Quantum wakes in lattice fermions

The wake following a vessel in water is a signature interference effect of moving bodies, and, as described by Lord Kelvin, is contained within a constant universal angle. However, wakes may accompany different kinds of moving disturbances in other situations and even in lattice systems. Here, we investigate the effect of moving disturbances on a Fermi lattice gas of ultracold atoms and analyze the novel types of wake patterns that may occur. We show how at half-filling, the wake angles are dominated by the ratio of the hopping energy to the velocity of the disturbance and on the angle of motion relative to the lattice direction. Moreover, we study the difference between wakes left behind a moving particle detector versus that of a moving potential or a moving particle extractor. We show that these scenarios exhibit dramatically different behavior at half-filling, with the “measurement wake” following an idealized detector vanishing, though the motion of the detector does still leaves a trace through a “fluctuation wake.” Finally, we discuss the experimental requirements to observe our predictions in ultracold fermionic atoms in optical lattices.

Figure 1

A lattice of cold atoms interacting with a moving disturbance. The disturbance can be an applied potential, a detector, or an extractor. The blue dots represent the fermionic atoms and the red focused laser beam illustrates the disturbance moving into the direction of the green arrow.

Figure 2

Snapshots of the wake developing following a moving particle detector at quarter filling. Plotted are the local particle densities of each lattice site as given by the color bar on the right. The location of the disturbance is marked by a red dot. The simulations are done by iteration of free evolution [Eq. (4)] interspersed with interactions with a particle detector [Eq. (6)] beginning from the ground state of free fermions on a $30\times 30$ lattice. The detector was initialized at position (8,15) and moves horizontally to the next site during time $1/\left(3.4t_{\text{hop}}\right)$, where ıns is the hopping parameter of the free evolution.

Figure 3

Comparison of density plots for wakes of a moving point potential (left) with a moving detector (right). The detector/point potential was initialized at position (8,15) and moves horizontally to the next site during time $1/\left(3.4t_{\text{hop}}\right)$, where $t_{\text{hop}}$ is the hopping parameter of the free evolution. Snapshots are taken after 18 time steps, when the full wake pattern has been sufficiently developed. At half-filling the difference is most dramatic (top) but differences remain at quarter-filling (bottom).

Figure 4

Density plots for varying velocities of a moving potential. From left to right, velocities $\alpha =1.7$, 1.0, and 0.7. Red lines represent the angles given by Eq. (35). Each line corresponds to the solution for a given quadrant in Fig. 6. Note the forward pointing cone is a result of a forward $d$-wave radiation when the source is moving slowly.

Figure 5

Density plots for varying the angle of a moving potential compared to the lattice vectors. A potential moving at $0^{\circ }$, ${23}^{\circ }$, and ${45}^{\circ }$ with respect to the lattice. Here $\alpha =1.7$. A smeared potential (Gaussian with half a lattice spacing width) is used instead of a point potential to include effects when the tip is not precisely on a lattice site. It is argued in the Appendix that the wake geometry is unaffected by this change away from a point potential. Red lines represent the angles given by Eq. (35). Note that for motion at ${45}^{\circ }$ the angles for quadrants 1 and 4 in Eq. (35) coincide reducing the number of lines to 3.

Figure 6

Fermi surface at half-filling. At half-filling all states with momenta $k_x$ and $k_y$ within the diamond shape are occupied. For the calculation we consider the four quadrants separately.

Figure 7

Density plot of a wake developing following a fermion extraction site moving at $\alpha =1.7$. The pictures show a steady state in the comoving frame.

Figure 8

Density plot for varying speeds of a moving particle extractor at half-filling. From left to right, speed $\alpha =1.7$, 1.0, and 0.7, respectively.

Figure 9

Density plots of a detector moving through a Fermi sea at several filling fractions. From left to right, the detector is moving at $\alpha =1.7$ for quarter-filling, half-filling, and three-quarter-filling, respectively. The color bar has the same scale but different offset (centered around the initial filling fraction).

Figure 10

Density of particles in a spatially spread mode centered around each lattice site [given by $n_A\left(\mathbf{r}\right)$ as defined in Eq. (44)] following a moving detector at half-filling. Here $\alpha =1.7$.

Figure 11

Density plots of potential (top) and extraction (bottom) wakes at half-filling over varying temperature. From left to right $\frac{T}{t_{\text{hop}}}=0$, 1, and 10.

Figure 12

Density plots of detection wakes at quarter-filling and varying initial temperature. From left to right $\frac{T}{t_{\text{hop}}}=0$, 1, and 10.

Figure 13

A Gaussian potential at half-filling and $\alpha =1.7$. From left to right, the standard deviation of the Gaussian in terms of lattice spacing $a$ is point potential $0.5a$ and $a$.

Figure 14

A Gaussian potential at half-filling, $\alpha =1.7$, and $0.5a$ standard of deviation where $a$ is the lattice spacing. Left is a Gaussian starting directly on lattice site (8,15). Right is a Gaussian starting in-between lattice sites at (8.3,15.6).