Excess energy and countercurrents after a quantum kick

A quantum system of interacting particles under the effect of a static external potential is hereby described as kicked when that potential suddenly starts moving with a constant velocity <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi mathvariant="bold">v</a:mi></a:math>. If initially in a stationary state, the excess energy at any time after the kick equals <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi mathvariant="bold">v</c:mi><c:mo>·</c:mo><c:mo>〈</c:mo><c:mi mathvariant="bold">P</c:mi><c:mo>〉</c:mo><c:mo>(</c:mo><c:mi>t</c:mi><c:mo>)</c:mo></c:mrow></c:math>, with <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi mathvariant="bold">P</f:mi></f:math> being the total momentum of the system. If the system is finite and remains bound, the long-time average of the excess energy tends to <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mi>M</h:mi><h:msup><h:mi>v</h:mi><h:mn>2</h:mn></h:msup></h:mrow></h:math>, with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mi>M</i:mi></i:math> the system’s total mass, or a related expression if there is particle emission. <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>M</j:mi><j:msup><j:mi>v</j:mi><j:mn>2</j:mn></j:msup></j:mrow></j:math> is twice what expected from an infinitely smooth onset of motion, and any monotonic onset is expected to increase the average energy to a value within both limits. In a macroscopic system, a particle flow emerges countering the potential’s motion when the particles stay partially behind. For charged particles, the described kinetic kick is equivalent to the kick given by the infinitely short electric-field pulse <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mrow><k:mi mathvariant="bold-script">E</k:mi></k:mrow><k:mo>=</k:mo><k:mfrac><k:mi>m</k:mi><k:mi>q</k:mi></k:mfrac><k:mi mathvariant="bold">v</k:mi><k:mspace width="0.16em"/><k:mi>δ</k:mi><k:mrow><k:mo>(</k:mo><k:mi>t</k:mi><k:mo>)</k:mo></k:mrow></k:mrow></k:math> to the system at rest, useful as a formal limit in ultrafast phenomena. A linear-response analysis of low-<o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mi>v</o:mi></o:math> countercurrents in kicked metals shows that the coefficient of the linear term in <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"><p:mi>v</p:mi></p:math> is the Drude weight. Nonlinear in <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:mi>v</q:mi></q:math> countercurrents are expected for insulators through the electron-hole excitations induced by the kick, going as <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"><r:msup><r:mi>v</r:mi><r:mn>3</r:mn></r:msup></r:math> at low <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"><s:mi>v</s:mi></s:math> for centrosymmetric ones. First-principles calculations for simple solids are used to ratify those predictions, although the findings apply more generally to systems such as Mott insulators or cold lattices of bosons or fermions.