Single-Atom Verification of the Information-Theoretical Bound of Irreversibility at the Quantum Level

Quantitative measure of disorder or randomness based on the entropy production characterizes thermodynamical irreversibility, which is relevant to the conventional second law of thermodynamics. Here we report, in a quantum mechanical fashion, the first theoretical prediction and experimental exploration of an information-theoretical bound on the entropy production. Our theoretical model consists of a simplest two-level dissipative system driven by a purely classical field, and under the Markovian dissipation, we find that such an information-theoretical bound, not fully validating quantum relaxation processes, strongly depends on the drive-to-decay ratio and the initial state. Furthermore, we carry out experimental verification of this information-theoretical bound by means of a single spin embedded in an ultracold trapped $^{40}$Ca$^{+}$ ion. Our finding, based on a two-level model, is fundamental to any quantum thermodynamical process and indicates much difference and complexity in quantum thermodynamics with respect to the conventionally classical counterpart.


I. INTRODUCTION
Unitary operations, demonstrating reversibility, are the key concept in every textbook of quantum mechanics. However, perfect isolation of quantum system is not practically possible due to the fact that any realistic system is subject to the coupling to an uncontrollable reservoir. So quantum systems must be regarded as open systems, in which irreversibility of state evolution is overwhelming. On the other hand, the classical concept of irreversibility is mainly associated with the second law of thermodynamics (SLT), arguing that in an isolated system the thermodynamic entropy, behaving as arrow of time, never decreases.
Over the past two decades, the nonequilibrium processes in thermodynamics have drawn much attention since the conventional equilibrium thermodynamics cannot reasonably treat most natural or engineered processes that occur far from equilibrium. One of the typical approaches for treating nonequilibrium processes is the fluctuation theorem [1,2], which compares the forward process to its time reverse, connecting the fundamental timereversal symmetry of the underlying microscopic dynamics to the thermodynamics. For example, the fluctuation theorem [3] has reproduced the Jarzyski equality [4], the only equality so far in nonequilibrium thermodynamics, based on the assumption of microscopically reversible and thermostated dynamics. In addition, the fluctuation theorem has also derived the result of nonnegativity of the entropy production, i.e., the essence of * Electronic address: llyan@zzu.edu.cn † Electronic address: zhoufei@wipm.ac.cn ‡ Electronic address: mangfeng@wipm.ac.cn the SLT. Based on this idea, some recent publications [5][6][7][8][9] have tried to develop theories to further understand the thermodynamic irreversibility inherent to nonequilibrium processes. In particular, a very recent work focusing on thermal relaxation processes acquired an informationtheoretical bound of irreversibility [10], which imposes a stronger constraint on the entropy production than the conventional SLT.
In addition to the study of nonequilibrium thermodynamics, there has been a parallel line of development exploring thermodynamics on the microscopic scale [11][12][13][14]. Results obtained so far have shown that conventional thermodynamics, describing how large numbers of particles behave, operates differently in quantum regime. Thus the thermodynamic quantities need to be redefined (or re-understood) and the conventional SLT should be modified [15][16][17][18]. For example, from the quantum perspective, the origin of fluctuations is not just thermal but also quantum, and the von Neumann entropy is irrelevant to the thermodynamic arrow of time, but the characteristic of the state [19].
In the present paper, we explore, both theoretically and experimentally, the information-theoretical bound of irreversibility studied classical mechanically in [10], in quantum regime. By introducing quantum information concepts, we first deliberate on this bound of irreversibility, under different conditions, in a dissipative process including both the thermal relaxation and decoherence. Amazingly, counter-intuitive scenarios are found, for some values of drive-to-decay ratio (DDR) and initial states of the system, that the bound can be violated. Particularly, by precisely manipulating a single ultracold trapped 40 Ca + ion, we demonstrate experimentally a single-spin verification of this bound, witnessing the predicted violation.

II. THEORY
We first review briefly the main points in [10]. A typical thermal relaxation process is treated by the variational principle using two quantities, i.e., the entropy production from thermodynamics and the relative entropy (also called Kulback-Leibler divergence) from information theory. Considering variation of the entropy production σ in a duration from τ = 0 to τ = t, one may obtain the following inequality, where D(ρ sys (0) ρ sys (t)) is the relative entropy representing the Kulback-Leibler divergence of the system's state ρ sys (t) from the state ρ sys (0). As this divergence is non-negative, Eq. (1) imposes a more strict bound than the conventional SLT with the bound of zero. Compared to classical dissipation, a quantum dissipative process involves decoherence in addition to the thermal relaxation. However, following the idea in [10] that focuses on a dissipative process with a timeindependent Hamiltonian of the system, we find that, Eq. (1) still validates the variation of the entropy production in quantum dissipative processes under the condition of no system-bath correlation. As such, we may redefine the entropy production and relative entropy in quantum way, i.e., by von Neumann entropy. The entropy production σ [0,t] = t 0 dτ σ(τ ), with definition of σ(τ ) = −Tr[ρ(τ ) ln ρ(τ )], and the relative entropy is defined as D(ρ 1 ρ 2 ) = Tr[ρ 1 ln(ρ 1 /ρ 2 )]. Considering a classical bath with much bigger size than the system, we focus our study on the system, which has no work change due to the time-independent Hamiltonian and has no correlation with the bath. So we have the total entropy σ(t) = σ sys (t) + σ bath (t) [20], where the entropies of the system and bath reflect, respectively, the heat change in the system and the heat flow into the bath. Their von Neumann entropies are given by σ sys = −Tr[ρ sys ln ρ sys ] and σ bath = −β i E i λ i , respectively, in which β is the usually defined inverse temperature, and the parameters λ i and E i correspond to the ith eigenvalue and the energy of ith eigenstate of ρ sys . To reveal whether Eq. (1) really holds or not, we introduce a balance parameter Υ(t) = σ [0,t] − D(ρ sys (0) ρ sys (t)) for estimate.

III. THE SCHEME AND THE SYSTEM
For the sake of generality, we consider a driven twolevel system, which is the fundamental element of any quantum system. The Hamiltonian in interaction representation is given by H = Ωσ x /2, where Ω is the Rabi frequency, σ x = |2 1| + |1 2| is the usual Pauli operator with the upper (lower) level |2 (|1 ), and we assume = 1. To understand the quantum relaxation process, we solve the Lindblad master equation of  with Γ the decay rate from the excited state to the ground state. When t → ∞, the system reaches the steady state whose eigenspectrum decomposes as ρ s = λ − |φ − φ| + λ + |φ + φ| with λ − > λ + (See Appendix A). In the subspace spanned by |φ − and |φ + , we find that |φ + is of the higher energy than |φ − , and the effective inverse temperature of the steady state is given by β = (ln λ − −ln λ + )/∆E with ∆E denoting the energy difference between |φ + and |φ − . Thus the corresponding Gibbs state is ρ g = e −βHi /Z g , where Z g is the partition function and the effective Hamiltonian H i = ∆Eσ int z /2 with σ int z = |φ + φ| − |φ − φ|. A nonequilibrium state of the system evolves to the Gibbs state ρ g under a quantum relaxation process, implying ρ s = ρ g at the end. In this case, we obtain the partition function Z g = 1/ λ − λ + .
In terms of the restricted conditions as listed in Appendix A, we segment the problem into four regimes based on the DDR values: small (i.e., Ω/Γ < 0.15), intermediate (i.e., 0.15 ≤ Ω/Γ < 6.25), large (i.e., 6.25 ≤ Ω/Γ < 25) and ultra-large (i.e., Ω/Γ ≥ 25). Since the regime is segmented by Ω/Γ, instead of Ω or Γ alone, the values of Ω and Γ can be very small even in the ultra-large DDR case. This implies that the Lindblad master equation under Markovian approximation validates all these segments [21]. For a comparison with the classical situation, we first consider, both theoretically and experimentally, the system initialized from a well-polarized state, i.e., ρ i = |2 2|. For such a non-equilibrium process undergoing a quantum relaxation until the Gibbs state ρ g , Fig. 1(a) presents a numerical check of the balance parameter over the whole regime, where two extreme cases can be treated analytically as below. (I) Ω/Γ ≪1, with the time unit τ a ≈ Γ −1 . For t ≪ τ a , we obtain β = 4(ln Γ − ln Ω)/∆Ẽ with the energy difference ∆Ẽ between |2 and |1 . Thus the balance parameter Υ(t) in the short-time relaxation process, i.e., t ≪ τ a , is Υ(t) = H(Γt) + 4Γt ln(Γ/Ω) − Γt, where H(•) denotes the binary entropy of •. Due to H(Γt) > Γt for Γt ≪ 1, we have Υ(t) > 4Γt, implying that Eq. (1) fully holds in the very small DDR case, as plotted in Fig. 1(b). (II) Ω/Γ ≫1, with the time unit τ c ≈ Ω −1 . For t ≪ τ c , the inverse temperature is written as β = 4/∆Ẽ, independent of Γ and Ω. As a result, in the preliminary region of the evolution, Υ(t) is given by Υ(t) = [2Γ − Ω(ln 4 − 1 − 2 ln Ωt)]Ωt 2 /4, which turns to be Υ(t) ∼ −Ω 2 t 2 <0 in the short time limit. This implies violation of the information-theoretical bound of irreversibility in this ultra-large DDR case. However, the situation is more complicated than a simple occurrence of violation. If we observe the evolution for a time longer than t c with t c = 2e −(Ω+2Γ)/2Ω τ c , we may find the revival of the information-theoretical bound, see Fig. 1(c). The violation and validity of the bound could happen repeatedly, which is resulted from the competition between coherence and relaxation regarding the system.
For the cases sandwiched by the two extreme cases, we have to investigate numerically, along with the experimental execution as elucidated below. Our experiment is carried out on a single ultracold 40 Ca + ion confined in a linear Paul trap as employed previously [22,23]. Under the pseudo-potential approximation, the axial and radial frequencies of the trap potential are, respectively, ω z /2π = 1.0 MHz and ω r /2π = 1.2 MHz. For our purpose, we employ a magnetic field of 0.6 mT directed in axial orientation, yielding the ground state 4 2 S 1/2 and the metastable state 3 2 D 5/2 split into two and six hyperfine energy levels, respectively. We encode qubit |↓ in |4 2 S 1/2 , m J = +1/2 and |↑ in |3 2 D 5/2 , m J = +5/2 , where m J represents the magnetic quantum number, and for simplicity the two levels are labeled as |1 and |2 (See Fig. 2). Before our implementation, the z-axis motional mode of the ion is cooled down to average phonon number of 0.030 (7). The qubit is manipulated by a narrow-linewidth Ti:sapphire laser with wavelength around 729 nm, which irradiates the ultracold ion under the government of the carrier-transition Hamiltonian H c = Ω(σ + e iφL + σ − e −iφL )/2, with the Rabi frequency Ω as the laser-ion coupling strength in units of = 1. φ L is the laser phase and σ +,− are Pauli operators for the qubit levels.
The single ultracold trapped ion is an ideal platform to explore the thermodynamics due to flexible modeling and ultimate accuracy [24][25][26][27][28]. Here we intend to manifest a dissipative two-level system, and thus we employ additionally the excited level |4 2 P 3/2 , m J = +3/2 labeled as |3 , with which we have a closed cycle |1 → |2 → |3 → |1 , as presented in Fig. 2. The first step from |1 to |2 is achieved by the Ti:sapphire laser (729-nm) tuned exactly to the resonance transition. The second step from  |2 to |3 is a dipolar transition made by a semiconductor laser (854-nm) under the restriction of the selection rule. The third step |3 → |1 is a spontaneous emission, also restricted by the selection rule. Practically, the Rabi frequency Ω and the dissipative rate Γ of the effective two-level system are tuned by tuning the 729-nm and 854-nm lasers under the condition of Ω ≪Ω. This makes sure that we may check the dissipation each time, as exemplified in the right panel of Fig. 2, before making measurements for verifying Eq. (1). The observed linear decay of P D validates the Lindblad master equation in treating all the cases of the DDR as elucidated below.

IV. EXPERIMENTAL OBSERVATION
Our observations are based on different values of DDR and different initial states of the system. So we divide this section into two parts relevant to, respectively, the system initialized from a well-polarized state and from states with coherence. In each part, observations due to some typical DDR values are demonstrated and analyzed.

A. For initialization from a well-polarized state
We have experimentally measured three curves, belonging to, respectively, the cases of small, intermediate  and large DDRs. Here we first consider the intermediate DDR (Fig. 3), which displays dynamics under a comparable competition between the drive and dissipation, finally reaching an equilibrium at nearly half of the maximal P D , see panels (a,b). Other two panels demonstrate that Eq. (1) is always valid in the whole process. Both the Kullback-Leibler divergence and the entropy production are initially zero, and then increase in time evolution with the latter always larger than the former. Despite the robustness of Eq. (1), we have observed fluctuations of the entropies in the evolution, along with the population variation, which are very different from the entropies' monotonous increase in the classical counterpart [10]. Moreover, the system's entropy production maximizes at the mixed state with maximal off-diagonal terms, implying farthest away from the pure state. In contrast, the relative entropy maximizes at the state with largest Kullback-Leibler divergence from the initial one (i.e., a well-polarized state), which is relevant to certain occupations in the diagonal and off-diagonal terms of the system's state. In this context, the larger value of Υ reflects more weight of the mixed state involved in the system, which is in contrast to either the equilibrium described in [10] or the thermal equilibrium in the conventionally thermodynamic perception.
We have also carried out experiments for cases with small and large DDRs initialized from |2 , which also validate Eq. (1). As presented in Fig. 4, the results reflect the weight of mixed states, as represented by Υ, in these quantum thermodynamic processes. The different variations of Υ here from in Fig. 3 indicate the facts that (1) the system's entropy production in the small DDR case descent quickly due to dominant dissipation, conforming with the observation of P D fast falling down to nearly zero; (2) The level |2 keeps the population for a relatively long time due to the large DDR, and then decay to |1 in a drastically oscillating fashion, yielding a slow increase of Υ in the former part and then a rich dynamics in the later until a complete equilibration.
In this context, we can understand the breakdown of Eq. (1) under the ultra-large DDR, as shown in Fig.  1(c), which is due to the system's coherence temporarily dominating over the dissipative effect. Unfortunately, we have no way to reach this regime experimentally for the initial state in the excited state |2 . In the present case, violating the bound of irreversibility occurs in the case of Ω/Γ ≥ 36 (See Fig. 1), thus observing the invalidity of the bound, under the conditions of Γ e ≫Ω ≫ Ω ≫ Γ, requires very small values of the Rabi frequency and effective decay rate, i.e., Ω ≤ Γ e /36 3 and Γ ≤ Γ e /36 4 , which is evidently challenging in our experimental observations. As investigated in Appendices C and D, even if we could meet such stringent conditions, credible measurement is unavailable in this case.

B. For initial states with coherence
Quantum mechanically, the two-level system could be initialized in a state involving coherence, rather than simply a well-polarized state, for which we present below that Eq. (1) is not always valid for most DDR values. In a quantum two-level system, an arbitrary state can be written as where σ = (σ x , σ y , σ z ) and the vector r = (x, y, z) with |r| ≤ 1. |r| = 1 denotes a pure state and r = (0, 0, 1) indicates an excited state denoted by ρ = |2 2|. The eigenvalues of ρ i are written as The energies of the two states are given by with the difference ∆E = z |r| (E 2 − E 1 ). Since E 2 > E 1 , if z ≥ 0, then we have ∆E ≥ 0 and |ϕ + is the upper state; if z ≤ 0, then ∆E ≤ 0 and and |ϕ − is the upper state. The initial entropies of the system and the bath are then given by For clarity of description, we employ r and z as additional variables, with which the initial state is denoted by the vector r = (x, y, z) with |r| ≤ 1. We exemplify the intermediate DDR in Fig. 5(a) by considering different initial states. For the initial state with z < 0, the bound violation would possibly occur, which could be understood as that with the system evolving away from the initially mixed state, the total entropy production, of negative values in some cases, is always smaller than the Kullback-Leibler divergence. Experimentally we have prepared the initial state by evolving for τ 0 = 4 µs from the state |2 , and we observed the negative values of Υ in all the evolution, see Fig. 5(b). By this way, we have found that the invalidity of Eq. (1) could be witnessed more easily with larger values of the DDR. As a supplement of Fig. 5(b), Fig. 6 presents time evolutions of state populations and characteristic entropies, demonstrating the reason leading to negative values of the balance parameter Υ(t). With the populations oscillating, the total entropy production is always negative but the relative entropy is always positive. This leads to breakdown of the information-theoretical bound of irreversibility in Fig. 5.
Correspondingly, we may observe the violation of information-theoretical bound of irreversibility in the small and large DDR cases by modifying the initial states in Fig. 5. As shown in Fig. 7, the bound holds for most initial states in the small DDR case, with the violation happening in the limit of z → −|r|. In contrast, in the represent, respectively, the bath's entropy production regarding σ bath , the total entropy production σ [0,t] , the system's entropy production regarding σsys and the relative entropy D(ρsys(0) ρsys(τ )). The error bars are standard deviation covering the statistical errors of 10,000 measurements for each data point.
case of large DDR, the bound is violated for most initial states and holds only in the limit of z → |r|, as shown in Fig. 8. However, experimentally, the result in Fig. 7(b) is not convincing due to the weak violation influenced by the considerably larger errors (which are not resulted from measurement imprecision, but the calculation of entropy, as explained in the next section). In contrast, the experimental result in Fig. 8(b) clearly shows the violation of the bound.

V. DISCUSSION
In our implementation, the decay rate of |3 is Γ e /2π = 23.1 MHz, and the Rabi frequency Ω can be up to 200 kHz. The effective decay is controlled by the power of the 854-nm laser. As the effective two-level system is operated with population transfer from lower to upper state and decay from the upper to lower state, it is very important to have an exact control of the 729-nm and 854-nm lasers for their polarizations and resonant frequencies.
Specifically, since the resonance transition of 729-nm irradiation along x direction is sensitive to the polarization of the laser, we have finely tuned the half-wavelength plate to reach the maximal resonance strength. Meanwhile, we control the frequency of 854-nm laser by an electric-optic modulator which is locked to another ultra-low expansion cavity (with linewidth of 0.5 MHz). To have an effective decay rate Γ, we have first swept the frequency of the 854-nm laser to find the exact resonance between |2 and |3 , and then determined the expected value of Γ by tuning the power of the 854-nm laser. To make sure our measurements to be performed with high precision, we have tried to suppress quantum projection noise in single-qubit measurements by 10,000 repetition. Moreover, since the Zeeman sublevel |3 2 D 5/2 , m J = +5/2 is highly sensitive to the magnetic field fluctuation, unexpected imperfection is involved in our qubit in initial-state preparations and qubit operations. Other possible errors can also be from the laser instability, imperfect single-qubit pulses and heat noise, whose effects are assessed from the Rabi oscillations in our case. After calibration, we have estimated the total error contributed on the imperfection of the initial-state preparation and the final-state detection to be 0.7(2)% and 0.22(8)%, respectively. These imperfections, along with statistic errors, are involved in the standard deviation represented by the error bars. Before ending our discussion, we would like to emphasize the intrinsic causes leading to some large error bars regarding our experimental data of entropy. In a general two-level system, the spectrum decomposition of a quantum state ρ can be written as ρ = p g |ϕ g ϕ| + p e |ϕ e ϕ| with |ϕ e,g denoting the eigenstates of ρ and p e + p g = 1.
In this case, the von Neumann entropy is given by S(ρ) = −p g log p g − p e log p e . Theoretically, the error regarding quantum projection noise for detection is estimated as ǫ g = p g (1 − p g )/N with N denoting the detection repetition. Thus, the error estimate of the von Neumann entropy is formulated as where we have assumed ǫ g = ǫ e in the second equation.
In the limit p g → 0 and 1, the above equation yields both cases to be ǫ S (ρ) → 0. However, considering the experimental situation, we have, except for the quantum projection error, some other errors as mentioned above, which could not be fully eliminated by the standard calibration method. Therefore, the realistic error in the limit p g → 0 is given by, which, as seen in Fig. 9, is larger than the counterpart with only theoretical consideration. For the case of p g → 1, we have similar situation. So we have some of the error bars for entropies evidently larger than the others, e.g., in Figs. 3-8.

VI. CONCLUSION
We have explored a quantum mechanical informationtheoretical bound of irreversibility using a fundamental two-level system, which applies to any quantum thermodynamic process. Our experimental witness is the first single-spin evidence verifying such a novel and tighter bound for irreversibility, which is expected to be helpful in understanding the SLT and irreversibility subject to quantum effects. Since von Neumann entropy is not relevant to the thermodynamic arrow of time, violating Eq. (1), although counter-intuitive in the viewpoint of classical physics, is possible and reasonable in quantum systems, which means the essential role of coherence as well as the complicated and nontrivial feature in quantum thermodynamics. We believe that our result provides a deeper insight into irreversibility in quantum regime and in particular helps further understanding of thermodynamics at the microscopic scale.

APPENDIX
We provide below supplementary information to the main text. We first describe the real scenario of simplest two-level system and then present a thorough transformation of a three-level system to an effective two-level system. The subsequent discussions are based on this effective two-level system. For clarity, we plot Fig. 10 with denotations labeled for convenience of description in Appendices A and B as below. Moreover, the analytical deductions presented below are basically general, but some numerical results exemplified, such as in Figs. 12-15, are made based on the initial state |2 . Experimental feasibility for ultra-large DDR case discussed in Appendices C and D is also based on the initial state |2 .
Appendix A: Quantum relaxation process in a two-level system Considering a two-level system such that excited and ground states are, respectively, given as |2 and |1 . The where Ω and Γ are, respectively, the Rabi frequency coupling |1 to |2 and the decay rate from |2 . (Right panel) Three-level system discussed in Appendix B, where Rabi frequency Ω couples |1 to |2 , Rabi frequencyΩ couples |2 to |3 , and Γe is the decay rate regarding |3 . mathematical equations arguing the systematic time evolution of the density matrix elements ρ ij = i|ρ|j can be described as follows, Assuming the system initially in |2 and following the aforementioned analytical formulation, we may write, For our purpose, we investigate the above dynamic process by taking different time scales into account as below, and with τ a < τ b < 2τ a . When t ≪ min[τ a , τ c ] and Γ ∼ Ω, we obtain It is evident from the above that as the population in |2 descents the coherence terms appears. However, for t ≫ τ b , the system approaches the steady state (t → ∞), which is given by The eigenvalues of the steady state are , and the corresponding eigenstates are satisfying θ + > θ − and θ − + θ + = π/2. Thus, the steady state can be rewritten as Denoting the energies of |1 and |2 by E 1,2 with E 2 > E 1 , the corresponding energies of |φ ± are E + = cos 2 θ + E 1 +sin 2 θ + E 2 , E − = cos 2 θ − E 1 +sin 2 θ − E 2 , with their difference given by Therefore, in the picture of |φ ± , |φ + refers to the excited state and |φ − designates the ground state, constituting a new-basis two-level system. In the steady state, the inverse temperature β of this new-basis two-level system is given by Since λ − > λ + , we always have β > 0. From the expressions above, we may rewrite β as, which implies that different inverse temperature can be obtained by changing the values of two control parameters, i.e, the Rabi frequency Ω and the decay rate Γ. In the limit of Γ ≫ Ω, we obtain In contrast, under the condition of Γ ≪ Ω, the inverse temperature can be expressed as Therefore, for a given decay rate Γ, the inverse temperature β decreases with the increase of Ω, as plotted in Fig.  11. This also implies that temperature increases with Ω.

Now we consider a Gibbs state
with the partition function Z g and the Hamiltonian in the space spanned by |φ − and |φ + with σ int z = |φ + φ| − |φ − φ|. In the limit of t → ∞, the quantum relaxation process reaches a steady state as ρ s = ρ g .
As such, we obtain the partition function Z g = 1/ λ − λ + . If we assume the system to be initially in a non-equilibrium state ρ i , e.g., ρ i = |2 2|, then the ensuing system evolution encapsulating the thermal relaxation process will lead to the Gibbs state ρ s .
Due to the condition Γ ≪ Ω, we have λ + ∼ (1 − Γ/Ω)/2 and λ − ∼ (1 + Γ/Ω)/2 where λ − /λ + = 1 + 2Γ/Ω. So we find, σ bath = sin 2 (Ωt) sin 2 (Ωt) + (1 + cos Ωt) 2 Thus the condition Ωt ≪ 1 yields D(ρ 0 ρ) > Ω 2 t 2 . Using the fact of Ω ≫ Γ, we obtain D(ρ 0 ρ) ≫ σ bath + σ sys for t ≪ τ c . Therefore, the generalized informationtheoretical bound of irreversibility is violated in the ultralarge DDR case. Moreover, for a time t c comparable to τ c , we consider following condition which violates the information-theoretical bound of irreversibility in the case of t < t c . Solving it, we obtain Thus, in the ultra-large DDR case, the informationtheoretical bound of irreversibility can be violated in the first duration (≤ τ c ) of the evolution (See right panel of Fig. 12) since there is a relatively strong coherent oscillation and thus quantum coherent evolution dominates the information interchange between the system and the bath, implying that the relaxation process is quantum mechanical. With the evolution going on, the violation and validity of the bound occur repeatedly.
Using the condition Γ e ≫Ω, we have Γ 2 e − 4Ω 2 = Γ e (1 − 2Ω 2 /Γ 2 e ) and thus ρ 11 is reduced to At t = 0, ρ 11 = 0. For t ≫ 1/Γ e , the last term turns to be negligible, i.e., In the case of t ≪ Γ e /Ω 2 , ρ 11 is further reduced to ρ 11 ∼ Ω 2 Γe t. Similarly, the solution of ρ 33 is and ρ 22 is written as for t ≫ 1/Γ e . Therefore, the effective decay rate from |2 to |1 is given by In other words, consideringΩ 2 = Γ e Γ, we have the steady state ρ s in this case with the same form as the two-level system treated in above section. Based on the deductions in above sections, the accurate form of the steady state of the effective two-level system is written as, Under the condition Γ e ≫Ω ≫ Ω, i.e., Γ e ≫ √ ΓΓ e ≫ Ω, we obtainρ with the leakage parameter ξ = Ω 2 Γ/Γ e (2Ω 2 + Γ 2 ). In the case of a very small DDR, it is simplified as ξ = Ω 2 /ΓΓ e , while in the ultra-large DDR case, it reduces to ξ = Γ/2Γ e (See left panel of Fig. 13). Leakage from an effective two-level system, where blue and red curves represent the ultra-large and small cases, respectively. (Right panel) Inverse temperature regarding a real two-level system (blue curve) and an effective two-level system (red curve), respectively, where the decay rate of the excited state in a three-level system is set as Γe = 10 4 Γ.
Writingρ s by the eigenspectrum decompositionρ s = λ − |φ − φ|+λ + |φ + φ| with the eigenvalues of the steady statẽ with the decay ratio r = Γ/Γ e , we may write the corresponding eigenstates as satisfyθ + >θ − andθ − +θ + = π/2. The inverse temper-ature is given bỹ In the case of r = 0, we haveβ = β. Moreover, in the limit of Γ ≫ Ω, the above equation reduces tõ which is the same as that for a real two-level system, as shown in right panel of Fig. 13. However, in the Γ ≪ Ω, we findβ which is different from the case of a real two-level system. This difference is vanishing if rΩ 2 ≪ Γ 2 , implying the condition Ω 2 ≪ ΓΓ e . As a result, in the case of ultra-large DDR, we have to satisfy the condition Γ ≪ Ω ≪ √ ΓΓ e , i.e., Γ e /Ω ∼ (Ω/Γ) 3 . Unfortunately, this is a very challenging condition for an effective twolevel system since violating the bound of irreversibility requires Ω/Γ ≥ 36, indicating that Ω ≤ Γ e /36 3 and Γ ≤ Γ e /36 4 . Since Γ e is generally of the order of tens of MHz in atomic systems, both Ω and Γ required would be less than 500 Hz, which means impossibility to accomplish qualified optic measurements.
The above discussions are based on the assumption that the system is initialized from |2 . In fact, as a quantum system, the state can be initialized in any superposition or mixed state, rather than simply in a well-polarized state. If the system is initially prepared in a state with coherence, the violation of the bound can be observed in the regimes of the small, intermediate and large DDR, as investigated in section IV (B) of the main text.
On the other hand, if we only focus on the condition for a ultra-large DDR, we may consider a realistic two-level system with a fixed decay rate. As such, Rydberg atoms, with effective Rabi frequency possibly larger than the decay rate by more than 50 times, might be available to witness the bound violation we predicted. For example, in a recent publication Phys. Rev. Lett. 121, 123603 (2018) using 87 Rb atoms, the Rydberg state |70S, J = 1/2; m J = −1/2 is of the lifetime 27 µs, an effective Rabi frequency has reached 2π× 2 MHz, implying Ω/Γ > 54. A larger DDR has also been achieved in a recent neutral atom experiment [Science 365, 570 (2019)] based on a two-photon process, in which the Rabi frequency can be varied from 0 to 2π× 5 MHz and implies that any value of Ω/Γ in the interval (0, 135] is feasible. Nevertheless, to witness the bound violation as we predicted, we consider that more elaborate investigation is required for the experimental operation and measurement, such as the influence of the two-photon process involved, the coherence time remained, and qualified steady state to be reached.  The divergence of the inverse temperatureβ exemplified in the right panel of Fig. 13 occurs in the case of Γ e = 10 4 Γ and Ω > 20Γ, which implies that the condition Γ e /Ω ∼ (Ω/Γ) 3 is no longer satisfied, i.e., invalidity of an effective two-level system. As discussed in last section, in the case of the initial state in |2 , we require Γ ≤ Γ e /36 4 and Ω ≤ Γ e /36 3 in the ultra-large DDR case, which ensures an effective two-level system even for Ω/Γ ≥ 36 in order to witness the violation of the irreversibility bound.
Moreover, theoretically, the steady state is achieved at t → ∞. This is, however, impossible for practical operations experimentally. To assess how well for a long time waiting approaching the steady state, we consider, for a particular waiting time, the deviation from the steady state by defining ǫ t = D(ρ t ρ s ). As shown in the top inset of Fig. 14, smaller Rabi frequency corresponds to less oscillation, but has no change for the waiting time. In contrast, in the bottom inset, we find that larger decay rate produces smaller deviation with an approximate relation ǫ t ∼ e −Γtw . Therefore, for a given deviation, we obtain a relation for the waiting time t w ∼ −Γ −1 ln ǫ t . For example, for ǫ t = 10 −3 , the waiting time should be t w ∼ 7Γ −1 . In this context, we analyze here another reason regarding the waiting time for the impossibility to realistically witness the violation of Eq. (1) in the ultra-large DDR case. Assuming we can achieve the ultra-large DDR case, we find large inaccuracy in measuring Υ(t) experimentally, which originates from the steady state achieved. In terms of our numerical results in Fig. 15, we obtain completely different results of Υ(t) for different waiting times. For example, for t w =7, 10, 15Γ −1 , the inverse temperatureβ =-5.82, 4.92, 4.06, while the deviations from the steady are ǫ t = 1.4×10 −5 , 3.2×10 −7 and 4×10 −8 , respectively. Therefore, witnessing the violation of the bound in this case requires a much long waiting time, which is very challenging.

Appendix E: Some points for Experimental operations
All the experimental evolutions are carried out in a simplest system of single spin consisting of an ultra-cold 40 Ca + ion, confined in a linear Paul Trap. The 397-nm laser is used for three dimensional Doppler cooling, whereas lasers, at wavelengths 866 nm in z-and ydirections and 854 nm in z-direction, are purposed as repumping and state quenching light sources. The 866nm laser is always on throughout our experiments. The 729-nm laser, in x-direction, performs state manipulation and experimental evolution in addition to sideband cooling so as to bring the trapped ion down to vibrational ground state. The 729-nm laser is an ultra-stable narrow linewidth Ti:Sapphire laser, corresponding to linewidth (FWHM) of 7 Hz, as measured via the heterodyne beat note method with respect to another laser system. It is locked to a high-finesse ultra-low expansion cavity via PDH locking method which could reach a long-term drift less than 0.06 Hz/s. In order to provide laser cooling, initialize quantum state and perform experimental evolutions and final state detection, we need a well-ordered laser pulse train with polarization, phase, frequency and amplitude control. Therefore, the lasers are controlled via the acousto-optic modulators (AOMs) by passing all the lasers through the AOMs before irradiating the ion. The operational systematic RF signals applied to all the AOMs are being supplied by the direct digital synthesizer (DDS) which is controlled by a field programmable gate array in addition to a TTL architecture intended to provide control signals to the RF switches. The DDS has the ability to provide the AOM signals with real time phase, frequency and amplitude control of all the lasers during the consecutive experimental progresses. A typical experimental sequence involves more than 300 optical pulses within a time slot of about 40 ms.