Experimental test of exchange fluctuation relations in an open quantum system

Elucidating the energy transfer between a quantum system and a reservoir is a central issue in quantum non-equilibrium thermodynamics, which could provide novel tools to engineer quantum-enhanced heat engines. The lack of information on the reservoir inherently limits the practical insight that can be gained on the exchange process. Here, we investigate the energy transfer for an open quantum system in the framework of quantum fluctuation relations. As a novel toolbox, we employ a nitrogen-vacancy center spin qubit in diamond, subject to repeated quantum projective measurements accompanied by a tunable dissipation channel. When the system is tuned to be insensitive to dissipation, we verify the closed-system quantum Jarzynski equality. In the presence of competition between dissipation and quantum projective measurements, the experimental results suggest a formulation of the energy exchange fluctuation relation that incorporates the reservoir properties in the guise of an effective temperature of the final out-of-equilibrium steady-state. Our findings pave the way to investigate energy exchange mechanisms in higher-dimension open quantum systems.

The connection between statistical properties of out-ofequilibrium dynamical systems, thermodynamics quantities and information theory has been deeply investigated in classical and quantum systems and codified in terms of fluctuation relations [1][2][3][4][5][6][7] . However, in open quantum systems, despite several contributions [8][9][10][11][12][13] , such connection is far from being completely understood, especially regarding the competition between thermal and quantum fluctuations, which assume a paramount role at the nanoscale. Accounting for the statistical fluctuations is the key to reformulate the second law of thermodynamics, usually expressed as inequalities, in terms of equalities. The Jarzynski equality 14,15 , as a major example, relates the exponentiated negative work done on a system, averaged over a statistically relevant ensemble of realizations of the system dynamics, with the change in free energy ∆F between two equilibrium thermal states. This framework has been also extended to describe the transport of energy and matter between different systems with different temperatures and chemical potentials 16,17 . Remarkably, these relations hold for any kind of process driving the system arbitrarily far from equilibrium, provided that the initial and final state are in thermodynamic equilibrium. In quantum mechanical settings, the quantum fluctuation relations can be recast in terms of the so-called characteristic function G, the Fourier transform of the probability density function of the considered non-equilibrium quantity. This contains the full information on the statistics of fluctuations and is obtained from two-time quantum correlations rather than by a singletime expectation value, remarkably reflecting the fact that work is not a quantum observable 18 . Still, in the absence of a heat reservoir, the internal energy variation time FIG. 1. Conceptual illustration of the experiment. A twolevel system is subject to series of quantum projective measurements (QPMs), and controllable thermal contact with a reservoir. We investigate the the energy exchange fluctuations occurred in the system from the beginning to the end, over a statistically relevant ensemble of realizations of the protocol. ∆E is solely due to work yielding a formally-equivalent quantum version of the Jarzynski equality (QJE) [19][20][21] , e −(∆E−∆F )/ε = 1, with ε being the energy scale of the system, which at finite temperture takes the conventional form of ε = 1/β. This has been indeed verified in various experimental settings with no heat flux involved, ranging from single trapped ions 22 to liquid-state nuclear magnetic resonance platforms 23 , atom chips 24 , and superconducting Xmon qubits 25 . Opening an energy exchange channel from the quantum system to a heat reservoir poses new challenging problems in describing the nonequilibrium thermodynamics 4 . While a weak coupling can be effectively tracked back to the case of an isolated system 26 , in the case of strong coupling evaluating the fluctuation relation for the exchanged energy would require information on the reservoir 8, 16 that is often not available.
Here, we experimentally test the energy exchange fluc-tuation relation in a two-level quantum system that is intermittently connected to a heat reservoir and subject to repeated quantum projective measurements (QPMs), as depicted in Fig. 1. We realize this protocol with the use of a single nitrogen-vacancy (NV) center qubit in diamond at room temperature, in the presence of trains of short laser pulses. Each absorbed laser pulse results in a QPM, and in an energy redistribution that can be modeled as heat exchange with an effective thermal reservoir. The time intervals between QPMs follow a stochastic distribution due to the finite absorption probability. Tuning the laser power enables the control of the coupling strength of the quantum system to the reservoir, in analogy to a variation of the thermal conductivity of the dissipative channel (thermal conduction channel). The combined effect of QPMs and heat flux can create or destroy quantum coherence during the dynamics of the system, an effect that goes beyond the classical description. From a thermodynamic point of view, the process associated with a QPM has been described as work or equivalently as a quantum-heat flux [27][28][29][30][31] . While ideal QPMs affect the energy distribution of the system, they preserve the validity of the conventional QJE 27 , also for stochastic distributions of QPMs 32 . This is no longer true when adding heat exchange. Still, measuring the statistics of the exponentiated energy fluctuations through a two-point measurement (TPM) protocol 23,25,26 we demonstrate an operative physical meaning of the normalization factor ε in the QJE for an open twolevel quantum system in terms of its effective initial and final temperatures.

Protocol implementation
The experimental platform is based on a negatively charged NV center -a localized impurity in the diamond lattice based on a nitrogen substitutional atom and a nearby vacancy, which forms an electronic spin S = 1 in its orbital ground-state. The electronic spin can be optically initialized into m S = 0, where m S stands for the eigenvalues of the spin operator S z 33 . A strong symmetry-breaking magnetic bias field removes the degeneracy of levels with spin projections m S = ±1. Close to the excited-level anticrossing, the optical initialization of the NV electronic spin also polarizes the NV's nitrogen nuclear spin into m I = +1 34 , allowing the selective coherent manipulation of the transition between the |m S , m I states |0, 1 ≡ |0 and |1, 1 ≡ |1 that form an effective two-level system, while the other states remain unaffected.
A continuous nearly-resonant microwave driving field (with detuning δ mw ∈ [0, 1.3] MHz) sets the Hamiltonian H of the system in the rotating frame, so that with σ i Pauli matrices, tan α = −Ω/δ mw and ω = δ 2 mw + Ω 2 , Ω being the bare Rabi frequency. The Rabi driving prevents spin dephasing to occur on the experiment timescales (∼ µs) 35 . On top of the unitary evolu- Phononic band QPM D is si pa tio n ch an ne l Laser mw FIG. 2. Energy levels of the NV center and protocol. a) The two-level system is formed by the dressed states under a strong resonant Rabi driving. Upon absorption of green laser light, the system is subject to a QPM of σz. A nonradiative decay route induces spin amplitude damping to the |0 state (see text), we refer to this decay route as dissipation channel. b) Scheme of the experimental TPM protocol. The energy of the system is measured at the beginning, and at the end of the protocol. During the time t f , between the two energy measurements, the system evolves under unitary evolution and is perturbed by equidistant short laser pulses. The time intervals between the laser pulses belong to the range 270 ≤ τ ≤ 750 ns, and therefore are much longer than the laser pulses duration tL = 41 ns. c) Exemplary spin state evolution (red dots) on the Bloch sphere, for α = π/3 and τ = 2π/ω, for a single realization (left) and on average (right).
tion U = e −iHt/ , we apply to the NV center trains of short laser pulses at intervals τ , as depicted in Fig. 2(b). The laser pulses trigger cycles of almost perfectly spinpreserving radiative transitions from the ground to the excited spin states accompanied by coupling to lattice phonons, and intersystem crossing through an intermediate metastable state (see Fig. 2(a) and Methods). Any superposition or mixed spin state in the energy basis is projected back into the σ z eigenstates, denoted by |0 or |1 , while the state coherence imprinted by the microwave during the prior unitary evolution is destroyed in the σ z basis, resulting in a QPM of σ z 36 . Significantly, this mechanism may simultaneously produce coherence in other bases, such as the energy basis, as shown in Fig. 2(c). By varying the laser power we tune the photon absorption rate, leading to an absorption probability in the range (0.18 − 0.68). Thus, the application of a train of equidistant laser pulses entails a stochastic time distribution of QPMs for each single realization. Even short laser pulses (lasting t L = 41 ns) also induce a partial population transfer from |1 to |0 , due to the spinnonconserving, non-radiative transition through the NV metastable singlet state (see Methods), a mechanism that is routinely exploited for achieving high-fidelity initialization of the NV spin state 33 under long-lasting laser illumination. This spin amplitude damping can be described as a controlled heat exchange with a Markovian reservoir 37  (c), respectively, with an initial thermal state th such that the Hamiltonian, after an energy measurement, a completely damped state |0 corresponds to a thermal state with effective temperature ranging from zero (α = 0) to infinity (α = π/2) in the energy basis. Note that in the experiment the thermal conduction channel is repeatedly opened for short time intervals t L and stays closed otherwise. This picture effectively incorporates all the laser-induced NV photo-dynamics involving the ground and excited triplet states and the metastable singlet state (see Supplemental Material). The competing effects of QPMs and heat exchange of the system with the reservoir lead to a non-trivial dynamics, affecting the energy fluctuation distribution. In order to characterize the statistics of energy exchange fluctuations we apply a TPM protocol, as sketched in Fig. 2(b). The protocol consists essentially in (i) preparation of the initial state, (ii) projection into an energy eigenstate, (iii) evolution during a time t f (unitary plus short laser pulses), and (iv) measurement of the final energy E f , in order to compute the energy variation ∆E. We recast this protocol by exploiting the advantage of a two-level system. The system is initialized in a Hamiltonian eigenstate, with density matrix ↑ = |↑ ↑| (resp., ↓ = |↓ ↓|), and initial energy E i equal to the eigenvalue E ↑ (resp., E ↓ ). For a two-level system, any arbitrary thermal state is obtained as a combination of these pure states: th = p ↑ ↑ + (1 − p ↑ ) ↓ , where p ↑ describes the probability of measuring the initial energy E ↑ at the beginning of the protocol. By virtue of the NV spin-dependent fluorescence, the number of measured photons, averaged over a statistically-relevant number of experiment repetitions (16 × 10 5 ), gives the conditional probability P j|i = P (E f = E j |E i ) to measure E j as final energy for a given initial state. Measuring the conditional probabilities P ↑|i for each of the two initial eigenstates i = ↑ or ↓ , gives access to the full statistics of ∆E (see Methods). The energy variation occurred to the qubit after the process can assume one of the three values ∆E ∈ {− ω, 0, + ω}. Figs. 3(a) and 3(c) show the conditional probability P ↑|↓ (respectively, P ↑|↑ ), when starting from ↓ (respectively, ↑ ), as a function of the evolution time t f . To quantitatively support that a twolevel model with amplitude damping and projective measurements provides an accurate description of the system dynamics, we performed a numerical Montecarlo simulation of the dynamics and found excellent agreement with data. Note that the only fit parameter is the absorption probability, which depends on the laser power (see Methods), and quantifies the stochasticity of the protocol. In the absence of laser pulses, the spin is a closed system and the eigenstates do not evolve in time (usually referred to as spin lock ), while the laser pulses produce discrete energy jumps, whose asymptotic behavior depends on τ and α, while their relative amplitude relies on the absorption probabilities.
Discussion For an open quantum system, the energy transfer between a system and a reservoir can be described by the following exchange fluctuation theorem 16,17 which takes into account the energy fluctuations of both the system (∆E) and the reservoir (∆E B ), and ε S and ε B are respectively their energy scales. As usual in practical problems, the implemented TPM protocol allows us to gain information only about the system energy variation ∆E, enabling thus to extract exp (−∆E/ε) = f (ε), where the normalization factor ε effectively encloses all the unknown information on the reservoir and qubitreservoir interaction. This measurement would be complete in the case of either ∆E = −∆E B (e.g., no work applied 16 ), or ∆E B = 0 (as for closed systems). In the following we clarify what is the physical meaning of ε for a quantum two-level system in the presence of a dissipative channel. Fig. 4 shows the distribution of energy variation (P ∆E=0 , P ∆E=+2E , and P ∆E=−2E ), for a fixed initial mixed state when varying the value of α and the power of the laser pulses. This result shows that in the presence of QPMs and dissipation, the energy distribution of the quantum system is modified. The system jumps between states with different coherences in the energy basis -as sketched in Fig. 2(c), and finally reaches, for a large number of QPMs, an out-of-equilibrium steady Probability to obtain ∆E = 0, +2E, −2E, respectively, as a function of the number of laser pulses NL experienced by the qubit, with p ↑ = 1/(1 + e). Each point represents the average of several experimental points (∼ 10) between consecutive laser pulses. The plotted datasets and simulations correspond to τ = 2π/ω, and α = π/2 (green dots and line, respectively), α = π/3 (in red), α = π/4 (in blue). Circle, square and triangle symbols refer to absorption probability 18, 40, and 68 %, respectively. Error bars are due to photoluminescence errors propagation from Eq. (9).
state, independent of the initial state. The final projective energy measurement then produces, on average, a mixed state defined by the balance between the energy variation due to QPMs applied to the system and the heat exchanged through the dissipation channel. Being a two-level system, the diagonal mixed steady-state in the energy eigenbasis is formally equivalent to a thermal state with an effective temperature T ∞ , which is in general different from the reservoir temperature: where P ∞ ↑ = P ∞ ↑|↑ = P ∞ ↑|↓ is the asymptotic probability to find the spin in the |↑ state. For ideal equally-spaced QPMs (perfect absorption, and only spinpreserving transitions) P ∞ ↑ can be analytically computed by modeling the spin temporal evolution with a master equation with the Lindblad formalism, yielding with µ ≡ 1−2 (sinα sin ωτ 2 ) 2 (see Supplemental Material). Given the experimental dissipation rate Γ Q , the analytic prediction of the final effective temperature matches the numerical simulations for ideal equally-spaced QPMs. In the experiment, the stochasticity of the temporal distribution of QPMs -induced by the finite photon absorption -removes the strong dependence on τ . The analytical model is still a good approximation of the system dynamics, provided one replaces τ with an effective interpulse spacing.
In the absence of dissipation (Γ Q = 0), QPMs bring the system into an equilibrium mixed state with infinite temperature (P ∞ ↑ = 1/2). The cases α = 0, π and τ = 2π/ω, i.e. µ = 1, are exceptions 38 . The same behavior is observed also in the presence of dissipation, when QPMs and amplitude damping act in a direction orthogonal to the Hamiltonian (α = π/2). In both cases, the normalization factor ε c giving f (ε c ) = 1 coincides with the initial effective temperature of the system (see Methods): This is confirmed experimentally when we set α = π/2 (Γ Q > 0), whereby the Jarzynski equality for a closed system with no free energy variation ( e −∆E/k B T0 = 1) is demonstrated to hold, as shown in Fig. 3(b). In this configuration, dissipation is indistinguishable from QPMs during the protocol, since the state before the final energy measurement is always orthogonal to the energy measurement operator, thus on average the final state is a completely mixed state with infinite effective temperature. Not surprisingly, when the system is sensitive to the presence of a dissipation channel, f (ε c ) may significantly differ from 1, as shown in Fig. 3(d), where the positive deviation from unity indicates a heat flux from the qubit to the reservoir. A statistical increase of the system energy would lead instead to f (ε c ) < 1. The deviation of f (ε c ) from 1 increases as the Hamiltonian approaches the direction along which the QPMs occur (α = 0). For two systems initially prepared at different temperatures and placed in thermal contact for a finite lapse of time, the statistics of exchanged heat Q is known to be described by the exchange fluctuation relation e −∆β Q = 1, where Q = ∆E = −∆E B and ∆β is the difference between the initial inverse temperatures of the two bodies 16 . In our setting, the application of QPMs counteracts the heat flux, leading the quantum system to a non-equilibrium steady-state, independent of the initial (a) α = π/4 and τ /2π = 1/ω with absoprtion probability p abs = 0.6 (blue squares), p abs = 0.32 (black triangles), and p abs = 0.2 (green circles). The red diamonds correspond to α = π/4, τ /2π = 1/2ω, and p abs = 0.39. (b) α = π/3 with τ /2π = 5/6ω and p abs = 0.27 (blue squares), with τ /2π = 1/2ω and p abs = 0.27 (black triangles), and with τ /2π = 1/ω and p abs = 0.2 (green circles). The red diamonds correspond to α = π/2, τ /2π = 1/ω, and p abs = 0.2. Finally, the blue empty markers denote e −∆E/εc for each case in plots (a) and (b).
state. By analogy with the above treatment, we propose to normalize the energy difference ∆E by a factor that combines the information of the initial and final states of the quantum system in terms of their effective temperatures: 1/∆ε = (T −1 Note that ∆ε reduces to the conventional expression of Eq. (5) when the final state is completely mixed (T ∞ → ∞). In the experiment, T 0 is set by the choice of the initial state th , and T ∞ is extracted from the final energy measurement of the steady-state (see Methods). Fig. 5 remarkably shows that the experimental data and simulation always verify the relation exp (−∆E/∆ε) = 1, irrespective of the initial state and the applied protocol (relative orientation between the QPM operator and the system Hamiltonian, inter-pulse time intervals and photo-absorption probability). Quantum fluctuation relations are expected to hold in the steady state 17 , yet we observe Eq. (6) to be satisfied also in the transient regime. Note that tiny deviations are present for N L ≤ 5 when the time interval between laser pulses corresponds to a full Rabi period -which removes the stochasticity of the pulse distribution. The deviations increase with the photon-absorption probability. As a reference, we report in the plot of Fig. 5 two examples of exp (−∆E/ε c ) , which shows instead a significant deviation from 1.

Conclusions
We explored the quantum exchange fluctuation relation for an open quantum system. We investigated the interplay between quantum projective measurements and a dissipation channel, and we proposed a formulation of the energy exchange fluctuation relation that takes into account the final out-of-equilibrium steady-state in terms of the final effective temperature of the two-level system. This relation holds irrespective of the effective temperature of the initial state, and of the direction along which the quantum projective measurements are applied, and is robust against the presence of randomness in the time intervals between measurements. Our study is enabled by the use of a single NV center in diamond at room temperature -exploiting the high control on the spin degrees of freedom, under the effect of trains of short laser pulses that perform QPMs and controllably open the system, through a dissipation channel whose thermal conductivity can be tuned. This work, therefore, exploits NV centers in diamond as a quantum simulator to explore the physics of an out-of-equilibrium open quantum system, and to verify quantum fluctuation relations.
Our work paves the way for the investigation of Jarzynski-like equalities for general open quantum systems beyond the two-level approximation, and we hope that it will stimulate further research to experimentally test our findings with other physical realizations, ranging from ions 22 to superconducting devices 25,39 , and to ultracold gases 40 .

METHODS
Experimental platform and modeling. We used a single NV center hosted in an electronic grade diamond sample, with 14-N concentration < 5 ppb (Element Six). The color center is optically addressed at ambient conditions with a home-built confocal microscope and its electronic spin is manipulated via resonant microwave driving. The NV center is chosen to be free from proximal 13-C nuclear spins. The 14-N spin is polarized due to a static bias magnetic field of 394 G, combined with electronic spin pumping 34 . The long coherence time of the nuclear spin ensures that it remains unaffected during the experiment. A microwave coherently manipulate the effective two-level system, composed by the m s = 0 and m s = +1 levels of the ground state. Spin-lattice relaxation is negligible over the experiment timescales (T 1 ∼ ms).
The absorption of 532 nm laser light pulses excites the NV-center electronic spin from the ground to the excited triplet states. The decay involves (i) radiative transitions to the ground state, spin-preserving (∼ 96.5 %, see Supplemetal Material), generating a red-shifted photoluminescence with zero-phonon line at 637 nm, and (ii) non-radiative transitions through a singlet metastable state. The photodynamics of the NV center is thus well described with a seven-level model 36,41 . However, the described experiments are well-reproduced by a reduced two-level model. The neglected photodynamics occurring through the hidden physical states is re-absorbed through an effective photon absorption probability Γ W t L , an effective dissipation probability Γ W Γ Q t L and a correc-tion on the QPM measurement read-out that takes into account the non-spin conserving probability. The simulations shown in the main text were realized by using a two-level system with absorption probability in the range 18 -68 %, and 44 % conditional probability to move populations to |0 once projected in |1 . The results of the analysis with a seven-level model and its comparison with the two-level one are reported in the Supplementary Material.
Effective temperature and energy distribution. The initialization of the protocol consists in a 2 µs-long laser pulse, which sets the initial state in |0 . It is followed by an appropriate resonant microwave pulse which rotates the spin around σ y by an angle α + π (α) to move the spin in one of the two eigenstates |↑ (|↓ ). The second energy measurement is obtained by applying a second resonant microwave pulse around σ y that rotates back the spin by an angle −α, and then it is followed by optical readout.
Measuring the conditional probabilities P ↑|i for each of the two initial eigenstates i = ↑ or ↓ gives access to the full statistics of ∆E even without directly measuring the output energy for each experiment realization, which cannot be achieved due to low photon collection efficiency from the NV center. Denoting N as the number of times the experiment is repeated, we combine N ↑ measurements using as input state ↑ and N ↓ = N − N ↑ realizations for ↓ , so that p ↑ ≡ N ↑ /N (p ↓ = 1−p ↑ ). Note also that P ↓|i = 1−P ↑|i . Once defined the density matrix th T associated to a thermal state, we can determine an effective temperature k B T by exploiting the definition of the partition function Z ≡ Tr e −H/k B T . Indeed, for a two-level system we have: thus obtaining the general expression of Eq. (5), for each T associated to p ↑ (T ). In general, this picture of effective temperature does not contain the possible population inversion, and is therefore valid only for the half of the Bloch sphere containing the state at lower energy. We operatively evaluate the mean value of the exponentiated energy fluctuations of the qubit as where P ∆E is the probability to observe the energy variation ∆E. These probabilities are defined by the energy distribution as: where ∆E i,j ≡ E j − E i , p i denotes the probability of measuring E i at the beginning of the TPM protocol, P j|i the conditional probability of measuring E j at the end of the TPM protocol, and the sum i,j is performed over all the possible initial i and final j measured energies. The quantities P j|i involved in Eq. (9) are extracted from the experimental data. In the experiment, we combine the statistical energy fluctuations obtained by measuring the final energy considering ↑ as initial state (thus, E i = E ↑ ) with the ones obtained when initializing the system in In this way, we can evaluate P ∆E , and consequently the terms in the RHS of Eq. (8) for the general form of the Jarzynski relation.

Measurement of the asymptotic temperature.
To estimate the value of the asymptotic temperature we used the experimental data after a large enough number of QPMs -for which P ↑|↑ = P ↑|↓ , leading to an asymptotic thermal state (see Eq. (7)) with p ↑ (T ∞ ) = P ↑|↑ . Otherwise, for low photon-absorption probability, the asymptotic temperture can be extracted from the value p ↑ that defines an initial state for which ∆E = 0 (that is, P ↑|↑ + 1−p ↑ p ↑ P ↑|↓ = 1).

SUPPLEMENTAL MATERIAL
The effective two-level system that we consider is formed by the m S = 0 and m S = 1 sublevels of the ground spin state of a nitrogen-vacancy (NV) center in diamond. A more complete description of the photodynamics of the NV center involves a seven-level model 41 : Here, we show our characterization of the seven-level model and how we can describe the dynamics of the system with an effective two-level system for the purposes of our experimental protocol. In addition, we show an analytic derivation of the effective final temperature of the out-of-equilibrium steady-state, in the case of perfect quantum projective measurements. We also consider the case of a stochastic temporal distribution of projective measurements, as performed in our experiments, and compare the analytic solution with numerical results.

A. Characterization of the seven-level system
The electronic seven-level model is composed by the ground-state and excited-state spin triplets, and one metastable spin singlet state, as depicted in Fig. 6. In the following, the order of the levels is {g +1 , g 0 , g −1 , e +1 , e 0 , e −1 , m}, where g stands for ground, e for excited, and m for metastable. The set of parameters that completely describes the system is X = {Γ eg , Γ 1m , Γ 0m , Γ m0 , θ}, together with the excitation rate Γ abs . Γ eg is the spontaneous spin-conserving emission rate and Γ ge = p abs · Γ eg represents the rate for the electric dipole absorption and stimulated emission, with absorption probability p abs . Note that Γ abs ∼ Γ eg , the difference being given by corrections due to spin nonpreserving radiative transition with rates γ eg . Thus, tan 2 θ = γge Γge = γeg Γeg is the relative probability for a spin non-conserving radiative transition to occur. Additional non-radiative transitions involve the spin singlet state, with decay rates Γ 1m , Γ 0m from m S = +1 and 0, respectively, towards the metastable, and Γ m0 from the singlet towards the m S = 0 in the ground state. All the possible transitions are schematically represented in Fig. 6.
The global Hamiltonian describing the radiationmatter coherent interaction for the system can be represented through the matrix: The spontaneous decays are described by a Lindbladian super-operator, whereby the sum of all the decay m s =+1 m s =0 m s =-1 S=0 Excited states Ground states FIG. 6. Energy levels involved in the excitation and relaxation process during the absorption of green laser light (not to scale). Solid lines represent radiative transitions, while dotted lines represent non-radiative decays. All the solid black arrows are associated to the same decay (resp., excitation) rates Γeg (resp., Γge), while all the gray arrows are associated to the decay (resp., excitation) rates γeg (resp., γge).
For the characterization of the decay rates, we performed experiments where we measure the emitted red photo-luminescence (PL) in terms of the illumination time with green laser light. Results of the experiment are shown in Fig. 7, as well as the simulations after fitting the parameters in X, which values are reported in Tab. I. The excitation rate depends on the intensity of the green laser light, in particular for the three different experiments shown in Fig. 7, we found that 0.2Γ ge ≤ Γ abs ≤ 0.45Γ ge .

B. Comparison between models
Once evaluated the quantities responsible of the dynamics of the system, we discuss how they are included in a two-level picture composed by the m S = 0, +1 energy levels of the ground states. As introduced in the main text, a combined event of spin-conserving absorption and radiative decay corresponds to a quantum projective measurement (QPM) on the σ z basis, characterized by collapse of the wavefunction and coherence cancellation. The absorption rate is taken into account in the two-level model by considering a finite probability of having a QPM each time we apply a laser pulse. On the other hand, the non-radiative decay branch of the seven-level model is included in the two-level model as a dissipation channel, as follows: starting with a pure m S = +1 state, the probability of the measurement outcome to be |0 is Γ 1m /(Γ 1m + Γ eg ) = 44 %. Note that we consider QPM and dissipation processes to be instantaneous. Instead, spin non-preserving radiative transitions are taken into account as an additional read-out error, such that when projecting in m S = 0 (+1), the probability to measure the state m S = +1 (0) is tan 2 θ 3.8 × 10 −2 . The twolevel model is able to predict the dynamics of the system by averaging a large enough amount of simulated single trajectories. In Fig. 8 we compare results obtained for the two-point measurement (TPM) protocol with quantum projective measurements (QPM) both by considering the two-level and the seven-level models. In particular, in the two-level model an effective (higher) absorption probability is used to include the dynamics of the neglected levels and recover the same results (within the experimental error) to those of the seven-level model, as shown in Fig. 8. The protocol performed in the experimental runs is based on applying QPMs alternated with evolutions under the Hamiltonian H = ω(cos ασ z − sin ασ x ). The quantum system evolves under the composition of unitary dynamics and measurement processes, which on average can be modelled by Lindbladian jump operators. As we measure σ z , the two measurement projectors are |0 0| and |1 1| and thus the jump operators are just equal to √ Γ |0 0| and √ Γ |1 1| acting for the time t L . With this formalism, ideal projective measurements are obtained in the limit Γ t L → ∞, giving the superoperator with I 4 denoting the identity matrix in the 4 × 4 space.
We have here assumed to adopt super-operators that act not on the density operator ρ of the two-level system under investigation but on its vectorization. This choice justifies the need to define the measurement projector M (indeed, M 2 = M ) in an Hilbert space with dimension 4. In this mathematical space (identified by the (·) above the symbols), the unitary evolution of the system is thus equal to U ≡ e −i Ht ( set to 1), with where (·) * denotes complex conjugation. As a result, after have prepared the system in a thermal state at temperature β, the super-operator governing its evolution is given by the application for n times, with n number of measurements, of the composition U M , allowing us to introduce the super-operator S ≡ ( U M ) n . After the evolution, we measure the conditional probability for the NV-center to be in the energy eigenstate |↑ by starting from |↑ , where |· denotes the vectorization of (·), and the energy eigenstates |↑ and |↓ can be written as From P ↑ i , since we assume a two-level system, we can compute the full statistics of energy variation ∆E, as well as the QJE e −∆E/εc . Here, it is worth noting that, being M a projector, S = M ( U M ) n = ( M U M ) n . Therefore, by introducing for calculation purposes the quantities µ ≡ 1 − 2 sin 2 (α) sin 2 ( ωτ 2 ) and N ≡ 1 2 (σ x ⊗ σ x − σ y ⊗ σ y ), we can derive an analytical expression for S, i.e. We now introduce the additional dissipative channel associated to each absorbed laser pulse. The consequent decay in the |0 state can be modeled on average by the Lindbladian jump operator √ Γ |0 1| (within the space of the two-level system) acting during the same laser time. This gives the super-operatorQ (defined in the Hilbert space with dimension 4) allowing us to introduce a different measurement super-operator, i.e. M Q ≡ Q M = M Q, able to model the measurement process with dissipation. Therefore, the expression of the super-operator for the global dynamical evolution of the NV-center is equal to where in the r.h.s. of Eq. (18) we have not considered the last unitary evolution, since at the end of the protocol an energy measurement is applied. Moreover, by denoting with i and f the indices for the states of the system at the initial and final energy measurement of the protocol, the conditional probability P f |i can be written as where and z f , z i correspond to + or − for |f and |i equal to |↑ or |↓ . In conclusion, in the limit of n → ∞, we recover Eq. (4) of the main text.

D. Stochastic limit
The finite absorption probability can be accounted in the analytical solution by considering a mean timeinterval of free evolution between effective QPMs, instead of the fixed one used in Sec C. The probability of having k effective absorptions for n laser pulses, each with an absorption probability p abs , is obtained from a binomial distribution. From this information we can also extract the probability distribution f (n, p abs ), that provides the probabilities of having consecutive time intervals τ between laser absorptions. We then obtain the effective value of µ as: where µ = 1−2 sin 2 (α) sin 2 ( τ ω 2 ) and n =1 f (n, p abs ) = 1. In Fig. 9 we present the comparison between the numerical simulation and the analytical solution using both µ ef f (p abs ) (finite absorption probability) and µ (perfect absorption). Let us observe that this effective µ correctly takes into account the finite absorption probability, however the analytical solution is still missing a description for the non-preserving spin radiative transition process, which is more important as we reduce the value of α, as shown in Fig. 9. Comparison of numerical and analytical calculation of the asymptotic probability for the spin to be in the |↑ state, as a function of the time between laser pulses τ . Respectively in blue and gray: Numerical simulation with the complete model, and without non-preserving spin transitions, i.e. θ = 0 (see Sec. B). In green and purple: Analytical solution for perfect absorption, and finite absorption (µ ef f (p abs ), see text), respectively. Time τ is scaled in terms of energy ν = ω/2π ( = 1).