Work-distribution quantumness and irreversibility when crossing a quantum phase transition in finite time

TL;DR

This paper investigates how strong interactions and quantum phase transitions in finite Hubbard chains influence energy fluctuations, irreversibility, and quantum work distributions during out-of-equilibrium dynamics, with implications for quantum thermodynamics.

Contribution

It reveals the impact of quantum phase transitions on work fluctuations and irreversibility in finite many-body systems, highlighting the role of correlations in quantum thermodynamics.

Findings

Energy fluctuation statistics change across the transition.

Irreversibility and entropy production increase near the transition.

Quantum features in work distribution are enhanced in the Mott-insulating phase.

Abstract

The thermodynamic behavior of out-of-equilibrium quantum systems in finite-time dynamics encompasses the description of energy fluctuations, which dictates a series of system's physical properties. In addition, strong interactions in many-body systems strikingly affect the energy-fluctuation statistics along a non-equilibrium dynamics. By driving transient currents to oppose the precursor to metal-Mott insulator transition in a diversity of dynamical regimes, we show how increasing correlations dramatically affect the statistics of energy fluctuations and consequently the quantum work distribution of finite Hubbard chains. Statistical properties of such distributions, as its skewness, that changes dramatically across the transition, can be related to irreversibility and entropy production. Even close to adiabaticity, the quasi quantum phase transition hinders equilibration, increasing the process irreversibility, and inducing strong quantum features in the quantum work distribution. In the Mott-insulating phase the work fluctuation-dissipation balance gets modified, with the irreversible entropy production dominating over work fluctuations. The effect of an interaction-driven quantum-phase-transition on thermodynamics quantities and irreversibility has to be considered in the design of protocols in small scale devices for application in quantum technology. Eventually, such many-body effects can also be employed in work extraction and refrigeration protocols at quantum scale.

Equations

$$H(t)$$

$$P(W)$$

$${\bar{\mathcal{W}}}^k$$

$$⟨\Sigma ⟩$$

$$⟨\Sigma ⟩\le {\beta }^2{\bar{\mathcal{W}}}^2/2.$$

Full text

Introduction —

After more than a century, the well-established laws of thermodynamics have been challenged by the quantum nature of nanoscale systemsHorodecki and Oppenheim (2013); Kosloff (2013); Goold et al. (2016); Vinjanampathy and Anders (2016); Binder et al. (2018); Millen and Xuereb (2016); Alicki and Kosloff (2018); Binder et al. (2019); quantum thermodynamics is now extending concepts such as heat, work, and entropyHorodecki and Oppenheim (2013); Skrzypczyk et al. (2014); Del Rio et al. (2011) to this scale. At the same time, working conditions for quantum technology devices often correspond to finite-temperatures, non-equilibrium regimes, so that development of related formalism is in high demand. In quantum systems, thermodynamic probability distributions contain rich information about the possible transitions between eigenstates Campisi and Goold (2017) and, more interestingly, thermal and quantum fluctuations Dorosz et al. (2008); Batalhão et al. (2014); Landi and Karevski (2016), equilibration and irreversibility Batalhão et al. (2015); Hoang et al. (2016); Zhong and Tong (2015); Apollaro et al. (2015). For small quantum systems these distributions are not symmetric and narrow Gaussian distributions Horodecki and Oppenheim (2013).

Quantum phase transitions (QPTs) are an exquisitely quantum phenomenon, so there is interest in investigating their signature on quantum thermodynamic quantities and their distributions Sindona et al. (2014); Marino and Silva (2014); Apollaro et al. (2015); Mascarenhas et al. (2014); Xu et al. (2018); Wang et al. (2018); Hickey and Genway (2014); Landi and Karevski (2016); Hoang et al. (2016); Schmidtke et al. (2018); Li and Shi (2019). In addition, many-body interactions, which are ubiquitous and notoriously difficult to treat, assume an even more complex role in out-of-equilibrium quantum systems Dorner et al. (2012); Arrais et al. (2019), where, e.g., they may affect the way the system reaches or settles into different phases. Relevant questions are: what is the role of many-body correlations for quantum particles driven out of equilibrium, and how do they affect quantum thermodynamical quantities? Do they contribute or oppose reversibility Micadei et al. (2019) and thermalization ? What if many-body correlations induce a QPT, what signatures appear in thermodynamic distributions? And how do they depend on the system size?

Most of the previous studies of QPT signatures in quantum thermodynamics focused on QPTs driven by external fields and/or on the sudden quench regime. They analysed features of quantum thermodynamic quantities, sometimes up to the second moment of their distribution, and their evolution as the critical parameter, usually an external field, is (suddenly) driven across the transition.

In this Letter, we consider the above questions in the context of microscopic models for strongly correlated systems undergoing finite time processes at finite temperature. We study the non-homogeneous one-dimensional Hubbard model at half filling, as it is driven out of equilibrium. Finite Hubbard chains may undergo a precursor to the metal-Mott insulator transition, a QPT driven solely by many-body interactions. Our focus is the quantum work probability distribution and its statistics: we inspect the first three moments, related to the mean, variance, and skewness. The latter has been to a large extent overlooked, and we demonstrate that it allows us to characterize the transition between the different coupling regimes, including the precursor to the metal-Mott insulator QPT (pM-QPT), as well as the different dynamical regimes (sudden quench to nearly-adiabatic). Our results also demonstrate that by considering the sudden quench regime alone, one misses the contribution of the dynamics to the QPT signatures, which becomes dominant in finite-time regimes. Many-body interactions strikingly affect the shape of the quantum work probability distribution: while it acquires some classical features for increasing system size and weak correlations, these are completely dismantled by the pM-QPT, which also averts the system from equilibrium. Interestingly, we show that, in the Mott-insulating phase, entropy production dominates over work fluctuations, in contrast to the literatureHermans (1991); Wood et al. (1991); Miller et al. (2019). Finally, we relate the skewness with the entropy production, and propose its role as a witness of irreversibility for many-body systems out-of-equilibrium.

Driven Hubbard chains —

The Hubbard model allows for both itinerant electron spins (conduction band) and localized magnetic moments. It was initially designed to describe strongly correlated systems such as transition metals; more recently it has been utilized to describe systems of importance to quantum technologies, such as cold atoms in an optical lattice, chains of trapped ions, excitons and electrons in coupled quantum dots, or small molecules Drewes et al. (2016, 2017); Boll et al. (2016); Braun et al. (2015); Scherg et al. (2018); Coe et al. (2011). Even non-driven, short Hubbard chains are characterized by a very rich physical behaviour, with many-body interactions driving a precursor to the metal to Mott insulator transition, Greentree et al. (2006); Murmann et al. (2015); Zawadzki et al. (2017) and studies of a driven Hubbard dimer show promising resultsCarrascal et al. (2015); Herrera et al. (2017).

Here we consider half-filled fermionic chains undergoing a process in which a time-dependent electric field is applied for a finite-time. Their Hamiltonian is

[TABLE]

where, $\hat{c}_{j\sigma}^{\dagger}$ ($\hat{c}_{j\sigma}$) are the creation (annihilation) operators for a fermion with spin $\sigma=\uparrow,\downarrow$ in the $j$-th site, $n_{j\sigma}=\langle\hat{c}_{j\sigma}^{\dagger}\hat{c}_{j\sigma}\rangle$ represents the corresponding $j$-site occupation, $J$ is the hopping parameter, $U$ is the Coulomb on-site repulsion, and $V_{j}(t)=\Delta_{j}\,t/\tau$, with $\Delta_{j}=\frac{10J}{L-1}j$, is the time-dependent linear potential that drives an out-of-equilibrium transient current along the chain.

The system is initially in thermal equilibrium at temperature $\beta^{-1}=k_{B}T=2.5J$ (where $k_{B}$ is the Boltzmann constant and $T$ the absolute temperature), with $\rho(t=0)={e^{-\beta H(t=0)}}/{Z_{t=0}}$, and $Z_{t=0}=\text{Tr}\left[e^{-\beta H(t=0)}\right]$. The driving time $\tau$ controls the rate of the dynamics that steers $H_{0}=H(t=0)$ to $H_{f}=H(t=\tau)$. The final Hamiltonian $H_{f}$ is independent of $\tau$. Our results were obtained via exact diagonalization; the time-evolution calculated by a routine provided by the QuTip package Johansson et al. (2013).

Statistics of work and correlations —

The probability distribution characterizing the quantum work Skrzypczyk et al. (2014) performed on the closed system111We consider processes fast enough to be represented as (a unitary) closed system dynamic. In other words, the calculations correspond to the scenario where the time duration of the driving protocol, $\tau$, is much smaller than any decoherence or relaxation times. is given by

[TABLE]

where $p_{n}^{0}$ is the initial-state occupation probability of the n-th eigenstate $\ket{n}$ of energy $\epsilon_{n}^{0}$ of $H_{0}$, and $p_{m|n}^{\tau}$ is the conditional probability for $\ket{n}$ to make a transition to the m-th eigenstate $\ket{m}$ of $H_{f}$.

The complexity of $P(W)$ scales with the number of the possible energy transitions. In the systems we consider, half-filling with zero magnetization, the number of allowed transitions increases from 16 for $L=2$, to $5\times 10^{3}$ for $L=8$. 222There are 4900 spin configurations for $L=8$ having $S_{z}=0$. This is highlighted by Figs. 1a and 1c, where $P(W)$ is shown for $L=4$ and $L=8$ for the non-interacting case ($U=0$). The exponential increase in the number of transitions transforms the distribution from an irregular set of peaks to a bell shape; changes in the type of dynamics – from sudden quench ($\tau=0$) to close-to-adiabatic behaviour ($\tau=10/J$) – strongly affect the shape of the distribution, which becomes increasingly asymmetric as $\tau$ increases. On the contrary, when considering the strongly interacting regime ($U=10J$, Figs. 1b and 1d) the shape of $P(W)$ seems basically unaffectedsup . We attribute this behaviour to the insulating phase which de-facto substantially reduces the available Hilbert space.

This qualitative picture is quantified by the $k$-th central moments of the quantum work distribution $P(W)$,

[TABLE]

The moments $k=1$ (mean), $k=2$ (variance), $k=3$ (skewness) are shown in Fig. 2, $L=4$ left and $L=8$ right; the corresponding ‘heatmaps’ for $k=3$ in Fig. 3, where the white line indicates $\bar{\mathcal{W}}^{3}=0$ sup . The first three moments are strongly dependent on $\tau$ for weak interactions, $U\approx 0$, while almost $\tau$-independent for $U\approx 10J$, once interactions have driven the pM-QPT and the system becomes insulating. Regardless of the huge increase in the Hilbert space, the behaviour across the transition is qualitatively independent from the system size, hinting to a possible scaling behaviour. The most striking features appear in the skewness $\bar{\mathcal{W}}^{3}$. For sudden quenches, $\tau\ll J^{-1}$, the skewness is relatively small and depends only weakly on $U$ (see Fig. 3). However, for finite-time processes, $\tau\stackrel{{\scriptstyle>}}{{\sim}}0.5/J$, $\bar{\mathcal{W}}^{3}$ changes sign across the pM-QPT (white line in Fig. 3), with proper minima and maxima bracketing the transition when $\tau\stackrel{{\scriptstyle>}}{{\sim}}2.5/J$ (see Fig. 3 and Fig. 2, lower panels). As $U$ increases, the system suffers a dynamic competition between the transient current induced by the drive and the increasing on-site repulsion. This leads to a dramatic change in the shape of $P(W)$, with a marked asymmetry shifting from left (before pM-QPT) to right (after pM-QPT). As $\tau$ increases, the region in-between $\bar{\mathcal{W}}^{3}(U)$ extrema shifts towards larger $U$’s (see Fig. 2, lower panels). In classical thermodynamics, probability distributions tends to be symmetric, and so it is particularly fit that a strong asymmetry in the distribution, and a dramatic change of this asymmetry, signals an exquisitely quantum phenomenon such as a QPT.

Entropy production and irreversibility —

Together with the statistics of work, we can inspect how the pM-QPT affects irreversibility. We quantify this by considering the entropy production

[TABLE]

where, $S\left(\rho_{\tau}||\rho_{\tau}^{\text{eq}}\right)=\text{Tr}\rho_{\tau}\left(\ln\rho_{\tau}-\ln\rho_{\tau}^{\text{eq}}\right)$ defines the Kullback relative entropy between the final state $\rho_{\tau}=\mathcal{U}_{\tau}\rho_{0}^{\text{eq}}\mathcal{U}_{\tau}^{\dagger}$, and its equilibrium counterpart $\rho_{\tau}^{\text{eq}}={e^{-\beta H(t=\tau)}}/{Z_{t=\tau}}$, with $\mathcal{U}_{t}$ the time-evolution operator. We note that $\left\langle\Sigma\right\rangle/\beta$ corresponds also to the energy that would be dissipated if thermalization would follow the finite-time driven protocol. We examine the entropy production in our systems in various dynamical and coupling regimes, full results for $L=4$ and $L=8$ are reported in the Supplemental Material. sup

For a finite quantum system, adiabaticity does not imply equilibration, hence, to quantitatively investigate this discrepancy, we focus on large $\tau$ results, and use, in addition to $\left\langle\Sigma\right\rangle$, the trace distance between final and corresponding equilibrium state, $D_{Tr}(\rho_{\tau},\rho^{\mathrm{eq}}_{\tau})=Tr\left[\sqrt{\left(\rho_{\tau}-\rho^{\mathrm{eq}}_{\tau}\right)^{\dagger}\left(\rho_{\tau}-\rho^{\mathrm{eq}}_{\tau}\right)}\right]/2$. This is plotted in the top left panel of Fig. 4 together with the entropy (middle left) and the skewness (bottom left) as a function of $U/J$, for $\tau\times J=10$ and $L=4,6,8$. We find that these quantities seem to display a scaling trend with system size. In particular, all quantities similarly signal the pM-QPT, moving from a minimum to a maximum. These extrema all shift towards $U=0^{+}$ (the thermodynamic limit for the metal-Mott insulator QPT) as $L$ increases (see dashed lines in Fig. 4, left panels). The transition pulls the final state away from equilibrium as demonstrated by the corresponding increase of $D_{Tr}(\rho_{\tau},\rho^{\mathrm{eq}}_{\tau})$, and dramatically affects the work distribution shape, as witnessed by the change in sign of the skewness. After the pM-QPT, as interactions increase further, the final state draws nearer to equilibrium, as the system, now almost an insulator, poorly responds to the applied field. Indeed, the work distribution comprises here very few transitions (Fig. 1b and d).

We note that the value of the trace distance demonstrates that the final system remains always significantly far from equilibrium, even when the skewness is zero ($U/J\approx 5$) and the distribution becomes more akin to the linear response form, $\langle W\rangle=\Delta F+\frac{\bar{\mathcal{W}}^{2}}{2k_{B}T}$, which is valid close to equilibrium, with $\Delta F=\langle W\rangle-\langle\Sigma\rangle/\beta$ the free energy variation.

Fig. 4 shows that increasing the system size drives a system farthest from equilibrium and increases the quantity of energy $\langle\Sigma\rangle/\beta$ to be dissipated for reaching it. However, the system size affects the work distribution asymmetry in opposite ways before and after the quasi-QPT. Before the pM-QPT, availability of an exponentially increasing number of transitions ‘regularize’ the distribution (compare Fig. 1a and c) contributing to the decrease of its asymmetry, while, by de-facto restricting the available Hilbert space, the pM-QPT restores full quantum features in $P(W)$, even for increasing size (compare Figs. 1b and d).

Entropy production and work fluctuation-dissipation relation

Close to adiabaticity, classical processes satisfy the work fluctuation-dissipation relation $\left\langle\Sigma\right\rangle=\beta^{2}\bar{\mathcal{W}}^{2}/2$Hermans (1991); Wood et al. (1991); however, recent studiesMiller et al. (2019) suggest that, for slow quantum processes in open systems, this is governed by the inequality

[TABLE]

We examine the effect of the pM-QPT on the work fluctuation-dissipation relation in Fig. 4, right upper panel, and show that the transition is marked by a reversing of the inequality (5), with work fluctuations hence becoming smaller than dissipation. Most interestingly, after the pM-QPT, while increasing $U$ leads the dynamical process back to adiabaticity (Fig. 4, right lower panel, $U>10J$), dissipation remains dominant over work fluctuations, even for very small values of $D_{\text{Tr}}(\rho_{\tau},\rho_{\tau}^{\mathrm{adiab}})$. This reversing of (5) is a many-body effect: the pM-QPT dramatically reduces the system response to the applied field, and hence the width of the work distribution, for all rate of driving, including slow driving (see Fig. 2, middle panels).

Conclusion —

We discussed the effects of many-body interactions on the statistics of work in fermionic chains driven for finite times. We considered dynamics from sudden quench to quasi-adiabaticity, and observed the signatures of the precursor to the metal-Mott insulator quantum phase transition. Our results show that, when the system is weakly correlated, the work probability distribution $P(W)$ is highly sensitive to the rate of driving, whereas it remains almost unaffected when many-body interactions are strong.

If the chains’ length $L$ is increased and $U/J\stackrel{{\scriptstyle<}}{{\sim}}1$, $P(W)$ acquires some of the features of a classical distribution, such as a well-defined maximum and a bell shape. In contrast, after crossing the precursor to the QPT, for $U/J\stackrel{{\scriptstyle>}}{{\sim}}5$, the quantum nature of the system dominates at all the explored values of $L$, strongly hindering work extraction with, nonetheless, a price paid in a residual entropy production. The quasi-Mott-insulating phase is associated with a striking reduction of the number of transitions arising from the dynamics, so that $P(W)$ becomes almost independent on the rate of variation of the external field. This feature leads to entropy production dominating work fluctuations even for slow processes, in contrast to the classical work fluctuation-dissipation relation, and at difference with recent predictions for slowly-driven open quantum systems.

For dynamics beyond sudden quenches, a change in sign and a remarkable variation in value of the skewness characterize the precursor to the metal-Mott insulator transition. These features persist even when the number of degrees of freedom is exponentially increased. In the sudden quench regime, the precursor to the QPT affects $P(W)$ only through its effects on the initial and final Hamiltonians’ eigenstates; instead, for finite driving times, the precursor to the metal-to-Mott insulator transition affects $P(W)$ twice, through its effect on the eigenstates and by modifying the system response to the applied drive. This leads to qualitatively different signatures of the precursor to the QPT on the quantum work distribution, depending on the dynamical regime.

By comparing to the trace distance between the final and the corresponding equilibrium state, we conclude that the third moment of $P(W)$ also retains information about the entropy production and equilibration across the precursor to the QPT.

Experimental realizations of correlated quantum matter could be implemented by means of small molecules and NMR Batalhão et al. (2015); Micadei et al. (2019), coupled quantum dots and ion traps Roßnagel et al. (2016); An et al. (2015), or cold atoms platforms Chiara et al. (2015); Cerisola et al. (2017). Our findings may help to design time-dependent protocols which exploit many-body interactions for, e.g., tailoring work extraction or optimizing efficiency of a refrigeration cycle where the coolant is a strong correlated many-body system, yielding to novel applications of quantum thermodynamics

We thank Marcela Herrera for very usefull discussions during the early stages of the work. We acknowlege financial support from the Royal Society (grant no. NA140436); CNPq (grant no. PVE-401414/2014-0); CAPES (grant no. PDSE-88881.135185/2016-01); (INCT-IQ). KZ acknowledges the Schumblenger Foundation for sponsorship through the program Faculty for the Future.

Referências

• Horodecki and Oppenheim (2013)

M. Horodecki and J. Oppenheim, Nature communications 4, 2059 (2013).

• Kosloff (2013)

R. Kosloff, Entropy 15, 2100–2128 (2013).

• Goold et al. (2016)

J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, Journal of Physics A: Mathematical and Theoretical 49, 143001 (2016).

• Vinjanampathy and Anders (2016)

S. Vinjanampathy and J. Anders, Contemporary Physics 57, 545 (2016).

• Binder et al. (2018)

F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (2018).

• Millen and Xuereb (2016)

J. Millen and A. Xuereb, New Journal of Physics 18, 011002 (2016), eprint 1509.01086.

• Alicki and Kosloff (2018)

R. Alicki and R. Kosloff, arXiv e-prints (2018), eprint 1801.08314.

• Binder et al. (2019)

F. Binder, L. Correa, C. Gogolin, J. Anders, and G. Adesso, Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, Fundamental Theories of Physics (Springer International Publishing, 2019), ISBN 9783319990460, URL https://books.google.com/books?id=5uWPDwAAQBAJ.

• Skrzypczyk et al. (2014)

P. Skrzypczyk, A. J. Short, and S. Popescu, Nature communications 5, 4185 (2014).

• Del Rio et al. (2011)

L. Del Rio, J. Åberg, R. Renner, O. Dahlsten, and V. Vedral, Nature 474, 61 (2011).

• Campisi and Goold (2017)

M. Campisi and J. Goold, Phys. Rev. E 95, 062127 (2017).

• Dorosz et al. (2008)

S. Dorosz, T. Platini, and D. Karevski, Phys. Rev. E 77, 051120 (2008).

• Batalhão et al. (2014)

T. B. Batalhão, A. M. Souza, L. Mazzola, R. Auccaise, R. S. Sarthour, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, and R. M. Serra, Phys. Rev. Lett. 113, 140601 (2014).

• Landi and Karevski (2016)

G. T. Landi and D. Karevski, Phys. Rev. E 93, 032122 (2016).

• Batalhão et al. (2015)

T. B. Batalhão, A. M. Souza, R. S. Sarthour, I. S. Oliveira, M. Paternostro, E. Lutz, and R. M. Serra, Phys. Rev. Lett. 115, 190601 (2015).

• Hoang et al. (2016)

D.-T. Hoang, B. P. Venkatesh, S. Han, J. Jo, G. Watanabe, and M.-S. Choi, Scientific reports 6, 27603 (2016).

• Zhong and Tong (2015)

M. Zhong and P. Tong, Phys. Rev. E 91, 032137 (2015).

• Apollaro et al. (2015)

T. J. G. Apollaro, G. Francica, M. Paternostro, and M. Campisi, Physica Scripta T165, 014023 (2015).

• Sindona et al. (2014)

A. Sindona, J. Goold, N. L. Gullo, and F. Plastina, New Journal of Physics 16, 045013 (2014).

• Marino and Silva (2014)

J. Marino and A. Silva, Phys. Rev. B 89, 024303 (2014).

• Mascarenhas et al. (2014)

E. Mascarenhas, H. Bragança, R. Dorner, M. Fran ça Santos, V. Vedral, K. Modi, and J. Goold, Phys. Rev. E 89, 062103 (2014).

• Xu et al. (2018)

B.-M. Xu, J. Zou, L.-S. Guo, and X.-M. Kong, Phys. Rev. A 97, 052122 (2018).

• Wang et al. (2018)

Q. Wang, D. Cao, and H. T. Quan, Phys. Rev. E 98, 022107 (2018).

• Hickey and Genway (2014)

J. M. Hickey and S. Genway, Phys. Rev. E 90, 022107 (2014).

• Schmidtke et al. (2018)

D. Schmidtke, L. Knipschild, M. Campisi, R. Steinigeweg, and J. Gemmer, Phys. Rev. E 98, 012123 (2018).

• Li and Shi (2019)

X. Li and Y. Shi, EPL (Europhysics Letters) 125, 67003 (2019).

• Dorner et al. (2012)

R. Dorner, J. Goold, C. Cormick, M. Paternostro, and V. Vedral, Phys. Rev. Lett. 109, 160601 (2012).

• Arrais et al. (2019)

E. G. Arrais, D. A. Wisniacki, A. J. Roncaglia, and F. Toscano, arXiv preprint arXiv:1907.06285 (2019).

• Micadei et al. (2019)

K. Micadei, J. P. Peterson, A. M. Souza, R. S. Sarthour, I. S. Oliveira, G. T. Landi, T. B. Batalhão, R. M. Serra, and E. Lutz, Nature communications 10, 2456 (2019).

• Hermans (1991)

J. Hermans, The Journal of Physical Chemistry 95, 9029 (1991).

• Wood et al. (1991)

R. H. Wood, W. C. Muhlbauer, and P. T. Thompson, The Journal of Physical Chemistry 95, 6670 (1991).

• Miller et al. (2019)

H. J. Miller, M. Scandi, J. Anders, and M. Perarnau-Llobet, arXiv preprint arXiv:1905.07328v1 (2019).

• Drewes et al. (2016)

J. H. Drewes, E. Cocchi, L. A. Miller, C. F. Chan, D. Pertot, F. Brennecke, and M. Köhl, Phys. Rev. Lett. 117, 135301 (2016).

• Drewes et al. (2017)

J. H. Drewes, L. A. Miller, E. Cocchi, C. F. Chan, N. Wurz, M. Gall, D. Pertot, F. Brennecke, and M. Köhl, Phys. Rev. Lett. 118, 170401 (2017).

• Boll et al. (2016)

M. Boll, T. A. Hilker, G. Salomon, A. Omran, J. Nespolo, L. Pollet, I. Bloch, and C. Gross, Science 353, 1257 (2016).

• Braun et al. (2015)

S. Braun, M. Friesdorf, S. S. Hodgman, M. Schreiber, J. P. Ronzheimer, A. Riera, M. del Rey, I. Bloch, J. Eisert, and U. Schneider, Proceedings of the National Academy of Sciences 112, 3641 (2015).

• Scherg et al. (2018)

S. Scherg, T. Kohlert, J. Herbrych, J. Stolpp, P. Bordia, U. Schneider, F. Heidrich-Meisner, I. Bloch, and M. Aidelsburger, Phys. Rev. Lett. 121, 130402 (2018).

• Coe et al. (2011)

J. P. Coe, V. V. França, and I. d’Amico, EPL (Europhysics Letters) 93, 10001 (2011).

• Greentree et al. (2006)

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. Hollenberg, Nature Physics 2, 856 (2006).

• Murmann et al. (2015)

S. Murmann, A. Bergschneider, V. M. Klinkhamer, G. Zürn, T. Lompe, and S. Jochim, Phys. Rev. Lett. 114, 080402 (2015).

• Zawadzki et al. (2017)

K. Zawadzki, I. D’Amico, and L. N. Oliveira, Brazilian Journal of Physics 47, 488 (2017).

• Carrascal et al. (2015)

D. J. Carrascal, J. Ferrer, J. C. Smith, and K. Burke, Journal of Physics: Condensed Matter 27, 393001 (2015).

• Herrera et al. (2017)

M. Herrera, R. M. Serra, and I. D’Amico, Scientific reports 7, 4655 (2017).

• Johansson et al. (2013)

J. R. Johansson, P. D. Nation, and F. Nori, Computer Physics Communications 184, 1234 (2013), eprint 1211.6518.

• (45)

See also Supplemental Material at https://github.com/krissiazawadzki/work_distributions_hubbard for: animations of the evolution of $P(W)$ for increasing $\tau$, for fixed $U=0,\leavevmode\nobreak\ 1J,\leavevmode\nobreak\ ,2J,\dots 10J$ and $2\leq L\leq 8$; the heatmaps for the average quantum work, and for the second to fourth moments $\langle W-\bar{W}\rangle^{k}$, $k=2,3,4$, for all the chains’ length considered ($2\leq L\leq 8$); and the heatmaps for the entropy production for $L=4$ and $L=8$.

• Roßnagel et al. (2016)

J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler, and K. Singer, Science 352, 325 (2016), ISSN 0036-8075.

• An et al. (2015)

S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.-Q. Yin, H. Quan, and K. Kim, Nature Physics 11, 193 (2015).

• Chiara et al. (2015)

G. D. Chiara, A. J. Roncaglia, and J. P. Paz, New Journal of Physics 17, 035004 (2015).

• Cerisola et al. (2017)

F. Cerisola, Y. Margalit, S. Machluf, A. J. Roncaglia, J. P. Paz, and R. Folman, Nature communications 8, 1241 (2017).