Singularities in large deviations of work in quantum quenches

We investigate large deviations of the work performed in a quantum quench across two different phases separated by a quantum critical point, using as an example the Dicke model quenched from its superradiant to its normal phase. We extract the distribution of the work from the Loschmidt amplitude and compute for both the corresponding large-deviation forms. Comparing these findings with the predictions of the classification scheme put forward in Gambassi and Silva [Phys. Rev. Lett. 109, 250602 (2012)], we are able to identify a regime which is in fact distinct to the ones identified so far: here the rate function exhibits a nonanalytical point which is a strong indication of the existence of an out-of-equilibrium phase transition in the rare fluctuations of the work.

Figure 1

Diagram of the coordinate transformations that diagonalize the thermodynamic limit of the Dicke Hamiltonian both in the NP and SP. The original pair of canonical coordinates $(\mathbf{x},\mathbf{p})$ in Eqs. (4) and (5) are transformed via a rotation $R\left({\theta }_{\mathrm{NP}}\right)$ to diagonalize the Hamiltonian in the NP. For the SP the transformation is more involved, requiring us to compose a translation represented by $U\left(\mathbf{t}\right)$, a dilation $D\left(\mathbf{d}\right)$, and another rotation $R\left({\theta }_{\mathrm{SP}}\right)$. The overall transformation that links NP and SP coordinates is easily obtained by composing successive transformations (see also Appendixes B and C). The explicit parameters that enter in the transformations are $\mathbf{t}=\left(\frac{\sqrt{2}\mathrm{\Omega }}{\omega }\sqrt{{\frac{N(1-{\mu }^2)}{\omega }}^{\phantom{{\int }^0}}},-\sqrt{{\frac{N(1-\mu )}{\mathrm{\Delta }}}^{\phantom{\int }}}\right),tan\left(2{\theta }_{\mathrm{SP}}\right)=\frac{2\omega \mathrm{\Delta }{\mu }^2}{{\mu }^2{\omega }^2-{\mathrm{\Delta }}^2},tan\left(2{\theta }_{\mathrm{NP}}\right)=\frac{4\mathrm{\Omega }\sqrt{\omega \mathrm{\Delta }}}{{\omega }^2-{\mathrm{\Delta }}^2}$, and $\mathbf{d}=\left(1,\sqrt{{\frac{2\mu }{1+\mu }}^{\phantom{\int }}}\right)$.

Figure 2

Phase transitions in the work distribution $p\left(w\right)$ obtained as the Legendre transform of the SCGF $l(-is)$ Eq. (11) (at fixed $\omega =1$). (a) The SCGF $l(-is)$ diverges as $s\to s_{+}^{*}$. Correspondingly, $p\left(w\right)$, which is obtained by a Legendre transform (LT) from $l(-is)$, approaches asymptotically the linear regime with slope $-s_{+}^{*}$ for $w\gg \overline{w}$ [with $\overline{w}$ the typical value of the work, i.e., the one for which $p\left(\overline{w}\right)=0$]. This scenario is the one expected for the models that belong to class B introduced in Ref. [14]. (b) The singular behavior of $l(-is)$ is controlled by $s_{-}^{*}$ which is approached at finite value with finite derivative $l^{\prime }\left(s_{-}^{*}\right)$. The LT thus displays a nonanalytical point in $w_c=-l^{\prime }(-i{s}_{-}^{*})$ and $p\left(w\right)=-s_{-}^{*}w$ for $w>w_c$. This nonanalytical point of the rate function corresponds to a phase transition in the rare fluctuations of the work. Plots in (a) and (b) are obtained with the same $\mathrm{\Delta }=0.8$ and $\mathrm{\Omega }=0.3$ by tuning the superradiant coupling from ${\mathrm{\Omega }}_0=0.47$ to ${\mathrm{\Omega }}_0=0.9$. (c) Analytical (diamond) and nonanalytical (star) domains of $p\left(w\right)$ in the plane ${\mu }_{\mathrm{NP}}-{\mu }_{\mathrm{SP}}$ with ${\mu }_{\mathrm{NP}}={\mathrm{\Omega }}_c^2/{\mathrm{\Omega }}^2$ and ${\mu }_{\mathrm{SP}}={\mathrm{\Omega }}_c^2/{\mathrm{\Omega }}_0^2$ at $\mathrm{\Delta }=0.8$. At fixed $\mathrm{\Omega }$, the nonanalytical point appears only if the quench starts deep enough in the SP. The solid line denotes the boundary between the two areas and is where the nonanalyticity in $p\left(w\right)$ appears.