Qubit Heating Near a Hotspot

The hotspot is taken to contain an observable sector, modelled by a single real scalar field, $\phi(x)$, that lives in a spatial region, ${\cal R}_{+}$, that represents the exterior of the black hole. The degrees of freedom interior to the black hole is modelled by $N$ real massless scalar fields, $\chi^{a}$ with $a=1,\cdots,N$, that reside in a different spatial region ${\cal R}_{-}$ that is disjoint from the region ${\cal R}_{+}$ everywhere except for the surface of a small sphere, ${\cal S}_{\xi}$, with radius $\xi$. In practice this means that both ${\cal R}_{+}$ and ${\cal R}_{-}$ have a small sphere excised from the origin (for all time) and the surface of this sphere is identified in the two spaces (see Figure 1).

The fields are allowed to interact with one another locally only on ${\cal S}_{\xi}$, but to keep the model solvable this interaction is limited to a bilinear mixing term. Although we neglect the external gravitational fields of the hotspot in regions ${\cal R}_{+}$ and ${\cal R}_{-}$ there is no reason why this could not also be included in more sophisticated versions of the model.³³3Without a strong gravitational field the interaction surface ${\cal S}_{\xi}$ is generically not light-like and so is not a local horizon, unlike for an honest-to-God black hole. Our interest in this paper is in scales much larger than $\xi$ and so we further consider the idealization⁴⁴4The limit $\xi\to 0$ is not required for the hotspot model, allowing it also to explore the opposite regime where UV scales involve distances much smaller than $\xi$, such as for the near-horizon EFTs considered in Burgess:2018pmm ; Rummel:2019ads , motivated to systematize the treatment of both conventional Price:1986yy ; Thorne:1986iy ; Damour:1978cg ; Parikh:1997ma ; Donnay:2019jiz or more exotic Cardoso:2016rao ; Abedi:2016hgu ; Holdom:2016nek ; Cardoso:2017cqb ; Bueno:2017hyj ; Mark:2017dnq ; Conklin:2017lwb ; Berti:2018vdi ; Zhou:2016hsh kinds of near-horizon physics. where the radius $\xi\to 0$, in which case ${\cal S}_{\xi}$ reduces to a single point of contact between ${\cal R}_{+}$ and ${\cal R}_{-}$ (which we situate at the origin ${{\bf x}}=\mathbf{0}$ of both ${\cal R}_{\pm}$). In this limit the couplings between $\phi$ and $\chi^{a}$ are captured by an effective action localized at ${{\bf x}}=0$.

The action has the form $S=S_{+}+S_{-}+S_{\rm int}$ where $S_{\pm}$ describe the free fields $\phi$ and $\chi^{a}$

$$S_{+}=-\frac{1}{2}\int_{{\cal R}_{+}}{\hbox{d}}^{4}x\;\partial_{\mu}\phi\,\partial^{\mu}\phi\quad\hbox{and}\quad S_{-}=-\frac{1}{2}\int_{{\cal R}_{-}}{\hbox{d}}^{4}x\;\delta_{ab}\,\partial_{\mu}\chi^{a}\partial^{\mu}\chi^{b}\,,$$

(2.1)

and the lowest-dimension interaction (mixing, really) on the interaction surface is given by

$$S_{\mathrm{int}}=-\int_{\cal W}{\hbox{d}}t\;\left[g_{a}\,\chi^{a}(t,\mathbf{0})\,\phi(t,\mathbf{0})+\frac{\lambda}{2}\,\phi^{2}(t,\mathbf{0})\right]\,,$$

(2.2)

where the integration is over the proper time along the hotspot world-line ${\cal W}$, which we take to be ${{\bf x}}=0$ in both ${\cal R}_{+}$ and ${\cal R}_{-}$. In fundamental units the couplings $g_{a}$ and $\lambda$ and have dimensions of length.

In what follows we imagine both of these couplings turn on suddenly at $t=0$ — i.e. we assume $g_{a}(t)=\Theta(t)\,g_{a}$, where $\Theta(t)$ is the Heaviside step function — but remain constant thereafter. Because our applications focus on a qubit that couples only to $\phi$, the couplings $g_{a}$ often appear only through the combination

$$\tilde{g}^{2}:=\delta^{ab}g_{a}g_{b}=Ng^{2}\,,$$

(2.3)

where the second equality specializes to the case where all couplings are equal (as we typically do). Because we solve for the evolution of $\phi$ exactly we need not assume that $\tilde{g}$ be particularly small.

Ref. Hotspot computes the evolution of the system after the couplings $\lambda$ and $g_{a}$ are turned on, assuming the system’s initial state at $t=0$ is initially uncorrelated

$$\rho_{0}=\rho_{+}\otimes\rho_{-}\,,$$

(2.4)

with the $\phi$ sector initially in its vacuum and the $\chi^{a}$ fields initially in a thermal state:

$$\rho_{+}=\ket{\mathrm{vac}}\bra{\mathrm{vac}}\quad\hbox{and}\quad\rho_{-}=\varrho_{\beta}:=\frac{e^{-\beta{\cal H}_{-}}}{{\cal Z}_{\beta}}\,,$$

(2.5)

with inverse temperature $\beta=1/T>0$. Here ${\cal H}_{-}$ denotes the Hamiltonian constructed from $S_{-}$ and ${\cal Z}_{\beta}:=\underset{}{\mathrm{Tr^{\,{}^{\prime}}}}[e^{-\beta{\cal H}_{-}}]$ is the thermal partition function, with the prime on the trace indicating that it is only taken over the $\chi$ sector.

Ref. Hotspot computes the system response by solving explicitly the field equations

$$(-\partial_{t}^{2}+\nabla^{2})\phi_{{\scriptscriptstyle H}}(t,{{\bf x}})=\delta^{3}({{\bf x}})\bigg{[}\lambda\phi_{{\scriptscriptstyle H}}(t,\mathbf{0})+g_{a}\chi_{{\scriptscriptstyle H}}^{a}(t,\mathbf{0})\bigg{]}$$

(2.6)

and

$$(-\partial_{t}^{2}+\nabla^{2})\chi^{a}_{{\scriptscriptstyle H}}(t,{{\bf x}})=\delta^{3}({{\bf x}})\;g_{a}\phi_{{\scriptscriptstyle H}}(t,\mathbf{0})\,,$$

(2.7)

within the Heisenberg picture of time evolution. This can be done very explicitly because the field equations are linear in all of the fields. To compute the response of the $\phi$ field the field $\chi^{a}$ is eliminated by solving (2.7) for it as a function of $\phi$.

Because the elimination of $\chi^{a}$ involves Coulomb-like Greens functions proportional to $1/|{{\bf x}}|$ it introduces singularities into the solution for $\phi$ at $|{{\bf x}}|=0$ that are regulated by instead evaluating at $|{{\bf x}}|=\epsilon$ for a microscopic scale $\epsilon$. Ref. Hotspot shows that physical predictions remain independent of $\epsilon$ once the singular regularization dependence is absorbed by replacing $\lambda\to\lambda_{\scriptscriptstyle R}$ with

$$\lambda_{{\scriptscriptstyle R}}:=\lambda-\frac{\tilde{g}^{2}}{4\pi\epsilon}\ ,$$

(2.8)

and this is why this particular coupling is included in addition to $\tilde{g}$. In what follows we assume this replacement has been done, though we drop the subscript ‘$R$’ to avoid notational clutter.

These steps allow the calculation of the $\phi$-field Wightman function,

$$W_{\beta}(t,{{\bf x}};t^{\prime},{{\bf x}}^{\prime}):=\mathrm{Tr}\Bigl{[}\phi_{{\scriptscriptstyle H}}(t,{{\bf x}})\phi_{{\scriptscriptstyle H}}(t^{\prime},{{\bf x}}^{\prime})\rho_{0}\Bigr{]}=\frac{1}{Z_{\beta}}\mathrm{Tr}\Bigl{[}\phi_{{\scriptscriptstyle H}}(t,{{\bf x}})\phi_{{\scriptscriptstyle H}}(t^{\prime},{{\bf x}}^{\prime})\big{(}\ket{\mathrm{vac}}\bra{\mathrm{vac}}\otimes e^{-\beta{\cal H}_{-}}\big{)}\Bigr{]}\,.$$

(2.9)

The result computed to leading-order in $\tilde{g}^{2}$ and $\lambda$ turns out to be given by

			$\displaystyle W_{\beta}(t,{{\bf x}};t^{\prime},{{\bf x}}^{\prime})\simeq\frac{1}{4\pi^{2}\big{[}-(t-t^{\prime}-i\delta)^{2}+|{{\bf x}}-{{\bf x}}^{\prime}|^{2}\big{]}}$
			$\displaystyle\qquad+\frac{\lambda}{16\pi^{3}}\bigg{(}\frac{\Theta(t-|{{\bf x}}|)}{|{{\bf x}}|}\frac{1}{(t-t^{\prime}-|{{\bf x}}|-i\delta)^{2}-|{{\bf x}}^{\prime}|^{2}}+\frac{\Theta(t^{\prime}-|{{\bf x}}^{\prime}|)}{|{{\bf x}}^{\prime}|}\frac{1}{(t-t^{\prime}+|{{\bf x}}^{\prime}|-i\delta)^{2}-|{{\bf x}}|^{2}}\bigg{)}$
			$\displaystyle\qquad-\frac{\tilde{g}^{2}\Theta(t-|{{\bf x}}|)\Theta(t^{\prime}-|{{\bf x}}^{\prime}|)}{64\pi^{2}\beta^{2}|{{\bf x}}||{{\bf x}}^{\prime}|\sinh^{2}\left[\frac{\pi}{\beta}(t-|{{\bf x}}|-t^{\prime}+|{{\bf x}}^{\prime}|-i\delta)\right]}$
			$\displaystyle\qquad+\frac{\tilde{g}^{2}}{32\pi^{4}}\bigg{(}-\frac{\Theta(t-|{{\bf x}}|)}{|{{\bf x}}|}\frac{t-t^{\prime}-|{{\bf x}}|}{\big{[}(t-t^{\prime}-|{{\bf x}}|-i\delta)^{2}-|{{\bf x}}^{\prime}|^{2}\big{]}^{2}}+\frac{\Theta(t^{\prime}-|{{\bf x}}^{\prime}|)}{|{{\bf x}}^{\prime}|}\frac{t-t^{\prime}+|{{\bf x}}^{\prime}|}{\big{[}(t-t^{\prime}+|{{\bf x}}^{\prime}|-i\delta)^{2}-|{{\bf x}}|^{2}\big{]}^{2}}\bigg{)}$
			$\displaystyle\qquad+\frac{\tilde{g}^{2}}{64\pi^{4}}\bigg{(}\frac{\delta(t-|{{\bf x}}|)}{|{{\bf x}}|\big{[}-(t^{\prime}+i\delta)^{2}-|{{\bf x}}^{\prime}|^{2}\big{]}}+\frac{\delta(t^{\prime}-|{{\bf x}}^{\prime}|)}{|{{\bf x}}^{\prime}|\big{[}-(t-i\delta)^{2}-|{{\bf x}}|^{2}\big{]}}\bigg{)}\qquad\hbox{(perturbative)}\,,$

where the delta functions and step functions describe the passage of the transients from the switch-on of couplings at $t=|{{\bf x}}|=0$, followed by late-time evolution in the future light cone of the switch-on event (i.e. for $t>|{{\bf x}}|$ and $t^{\prime}>|{{\bf x}}^{\prime}|$). In this late-time regime the above perturbative expression becomes

	$\displaystyle W_{\beta}(t,{{\bf x}};t^{\prime},{{\bf x}}^{\prime})$	$\displaystyle\simeq$	$\displaystyle\frac{1}{4\pi^{2}\big{[}-(t-t^{\prime}-i\delta)^{2}+|{{\bf x}}-{{\bf x}}^{\prime}|^{2}\big{]}}+\frac{\lambda}{16\pi^{3}|{{\bf x}}||{{\bf x}}^{\prime}|}\bigg{[}\frac{|{{\bf x}}|+|{{\bf x}}^{\prime}|}{(t-t^{\prime}-i\delta)^{2}-(|{{\bf x}}+|{{\bf x}}^{\prime}|)^{2}}\bigg{]}$
			$\displaystyle\qquad-\frac{\tilde{g}^{2}}{64\pi^{2}\beta^{2}|{{\bf x}}||{{\bf x}}^{\prime}|\sinh^{2}\left[\frac{\pi}{\beta}(t-|{{\bf x}}|-t^{\prime}+|{{\bf x}}^{\prime}|-i\delta)\right]}$
			$\displaystyle\qquad+\frac{\tilde{g}^{2}}{32\pi^{4}}\bigg{(}-\frac{1}{|{{\bf x}}|}\frac{t-t^{\prime}-|{{\bf x}}|}{\big{[}(t-t^{\prime}-|{{\bf x}}|-i\delta)^{2}-|{{\bf x}}^{\prime}|^{2}\big{]}^{2}}+\frac{1}{|{{\bf x}}^{\prime}|}\frac{t-t^{\prime}+|{{\bf x}}^{\prime}|}{\big{[}(t-t^{\prime}+|{{\bf x}}^{\prime}|-i\delta)^{2}-|{{\bf x}}|^{2}\big{]}^{2}}\bigg{)}$
			(late times, perturbative)

In these expressions $\delta\to 0^{+}$ is a positive infinitesimal that is taken to zero at the end of the calculation.

But the simplicity of the model allows a more general determination of the Wightman function in the future light-cone of $t=|{{\bf x}}|=0$ that is not restricted to perturbatively small couplings. This more exact treatment gives (for $t>|{{\bf x}}|$ and $t^{\prime}>|{{\bf x}}^{\prime}|$)

$$W_{\beta}(t,{{\bf x}};t^{\prime},{{\bf x}}^{\prime})=\mathscr{S}(t,{{\bf x}};t,{{\bf x}}^{\prime})+\mathscr{E}_{\beta}(t,{{\bf x}};t,{{\bf x}}^{\prime})$$

(2.12)

with the temperature-independent part given by

	$\displaystyle\mathscr{S}(t,{{\bf x}};t,{{\bf x}}^{\prime})$	$\displaystyle=$	$\displaystyle\frac{1}{4\pi^{2}\left[-(t-t^{\prime}-i\delta)^{2}+|{{\bf x}}-{{\bf x}}^{\prime}|^{2}\right]}$
			$\displaystyle\qquad+\frac{2\epsilon^{2}}{\tilde{g}^{2}|{{\bf x}}||{{\bf x}}^{\prime}|}\bigg{[}I_{-}(t-t^{\prime}+|{{\bf x}}|+|{{\bf x}}^{\prime}|,c)-I_{-}(t-t^{\prime}-|{{\bf x}}|+|{{\bf x}}^{\prime}|,c)$
			$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-I_{+}(t-t^{\prime}-|{{\bf x}}|+|{{\bf x}}^{\prime}|,c)+I_{+}(t-t^{\prime}-|{{\bf x}}|-|{{\bf x}}^{\prime}|,c)\bigg{]}$
			$\displaystyle\qquad+\frac{\epsilon}{8\pi^{2}|{{\bf x}}||{{\bf x}}^{\prime}|}\bigg{[}-\frac{1}{t-t^{\prime}+|{{\bf x}}|+|{{\bf x}}^{\prime}|-i\delta}+\frac{1}{t-t^{\prime}-|{{\bf x}}|-|{{\bf x}}^{\prime}|-i\delta}\bigg{]}$
			$\displaystyle\qquad\quad-\frac{32\pi^{2}\epsilon^{4}(1+\frac{\lambda}{2\pi\epsilon})}{\tilde{g}^{4}|{{\bf x}}||{{\bf x}}^{\prime}|}\bigg{[}I_{-}(t-t^{\prime}-|{{\bf x}}|+|{{\bf x}}^{\prime}|,c)+I_{+}(t-t^{\prime}-|{{\bf x}}|+|{{\bf x}}^{\prime}|,c)\bigg{]}$
			$\displaystyle\qquad\qquad-\frac{\epsilon^{2}}{4\pi^{2}|{{\bf x}}||{{\bf x}}^{\prime}|(t-t^{\prime}-|{{\bf x}}|+|{{\bf x}}^{\prime}|-i\delta)^{2}}$

where

$$c:=\frac{16\pi^{2}\epsilon}{\tilde{g}^{2}}\left(1+\frac{\lambda}{4\pi\epsilon}\right)\ .$$

(2.14)

and the functions $I_{\mp}(\tau)$ are defined by

$$I_{\mp}(\tau,c)=e^{\pm c\tau}E_{1}\big{(}\pm c[\tau-i\delta]\big{)}=e^{\pm c\tau}\int_{z}^{\infty}{\hbox{d}}u\;\frac{e^{-u}}{u}\,,$$

(2.15)

and the limit $\delta\to 0^{+}$ is again understood. The temperature-dependent⁵⁵5Notice that although $\mathscr{E}_{\beta}$ contains all of the dependence on temperature it does not vanish in the $T\to 0$ limit. part is similarly given by

$$\mathscr{E}_{\beta}(t,{{\bf x}};t,{{\bf x}}^{\prime})=\frac{2\epsilon^{2}}{\tilde{g}^{2}|{{\bf x}}||{{\bf x}}^{\prime}|}\bigg{[}\Psi\bigg{(}e^{-\tfrac{2\pi(t-t^{\prime}-|{{\bf x}}|+|{{\bf x}}^{\prime}|-i\delta)}{\beta}},\frac{c\beta}{2\pi}\bigg{)}+\Psi\bigg{(}e^{+\tfrac{2\pi(t-t^{\prime}-|{{\bf x}}|+|{{\bf x}}^{\prime}|-i\delta)}{\beta}},\frac{c\beta}{2\pi}\bigg{)}\bigg{]}-\frac{2\pi}{c\beta}\bigg{]}$$

(2.16)

with $\Psi(z,a):=\Phi(z,1,a)$ where $\Phi(z,s,a)$ is the Lerch transcendent, defined by the series

$$\Phi(z,s,a):=\sum_{n=0}^{\infty}\frac{z^{n}}{(a+n)^{s}}$$

(2.17)

for complex numbers in the unit disc (with $|z|<1$), and by analytic continuation elsewhere in the complex plane. A convenient integral representation for $\Phi(z,s,a)$ is given by

$$\Phi(z,s,a)\ =\ \frac{1}{\Gamma(s)}\int_{0}^{\infty}{\hbox{d}}x\;\frac{x^{s-1}e^{-ax}}{1-ze^{-x}}\qquad\mathrm{valid\ for\ }\mathrm{Re}[s]>0,\ \mathrm{Re}[a]>0\ \&\ z\in\mathbb{C}\setminus[1,\infty)\,.$$

(2.18)

The full correlation function $W_{\beta}=\mathscr{S}+\mathscr{E}_{\beta}$ obtained using (2.2) and (2.16) reduces to the perturbative correlation function quoted in (2.2) once linearized in $\lambda$ and $\tilde{g}^{2}$, and seeing how this works clarifies the domain of validity of perturbative methods. The expansion in $\tilde{g}^{2}$ in particular is captured by the asymptotic form for $\mathscr{S}$ in the regime $c\tau\gg 1$ as well as the expansion of $\mathscr{E}_{\beta}$ in the regime $c\beta\gg 1$. These can be made explicit using the asymptotic expression

$$E_{1}(z)\ \simeq\ e^{-z}\bigg{[}\frac{1}{z}-\frac{1}{z^{2}}+{\cal O}\left(z^{-3}\right)\bigg{]}\qquad\qquad\mathrm{for}\ |z|\gg 1$$

(2.19)

which implies that the functions $I_{\mp}(\tau,c)$ for $|c\tau|\gg 1$ have the asymptotic expansion

$$I_{\mp}(\tau,c)\ \simeq\ \pm\frac{1}{c(\tau-i\delta)}-\frac{1}{c^{2}(\tau-i\delta)^{2}}+{\cal O}\left(|c\tau|^{-3}\right)\qquad\qquad\mathrm{for}\ |c\tau|\gg 1\ .$$

(2.20)

The behaviour of $\Psi(z,a)$ for $c\beta\gg 1$ is similarly given by the following asymptotic series for the Lerch transcendent for large positive $a$:

$$\Phi(z,s,a)\simeq\frac{a^{-s}}{1-z}+\sum_{n=1}^{N-1}\frac{(-1)^{n}\Gamma(s+n)}{n!\;\Gamma(s)}\cdot\frac{\mathrm{Li}_{-n}(z)}{a^{s+n}}+{\cal O}\left(a^{-s-N}\right)\qquad\quad\mathrm{for\ }a\gg 1$$

(2.21)

which applies for fixed $s\in\mathbb{C}$ and fixed $z\in\mathbb{C}\setminus[1,\infty)$, where $\mathrm{Li}_{-n}(z)=\left(z\partial_{z}\right)^{n}\frac{z}{1-z}$ are polylogarithm functions of negative-integer order. These and some other properties are explored in Appendix A.

Besides verifying that the apparent $\epsilon$-dependence cancels in $W_{\beta}$ in the perturbative limit, the above expressions show that the perturbative limit arises as an expansion in powers of

$$\frac{1}{c\tau}=\frac{\tilde{g}^{2}}{16\pi^{2}\epsilon\tau}\left(1+\frac{\lambda}{4\pi\epsilon}\right)^{-1}\ll 1\quad\hbox{and}\quad\frac{1}{c\beta}=\frac{\tilde{g}^{2}T}{16\pi^{2}\epsilon}\left(1+\frac{\lambda}{4\pi\epsilon}\right)^{-1}\ll 1\,,$$

(2.22)

which includes low temperatures ($T$) and long times ($\tau$) compared with the UV scale $\tilde{g}^{2}/4\pi\epsilon$. As shown in Hotspot the dependence of (2.22) on $\lambda/4\pi\epsilon$ is properly captured by renormalization-group methods in the world-line EFT for this system.

The free Hamiltonian for the qubit-field system is assumed to be described by the Hamiltonian

$${H}_{\mathrm{tot}}(t)={H}_{{\scriptscriptstyle S}}(t)\otimes\boldsymbol{I}+{\cal I}\otimes\boldsymbol{H}_{0}+{H}_{\mathrm{int}}^{{\scriptscriptstyle Q}}(t)\,,$$

(3.1)

where ${H}_{{\scriptscriptstyle S}}(t)$ is the Schrödinger-picture Hamiltonian for the fields $\phi$ and $\chi^{a}$ described in §2, ${\cal I}$ and $\boldsymbol{I}$ are (respectively) the unit operators acting on the Hilbert space for these fields and on the Hilbert space of the qubit. $\boldsymbol{H}_{0}$ denotes the free $2\times 2$ qubit Hamiltonian, and is assumed (in its rest frame) to be

$\displaystyle\boldsymbol{H}_{0}$

$\displaystyle=$

$\displaystyle\frac{\omega}{2}\,\boldsymbol{\sigma_{3}}\ =\ \frac{\omega}{2}\left[\begin{matrix}1&0\\ 0&-1\end{matrix}\right]\ ,$

(3.2)

where $\omega$ denotes the splitting between its two levels.

$H^{\scriptscriptstyle Q}_{\rm int}$ describes the Schrödinger-picture qubit-field interaction, in which the qubit couples only to the field $\phi$ evaluated at the local qubit position, taken to be at rest relative to the hotspot and displaced from it by ${{\bf x}}_{\scriptscriptstyle Q}$. This interaction is chosen to drive transitions between the qubit levels,

$${H}_{\mathrm{int}}^{{\scriptscriptstyle Q}}(t)=\hat{\lambda}_{{\scriptscriptstyle Q}}(t)\;{\phi}_{{\scriptscriptstyle S}}({{\bf x}}_{\scriptscriptstyle Q})\otimes{\cal I}_{-}\otimes\boldsymbol{\sigma_{1}}\quad\hbox{ where}\;\boldsymbol{\sigma_{1}}=\left[\begin{matrix}0&1\\ 1&0\end{matrix}\right]\,,$$

(3.3)

where ${\cal I}_{-}$ is the unit matrix in the $\chi^{a}$ sector of the Hilbert space, and the dimensionless coupling parameter $\hat{\lambda}_{\scriptscriptstyle Q}$ is assumed to be small so as to justify treating the qubit-field interaction perturbatively.

We imagine the qubit-field coupling to be turned on suddenly at time $t=t_{0}$,

$$\hat{\lambda}_{\scriptscriptstyle Q}(t)=\lambda_{{\scriptscriptstyle Q}}\,\Theta(t-t_{0})$$

(3.4)

with $t_{0}>0$ so that switch-on occurs after the fields have already begun to interact. Our focus is not on the transients associated with this turn-on, and instead on the qubit’s late-time approach to equilibrium and on how this approach depends on the other scales of the problem such as the distance $|{{\bf x}}_{\scriptscriptstyle Q}|$ between the qubit and the hotspot.

To compute the qubit evolution we adopt the interaction picture, with the interaction Hamiltonian including only $H^{\scriptscriptstyle Q}_{\rm int}(t)$. In this picture the interactions between the $\phi$ and $\chi^{a}$ fields are all regarded as being within the ‘unperturbed’ Hamiltonian. Interaction picture evolution of the fields $\phi$ and $\chi^{a}$ is therefore the same as what was considered Heisenberg-picture evolution in the absence of the qubit, and so is given by the same equations — i.e. eqs. (2.6) and (2.7) — that were solved in Hotspot .

In this interaction picture the system state evolves purely due to the qubit-field interaction and so does not evolve at all until the time $t=t_{0}$ when these turn on. As a consequence the system’s density matrix at $t=t_{0}$ remains unchanged from its initially uncorrelated configuration at $t=0$:

$$R_{{\scriptscriptstyle S}}(t_{0})=R_{{\scriptscriptstyle S}}(0)=\rho_{0}\otimes\varrho(0)\,,$$

(3.5)

where $\varrho(0)$ denotes the initial qubit state and $\rho_{0}$ is the field state given in (2.4), in which the $\phi$ sector starts in its vacuum while the $\chi^{a}$ are prepared in a thermal state.

The evolution of the system’s state for $t>t_{0}$ is given in the interaction picture by

$$\frac{\partial R(t)}{\partial t}=-i\Bigl{[}{V}_{\mathrm{tot}}(t),R(t)\Bigr{]}\ ,$$

(3.6)

where $V_{\rm tot}(t)$ is the interaction-picture interaction Hamiltonian for the field-qubit system coupling is defined by

$${V}_{\mathrm{tot}}(t)\ =\ \bigg{(}{U}^{\ast}(t,0)\otimes e^{+i\boldsymbol{H}_{0}t}\bigg{)}\;{H}_{\mathrm{int}}(t)\;\bigg{(}{U}(t,0)\otimes e^{-i\boldsymbol{H}_{0}t}\bigg{)}\ =\ \hat{\lambda}_{{\scriptscriptstyle Q}}(t)\;{\phi}_{{\scriptscriptstyle I}}(t,\mathbf{x}_{{\scriptscriptstyle Q}})\otimes\boldsymbol{{\mathfrak{m}}}(t)$$

(3.7)

where

$${U}(t,s)=\mathcal{T}\,\exp\left(-i\int_{s}^{t}{\hbox{d}}\tau\;{H}_{{\scriptscriptstyle S}}(\tau)\right)\,,$$

(3.8)

is the evolution operator for the fields $\phi$ and $\chi^{a}$. The second equality in eq. (3.7) uses the time-evolution property for interaction-picture fields: $\phi_{\scriptscriptstyle I}(t,{{\bf x}})={U}^{\ast}(t,0){\phi}_{{\scriptscriptstyle S}}(\mathbf{x}_{{\scriptscriptstyle Q}}){U}(t,0)$, where $\phi_{\scriptscriptstyle S}$ is the Schrödinger-picture operator. Notice that the interaction-picture field $\phi_{\scriptscriptstyle I}$ appearing in (3.7) is precisely the same as what was called the Heisenberg-picture field $\phi_{\scriptscriptstyle H}$ in §2, since that section did not include the field-qubit interaction being considered here. As a result the Wightman function for $\phi_{\scriptscriptstyle I}(t,{{\bf x}})$ is given by the expressions (2.12), (2.2) and (2.16) computed in Hotspot . Finally, the matrix $\boldsymbol{{\mathfrak{m}}}(t)$ is similarly defined as

$$\boldsymbol{{\mathfrak{m}}}(t):=e^{+i\boldsymbol{H}_{0}t}\boldsymbol{\sigma_{1}}e^{-i\boldsymbol{H}_{0}t}\ =\ \left[\begin{matrix}0&e^{+i\omega t}\\ e^{-i\omega t}&0\end{matrix}\right]\ .$$

(3.9)

Since our goal is to follow only the dynamics of the qubit keeping track of the entire density matrix $R(t)$ carries too much information. For qubit measurements it suffices instead to follow the evolution of the qubit’s reduced interaction-picture density matrix

$$\varrho(t):=\underset{\rm fields}{\mathrm{Tr}}\left[R(t)\right]\ ,$$

(3.10)

since this carries the information required to predict measurements in this sector. At first sight the evolution of this reduced density matrix is obtained simply by tracing over (3.6), though the resulting equation has the disadvantage that its right-hand side is not expressed purely in terms of $\varrho$ without reference to the field part of the system’s density matrix.

A more useful equation would compute the evolution of both $\varrho(t)$ and the density matrix for the field sector, and then use the evolution equations to eliminate the field density matrix as a function of $\varrho$. Doing so is a solved problem in the theory of open quantum systems, and the resulting self-contained equation for the evolution of $\varrho(t)$ is the so-called Nakajima-Zwanzig equation Nak ; Zwan (for a review see e.g. EFTBook ). The logic of its derivation is to exploit the linearity of the evolution equation (3.6) and to compute how it commutes with a projection super-operator $\mathscr{P}\{\cdot\}$ that maps operators acting on the full Hilbert space onto operators acting only within the qubit (in this case) sector, defined in such a way as to ensure that $\mathscr{P}[R(t)]=\varrho(t)$.

The idea then is to compute what (3.6) predicts for both $\mathscr{P}\{R(t)\}$ and for its complement $\mathscr{Q}\{R(t)\}$, where $\mathscr{Q}\{R(t)\}={\cal I}-\mathscr{P}\{R(t)\}$. The result is a coupled set of linear evolution equations, which can be formally integrated to obtain $\mathscr{Q}\{R\}$ as a function of time and of $\varrho(t)$. Substituting the result back into the equation for $\mathscr{P}\{R(t)\}$ provides the equation we seek. Because of the elimination of $\mathscr{Q}\{R(t)\}$ the equation for $\varrho(t)$ that results is generically nonlocal in time, making $\partial_{t}\varrho(t)$ depend on $\varrho(t)$ but also on the entire history $\varrho(s)$ for $s<t$.

The application of this equation to qubit systems in various environments is studied in detail in Kaplanek:2019dqu ; Kaplanek:2019vzj ; Kaplanek:2020iay so we here simply quote what the Nakajima-Zwanzig equation gives for $\varrho(t)$ in the present instance. The result can be computed explicitly as a series in the coupling $\lambda_{\scriptscriptstyle Q}$, and when evaluated for $t>t_{0}$ to second order in $\lambda_{{\scriptscriptstyle Q}}$ takes the form

	$\displaystyle\partial_{t}\boldsymbol{\varrho}(t)$	$\displaystyle=$	$\displaystyle-i\bigg{[}\frac{\lambda_{{\scriptscriptstyle Q}}^{2}\,\omega_{\rm ct}}{2}\,\boldsymbol{\sigma_{3}},\boldsymbol{\varrho}(t)\bigg{]}$
			$\displaystyle\qquad+\lambda_{{\scriptscriptstyle Q}}^{2}\int_{t_{0}}^{t}{\hbox{d}}s\;\bigg{(}\Bigl{[}\boldsymbol{{\mathfrak{m}}}(s)\boldsymbol{\varrho}(s),\boldsymbol{{\mathfrak{m}}}(t)\Bigr{]}{\cal W}(t,s)+\Bigl{[}\boldsymbol{{\mathfrak{m}}}(t),\boldsymbol{\varrho}(s)\boldsymbol{{\mathfrak{m}}}(s)\Bigr{]}{\cal W}^{\ast}(t,s)\bigg{)}$

where $\omega_{\rm ct}$ of the first term is a counter-term for the qubit frequency, which is written $\omega=\omega_{\rm phys}+\lambda_{\scriptscriptstyle Q}^{2}\omega_{\rm ct}$ with $\omega_{\rm ct}$ chosen to ensure that $\omega_{\rm phys}$ remains the physically measured qubit energy to the order we work in $\lambda_{\scriptscriptstyle Q}$. This shift is required because corrections to the qubit energy arise at order $\lambda_{\scriptscriptstyle Q}^{2}$, which $\omega_{\rm ct}$ is chosen to cancel.⁶⁶6As it happens these corrections also diverge and so $\omega_{\rm ct}$ provides the counterterm that cancels this divergence. Because $\omega_{\rm ct}$ is of order $\lambda_{\scriptscriptstyle Q}^{2}$, within the interaction picture it is included into the perturbing Hamiltonian by writing

$${V}_{\mathrm{tot}}(t)=\hat{\lambda}_{{\scriptscriptstyle Q}}(t)\;{\phi}_{{\scriptscriptstyle I}}(t,\mathbf{x}_{{\scriptscriptstyle Q}})\otimes\boldsymbol{{\mathfrak{m}}}(t)+\frac{\hat{\lambda}_{{\scriptscriptstyle Q}}^{2}(t)\,\omega_{\rm ct}}{2}\;{\cal I}\otimes\boldsymbol{\sigma_{3}}\ .$$

(3.12)

The convolution in the second line of (3.2) reveals how $\partial_{t}\varrho$ depends on the previous history of the qubit’s evolution, and the kernel ${\cal W}(t_{1},t_{2})$ appearing in this convolution is the Wightman function for the field $\phi_{\scriptscriptstyle I}$ — i.e. precisely the quantity quoted above in §2 that is calculated explicitly in Hotspot — evaluated at two points along the qubit world-line:

$${\cal W}(t_{1},t_{2}):=\mathrm{Tr}\Bigl{[}\phi_{{\scriptscriptstyle I}}(t_{1},{{\bf x}}_{\scriptscriptstyle Q})\phi_{{\scriptscriptstyle I}}(t_{2},{{\bf x}}_{\scriptscriptstyle Q})\rho_{0}\Bigr{]}\,.$$

(3.13)

Because (3.2) is a $2\times 2$ matrix equation it looks harder to solve than it really is. In particular, the properties $\mathrm{tr}\,\boldsymbol{\varrho}=1$ and $\boldsymbol{\varrho}^{\dagger}=\boldsymbol{\varrho}$ can be used to eliminate $\varrho_{22}=1-\varrho_{11}$ and $\varrho_{21}=\varrho_{12}^{\ast}$ from these equations, so it suffices to know how $\varrho_{12}$ and $\varrho_{11}$ evolve. Using (3.2) to evaluate the evolution for these two components reveals that they decouple from one another, and so evolve independently with

	$\displaystyle\frac{\partial\varrho_{11}(t)}{\partial t}$	$\displaystyle=$	$\displaystyle 2\lambda_{{\scriptscriptstyle Q}}^{2}\int_{t_{0}}^{t}{\hbox{d}}s\;\bigg{(}\mathrm{Re}[{\cal W}(t,s)]\cos(\omega[t-s])+\mathrm{Im}[{\cal W}(t,s)]\sin(\omega[t-s])\bigg{)}$		(3.14)
			$\displaystyle\quad\quad\quad\quad\quad-4\lambda_{{\scriptscriptstyle Q}}^{2}\int_{t_{0}}^{t}{\hbox{d}}s\;\mathrm{Re}[{\cal W}(t,s)]\cos(\omega[t-s])\varrho_{11}(s)$

and (for $t>t_{0}$)

	$\displaystyle\frac{\partial\varrho_{12}(t)}{\partial t}$	$\displaystyle=$	$\displaystyle-i\lambda_{{\scriptscriptstyle Q}}^{2}\,\omega_{\rm ct}\;\varrho_{12}(t)-2\lambda_{{\scriptscriptstyle Q}}^{2}\int_{t_{0}}^{t}{\hbox{d}}s\;\mathrm{Re}[{\cal W}(t,s)]e^{+i\omega[t-s]}\varrho_{12}(s)$		(3.15)
			$\displaystyle\quad\quad\quad\quad\quad+2\lambda_{{\scriptscriptstyle Q}}^{2}e^{+2i\omega t}\int_{t_{0}}^{t}{\hbox{d}}s\;\mathrm{Re}[{\cal W}(t,s)]e^{-i\omega[t-s]}\varrho^{\ast}_{12}(s)\,.$

The evolution equations normally used (for instance in Sciama:1981hr ) when perturbatively treating Unruh-DeWitt detectors are obtained from (3.14) and (3.15) by replacing all appearances of $\varrho_{ij}(t)$ on the right-hand side with the initial condition $\varrho_{ij}(t_{0})$. (In particular these terms would all vanish if the intial state was the ground state, which corresponds to $\varrho_{22}(t_{0})=1$ with all others zero in the present notation.) Indeed replacing $\varrho_{ij}(t)$ with $\varrho_{ij}(t_{0})$ seems very reasonable at first sight because the difference between $\varrho_{ij}(t)$ and $\varrho(t_{0})_{ij}$ is higher order in $\lambda_{\scriptscriptstyle Q}$ and so straight-up perturbation theory should drop this difference. Eqs. (3.14) and (3.15) are nonetheless better approximations at late times and disagree with naive perturbation theory precisely because perturbative methods break down at late times.

Our intended application of these expressions is to understand whether (and how quickly) the qubit thermalizes due to its indirect interaction with the hotspot, and we have no interest in the transients associated with the turn-on of couplings. This makes very late times our focus of interest, and so we restrict our attention to qubit positions and times that satisfy

$$t\ >\ t_{0}\ >\ |{{\bf x}}_{{\scriptscriptstyle Q}}|\,,$$

(3.16)

where $t_{0}$ is the turn-on time for the qubit interaction appearing in (3.4). The choice $t_{0}>|{{\bf x}}_{{\scriptscriptstyle Q}}|$ ensures that the outgoing wave caused by the $t=0$ turn-on of the $\phi$-$\chi^{a}$ field couplings have had time to have passed the location of the qubit. In practice we choose times in the far future for which the full correlation function (2.12) (without the step- and delta-functions) can be used. Since in this limit ${\cal W}(t,s)={\cal W}(t-s)$ it is convenient to define $\widetilde{\cal W}$ by

$$\widetilde{{\cal W}}(\tau):={\cal W}(\tau,0)=\mathscr{S}(\tau,{{\bf x}}_{{\scriptscriptstyle Q}};0,{{\bf x}}_{{\scriptscriptstyle Q}})|_{\lambda=0}+\mathscr{E}_{\beta}(\tau,{{\bf x}}_{{\scriptscriptstyle Q}};0,{{\bf x}}_{{\scriptscriptstyle Q}})|_{\lambda=0}\,,$$

(3.17)

where $\mathscr{S}$ and $\mathscr{E}_{\beta}$ are the functions defined in (2.2) and (2.16). Explicitly

	$\displaystyle\widetilde{{\cal W}}(\tau)$	$\displaystyle=$	$\displaystyle-\frac{1}{4\pi^{2}(\tau-i\delta)^{2}}+\frac{2\epsilon^{2}}{\tilde{g}^{2}|{{\bf x}}_{{\scriptscriptstyle Q}}|^{2}}\bigg{[}I_{-}(\tau+2|{{\bf x}}_{{\scriptscriptstyle Q}}|,c_{0})-I_{-}(\tau,c_{0})-I_{+}(\tau,c_{0})+I_{+}(\tau-2|{{\bf x}}_{{\scriptscriptstyle Q}}|,c_{0})\bigg{]}$
			$\displaystyle\qquad+\frac{\epsilon}{8\pi^{2}|{{\bf x}}_{{\scriptscriptstyle Q}}|^{2}}\bigg{[}-\frac{1}{\tau+2|{{\bf x}}_{{\scriptscriptstyle Q}}|-i\delta}+\frac{1}{\tau-2|{{\bf x}}_{{\scriptscriptstyle Q}}|-i\delta}\bigg{]}$
			$\displaystyle\qquad\qquad-\frac{32\pi^{2}\epsilon^{4}}{\tilde{g}^{4}|{{\bf x}}_{{\scriptscriptstyle Q}}|^{2}}\bigg{[}I_{-}(\tau,c_{0})+I_{+}(\tau,c_{0})\bigg{]}-\frac{\epsilon^{2}}{4\pi^{2}|{{\bf x}}_{{\scriptscriptstyle Q}}|^{2}(\tau-i\delta)^{2}}$
			$\displaystyle\qquad\qquad\qquad+\frac{2\epsilon^{2}}{\tilde{g}^{2}|{{\bf x}}_{{\scriptscriptstyle Q}}|^{2}}\bigg{[}\Phi\bigg{(}e^{-\tfrac{2\pi(\tau-i\delta)}{\beta}},1,\frac{c_{0}\beta}{2\pi}\bigg{)}+\Phi\bigg{(}e^{+\tfrac{2\pi(\tau-i\delta)}{\beta}},1,\frac{c_{0}\beta}{2\pi}\bigg{)}-\frac{2\pi}{c_{0}\beta}\bigg{]}$

with the functions $\Phi$ and $I_{\pm}(\tau,c_{0})$ as defined in (2.15) and (2.17). The parameter $c_{0}$ here denotes

$$c_{0}\ :=\ c\;|_{\lambda=0}\ =\ \frac{16\pi^{2}\epsilon}{\tilde{g}^{2}}$$

(3.19)

which arises because we set $\lambda=0$ in this section (c.f. (2.14)).

After a change of variable $s\to t-s$ on the right-hand side of (3.14) and (3.15), and using the symmetry $\widetilde{{\cal W}}^{\ast}(\tau)=\widetilde{{\cal W}}(-\tau)$ of the Wightman functions that follows from the hermiticity of the field ${\phi}$, we get the evolution equations in the form we ultimately solve

$\displaystyle\frac{\partial\varrho_{11}(t)}{\partial t}$

$\displaystyle=$

$\displaystyle\lambda_{{\scriptscriptstyle Q}}^{2}\int_{-(t-t_{0})}^{t-t_{0}}{\hbox{d}}s\;\widetilde{{\cal W}}(s)\,e^{-i\omega s}-4\lambda_{{\scriptscriptstyle Q}}^{2}\int_{0}^{t-t_{0}}{\hbox{d}}s\;\mathrm{Re}[\widetilde{{\cal W}}(s)]\cos(\omega s)\varrho_{11}(t-s)$

(3.20)

and

	$\displaystyle\frac{\partial\varrho_{12}(t)}{\partial t}$	$\displaystyle=$	$\displaystyle-i\lambda_{{\scriptscriptstyle Q}}^{2}\,\omega_{\rm ct}\varrho_{12}(t)-2\lambda_{{\scriptscriptstyle Q}}^{2}\int_{0}^{t-t_{0}}{\hbox{d}}s\;\mathrm{Re}[\widetilde{{\cal W}}(s)]\,e^{+i\omega s}\varrho_{12}(t-s)$
			$\displaystyle\qquad\qquad\qquad\quad\quad+2\lambda_{{\scriptscriptstyle Q}}^{2}\,e^{+2i\omega t}\int_{0}^{t-t_{0}}{\hbox{d}}s\;\mathrm{Re}[\widetilde{{\cal W}}(s)]\,e^{-i\omega s}\varrho^{\ast}_{12}(t-s)\,.$

Equations (3.20) and (3.2) are in general difficult to solve, largely due to the convolutions appearing on their right-hand sides. We seek here approximate solutions in the special situation where the kernel $\widetilde{\cal W}(s)$ varies over some time-scale $\tau_{c}$, say, that is much shorter than the scale $\tau_{\rho}$ over which $\varrho(t-s)$ varies. In such a case the integrands of eqs. (3.20) and (3.2) can be usefully expanded in powers of $s$,

$$\boldsymbol{\varrho}(t-s)\simeq\boldsymbol{\varrho}(t)-s\,\partial_{t}{\boldsymbol{\varrho}}(t)+\ldots\ ,$$

(3.22)

with successive terms suppressed by powers of $\tau_{c}/\tau_{\rho}$ after the integration over $s$ is performed. Once derivatives of $\varrho$ can be neglected then equations (3.20) and (3.2) become Markovian because they give $\partial_{t}\varrho(t)$ directly in terms of $\varrho(t)$ (without a convolution over earlier times) and can be integrated with little difficulty.