Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: The method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain

The tight-binding model (TBM) [9, 10] is a textbook description of a quantum solid in which an electron is assumed mostly localized on the sites of a lattice but because of spontaneous quantum fluctuations makes occasional tunnelling to nearest-neighbour sites. Specifically, the tight-binding chain (TBC), referring to the TBM in one dimension, is characterized by a Hamiltonian modelling the motion of a quantum particle, e.g., an electron, between the nearest-neighbour sites of a one-dimensional lattice. The Hamiltonian reads

$\displaystyle H=-\frac{\Delta}{2}\sum_{n=-\infty}^{\infty}\Big{(}|n\rangle\langle n+1|+|n+1\rangle\langle n|\Big{)},$

(13)

where $\Delta>0$ is the nearest-neighbour intersite hopping integral, and $|n\rangle$ is the so-called Wannier state denoting the state of the particle when located on site $n$. The Wannier states form a set of complete basis states satisfying $\langle m|n\rangle=\delta_{mn}$ and $\sum_{m=-\infty}^{\infty}|m\rangle\langle m|=\mathbb{I}$, the identity operator. Here, we have taken the lattice spacing to be unity, without loss of generality.

The TBM plays a critical role in the band theory of solids. With the advent of mesoscopic physics, the TBM finds relevance in discussing one-dimensional nanowires [15, 16, 17]. In this subsection, we collect relevant results for the quantum mechanics of a tight-binding chain. Although most of these results are already documented in the literature vide the work of Dunlap and Kenkre [9], we nevertheless systematically collate them here in order to apply the formalism developed in Sec. 2 later in the paper.

A particle moving under the Hamiltonian (13) has a motion akin to the ballistic motion of a classical particle. To see this explicitly, we introduce the operators

$\displaystyle K\equiv\sum_{n=-\infty}^{\infty}|n\rangle\langle n+1|,~{}~{}K^{\dagger}\equiv\sum_{n=-\infty}^{\infty}|n+1\rangle\langle n|,$

(14)

in terms of which we have

$\displaystyle H=-\frac{\Delta}{2}(K+K^{\dagger}).$

(15)

Note that $K$ and $K^{\dagger}$ commute with each other, with their product being $KK^{\dagger}=K^{\dagger}K=\mathbb{I}$. Also, one has $K^{2}=\sum_{n=-\infty}^{\infty}|n\rangle\langle n+2|$ and $(K^{\dagger})^{2}=\sum_{n=-\infty}^{\infty}|n+2\rangle\langle n|$. That $K$ and $K^{\dagger}$ commute implies that we can find a representation in which $K$ and $K^{\dagger}$ can be simultaneously diagonalized. Such a representation is elicited through the so-called Bloch states, defined in the reciprocal momentum-space as

$\displaystyle|k\rangle\equiv\frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}\mathrm{e}^{-{\rm i}nk}|n\rangle,$

(16)

so that we have $|n\rangle=(1/\sqrt{2\pi})\int_{-\pi}^{\pi}{\rm d}k\,\exp({\rm i}nk)|k\rangle$. It is easily verified that

$\displaystyle K|k\rangle=\mathrm{e}^{-{\rm i}k}|k\rangle,~{}~{}K^{\dagger}|k\rangle=\mathrm{e}^{{\rm i}k}|k\rangle,$

(17)

thus proving that the Bloch state $|k\rangle$ is indeed an eigenstate of $K$ and $K^{\dagger}$, with eigenvalues $\exp(-{\rm i}k)$ and $\exp({\rm i}k)$, respectively. We therefore have

$\displaystyle\mathrm{e}^{\mathrm{i}Ht}|m^{\prime}\rangle$

$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{-\pi}^{\pi}{\rm d}k~{}\mathrm{e}^{\mathrm{i}km^{\prime}}\mathrm{e}^{\mathrm{i}Ht}|k\rangle=\frac{1}{\sqrt{2\pi}}\int_{-\pi}^{\pi}{\rm d}k~{}\mathrm{e}^{\mathrm{i}km^{\prime}}\mathrm{e}^{-\mathrm{i}\Delta t\cos k}|k\rangle,$

(18)

where in obtaining the last equality, one uses Eq. (17). We therefore have on using $\langle k^{\prime}|k\rangle=\delta(k-k^{\prime})$ that

$\displaystyle\langle m|\mathrm{e}^{\mathrm{i}Ht}|m^{\prime}\rangle=\frac{1}{2\pi}\int_{-\pi}^{\pi}{\rm d}k~{}\mathrm{e}^{-\mathrm{i}k(m-m^{\prime})-\mathrm{i}\Delta t\cos k}.$

(19)

Let us now ask the question: what is the probability $P_{m}(t)$ that the particle is on site $m$ at time $t>0$, given that the particle was on site $n_{0}$ at time $t=0$? By definition, we have

$\displaystyle P_{m}(t)=\langle m|\rho(t)|m\rangle.$

(20)

Here, $\rho(t)$ is the density operator of the system, whose time evolution follows the usual dynamics

$\displaystyle\rho(t)=\mathrm{e}^{-\mathrm{i}\mathcal{L}t}\rho(0)=\mathrm{e}^{-\mathrm{i}Ht}\rho(0)\mathrm{e}^{\mathrm{i}Ht}.$

(21)

Equation (21) implies unitary evolution of the density operator that preserves its trace: ${\rm Tr}[\rho(t)]=1~{}\forall~{}t$. In our case, we have $\rho(0)=|n_{0}\rangle\langle n_{0}|$, so that it may be shown on using Eq. (19) that

$$P_{m}(t)=\frac{1}{(2\pi)^{2}}\int_{-\pi}^{\pi}{\rm d}k\int_{-\pi}^{\pi}{\rm d}k^{\prime}~{}\mathrm{e}^{{\rm i}(m-n_{0})(k-k^{\prime})}~{}\mathrm{e}^{-{\rm i}\Gamma_{kk^{\prime}}t},$$

(22)

with

$\displaystyle\Gamma_{kk^{\prime}}\equiv\Delta(\cos k^{\prime}-\cos k).$

(23)

It may be checked that as desired, $\sum_{m=-\infty}^{\infty}P_{m}(t)=1$. Using the identities $\exp(-\mathrm{i}\Delta t\cos k)=\sum_{n}\exp(-\mathrm{i}n\pi/2+\mathrm{i}nk)J_{n}(\Delta t)$ [9], with $J_{n}(x)$ the Bessel function of the first kind, and $\int_{-\pi}^{\pi}{\rm d}k~{}\exp(\mathrm{i}(m-n)k)=2\pi\delta_{mn}$, one may straightforwardly show that

$\displaystyle\frac{1}{(2\pi)^{2}}\int_{-\pi}^{\pi}{\rm d}k\int_{-\pi}^{\pi}{\rm d}k^{\prime}~{}\mathrm{e}^{\mathrm{i}m(k-k^{\prime})}\mathrm{e}^{-\mathrm{i}\Gamma_{kk^{\prime}}t}=J_{m}^{2}(\Delta t).$

(24)

Then, from Eq. (22), we get [18]

$\displaystyle P_{m}(t)=J_{m-n_{0}}^{2}(\Delta t),$

(25)

implying that $\Delta$ sets the intrinsic time scale in the model. Using the result that in the limit $\nu\to\infty$ we have for $x\neq 0$ and fixed that $J_{\nu}(x)\sim(1/\sqrt{2\pi\nu})(\mathrm{e}\,x/(2\nu))^{\nu}$ [19, Eq. 10.19.1], we get

$\displaystyle P_{m}(t)\sim\left(\frac{\mathrm{e}\,\Delta~{}t}{2|m-n_{0}|}\right)^{2|m-n_{0}|};\quad|m-n_{0}|\to\infty.$

(26)

The discrete Fourier transform of the function $P_{m}(t)$, given by $\widetilde{P}(k,t)\equiv 1/(\sqrt{2\pi})\sum_{m=-\infty}^{\infty}\exp(-{\rm i}mk)P_{m}(t)$, is obtained as

$\displaystyle\widetilde{P}(k,t)$

$\displaystyle=\frac{1}{(2\pi)^{3/2}}\int_{-\pi}^{\pi}{\rm d}k^{\prime}~{}\mathrm{e}^{-{\rm i}n_{0}k}~{}\mathrm{e}^{{\rm i}\Delta t\left(\cos(k+k^{\prime})-\cos k^{\prime}\right)},$

(27)

using $\sum_{m=-\infty}^{\infty}\exp(-{\rm i}(k+k_{1}-k_{2})m)=2\pi\delta(k+k_{1}-k_{2})$.

Equation (22) implies that the particle while starting from the site $n_{0}$ eventually spreads out to other sites of the lattice. The average displacement from the initial location $n_{0}$ is given by

$\displaystyle\mu\equiv\sum_{m=-\infty}^{\infty}(m-n_{0})P_{m}(t)=0,$

(28)

since Eq. (22) implies that $P_{m}(t)$ is invariant under $m-n_{0}\to-(m-n_{0})$. Let us then compute the mean-squared displacement (MSD) of the particle about $n_{0}$:

$\displaystyle S(t)\equiv\sum_{m=-\infty}^{\infty}(m-n_{0})^{2}P_{m}(t)=\sum_{m=-\infty}^{\infty}m^{2}P_{m}(t)+n_{0}^{2}-2n_{0}\sum_{m=-\infty}^{\infty}mP_{m}(t),$

(29)

where we have used $\sum_{m=-\infty}^{\infty}P_{m}(t)=1$. Now, we have $\sum_{m=-\infty}^{\infty}mP_{m}(t)=\sum_{m=-\infty}^{\infty}(m-n_{0}+n_{0})P_{m}(t)=n_{0}$, so that $S(t)=\sum_{m=-\infty}^{\infty}m^{2}P_{m}(t)-n_{0}^{2}$. In terms of $\widetilde{P}(k,t)$, we get $\sum_{m=-\infty}^{\infty}m^{2}P_{m}(t)=-\sqrt{2\pi}{\rm d}^{2}\widetilde{P}(k,t)/{\rm d}k^{2}|_{k=0}$, using which it may be shown straightforwardly that

$\displaystyle S(t)=\frac{\Delta^{2}t^{2}}{2}.$

(30)

Thus, we see that the MSD has a quadratic time dependence similar to the case of the ballistic motion of a free classical particle but with a ‘quantal’ speed equal to $\Delta/\sqrt{2}$. In interpreting Eq. (30), note that we have set the lattice spacing to unity at the outset.

Here, we will consider the form of $T$ that implements stochastic resets of the TBC system at random times. We formulate our problem thus: The quantum particle is taken to begin its journey at time $t=0$ from an arbitrary site $n_{0}$ of the TBC, so that the corresponding density operator is $\rho(0)=|n_{0}\rangle\langle n_{0}|$. It then streams unitarily under the Hamiltonian (13) until a random time $t_{1}$ at which an instantaneous reset takes place of the density operator onto the form $|{\cal N}\rangle\langle{\cal N}|$ corresponding to a complete collapse of the state vector of the system onto the state $|{\cal N}\rangle$ corresponding to an arbitrary site ${\cal N}$. Subsequent to the first reset, the sojourn of the particle begins anew from the site ${\cal N}$ and an unitary evolution follows until time $t_{2}\,(t_{2}>t_{1})$, again random, when a second reset takes place and subsequent to which the particle evolution is renewed a second time.This ongoing ‘renewal’ process, much akin to the random walk scenario of Montroll et al. [20, 21, 22], continues at a sequence of random times, and finally, we may ask at time $t$ about the probability $P_{m}(t)$ for the particle to be on an arbitrary site $m$ of the lattice. We note that for exponential $p(t)$, we may view the dynamics in a manner similar to the one discussed in Ref. [11]: in a small time interval ${\rm d}t$, the wave vector of the system becomes either the state $|{\cal N}\rangle$ with probability $\lambda{\rm d}t$ (i.e., there is a reset to the state $|{\cal N}\rangle$), or, evolves unitarily under the Hamiltonian $H$ with probability $1-\lambda{\rm d}t$ [23]; consequently, reset to the state $|{\cal N}\rangle$ is indeed a probabilistic event in time.

Stochastic reset of quantum dynamics, involving quantum dynamics interrupted at random times with reset to a given state, has been a topic of much current interest [11], following recent extensive research on resetting in the classical domain. This emerging field of research was initialized in Ref. [24] in the context of classical diffusion of a single particle that is reset to its initial location at random times. A remarkable revelation of the study is the following. In the absence of resetting, the distribution of the particle location is an ever-spreading Gaussian centered at the initial location of the particle and so does not have a stationary state. By contrast, introducing resetting into the dynamics leads to a time-independent probability distribution at long times and the emergence of a nonequilibrium stationary state. Reference [24] is considered a landmark contribution in recent times, which has really ushered in a new beginning in studies of stochastic processes. Starting with the example of a single diffusing particle, subsequent focus has been on a wide variety of dynamical scenarios of both theoretical and practical relevance. To name a few, we may cite in the context of a single particle the example of its diffusive motion in a bounded domain [25] and in a potential [26], and the examples of continuous-time random walk [27, 28], Lévy [29] and exponential constant-speed flights [30]. Resetting has also been studied in interacting particle systems such as fluctuating interfaces [31, 32] and reaction-diffusion models [33]. Stochastic resets of quantum dynamics studied in Ref. [11] considered resetting of several representative closed quantum systems at constant rate $r$, to show that any generic observable never thermalizes under such dynamics for any $r$. We refer the reader to the recent review [34] on the topic of resetting.

We now set out to obtain the probability $\overline{P}_{m}(t)$ for the particle to be on site $m$ at time $t$ while evolving according to our scheme of repeated resets at random times. The overline denotes as in Eq. (1) an average over different realizations of the dynamical protocol. To this end, we will employ the formalism developed in Sec. 2. In our case, we have

$\displaystyle\rho(0)=|n_{0}\rangle\langle n_{0}|,$

(31)

while the projection superoperator $T$ projects the instantaneous density operator onto the form $|{\cal N}\rangle\langle{\cal N}|$. The matrix elements of the superoperator $\widetilde{U}_{0}(s)$ defined in Eq. (12) may be obtained by referring to Appendix 6, and by using Eqs. (9) and (19) as

	$\displaystyle(mn|\widetilde{U}_{0}(s)|m^{\prime}n^{\prime})$	$\displaystyle=(mn|\int_{0}^{\infty}{\rm d}t~{}\mathrm{e}^{-(s+\lambda){\mathbb{I}}t-\mathrm{i}\mathcal{L}t}|m^{\prime}n^{\prime})=\int_{0}^{\infty}{\rm d}t~{}\mathrm{e}^{-(s+\lambda)\mathbb{I}t}\langle m|\mathrm{e}^{-\mathrm{i}Ht}|m^{\prime}\rangle\langle n^{\prime}|\mathrm{e}^{\mathrm{i}Ht}|n\rangle$
		$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{-\pi}^{\pi}{\rm d}k\int_{-\pi}^{\pi}{\rm d}k^{\prime}~{}\mathrm{e}^{\mathrm{i}(m-m^{\prime})k-\mathrm{i}(n-n^{\prime})k^{\prime}}\frac{1}{s+\lambda+\mathrm{i}\Gamma_{kk^{\prime}}}.$		(32)

It then follows that

$\displaystyle\sum_{m}(mm|\widetilde{U}_{0}(s)|nn)=\frac{1}{s+\lambda},$

(33)

which may be shown by using $\sum_{m=-\infty}^{\infty}\exp(\mathrm{i}m(k-k^{\prime}))=2\pi\delta(k-k^{\prime})$.

We want the form of $T$ to be such that subsequent to a reset at any time instant $t$, the density operator becomes $\rho_{+}(t)=T\rho_{-}(t)=\mathinner{|{{\cal N}}\rangle}\mathinner{\langle{{\cal N}}|}$, with the trace of the operators satisfying ${\rm Tr}[\rho_{+}(t)]={\rm Tr}[\rho_{-}(t)]=1~{}\forall~{}t$ and $\rho_{-}(t)$ denoting the density operator prior to the reset. This is guaranteed with the form

$\displaystyle(n_{1}n_{1}^{\prime}|T|n_{2}n_{2}^{\prime})=\delta_{n_{1}n_{1}^{\prime}}\delta_{n_{2}n_{2}^{\prime}}\delta_{n_{1}{\cal N}}.$

(34)

Indeed, then, we have

$\displaystyle\langle n|T\rho_{-}(t)|n^{\prime}\rangle$

$\displaystyle=\sum_{n_{3},n_{4}}(nn^{\prime}|T|n_{3}n_{4})\mathinner{\langle{n_{3}}|}\rho_{-}(t)\mathinner{|{n_{4}}\rangle}=\delta_{nn^{\prime}}\delta_{n{\cal N}}\sum_{n_{3}}\mathinner{\langle{n_{3}}|}\rho_{-}(t)\mathinner{|{n_{3}}\rangle}=\delta_{nn^{\prime}}\delta_{n{\cal N}},$

(35)

implying that $T\rho_{-}(t)=\mathinner{|{\cal N}\rangle}\mathinner{\langle{\cal N}|}$; here we have used the fact that ${\rm Tr}[\rho_{-}(t)]=1~{}\forall~{}t$.

Now, by definition, we have

$\displaystyle\overline{P}_{m}(t)=\langle m|\overline{\rho}(t)|m\rangle.$

(36)

Denoting $\overline{P}_{m}(t)$ in the Laplace domain by $\widetilde{\overline{P}}_{m}(s)$, one obtains from Eqs. (4) and (11) that

	$\displaystyle\widetilde{\overline{P}}_{m}(s)$	$\displaystyle=\langle m|\widetilde{U}_{0}(s)\rho(0)+\lambda\widetilde{U}_{0}(s)T\widetilde{U}_{0}(s)\rho(0)+\lambda^{2}\widetilde{U}_{0}(s)T\widetilde{U}_{0}(s)T\widetilde{U}_{0}(s)\rho(0)+\ldots|m\rangle$
		$\displaystyle\equiv\sum_{p=0}^{\infty}\widetilde{\overline{P}}_{m}^{(p)}(s),$		(37)

where $\widetilde{\overline{P}}_{m}^{(p)}(s);~{}p\in[0,\infty)$ is the $p$-th term of the infinite series, which corresponds to $p$ instantaneous interactions.

From Eq. (37), we get

	$\displaystyle\widetilde{\overline{P}}_{m}^{(0)}(s)$	$\displaystyle=\langle m|\widetilde{U}_{0}(s)\rho(0)|m\rangle=\sum_{n,n^{\prime}}(mm|\widetilde{U}_{0}(s)|nn^{\prime})\mathinner{\langle{n}|}\rho(0)\mathinner{|{n^{\prime}}\rangle}=(mm|\widetilde{U}_{0}(s)|n_{0}n_{0})$
		$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{-\pi}^{\pi}{\rm d}k\int_{-\pi}^{\pi}{\rm d}k^{\prime}~{}\mathrm{e}^{\mathrm{i}(m-n_{0})(k-k^{\prime})}\frac{1}{s+\lambda+\mathrm{i}\Gamma_{kk^{\prime}}},$		(38)

where we have used Eqs. (31) and (32). The quantity $\overline{P}_{m}^{(0)}(t)$ may be read off from Eq. (38) to be

$\displaystyle\overline{P}_{m}^{(0)}(t)$

$\displaystyle=\frac{\mathrm{e}^{-\lambda t}}{(2\pi)^{2}}\int_{-\pi}^{\pi}{\rm d}k\int_{-\pi}^{\pi}{\rm d}k^{\prime}~{}\mathrm{e}^{\mathrm{i}(m-n_{0})(k-k^{\prime})}\mathrm{e}^{-\mathrm{i}\Gamma_{kk^{\prime}}t}=\mathrm{e}^{-\lambda t}J_{m-n_{0}}^{2}(\Delta t),$

(39)

where we have used Eq. (24). From Eqs. (38) and (39), we get

$\displaystyle\mathfrak{L}((mm|\widetilde{U}_{0}(s)|n_{0}n_{0}))=\mathrm{e}^{-\lambda t}J_{m-n_{0}}^{2}(\Delta t).$

(40)

Similarly, from Eq. (37), we get

	$\displaystyle\widetilde{\overline{P}}_{m}^{(1)}(s)$	$\displaystyle=\lambda\sum_{\begin{subarray}{c}n_{1},n_{2},n_{3},\\ n_{4},n_{5},n_{6}\end{subarray}}(mm|\widetilde{U}_{0}(s)|n_{1}n_{2})(n_{1}n_{2}|T|n_{3}n_{4})(n_{3}n_{4}|\widetilde{U}_{0}(s)|n_{5}n_{6})\langle n_{5}|\rho(0)|n_{6}\rangle$
		$\displaystyle=\lambda\sum_{n_{3}}(mm|\widetilde{U}_{0}(s)|{\cal N}{\cal N})(n_{3}n_{3}|\widetilde{U}_{0}(s)|n_{0}n_{0})$
		$\displaystyle=\frac{\lambda}{(2\pi)^{2}}\int_{-\pi}^{\pi}{\rm d}k_{1}\int_{-\pi}^{\pi}{\rm d}k_{2}~{}\mathrm{e}^{\mathrm{i}(m-{\cal N})(k_{1}-k_{2})}\frac{1}{(s+\lambda)(s+\lambda+\mathrm{i}\Gamma_{k_{1}k_{2}})},$		(41)

where in obtaining the last step, we have used Eqs. (32) and (33). Using the fact that the Laplace transform of a convolution of two functions is given by the product of their individual Laplace transforms, we may write

	$\displaystyle\widetilde{\overline{P}}_{m}^{(1)}(s)$	$\displaystyle=\lambda\mathfrak{L}\Big{(}\int_{0}^{t}{\rm d}t^{\prime}\,\mathrm{e}^{-\lambda(t-t^{\prime})}\frac{1}{(2\pi)^{2}}\int_{-\pi}^{\pi}{\rm d}k_{1}\int_{-\pi}^{\pi}{\rm d}k_{2}~{}\mathrm{e}^{\mathrm{i}(m-{\cal N})(k_{1}-k_{2})}\mathrm{e}^{-(\lambda+\mathrm{i}\Gamma_{k_{1}k_{2}})t^{\prime}}\Big{)}$
		$\displaystyle=\lambda\mathfrak{L}\Big{(}\mathrm{e}^{-\lambda t}\int_{0}^{t}{\rm d}t^{\prime}\,J_{m-{\cal N}}^{2}(\Delta t^{\prime})\Big{)},$		(42)

where we have used Eq. (24). We thus arrive at the result

$\displaystyle\overline{P}_{m}^{(1)}(t)=\lambda\mathrm{e}^{-\lambda t}\int_{0}^{t}{\rm d}t^{\prime}~{}J^{2}_{m-{\cal N}}(\Delta t^{\prime}).$

(43)

Proceeding in a similar manner, we get

$\displaystyle\widetilde{\overline{P}}_{m}^{(2)}(s)$

$\displaystyle=\frac{\lambda^{2}}{(2\pi)^{2}}\int_{-\pi}^{\pi}{\rm d}k_{1}\int_{-\pi}^{\pi}{\rm d}k_{2}~{}\mathrm{e}^{\mathrm{i}(m-{\cal N})(k_{1}-k_{2})}\frac{1}{(s+\lambda)^{2}(s+\lambda+\mathrm{i}\Gamma_{k_{1}k_{2}})}.$

(44)

Using the result on Laplace transform of a convolution and Eq. (24), the last equation may be written as

$\displaystyle\widetilde{\overline{P}}_{m}^{(2)}(s)=\lambda^{2}\mathfrak{L}\Big{(}\mathrm{e}^{-\lambda t}\int_{0}^{t}{\rm d}t^{\prime}\,(t-t^{\prime})~{}J_{m-{\cal N}}^{2}(\Delta t^{\prime})\Big{)}$

(45)

yielding

$\displaystyle\overline{P}_{m}^{(2)}(t)=\lambda^{2}~{}\mathrm{e}^{-\lambda t}\int_{0}^{t}{\rm d}t^{\prime}~{}(t-t^{\prime})J^{2}_{m-{\cal N}}(\Delta t^{\prime}).$

(46)

The general expression may thus be written as

$\displaystyle\overline{P}_{m}^{(p)}(t)=\lambda^{p}~{}\mathrm{e}^{-\lambda t}\int_{0}^{t}\!\!{\rm d}t^{\prime}~{}\frac{(t-t^{\prime})^{p-1}}{(p-1)!}~{}J^{2}_{m-{\cal N}}(\Delta t^{\prime});\quad p\in[1,\infty),$

(47)

so that inverting Eq. (37) back to the time domain and using the above equation, one gets [18]

$\displaystyle\overline{P}_{m}(t)=\mathrm{e}^{-\lambda t}J_{m-n_{0}}^{2}(\Delta t)+\lambda\int_{0}^{t}{\rm d}t^{\prime}~{}\mathrm{e}^{-\lambda t^{\prime}}J_{m-{\cal N}}^{2}(\Delta t^{\prime}).$

(48)

Equation (48) is a key result of the paper, yielding the average probability at time $t$ to be on site $m$ for a quantum particle starting at site $n_{0}$ and evolving under Hamiltonian (13) subject to repeated resets at random times. This result may be contrasted with Eq. (25) that holds in the absence of resets. Summing over $m$ on both sides of Eq. (48), and using the results $J_{-m}(x)=(-1)^{m}J_{m}(x)$ and $1=J^{2}_{0}(x)+2\sum_{m=1}^{\infty}J^{2}_{m}(x)$, we get $\sum_{m=-\infty}^{\infty}\overline{P}_{m}(t)=\exp(-\lambda t)+\lambda\int_{0}^{t}{\rm d}t^{\prime}\exp(-\lambda t^{\prime})=1$, as desired.

The result (48), which may be obtained as a simple extension of formula (3) in Ref. [31], has a simple physical interpretation. The first term on the right hand side arises from those realizations of evolution for time $t$ that did not involve a single reset (the probability for which is $\exp(-\lambda t)$) [23], so that the corresponding contribution to $\overline{P}_{m}(t)$ would be given by the product of the probability of no reset for time $t$ with the probability to be on site $m$ at time $t$ in the absence of any reset and while starting from site $n_{0}$ at time $t=0$. The latter quantity is obtained from our analysis in 2 as equal to $J_{m-n_{0}}^{2}(\Delta t)$, see Eq. (25). In order to understand the second term on the right hand side, let us appreciate that every time there is a reset, the system collapses to the state $|{\cal N}\rangle$ and its evolution starts afresh from site ${\cal N}$. Consequently, at time $t$, what should matter is when has been the last reset, and the corresponding contribution to $\overline{P}_{m}(t)$ would be given by the product of the probability at time $t$ of the last reset to occur in the interval $[\,t^{\prime},\,t^{\prime}-{\rm d}t^{\prime}\,]$, with $t^{\prime}\in[\,0,\,t\,]$, with the probability in the absence of any reset for the particle to be on site $m$ owing to evolution for time duration $t-t^{\prime}$ while starting from site ${\cal N}$. The former probability is given by $\lambda{\rm d}t^{\prime}\,\exp(-\lambda(t-t^{\prime}))$ [23]. We thus get the contribution as equal to $\lambda\int_{0}^{t}{\rm d}t^{\prime}\,\exp(-\lambda(t-t^{\prime}))~{}J_{m-{\cal N}}^{2}(\Delta(t-t^{\prime}))$, to which effecting a change of integration variable directly yields the second term on the right hand side of Eq. (48). An interpretation similar to the foregoing leads in the case of resetting in the classical domain [34] to a result for the probability distribution similar in form to the one in Eq. (48). We note in passing that the result (48) may also be obtained by using the formalism of Ref. [11]. However, our motivation here is not to obtain only the reset result (48), but to put forward a general formalism as in Sec. 2 to treat unitary evolution interspersed with non-unitary interactions at random times modelled by any interaction superoperator $T$, and the reset case is but a special case of this scenario.

Now, using the result $\sum_{m=-\infty}^{\infty}m\,J^{2}_{m-n_{0}}(x)=n_{0}$ (see the paragraph following Eq. (29)), one obtains from Eq. (48) the average displacement of the particle from the initial position $n_{0}$ as

$\displaystyle\overline{\mu}(t)\equiv\sum_{m=-\infty}^{\infty}(m-n_{0})~{}\overline{P}_{m}(t)=({\cal N}-n_{0})\Big{[}1-\mathrm{e}^{-\lambda t}\Big{]},$

(49)

which in the limit $\lambda\to 0$ reproduces as expected the result (28). Next, by using the result $\sum_{m=-\infty}^{\infty}m^{2}\,J^{2}_{m}(x)=x^{2}/2$ (see Eqs. (29) and (30)), the MSD of the particle about the initial location $n_{0}$ is obtained from Eq. (48) as

$\displaystyle\overline{S}(t)$

$\displaystyle\equiv\sum_{m=-\infty}^{\infty}(m-n_{0})^{2}~{}\overline{P}_{m}(t)=\frac{\Delta^{2}}{\lambda^{2}}\bigg{[}1-\mathrm{e}^{-\lambda t}(1+\lambda t)\bigg{]}+({\cal N}-n_{0})^{2}(1-\mathrm{e}^{-\lambda t}),$

(50)

which in the limit $\lambda\to 0$ reproduces correctly the result (30).