---
abstract: 'Stochastic resetting is prevalent in natural and man-made systems giving rise to a long series of non-equilibrium phenomena. Diffusion with stochastic resetting serves as a paradigmatic model to study these phenomena, but the lack of a well-controlled platform by which this process can be studied experimentally has been a major impediment to research in the field. Here, we report the experimental realization of colloidal particle diffusion and resetting via holographic optical tweezers. This setup serves as a proof-of-concept which opens the door to experimental study of resetting phenomena. It also vividly illustrates why existing theoretical models must be improved and revised to better capture the real-world physics of stochastic resetting.'
author:
- 'Ofir Tal-Friedman$^{1}$'
- 'Arnab Pal$^{2,3}$'
- 'Amandeep Sekhon$^{2}$'
- 'Shlomi Reuveni$^{2,3}$'
- 'Yael Roichman$^{1,2}$'
title: Experimental realization of diffusion with stochastic resetting
---

=1

Stochastic resetting is ubiquitous in nature, and has recently been the subject of vigorous studies in physics [@Evans2011_1; @Evans2011_2; @Evans2011_3], chemistry [@Restart-Biophysics1; @Restart-Biophysics2; @Restart-Biophysics6], biological physics [@Restart-Biophysics3; @Restart-Biophysics8], computer science [@restart-CS1; @restart-CS2], queuing theory [@queue1; @queue2] and other cross-disciplinary fields (see [@review] for extensive account of recent developments). A stylized model to study resetting phenomena was proposed by Evans and Majumdar in 2011 [@Evans2011_1]. The model, which considers a diffusing particle subject to stochastic resetting, exhibits many rich properties e.g., the emergence of a non-equilibrium steady state and interesting relaxation dynamics [@Evans2011_1; @Evans2011_2; @Evans2011_3; @Evans2014_3; @Pal2016_1; @relaxation1; @relaxation2; @local] which were also observed in other systems subject to stochastic resetting [@restart_conc3; @SEP; @return1; @return2; @return3; @return4; @return5; @Bod1; @Bod2]. The model is also pertinent to the study of search and first-passage time (FPT) questions [@RednerBook; @Schehr-review]. In particular, it was used to show that resetting can significantly reduce the mean FTP of a diffusing particle to a target by mitigating the deleterious effect of large FPT fluctuations that are intrinsic to diffusion in the absence of resetting [@Evans2011_1; @Evans2011_2; @Evans2011_3; @review; @Pal2016_1; @Ray; @interval]. Interestingly, this beneficial effect of resetting also extends beyond free diffusion and applies to many other stochastic processes [@review; @return5; @Bod1; @Bod2; @return3; @return4; @ReuveniPRL; @PalReuveniPRL; @branching_II; @Restart-Search1; @Restart-Search2; @Chechkin; @Landau; @HRS]; and further studies moreover revealed a genre of universality relations associated with optimally restarted processes as well as the existence of a globally optimal resetting strategy [@Restart-Biophysics1; @Restart-Biophysics2; @ReuveniPRL; @PalReuveniPRL; @Chechkin; @branching_II; @Landau; @HRS].

![Experimental realization of diffusion with stochastic resetting. a) A sample trajectory of a silica particle diffusing (blue) near the bottom of a sample cell. The particle sets off from the origin and is stochastically reset at a rate $r=0.05 s^{-1}$. Following a resetting epoch, the particle is driven back to the origin at a constant radial velocity $v=0.8 \mu m/s$ using HOTs (red). After the particle arrives at the origin it remains trapped there for a short period of time to improve localization (green). Inset shows a schematic illustration of the experiment. b) Projection of the particle’s trajectory onto the $x$-axis.[]{data-label="Fig:expt"}](Fig1_slow.pdf)

Despite a long catalogue of theoretical studies dedicated to stochastic resetting, no attempt to experimentally study resetting in a controlled environment has been made to date. This is needed as resetting in the real world is never ‘clean’ as in theoretical models which glance over physical complications for the sake of analytical tractability and elegance. In this letter, we report the experimental realization of diffusion with stochastic resetting ([Fig. \[Fig:expt\]]{}). Our setup comprises of a colloidal particle suspended in fluid (in quasi-two dimensions) and resetting is implemented via holographic optical tweezers (HOTs) [@Dufresne2001; @Polin2005; @Grier06; @Crocker1996]. We study two, physically amenable, resetting protocols in which the particle is returned to the origin: (i) at a constant velocity, and (ii) within a constant time. In both cases, resetting is stochastic and time intervals between resetting events come from an exponential distribution with mean $1/r$.

In what follows, we utilize the setup in [Fig. \[Fig:expt\]]{} to study two different statistical measures of diffusion with stochastic resetting. First, we study the long time position distribution of a tagged particle and how it depends on the resetting protocol. Then, we study the mean FPT of a tagged particle to a region in space. Finally, we also consider the work and energy required to implement resetting in our system. In all cases, we discover that existing theoretical models must be extended and revised to better capture the physics of stochastic resetting in the real world. We conclude with discussion and outlook on the future of experimental studies of stochastic resetting.

Our experimental setup is based on a home built holographic optical tweezers (HOTs) system. It uses a spatial light modulator (Hamamtsu, X10468-04) to imprint a computer generated phase pattern on an expanded laser beam (Coherent, Verdi $\lambda=532$nm). The beam is then projected on the back aperture of a 100x objective (oil immersion, NA = 1.4) mounted on an Olympus microscope (IX71). Samples consist of a dilute colloidal suspension of spherical silica particles with a diameter of $d=1.5\pm 0.02\mu m$ and a refractive index of $n_p=1.46$ (Kisker Biotech, lot\# GK0611140 02) in double distilled water sealed between a glass slide and a coverslip with a sample thickness of approximately $20\mu$m. Motion of the particle (confined to a quasi two dimensional geometry) is recorded by a CMOS camera (Grasshopper 3, Point Gray) at a rate of 20 fps. Particle position is extracted using conventional video microscopy algorithms [@Crocker1996] with an accuracy of approximately 30nm. A laser power of $1$W was used to ensure sufficient trapping of the particle. We utilize in-house developed programs for hardware control and data analysis.

We start our experiments by realizing diffusion with stochastic resetting in the following manner. Every experiment starts by drawing a series of random resetting times $\{t_1,t_2,t_3,...\}$ taken from an exponential distribution with mean $1/r$. At time zero, the particle is trapped at the origin and the experiment, which consists of a series of statistically identical steps, begins. At the $i$-th step of the experimental protocol, the particle is allowed to diffuse for a time $t_i$ eventually arriving at a position $(x_i,y_i)$. At this time, an optical trap is projected onto the particle and the particle is dragged by the trap to its initial position. A typical trajectory of a colloidal particle performing diffusion under stochastic resetting with $r=0.05s^{-1}$ is shown in [Fig. \[Fig:expt\]]{}a (see also Supplementary movie 1). Note that the trajectory is composed of three phases of motion: diffusion, return, and a short waiting time to allow for optimal localization at the origin ([Fig. \[Fig:expt\]]{}b). To collect sufficient statistics, we perform approximately 450 resetting events for each constant velocity experiment, and 305 events for the constant return time experiment.

*Stochastic resetting with instantaneous returns.—* We first utilize our setup to study a canonical (yet non-physical) case in which upon resetting the particle is teleported back to the origin in zero time. This case was the first to be analysed theoretically [@Evans2011_1], thus providing a benchmark for experimental results. To obtain trajectories of diffusion with stochastic resetting and instantaneous returns we digitally remove the return (red) and wait (green) phases of motion from the experimentally measured trajectories ([Fig. \[Fig:expt\]]{}b). A sample trajectory obtained via this procedure is shown in Fig. S1.

![Steady-state distribution of diffusion with stochastic resetting and instantaneous returns. a) Distribution of the position along the $x$-axis. Markers come from experiments and the dashed line is the theoretical prediction of Eq. (1). b) The radial position distribution. Markers come from experiments and the dashed line is the theoretical prediction $\rho(R)=\alpha_0^2RK_0(\alpha_0R)$ [@SM] with $K_n(z)$ standing for the modified Bessel function of the second kind [@Stegun]. In both panels no fitting procedure was applied: $D=0.18\pm 0.02 \mu m^2/s$ was measured independently and $r=0.05s^{-1}$ was set by the operator.[]{data-label="Fig:ssDist"}](1DinstantPDFv2.pdf "fig:"){width="4.25cm" height="3.25cm"} ![Steady-state distribution of diffusion with stochastic resetting and instantaneous returns. a) Distribution of the position along the $x$-axis. Markers come from experiments and the dashed line is the theoretical prediction of Eq. (1). b) The radial position distribution. Markers come from experiments and the dashed line is the theoretical prediction $\rho(R)=\alpha_0^2RK_0(\alpha_0R)$ [@SM] with $K_n(z)$ standing for the modified Bessel function of the second kind [@Stegun]. In both panels no fitting procedure was applied: $D=0.18\pm 0.02 \mu m^2/s$ was measured independently and $r=0.05s^{-1}$ was set by the operator.[]{data-label="Fig:ssDist"}](1DinstantPDFradialv2.pdf "fig:"){width="4.25cm" height="3.25cm"}

A particle undergoing free Brownian motion is not bound in space. It has a Gaussian position distribution with a variance that grows linearly with time. Repeated resetting of the particle to its initial position will, however, result in effective confinement and in a non-Gaussian steady state distribution [@Evans2011_1; @Evans2011_2]. Estimating the steady state distribution of the particle’s position along the $x$-axis from recorded trajectories (see SI for details), we find (Fig. 2a) that the experimentally measured results conform with the theoretical result derived by Evans and Majumdar [@Evans2011_1; @Evans2011_2] $$\rho(x)=\frac{\alpha_0}{2}e^{-\alpha_0|x|}~,
 \label{Eq:SS}$$ where $\alpha_0=\sqrt{r/D}$ is an inverse length scale corresponding to the typical distance diffused by the particle in the time between two resetting events, and $D$ is the diffusion constant. The steady state radial density of the particle can also be extracted from the experimental trajectories by looking at the steady-state distribution of the distance $R=\sqrt{x^2+y^2}$ from the origin. Here too, we find excellent agreement with the theoretical result (Fig. 2b).

*Stochastic resetting with non-instantaneous returns.—* We now turn our attention to more realistic pictures of diffusion with stochastic resetting. These have just recently been considered theoretically in attempt to account for the non-instantaneous returns and waiting times that are seen in all physical systems that include resetting [@Restart-Biophysics1; @Restart-Biophysics2; @Restart-Biophysics6; @HRS; @return1; @return2; @return3; @return4; @return5]. First, we consider a case where upon resetting HOTs are used to return the particle to the origin at a constant radial velocity $v=\sqrt{v_x^2+v_y^2}$ ([Fig. \[Fig:expt\]]{}). This case naturally arises for resetting by constant force in the over-damped limit. We find that the radial steady state density is then given by [@SM] (R)=p\_D\^[c.v.]{}\_(R)+(1-p\_D\^[c.v.]{})\_(R), \[radial-constant-velocity\] where $p_D^{c.v.}=\left(1+\frac{\pi r}{2 \alpha_0 v} \right)^{-1}$ is the steady-state probability to find the particle in the diffusive phase. $\rho_{\text{diff}}(R)=\alpha_0^2 R K_0(\alpha_0R)$ and $\rho_{\text{ret}}(R)=\frac{2\alpha_0^2}{\pi}RK_1(\alpha_0R)$ stand for the conditional probability densities of the particle’s position when in the diffusive and return phases respectively. Here $K_{n}(z)$ is once again the modified Bessel function of the second kind [@Stegun]. The result in [Eq. (\[radial-constant-velocity\])]{} is in very good agreement with experimental data as shown in [Fig. \[non-inst\]]{}a and Fig. S3.

![Steady-state distributions of diffusion with stochastic resetting and non-instantaneous returns. a) The radial position distribution, $\rho(R)$, as a function of the distance $R$ and the radial return velocity $v$ as given by [Eq. (\[radial-constant-velocity\])]{}. Experimental results of a realization with $v=0.8\mu m/s$ are superimposed on the theoretical prediction (black spheres). b) The radial position distribution as a function of $R$ and the return time $\tau_0$ as given by [Eq. (\[radial-constant-time\])]{}. Experimental results of a realization with $\tau_0=3.79 s$ are superimposed on the theoretical prediction (black spheres).[]{data-label="non-inst"}](3Dradialjoint1.pdf){width="8.5cm"}

Next, we consider a case where upon resetting HOTs are used to return the particle to the origin at a constant time $\tau_0$ — irrespective of the particle’s position at the resetting epoch. This case is appealing due to its simplicity and ease of experimental implementation. Here too, we find that the radial steady-state position distribution can be put in a closed form which reads [@SM] (R)=p\_D\^[c.t.]{}\_(R)+(1-p\_D\^[c.t.]{})\_(R), \[radial-constant-time\] where $ p_D^{c.t.}=(1+r \tau_0)^{-1}$ is the steady-state probability to find the particle in the diffusive phase, and with $\rho_{\text{diff}}(R)=\alpha_0^2 R K_0(\alpha_0R)$ and $\rho_{\text{ret}}(R)=\frac{\pi \alpha_0^2}{2}\left[ \frac{1}{\alpha _0}-R \left[K_0\left( \alpha _0 R \right) \pmb{L}_{-1}\left( \alpha _0 R \right) 
+K_1\left( \alpha _0 R \right) \pmb{L}_0\left( \alpha _0 R \right)\right] \right]$, standing for the conditional probability densities of the particle’s radial position when in the diffusive and return phases respectively. Here, $\pmb{L}_n$ is the modified Struve function of order $n$ [@Stegun]. The result in [Eq. (\[radial-constant-velocity\])]{} is in very good agreement with experimental data as shown in [Fig. \[non-inst\]]{}b and Fig. S5.

Comparing the steady-state distributions for the constant time and constant velocity cases, we find that they are almost identical for short return times and high return speeds. Indeed, in these limits the two protocols are virtually indistinguishable as returns are effectively instantaneous. On the other extreme, i.e., for long return times and slow return speeds, marked differences are found between the distributions (Fig. S4 and S6).

*First Passage under stochastic resetting.—* Having realized diffusion with stochastic resetting and analyzed its stationary properties, we now turn to study how resetting affects the first-passage statistics of a Brownian particle. First-passage processes have numerous applications in natural sciences as they are used to describe anything from chemical reactions to single‐cell growth and division, and everything from transport dynamics to search and animal foraging [@Evans2011_3; @Restart-Biophysics1; @Restart-Biophysics2; @Restart-Biophysics6; @Restart-Biophysics3; @Restart-Biophysics8; @RednerBook; @Schehr-review; @ReuveniPRL; @PalReuveniPRL; @branching_II; @Restart-Search1; @Restart-Search2; @Chechkin; @Landau; @HRS; @Frinkes2010; @Branton2010; @Tu2013; @Bezrukov2000; @Grunwald2010; @Ghale2014; @Ma2013; @Iyer2014; @Ingraham1983; @Amir2014; @Osella2014; @cooper1991; @MetzlerBook]. To this end, it is known that while the mean first-passage time (MFPT) of a Brownian particle to a stationary target diverges [@RednerBook; @Schehr-review], resetting will render it finite [@Evans2011_1] even if the returns are non-instantaneous [@Restart-Biophysics1; @Restart-Biophysics2; @Restart-Biophysics6; @HRS; @return3; @return5].

To experimentally study first-passage under stochastic resetting, we consider the setup illustrated in [Fig. \[Fig:MFPT\]]{}a. In this set of experiments, similarly to the previous set, the experiment starts at time zero with the particle positioned at the origin. Resetting is conducted stochastically with rate $r$, and HOTs are used to return the particle to the origin at a constant return time $\tau_0$. However, we now also define a target, set to be a virtual infinite absorbing wall located at $x=L$, i.e., parallel to the $y$-axis. The particle is allowed to diffuse with stochastic resetting until it hits the target, and the hitting times (first-passage times) are recorded. Experiments were performed at 5 different resetting rates: $r=0.05s^{-1}$, $0.0667s^{-1}$, $0.125s^{-1}$, $0.5s^{-1}$, and $1s^{-1}$ with a constant return time of $\tau_0= 3.79s$. A typical trajectory extracted from such an experiment with $L = 1 \mu$m and $r=0.05s^{-1}$ is shown in Fig. \[Fig:MFPT\]b, Fig. S7, and Supplementary movie 2. We extract the FPTs from this and other trajectories from the duration of paths that start at the origin and end at the first crossing of the virtual wall (Fig. \[Fig:MFPT\]b). Several hundreds of resetting events were performed to gather enough FPT statistics (see SI for details).

To check agreement between data coming from FPT experiments and theory, we derived a formula for the mean FPT of diffusion with stochastic resetting and constant time returns to the origin. This is given by [@SM] T\_r =( +\_0 ) . \[Eq:MFPT\_return\] Equation \[Eq:MFPT\_return\] is in excellent agreement with the experimental data as shown in [Fig. \[Fig:MFPT\]]{}c, including accurate prediction of the optimal resetting rate which minimizes the mean FPT of the particle to the target.

*Work and energy.—*A central, and previously unexplored, aspect of stochastic resetting in physical systems concerns the energetic cost associated with the resetting process itself. As discussed above, stochastic resetting prevents a diffusing particle from spreading over the entire available space as it normally would. Instead, a localized, non-equilibrium, steady-state is formed; but the latter can only be maintained by working on the system continuously.

![Energetic cost of resetting. a) The radial distance from the origin vs. time for a particle diffusing with stochastic resetting at rate $r=0.05s^{-1}$ and constant radial return velocity $v=0.8\mu m/s$. b) The cumulative energy expenditure for the trajectory in panel a) (neglecting the cost of the wait period). c) The distribution of energy spent per resetting event. Red disks come from experiments and the theoretical prediction of [Eq. (\[Eq:Energy\])]{} is plotted as a solid blue line. d) Normalized energy spent per resetting event at constant power vs. the normalized radial return velocity as given by [Eq. (\[minmax\])]{}. The minimal energy is attained at a maximal velocity for which the trap is just barely strong enough to overcome the fluid drag force and prevent the particle from escaping the trap.[]{data-label="Fig:work"}](work_const_v.pdf)

In our experiments, work is done by the laser to capture the particle in an optical trap and drag it back to the origin. The total energy spent per resetting event is then simply given by $E=\mathcal{P}\tau(R)$, where $\mathcal{P}$ is the laser power fixed at 1W and $\tau(R)$ is the time required for the laser to trap the particle at a distance $R$ and bring it back to the origin. As the particle’s distance at the resetting epoch fluctuates randomly from one resetting event to another ([Fig. \[Fig:work\]]{}a), the energy spent per resetting event is also random ([Fig. \[Fig:work\]]{}b). To compute its distribution, we note that $E$ is proportional to the return time whose probability density function is in turn given by [@SM] (t)=\_[-]{}\^ d\_0\^ dR R \_0\^dt’ G\_0(R,t’)f(t’) , where $f(t)=re^{-rt}$ is the resetting time density and $G_0(R,t)=\frac{1}{4\pi Dt}e^{-R^2/4Dt}$ is the diffusion propagator in polar coordinates. For the case of constant radial return velocity, $v$, we have $\tau(R)=R/v$. A simple derivation then yields the probability density of the energy spent per resetting event [@SM] (E)= K\_0(E/E\_0) ,\[Eq:Energy\] with $E_0=\alpha_0^{-1}v^{-1}\mathcal{P}$; and note that this is a special case of the K-distribution [@Redding; @Long]. The mean energy spent per resetting event can be computed directly from [Eq. (\[Eq:Energy\])]{} and is given by $\langle E \rangle=\pi E_0/2$. Equation (\[Eq:Energy\]) demonstrates good agreement with experimental data ([Fig. \[Fig:work\]]{}c).

As $\langle E \rangle \propto v^{-1}$, the average energy spent per resetting event in our experiment can be made smaller by working at higher return velocities. Note, however, that the stiffness of the optical trap should be strong enough to oppose the drag force acting on the particle so as to keep it in the trap. Assuming that the maximum allowed displacement of a particle in the trap is $\approx0.5\mu m$ [@Roichman07], we find that working conditions must obey $k\ge2\gamma v$. As the stiffness is proportional to the laser power, $k=\mathcal{C} \mathcal{P}$ (where $\mathcal{C}$ is the conversion factor), the maximal working velocity is given by $v_{\text{max}}\approx \frac{1}{2} \mathcal{C} \mathcal{P}/\gamma$ which—independent of laser power— minimizes energy expenditure to $E_{\text{min}} \approx \pi \gamma \mathcal{C}^{-1} \alpha_0^{-1}$. Going to dimensionless variables we find E /E\_=v\_/v \[minmax\] for $v<v_{\text{max}}$ ([Fig. \[Fig:work\]]{}d).

*Discussion and future outlook.—*In this study, we have demonstrated a unique and versatile method to realize experimentally a resetting process in which many parameters can be easily controlled. We have used this technique to verify an array of theoretical predictions, and further motivate the derivation of new results which come to address novel consideration that arise from experimental realization of diffusion with stochastic resetting. Of prime importance in this regard is the energetic cost of resetting [@thermo1; @thermo2; @thermo3].

The optical trapping method used herein is far from being the most efficient way to apply force to a colloidal particle. In fact, in our experiments we used $1W$ of power at the laser output to create a trap of $k=30pN/\mu m$ for a silica bead of radius $a=0.75\mu m$. For experiments with a constant return velocity $v=0.8\mu m/s$ and resetting rate $r=0.05s^{-1}$, the average return time was $ \langle \tau(R) \rangle=3.68s$. This translates to an average energy expenditure of $\langle E\rangle=\mathcal{P}\langle \tau(R) \rangle=3.68\pm0.05J$ per resetting event. In contrast, the work done against friction to drag the particle at a constant velocity $v$ for a distance $R$ is given by $W_{\text{drag}}=\gamma v R$ where $\gamma=6\pi\eta a$ is the the Stokes drag coefficient. Taking averages, we find that the work required per resetting event is given by $\langle W_{\text{drag}} \rangle= \gamma v \langle R  \rangle = \pi \alpha_0^{-1} \gamma v/2$, which translates into $3.4\cdot10^{-20}J$ or $8.3 k_BT$ per resetting event. We thus see that $\langle W_{\text{drag}} \rangle \ll \langle E \rangle$, i.e., that the work required to reset the particle’s position is orders of magnitude smaller than the actual amount of energy spent when resetting is done using HOTs.

Concluding, we note that experimental research of stochastic resetting is still in its infancy with many open questions left to be answered by consecutive studies. To this end, the setup described above along with its future extensions provide a promising platform.

**Acknowledgments** The authors acknowledge Gilad Pollack for his help in coding the resetting protocol of the HOTs. Arnab Pal acknowledges support from the Raymond and Beverly Sackler Post-Doctoral Scholarship at Tel-Aviv University; and Somrita Ray for many fruitful discussions. Amandeep Sekhon acknowledges support from the Ratner center for single molecule studies. Shlomi Reuveni acknowledges support from the Azrieli Foundation, from the Raymond and Beverly Sackler Center for Computational Molecular and Materials Science at Tel Aviv University, and from the Israel Science Foundation (grant No. 394/19). Yael Roichman acknowledges support from the Israel Science Foundation (grant No. 988/17).

[99]{}

Evans, M.R. and Majumdar, S.N., 2011. Diffusion with stochastic resetting. Physical review letters, 106(16), p.160601.

Evans, M.R. and Majumdar, S.N., 2011. Diffusion with optimal resetting. Journal of Physics A: Mathematical and Theoretical, 44(43), p.435001.

Evans, M.R., Majumdar, S.N. and Mallick, K., 2013. Optimal diffusive search: nonequilibrium resetting versus equilibrium dynamics. Journal of Physics A: Mathematical and Theoretical, 46(18), p.185001.

Reuveni, S., Urbakh, M. and Klafter, J., 2014. Role of substrate unbinding in Michaelis-Menten enzymatic reactions. Proceedings of the National Academy of Sciences, 111(12), pp.4391-4396.

Rotbart, T., Reuveni, S. and Urbakh, M., 2015. Michaelis-Menten reaction scheme as a unified approach towards the optimal restart problem. Physical Review E, 92(6), p.060101.

Robin, T., Reuveni, S. and Urbakh, M., 2018. Single-molecule theory of enzymatic inhibition. Nature communications, 9(1), p.779.

Roldan, E., Lisica, A., Sanchez-Taltavull, D. and Grill, S.W., 2016. Stochastic resetting in backtrack recovery by RNA polymerases. Physical Review E, 93(6), p.062411.

Budnar, S., Husain, K.B., Gomez, G.A., Naghibosadat, M., Varma, A., Verma, S., Hamilton, N.A., Morris, R.G. and Yap, A.S., 2019. Anillin promotes cell contractility by cyclic resetting of RhoA residence kinetics. Developmental cell, 49(6), pp.894-906.

Luby, M., Sinclair, A. and Zuckerman, D., 1993. Optimal speedup of Las Vegas algorithms. Information Processing Letters, 47(4), pp.173-180.

Gomes, C.P., Selman, B. and Kautz, H., 1998. Boosting combinatorial search through randomization. AAAI/IAAI, 98, pp.431-437.

Di Crescenzo, A., Giorno, V., Nobile, A.G. and Ricciardi, L.M., 2003. On the M/M/1 queue with catastrophes and its continuous approximation. Queueing Systems, 43(4), pp.329-347.

Kumar, B.K. and Arivudainambi, D., 2000. Transient solution of an M/M/1 queue with catastrophes. Computers & Mathematics with applications, 40(10-11), pp.1233-1240.

Evans, M.R., Majumdar, S.N. and Schehr, G., 2020. Stochastic resetting and applications. Journal of Physics A: Mathematical and Theoretical.

Evans, M.R. and Majumdar, S.N., 2014. Diffusion with resetting in arbitrary spatial dimension. Journal of Physics A: Mathematical and Theoretical, 47(28), p.285001.

Majumdar, S.N., Sabhapandit, S. and Schehr, G., 2015. Dynamical transition in the temporal relaxation of stochastic processes under resetting. Physical Review E, 91(5), p.052131.

Pal, A., 2015. Diffusion in a potential landscape with stochastic resetting. Physical Review E, 91(1), p.012113.

Pal, A., Kundu, A. and Evans, M.R., 2016. Diffusion under time-dependent resetting. Journal of Physics A: Mathematical and Theoretical, 49(22), p.225001.

Pal, A., Chatterjee, R., Reuveni, S. and Kundu, A., 2019. Local time of diffusion with stochastic resetting. Journal of Physics A: Mathematical and Theoretical, 52(26), p.264002.

Eule, S. and Metzger, J.J., 2016. Non-equilibrium steady states of stochastic processes with intermittent resetting. New Journal of Physics, 18(3), p.033006.

Evans, M.R. and Majumdar, S.N., 2018. Effects of refractory period on stochastic resetting. Journal of Physics A: Mathematical and Theoretical. 52 01LT01.

Bodrova, A.S., Chechkin, A.V. and Sokolov, I.M., 2019. Scaled Brownian motion with renewal resetting. Physical Review E, 100(1), p.012120.

Bodrova, A.S., Chechkin, A.V. and Sokolov, I.M., 2019. Nonrenewal resetting of scaled Brownian motion. Physical Review E, 100(1), p.012119.

Basu, U., Kundu, A. and Pal, A., 2019. Symmetric exclusion process under stochastic resetting. Physical Review E, 100(3), p.032136.

Pal, A., Kusmierz, L. and Reuveni, S., 2019. Time-dependent density of diffusion with stochastic resetting is invariant to return speed. Physical Review E, 100(4), p.040101.

Pal, A., Kusmierz, L. and Reuveni, S., 2019. Invariants of motion with stochastic resetting and space-time coupled returns. New Journal of Physics, 21(11), p.113024.

Bodrova, A.S. and Sokolov, I.M., 2019. Resetting processes with non-instanteneous return. arXiv preprint arXiv:1907.12326.

Maso-Puigdellosas, A., Campos, D. and Mendez, V., 2019. Transport properties of random walks under stochastic noninstantaneous resetting. Physical Review E, 100(4), p.042104.

Redner, S., 2007. A Guide to First-Passage Processes. A Guide to First-Passage Processes, by Sidney Redner, Cambridge, UK: Cambridge University Press, 2007.

Bray, A.J., Majumdar, S.N. and Schehr, G., 2013. Persistence and first-passage properties in nonequilibrium systems. Advances in Physics, 62(3), pp.225-361.

Ray, S., Mondal, D. and Reuveni, S., 2019. Peclet number governs transition to acceleratory restart in drift-diffusion. Journal of Physics A: Mathematical and Theoretical, 52(25), p.255002.

Pal, A. and Prasad, V.V., 2019. First passage under stochastic resetting in an interval. Physical Review E, 99(3), p.032123.

Kusmierz, L., Majumdar, S.N., Sabhapandit, S. and Schehr, G., 2014. First order transition for the optimal search time of Lévy flights with resetting. Physical review letters, 113(22), p.220602.

Kusmierz, L., & Gudowska-Nowak, E. (2015). Optimal first-arrival times in Lévy flights with resetting. Physical Review E, 92(5), 052127.

Reuveni, S., 2016. Optimal stochastic restart renders fluctuations in first passage times universal. Physical review letters, 116(17), p.170601.

Pal, A. and Reuveni, S., 2017. First Passage under Restart. Physical review letters, 118(3), p.030603.

Chechkin, A. and Sokolov, I.M., 2018. Random search with resetting: a unified renewal approach. Physical review letters, 121(5), p.050601.

Pal, A., Eliazar, I. and Reuveni, S., 2019. First passage under restart with branching. Physical review letters, 122(2), p.020602.

Pal, A. and Prasad, V.V., 2019. Landau-like expansion for phase transitions in stochastic resetting. Physical Review Research, 1(3), p.032001.

Pal, A., Kusmierz, L. and Reuveni, S., 2019. Home-range search provides advantage under high uncertainty. arXiv preprint arXiv:1906.06987.

Dufresne, E.R., Spalding, G.C., Dearing, M.T., Sheets, S.A. and Grier, D.G., 2001. Computer-generated holographic optical tweezer arrays. Review of Scientific Instruments, 72(3), pp.1810-1816.

Polin, M., Ladavac, K., Lee, S.H., Roichman, Y. and Grier, D.G., 2005. Optimized holographic optical traps. Optics Express, 13(15), pp.5831-5845.

Grier, D.G. and Roichman, Y., 2006. Holographic optical trapping. Applied optics, 45(5), pp.880-887.

Crocker, J.C. and Grier, D.G., 1996. Methods of digital video microscopy for colloidal studies. Journal of colloid and interface science, 179(1), pp.298-310.

See supplemental material for the detailed derivation.

Abramowitz, M. and Stegun, I.A. eds., 1948. Handbook of mathematical functions with formulas, graphs, and mathematical tables (Vol. 55). US Government printing office.

Firnkes, M., Pedone, D., Knezevic, J., Doblinger, M. and Rant, U., 2010. Electrically facilitated translocations of proteins through silicon nitride nanopores: conjoint and competitive action of diffusion, electrophoresis, and electroosmosis. Nano letters, 10(6), pp.2162-2167.

Branton, D., Deamer, D.W., Marziali, A., Bayley, H., Benner, S.A., Butler, T., Di Ventra, M., Garaj, S., Hibbs, A., Huang, X. and Jovanovich, S.B., 2010. The potential and challenges of nanopore sequencing. In Nanoscience and technology: A collection of reviews from Nature Journals (pp. 261-268).

Tu, L.C., Fu, G., Zilman, A. and Musser, S.M., 2013. Large cargo transport by nuclear pores: implications for the spatial organization of FG‐nucleoporins. The EMBO journal, 32(24), pp.3220-3230.

Bezrukov, S.M., Kullman, L. and Winterhalter, M., 2000. Probing sugar translocation through maltoporin at the single channel level. FEBS letters, 476(3), pp.224-228.

Ghale, G., Lanctot, A.G., Kreissl, H.T., Jacob, M.H., Weingart, H., Winterhalter, M. and Nau, W.M., 2014. Chemosensing ensembles for monitoring biomembrane transport in real time. Angewandte Chemie International Edition, 53(10), pp.2762-2765.

Grünwald, D. and Singer, R.H., 2010. In vivo imaging of labelled endogenous $\beta$-actin mRNA during nucleocytoplasmic transport. Nature, 467(7315), p.604.

Ma, J., Liu, Z., Michelotti, N., Pitchiaya, S., Veerapaneni, R., Androsavich, J.R., Walter, N.G. and Yang, W., 2013. High-resolution three-dimensional mapping of mRNA export through the nuclear pore. Nature communications, 4, p.2414.

Iyer-Biswas, S., Wright, C.S., Henry, J.T., Lo, K., Burov, S., Lin, Y., Crooks, G.E., Crosson, S., Dinner, A.R. and Scherer, N.F., 2014. Scaling laws governing stochastic growth and division of single bacterial cells. Proceedings of the National Academy of Sciences, 111(45), pp.15912-15917.

Ingraham, J.L., Maaloe, O. and Neidhardt, F.C., 1983. Growth of the bacterial cell. Sinauer Associates.

Amir, A., 2014. Cell size regulation in bacteria. Physical Review Letters, 112(20), p.208102.

Osella, M., Nugent, E. and Lagomarsino, M.C., 2014. Concerted control of Escherichia coli cell division. Proceedings of the National Academy of Sciences, 111(9), pp.3431-3435.

Cooper, S., 1991. Bacterial growth and division: biochemistry and regulation of the division cycle of prokaryotes and eukaryotes.

Metzler, R., Redner, S. and Oshanin, G., 2014. First-Passage Phenomena and Their Applications (Vol. 35). Singapore: World Scientific.

Redding, N.J., 1999. Estimating the parameters of the K distribution in the intensity domain (No. DSTO-TR-0839). ELECTRONICS RESEARCH LAB SALISBURY (AUSTRALIA).

Long, M.W., 1975. Radar reflectivity of land and sea. Lexington, Mass., DC Heath and Co., 1975. 390 p. Roichman, Y., Wong, V. and Grier, D., 2007. Colloidal transport through optical tweezer arrays. Phys. Rev. E., 75(1), p011407. Fuchs, J., Goldt, S. and Seifert, U., 2016. Stochastic thermodynamics of resetting. EPL (Europhysics Letters), 113(6), p.60009.

Pal, A. and Rahav, S., 2017. Integral fluctuation theorems for stochastic resetting systems. Physical Review E, 96(6), p.062135.

Gupta, D., Plata, C.A. and Pal, A., 2020. Work fluctuations and Jarzynski equality in stochastic resetting. Physical Review Letters, 124(11), p.110608.
