Engineering negative stress-energy densities with quantum energy teleportation

We use quantum energy teleportation in the light-matter interaction as an operational means to create quantum field states that violate energy conditions and have negative local stress-energy densities. We show that the protocol is optimal in the sense that it scales in a way that saturates the quantum interest conjecture.

Figure 1

Progression of the $1+1$ D energy density wave is shown with each time slice offset in the $y$ axis. As time progresses the wave can be seen propagating in both directions until $t=T=15.29$ when Bob’s interaction introduces the negative energy density, obscured by a left-moving positive energy contribution. Later propagation shows Bob’s contribution splitting into left and right propagators, leaving the desired significant negative energy density. Here a Lorentzian smearing was used.

Figure 2

Energy density at $x=T+\mathrm{\Delta }T$, where Bob’s interaction takes place at $t=T$. The additional time $\mathrm{\Delta }T$ is so that Bob’s left-moving energy packet moves away, as seen in Fig. 1, increasing the amount of negative energy present. These 3 plots (from the left $f$, $g$ and $h$ smearings) support the notion of large second derivatives relative to the first derivative of Alice’s smearing $\lambda$. Also note the differing energy packet forms of each smearing function. Certain smearings are inherently more effective at maximizing negative energy yields.

Figure 3

Radial slice of the (normalized) smearing functions used for $3+1$ D Gaussian smearing QET. With $r_0^{\mathrm{B}}\ne 0$ and ${\mathit{x}}_{\mathrm{A}}={\mathit{x}}_{\mathrm{B}}$, $\mu (r)$ resembles a shell surrounding Alice.

Figure 4

The $x$-axis slice of the progression of the $3+1$ D energy density wave at various times with Bob’s interaction taking place at $t=T=18.85$. The waves are “ingoing” and “outgoing,” with the spherical nature of these waves apparent by their decay with increasing $|x|$. Immediately following Bob’s interaction, there is a small amount of negative energy that is being partially obscured by Bob’s ingoing positive energy contribution. Here a Gaussian smearing was used. We also refer the reader to Fig. 5, which shows the energy distribution following the separation of ingoing and outgoing waves.

Figure 5

Energy density at $x=T+\mathrm{\Delta }T$, where Bob’s interaction takes place at $t=T$. The additional time $\mathrm{\Delta }T$ is to allow the ingoing contribution ($r\approx 5$) to move away from the outgoing wave packet ($r\approx 32$), revealing the negative energy density that propagates radially outward. Unlike the $1+1$ D case the ingoing and outgoing wave packets will respectively increase and decay according to the inverse square law. As the ingoing wave packet diverges, this becomes a high energy scattering problem, which is beyond the scope of this paper. The divergences at $r=0$ come as an artifact of the strict spherical symmetry of the problem and, furthermore, will not affect the outgoing wave packet as this is a local field theory. However, if multiple sequential QET protocols are employed, then the aftermath of this scattering may become important. In the outgoing wave packet the negative energy well will then decay with time; however, the ratio between the depth of the negative energy well and its neighboring positive energy peak will remain constant. Here a contour plot of the energy density for the $z=0$ surface is shown, accompanied by a radial plot. These three plots (from the left $f$, $g$ and $h$ smearings) have been truncated so that the divergent nature of the ingoing wave at $r=0$ does not shadow the details of the outgoing wave packet and its negative energy density.