop a general formalism for the scattering of these Floquet-Bloch waves. An important role is played by the conservation of Floquet quasienergy, which is defined only up to the addition of integer multiples of $\hslash \omega$ for a Hamiltonian with period $T=2\pi /\omega$. We discuss the consequences of this for the interpretation of “elastic” and “inelastic” scattering in cases of physical interest. We illustrate our general results with applications to the scattering of a single particle in a Floquet-Bloch state from a static potential and the scattering of two bosonic particles in Floquet-Bloch states through their interparticle interaction. We analyze examples of these scattering processes that are closely related to the schemes used to generate artificial gauge fields in cold-atom experiments, through optical dressing of internal states, or through time-periodic modulations of tight-binding lattices. We show that the effects of scattering cannot, in general, be understood by an effective time-independent Hamiltonian, even in the limit $\omega \to \infty$ of rapid modulation. We discuss the relative sizes of the elastic scattering (required to stabilize many-body phases) and of the inelastic scattering (leading to deleterious heating effects). In particular, we describe how inelastic processes that can cause significant heating in the current experimental setup can be switched off by additional confinement of transverse motion.

Figure 1

Unbounded single-particle dispersion ${\epsilon }_k^0/\left(\hslash \omega \right)\propto k^2$ as a function of dimensionless wave vector $k$ reduced to the first BZ (bold lines) and continuous dispersion with periodically repeated images (dashed lines). Scattering an initial state [red (middle) circle] to final state [green (left) circle] would be considered elastic and scattering from initial state [red (middle) circle] to a different branch [blue (right) circle] would be considered inelastic. When regarding these processes with respect to the continuously defined dispersions, elastic scattering corresponds to no change in the Floquet index $m$, whereas inelastic scattering changes $m$.

Figure 2

Bounded single-particle dispersion $\epsilon \left(k\right)/\left(\hslash \omega \right)=-0.4cos\left(k\right)$ as a function of dimensionless wave vector $k$ in the energy BZ with periodically repeated images (left) and contours of the reduced two-particle quasienergy (right). Depicted is a two-particle scattering process during which two particles initially in the band minimum [red (middle) circles] scatter into higher quasienergy states [blue (left and right) circles]. This process conserves the reduced two-particle quasienergy or equivalently can be viewed as one-particle scattering into the lower shifted dispersion (gray circle).

Figure 3

Dispersion $E/E_r$ as a function of $k/k_r$ [see Eq. (50)], including an energy offset to have zero minimum. The top row shows the dispersion for $\hslash \delta /E_r=0$ and $\hslash \Omega /E_r=1$ on the left and $\hslash \Omega /E_r=4$ on the right and the bottom row for the same parameters in the case of $\hslash \delta /E_r=1$.

Figure 4

Stability diagram for (a) $\hslash \delta /E_r=0$ and (b) $\hslash \delta /E_r=16$ with initial state of two particles with quasimomentum $k=k_0$ in the lower band. For no detuning and $\hslash \Omega /E_r<4$ both cases of $k=+|k_0|$ (dashed lines) and $k=-|k_0|$ (solid lines) are shown, where detuning the minimum is unique. Shaded regions correspond to parameter regimes in which inelastic scattering is allowed. The bottom region (A) in light blue corresponds to inelastic scattering where both particles remain in the lower bands, i.e., the $(-,-)$ final state, in the middle region (B) particles can scatter into either the $(-,-)$ or the $(+,-)$ final state, and in the top region (C) scattering into all states $(-,-),(+,-)$ and $(+,+)$ is allowed. The thick dashed lines of constant $\omega \hslash /E_r$ and of constant $\Omega \hslash /E_r$ correspond to the cuts along which the scattering rate is shown in Figs. 5 and 6.

Figure 5

Dimensionless scattering rate ${\Gamma }^{1\mathrm{D}}$ [see Eq. (54)] for an initial state with particles in the lower band with momentum $k=\pm |k_0|$ getting inelastically scattered. In the top row, for no detuning, $\hslash \delta /E_r=0$ as a function of $\omega \hslash /E_r$ for fixed $\hslash \Omega /E_r=4$ on the left and as a function of $\hslash \Omega /E_r$ for fixed $\omega \hslash /E_r=7$ on the right (dashed $k=+|k_0|$, full $k=-|k_0|$), and at the bottom, for $\hslash \delta /E_r=16$ as a function of $\omega \hslash /E_r$ for fixed $\hslash \Omega /E_r=16$ on the left and as a function of $\hslash \Omega /E_r$ for fixed $\omega \hslash /E_r=20$ on the right, as indicated in Fig. 4. The rate shows divergences at the opening and closing of scattering channels corresponding to the borders in Fig. 4, at which the density of states of the final states diverges.

Figure 6

Dimensionless scattering rate ${\Gamma }^{2\mathrm{D}}$ [see Eq. (59)] for an initial state with particles in the lower band with momentum $k=\pm |k_0|$ and relative momentum $k_y=0$ getting inelastically scattered for the extension to a 2D setting with free motion in a transverse direction. In the top row, for no detuning, $\hslash \delta /E_r=0$ as a function of $\omega \hslash /E_r$ for fixed $\hslash \Omega /E_r=4$ on the left and as a function of $\hslash \Omega /E_r$ for fixed $\omega \hslash /E_r=7$ on the right (dashed $k=+|k_0|$, full $k=-|k_0|$), and at the bottom, for $\hslash \delta /E_r=16$ as a function of $\omega \hslash /E_r$ for fixed $\hslash \Omega /E_r=16$ on the left and as a function of $\hslash \Omega /E_r$ for fixed $\omega \hslash /E_r=20$ on the right, as indicated in Fig. 4. The rate shows jumps at the opening and closing of scattering channels corresponding to the borders in Fig. 4.

Figure 7

Sketch of the 1D time-periodically driven lattice potential given in Eq. (60). The lattice is staggered with an energy offset $V_2$ between neighboring sites, which suppresses tunneling along the lattice. Tunneling is then restored by resonantly modulating the site energies with a modulation strength $V_{\omega }$ at frequency $\hslash \omega =V_2$.

Figure 8

Quasienergies of the resonantly modulated lattice, Eqs. (70) and (71), as a function of the quasimomentum $k$ in arbitrary energy units. The two lowest bands of the original lattice ($a$) and ($b$) both split into two subbands $\tau =\pm$, which are degenerate at the BZ boundaries. Depicted is a typical situation in which the energy of the periodic modulation $\hslash \omega$ is larger than the bandwidth of the lowest band and smaller than the band gap ${\Delta }_g$. The circles show our initial state with two particles in the lowest band and a possible final state with two particles in the upper band after scattering. For this plot a nearest-neighbor tight-binding dispersion ${\epsilon }^{\left(n\right)}\left(k\right)={\mathrm{t}}^{\left(n\right)}cos\left(k\right)$ is assumed with parameters ${\mathrm{t}}^a=1.1$, ${\mathrm{t}}^b=2.3$, $\hslash \omega =4.8$, ${\Delta }_g=10$, and $\kappa =1$.

Figure 9

Stability diagram of the lowest band in a time-periodically modulated 1D lattice of depth $V_1=6E_r$ for which the band gap is $4.75E_r$. Shaded regions correspond to energetically allowed scattering from the ground state into the first excited band. Different lobes correspond to different orders of the instability $m$ starting with $m=1$ at the top and increasing downwards. In (a) only the first four such lobes are shown for clarity; in (b) the region $1\le \omega \hslash /E_r\le 3$ with the $m=4,5,6,7$ lobes which overlap the $m=5$ lobe are shown. The dashed lines of constant $\omega \hslash /E_r$ correspond to the cuts along which the scattering rate is shown in Fig. 10.

Figure 10

Dimensionless scattering rate ${\Gamma }^{1\mathrm{D}}$ [see Eq. (79)] along the cuts $\hslash \omega /E_r=\text{const}.$, as indicated in Fig. 9 for particles in the first band ($a$) with momentum $k=0$ scattering into the second band ($b$). The left panel corresponds to $\hslash \omega /E_r=10$ for which $m=1$ is the only scattering channel, whereas the right panel corresponds to $\hslash \omega /E_r=2$ for which $m=5$ transitions are allowed for $0.6\le V_{\omega }/E_r\le 6.9$ and both $m=4,5$ for $2.1\le V_{\omega }/E_r\le 5.3$.

Figure 11

Dimensionless scattering rate ${\Gamma }_{a\to b}^{2\mathrm{D}}$ for the extension to a weakly confined system with a free transverse degree of motion [see Eq. (80)] along the cuts $\hslash \omega /E_r=\text{const}.$ as indicated in Fig. 9 for particles in the first band with quasimomentum $k=0$ and relative momentum $k_y=0$ scattering into the second band, integrated over the final states with crystal momentum $q$ and relative momentum $q_y$. The left panel corresponds to $\hslash \omega /E_r=10$, for which $m\ge 1$ are the available inelastic scattering channel, whereas the right panel corresponds to $\hslash \omega /E_r=2$, for which $m\ge 5$ transitions are allowed for $0.6\le V_{\omega }/E_r\le 6.9$ and $m\ge 4$ for $2.1\le V_{\omega }/E_r\le 5.3$.

Figure 12

Dimensionless scattering rate ${\Gamma }_{a\to a}^{2\mathrm{D}}$ for the extension to a weakly confined system with a free transverse degree of motion [see Eq. (80)] along the cuts $\hslash \omega /E_r=\text{const}.$, as indicated in Fig. 9 for particles in the first band with quasimomentum $k=0$ and relative momentum $k_y=0$ scattering inelastically and remaining in the first band, integrated over the final states with crystal momentum $q$ and relative momentum $q_y$. The left panel corresponds to $\hslash \omega /E_r=10$ and the right panel corresponds to $\hslash \omega /E_r=2$. In both cases all processes with $m\ge 1$ are inelastic and allowed as the particles remain in the same band and the energy in $y$ direction is not gapped.