Quantum thermodynamics in adiabatic open systems and its trapped-ion experimental realization

Quantum thermodynamics aims at investigating both the emergence and the limits of the laws of thermodynamics from a quantum mechanical microscopic approach. In this scenario, thermodynamic processes with no heat exchange, namely, adiabatic transformations, can be implemented through quantum evolutions in closed systems, even though the notion of a closed system is always an idealization and approximation. Here, we first theoretically discuss thermodynamic adiabatic processes in open quantum systems, which evolve non-unitarily under decoherence due to its interaction with its surrounding environment. From a general approach for adiabatic non-unitary evolution, we establish heat and work in terms of the underlying Liouville superoperator governing the quantum dynamics. As a consequence, we derive the conditions that an adiabatic open-system quantum dynamics implies in the absence of heat exchange, providing a connection between quantum and thermal adiabaticity. Moreover, we determine families of decohering systems exhibiting the same maximal heat exchange, which imply in classes of thermodynamic adiabaticity in open systems. We then approach the problem experimentally using a hyperfine energy-level quantum bit of an Ytterbium $^{171}$Yb$^+$ trapped ion, which provides a work substance for thermodynamic processes, allowing for the analysis of heat and internal energy throughout a controllable engineered dynamics.

Quantum thermodynamics aims at investigating both the emergence and the limits of the laws of thermodynamics from a quantum mechanical microscopic approach. In this scenario, thermodynamic processes with no heat exchange, namely, adiabatic transformations, can be implemented through quantum evolutions in closed systems, even though the notion of a closed system is always an idealization and approximation. Here, we first theoretically discuss thermodynamic adiabatic processes in open quantum systems, which evolve non-unitarily under decoherence due to its interaction with its surrounding environment. From a general approach for adiabatic non-unitary evolution, we establish heat and work in terms of the underlying Liouville superoperator governing the quantum dynamics. As a consequence, we derive the conditions that an adiabatic open-system quantum dynamics implies in the absence of heat exchange, providing a connection between quantum and thermal adiabaticity. Moreover, we determine families of decohering systems exhibiting the same maximal heat exchange, which imply in classes of thermodynamic adiabaticity in open systems. We then approach the problem experimentally using a hyperfine energy-level quantum bit of an Ytterbium 171 Yb + trapped ion, which provides a work substance for thermodynamic processes, allowing for the analysis of heat and internal energy throughout a controllable engineered dynamics.
The notion of adiabaticity is a fundamental concept in a number of different areas in physics, including quantum information processing [1][2][3][4] and quantum thermodynamics [5][6][7] . In the context of closed quantum systems, adiabaticity is understood as the phenomenon in which the Hilbert space of the system can be (quasi-)perfectly decomposed into decoupled Schrodinger-eigenspaces, composed by the eigenvectors of the Hamiltonian with distinct non-crossing instantaneous energies [8][9][10] . Then, by initially preparing a quantum system in an energy eigenstate, the system undergoes a decoupled evolution to the corresponding energy eigenstate at later times. However, the concept of a closed system is always an idealization and approximation. Indeed, real quantum systems are always coupled to a surrounding environment. In open quantum systems described by convolutionless master equations, the definition of adiabaticity can be naturally extended to the decomposition of the Hilbert-Schmidt space into Lindblad-Jordan eigenspaces associated with distinct eigenvalues of the generator of the dynamics 11-14 . In thermodynamics, adiabaticity is associated to a process with no heat exchange between the system and its reservoir. In general, it is not possible to associate an observable for the thermodynamic definition of heat and of work 15 . Then, the starting point widely used to define such physical quantities in quantum systems is from the definition of internal energy given as U(t) = H(t) 5,16 . From this definition, we obtain the work (δW) and exchanged heat (δQ) between the reservoir and system as δW = Tr{ρ(t)Ḣ(t)}dt and δQ = Tr{ρ(t)H(t)}dt , respectively. These quantities are well-defined when at least one of them vanishes, thus the non-vanishing quantity can be identified with the internal energy variation ∆U(t) during the entire process. For example, for a unitary transformation associated with a closed quantum system, we necessarily have δQ cl = 0, so that any variation ∆U(t) is due some work performed on/by the system 5,17 . Eq. (1) can be directly employed to analyze quantum thermodynamical cycles, as an efficient way of assuring that no heat is exchanged in intermediate steps [18][19][20] or to minimize quantum friction in a nonequilibrium setup [21][22][23] .
Here, we theoretically and experimentally discuss thermodynamical adiabatic processes in real (open) quantum systems evolving under decoherence. To this end, we address the problem from a general approach for adiabatic dynamics in decohering systems. In contrast with closed systems, some amount of heat may be exchanged in the case of non-unitary evolution. In particular, we will establish a sufficient condition to ensure that an adiabatic open-system dynamics (associated with Lindblad-Jordan decoupled eigenspaces) leads to an adiabatic thermodynamical process (associated with no heat exchange). Moreover, for thermodynamically non-adiabatic processes, we discuss how to minimize the entropy production as a function of their total evolution time. Our results are then experimentally implemented by using a hyperfine energylevel quantum bit (qubit) of an Ytterbium 171 Yb + trapped ion, where reservoir engineering is performed to achieve a controllable adiabatic dynamics. Due to requirements of the usual definitions of heat and work, the investigation of thermodynamic quantities in adiabatic dynamics is achieved with timedependent decoherence effects. To this end, we introduce an efficient control to a Gaussian noise with time-dependent amplitude, which is then used to simulate a dephasing channel with a time-dependent decoherece rate γ(t).

arXiv:1902.01145v1 [quant-ph] 4 Feb 2019
We start by introducing heat and work in a general formalism for adiabaticity in open quantum system, namely, the superoperator formalism 11 . In this work, we will consider a discrete quantum system S defined over a d-dimensional Hilbert space. The system S interacts with its surrounding environment A. The dynamics is assumed to be described by a timelocal master equationρ(t) = L t [ρ(t)], where ρ(t) is the density operator associated with S and L t [•] is a time-dependent Liouville operator. The Liouville operator takes the form is the unitary part of the dynamics and R t [•] describes the decohering effects of A over S.
In the superoperator formalism, the open-system dynamics can be provided from a Schrödinger-like equation |ρ(t) = L(t)|ρ(t) , where the density operator |ρ(t) is represented by a D 2 -dimensional vector (hence the double ket notation), whose components k (t) can be suitably expanded in terms of tensor products of the Pauli basis {1, σ 1 , σ 2 , σ 3 } 11 . For instance, for the case of a single qubit (D = 2), we have with the components h k (t) of h(t)| defined by h k (t) = Tr{H(t)σ k }. In this notation, the inner product of vectors |u and |v associated with operators u and v, respectively, is defined as u|v = (1/D)Tr(u † v). Since we are analyzing farfrom-equilibrium thermodynamical processes, we also consider the entropy production during the dynamics. The entropy production Σ during a non-equilibrium process is defined as Σ = Tr{ρ(t)[log ρ(t) − log ρ th ]}, where ρ th describes the equilibrium state at inverse temperature β th 22 . In the superoperator formalism it is possible to show that (see Supplemental Material 24 ) with the components of ρ log (t)| and ρ th log | given by log n (t) = Tr{σ n log ρ(t)} and log,th n = Tr{σ n log ρ th }, respectively. For a general process, Eq. (3) may be hard to be computed. However, as it will be shown, it can be analytically derived for a general adiabatic quantum dynamics.
Because L(t) is non-Hermitian, it cannot always be diagonalized. Then, the definition of adiabaticity in this scenario is subtler than in the case of closed systems. For open systems, the adiabatic dynamics can be defined in terms of the Jordan decomposition of L(t) 11 . More specifically, adiabaticity is associated with a completely positive trace-preserving dynamics that can be decomposed into decoupled Lindblad-Jordan eigenspaces associated with distinct non-crossing instantaneous eigenvalues λ i (t) of L(t). We notice here that some care is required in order to find a basis for describing the density operator. The standard technique is to start from the instantaneous right and left eigenstates of L(t), completing these eigensets in order to compose right | are the k i -th right and left vector, respectively, associated with the eigenspace with eigenvalue λ i (t) in the Jordan decomposition of L(t). These Jordan-preserving left and right bases can always be built such that they satisfy a bi-orthonormal relation- Assuming an open-system adiabatic dynamics, we can analytically derive work, heat, and entropy production. Indeed, by taking the initial density operator as , we obtain that work, heat, and entropy production are provided by where , with ρ ad log (t)| standing for the adiabatic evolved state associated with ρ log (t)| and δW ad (δQ ad ) being identified to the amount of work (heat) performed on/by the system when δQ ad = 0 (δW ad = 0).
The validity of Eqs. (4), (5), and (6) is shown in the Supplemental Material 24 . They provide general expressions for work, heat, and entropy production during an open-system adiabatic dynamics. In particular, as a consequence we can obtain a sufficient condition for avoiding heat exchange in a quantum mechanical adiabatic evolution. More specifically, if the initial state ρ(0) of the system can be written as a superposition of the eigenstate set {|D (k i ) i (0) } with eigenvalue λ k (t) = 0, for every t ∈ [0, τ], the adiabatic dynamics implies in no heat exchange. Therefore, we can establish that an adiabatic dynamics in quantum mechanics is not in general associated with an adiabatic process in quantum thermodynamics, with a sufficient condition for thermal adiabaticity being the evolution within an eigenstate set with vanishing eigenvalue of L(t).
Heat exchange for a qubit adiabatic dynamics. As an illustration, let us begin by considering a two-level system initialized in a thermal equilibrium state ρ th (0) for the Hamiltonian H(0) at inverse temperature β = 1/k B T , where k B and T are the Boltzmann's constant and the absolute temperature, respectively. Let the system be governed by a Lindblad equation, where the environment acts as a dephasing channel in the energy eigenstate basis {|E n (t) } of H(t). Thus, we describe the coupling between the system and its reservoir through R dp In this case, the set of eigenvectors of L(t) can be obtained from set of opera- In the superoperator formalism, the initial state ρ th (0) is written as |ρ th (0) = Z −1 (0) n e −βE n (0) |D nn (0) , where Z(t) = Tr{e −βH(t) } is the partition function of the system. Therefore, since |ρ th (0) is given by a superposition of eigenvectors of L(t) with eigenvalue λ nn (t) = 0, we obtain from Eq. (5) that δQ ad = 0. Therefore, thermal adiabaticity is achieved for an arbitrary open-system adiabatic dynamics subject to dephasing in the energy eigenbasis. Hence, any internal energy variation for this situation should be identified as work.
In contrast, we can use a similar qubit system to find a process in which heat can be exchanged, i.e., δQ ad 0. To this end, let us consider dephasing in the computational basis, with the coupling between the system and its reservoir through In order to guarantee that any internal energy variation is associated to heat exchange, we consider a constant Hamiltonian during the entire nonunitary evolution (so that δW ad = 0). Since R z t [•] must not be written in the eigenbasis of the Hamiltonian, we assume a Hamiltonian H x = ωσ x , where the system is initialized in the typical initial state of a thermal machine, namely, the thermal state of the Hamiltonian H x at inverse temperature β. At this stage, the system is unitarily driven by the time-dependent Hamiltonian H(t), which varies from H x toH x = ωσ x , with ω ω, so that no heat is exchanged. After this stage we have the contact with the reservoir, which is then governed byH x . By letting the system undergo a non-unitary adiabatic dynamics, the evolved state is 24 so that we can compute the exchanged heat during an infinitesimal time interval dt as 24 The negative argument in the exponential of Eq. (8) shows that the higher the mean-value of γ(t) the faster the heat exchange ends. In fact, we can use the well-known mean-value theorem for real functions to write e −2 t 0 γ(ξ)dξ = e −2γ∆t , wherē γ = (1/∆t) t 0 γ(ξ)dξ is the mean-value of γ(t) within the interval ∆t. Thus, if we define the amount of exchanged heat during the entire evolution as ∆Q(τ dec ) = τ dec 0 δQ ad (t), where τ dec is the total evolution time of the nonunitary dynamics, we get Notice that ∆Q(τ dec ) > 0 for any value of the average dephasing rateγ. Therefore, the dephasing channel considered here works like an artificial thermal reservoir at inverse temperature β deph < β. We can further compute the maximum exchanged heat from Eq. (9) as a quantity independent on the environment parameters and given by ∆Q max = ω tanh[β ω]. It would be worth to highlight that, for quantum thermal machines weakly coupled to thermal reservoirs at different temperatures 16 , the maximum heat ∆Q max is obtained with high-temperature hot reservoirs 18,25,26 .
The Hamiltonian H is taken as a constant operator so that no work is realized by/on the system. Assume that the heat exchange between S and its reservoir during the quantum evolution is given by ∆Q. Then, any unitarily related adiabatic dynamics driven A proof of Theorem 1 can be found in the Supplemental Material 24 . It guarantees that there is an infinite class of models with a maximum heat exchanged given by ∆Q max , providing a procedure to inversely engineer environments to extract ∆Q max . As an example of application of Theorem 1, let us consider a system-reservoir interaction governed by . We can then use Theorem 1 to show that the results previously obtained for dephasing can be reproduced if the quantum system is initially prepared in thermal state of H 0 y = ωσ y . Such result is clear if we choose U = R x (π/2)R z (π/2). Then, it follows that where R z(x) (θ) are rotation matrices with angle θ around z(x)-axes for the case of a single qubit. Thus, Theorem 1 assures that the maximum exchanged heat will be ∆Q max = ω tanh[β ω].
Concerning the entropy production rate, it can be obtained from Eq. (6), yielding with such that, in limitγ∆t → ∞, we get Σ → 0. Therefore Σ achieves its steady value Σ s . It is important to highlight that the sign of Σ can change during the adiabatic evolution. In fact, if the initial thermal state is such that ω/ω > T/T th , the entropy production starts exhibiting a decreasing behavior. At an instant t > t 0 , such that 2γ∆t = log[tanh β ω/ tanh β th ω], the entropy production increases, achieving the value Σ s . Notice that it is possible to find a vanishing entropy production by suitably adjusting the evolution time interval ∆t.
Experimental realization. We now discuss an experimental realization to test the thermodynamics of adiabatic processes in an open-system evolution. This is implemented using the hyperfine energy levels of an Ytterbium ion 171 Yb + confined by a six-needles Paul trap, with a qubit encoded into the 2 S 1/2 ground state, |0 ≡ | 2 S 1/2 ; F = 0, m F = 0 and |1 ≡ | 2 S 1/2 ; F = 1, m F = 0 , as shown in Fig. (1a) 28 . The target HamiltonianH x can be realized using a resonant microwave with Rabi frequency adjusted toω. To this end, the channel 1 (CH1) waveform of a programmable two-channel arbitrary waveform generator (AWG) is used, which has been programmed to the angular frequency 2π × 200 MHz. As depicted in Fig. (1b), to implement the dephasing channel we use the Gaussian noise frequency modulation (FM) microwave technique, which has been developed in a recent previous work and shows high controllability 29 . Since we need to implement a time-dependent decohering quantum channel, we use the channel 2 (CH2) waveform as amplitude modulation (AM) source to achieve high control of the Gaussian noise amplitude, consequently, to optimally control of the dephasing rate γ(t). See the Supplemental Material for a detailed description of the experimental setup, including the implementation of the quantum channel and the quantum process tomography 24 .
To begin with, we need to guarantee that the dynamics of the system is really adiabatic 11 . Then, we compute the fidelity F (τ dec ) of finding the system in a path given by Eq. (7), where F (t) = Tr{[ρ 1/2 exp (t)ρ ad (t)ρ 1/2 exp (t)] 1/2 }, with ρ ad (t) the density matrix provided Eq. (7) and ρ exp (t) the experimental density ma-  (9) trix obtained from quantum tomography. In Table I we show the minimum experimental fidelity F min = min τ dec F (τ dec ) for several choices of the parameter γ 0 . This result shows that the system indeed evolves as predicted by the adiabatic solution for every γ 0 and τ dec with excellent experimental agreement. As a further development, we analyze in Fig. 2a the experimental results for the heat exchange ∆Q(τ dec ) as a function of τ dec , where we have chosen ω = ω 0 ,ω = 2ω 0 and γ(t) = γ 0 (1 + t/τ dec ). The solid curves in Fig. 2a are computed from Eq. (9), while the experimental points are directly computed through the variation of internal energy as ∆Q(τ dec ) = U fin − U ini , where U fin(ini) = Tr{ρ fin(ini) H(τ)}. The computation of U fin(ini) is directly obtained from quantum state tomography of ρ fin(ini) for each value of τ dec . Although the maximum exchanged heat is independent of γ 0 , the initial dephasing rate γ 0 affects the power for which the system exchanges heat with the reservoir for a given evolution time τ dec . By defin-ingP(τ dec ) = |∆Q(τ dec )|/τ dec , we can quantify the average power for extracting/introducing the amount |Q(τ dec )| in the time interval τ dec . We then obtainP(τ dec ) = |∆Q max |η(τ dec ,γ), where η(τ dec ,γ) = (1 − e −2γτ dec )/τ dec . This result is illustrated in Fig. 2b, where we have plottedP(τ dec ) during the entire heat exchange (within the interval τ dec ) as a function of τ dec . Notice that, as in the case of ∆Q(τ dec ), the asymptotic behavior of the average power is independent of γ 0 . Moreover, as predicted by Eq. (10), we can control the entropy production along an adiabatic quantum evolution. As shown in Fig. 2c, the entropy production varies as a function of τ dec . In particular, it can be optimized to be vanishing by for a suitable time τ dec for which the system is kept in touch with its surrounding environment.

Conclusions
From a general approach for adiabaticity in open quantum systems, we provided a relationship between adiabaticity in quantum mechanics and in quantum thermodynamics. In particular, we derived a sufficient condition for which the adiabatic dynamics in open quantum systems leads to adiabatic processes in thermodynamics. By using a particular example of a single qubit undergoing an open-system adiabatic evolution path, we have illustrated the existence of both adiabatic and diabatic regimes in quantum thermodynamics, computing the associated heat fluxes in the processes. As a further result, we also proved the existence of an infinite family of decohering systems exhibiting the same maximum heat exchange. From the experimental side, we have realized adiabatic opensystem evolutions using an Ytterbium trapped ion, with its hyperfine energy level encoding a qubit (work substance). In particular, heat exchange and entropy production can be optimized along the adiabatic path as a function of the total evolution time. Our implementation exhibits high controllability, opening perspectives for analyzing thermal machines (or refrigerators) in open quantum systems under adiabatic evolutions. The associated effects of the engineered reservoirs on the thermal efficiencies are left for future research.

Acknowledgment
We thank Yuan-Yuan Zhao, Zhibo Hou, Jun-Feng Tang, and Yue-xin Huang for valuable discussion. This work was sup-

Supplemental material for: Quantum thermodynamics in adiabatic open systems and its trapped-ion experimental realization
Chang-Kang Hu, 1,2 , * Alan C. Santos, 3

Heat and work in superoperator formalism
Let us consider the heat exchange as where we have used the equationρ(t) = L[ρ(t)]. To derive the corresponding expression in the superoperator formalism we first define the basis of operators given by In this basis, we can write ρ(t) and H(t) generically as where we have h n (t) = Tr{H(t)σ n } and n (t) = Tr{ρ(t)σ † n }. Then, we get Now, we use the definition of the matrix elements of the superoperator L(t), associated with L[•], which reads L mn = (1/D)Tr{σ † m L[σ n ]}, so that we write In conclusion, by defining the vector elements we can rewrite Eq. (A4), yielding Equivalently, where we have used Eq. (A2) to writeḢ(t) = (1/D) D 2 −1 n=0ḣ n (t)σ † n and, consequently, so that we use the definition of the coefficients n (t) to get By using Eqs. (A5) and (A6) into Eq. (A10), we conclude that

Entropy production in superoperator formalism
Our starting point is the definition where ρ th is a thermal reference state. Now, let us study the dynamics of Σ(t) by taking its time derivative Then, we find By using that Tr{ρ(t)} = 1, we get Tr{ρ(t)} = 0. Therefore or equivalently (by usingρ(t) = L t [ρ(t)]) Now, let us to write so that we can define the vectors ρ log (t)| and ρ th log | associated to log ρ(t) and ρ th , where their components log n (t) and log,th n are obtained as log n (t) = Tr{σ n log ρ(t)} and log,th n = Tr{σ n log ρ th }. Thus, we get so that we get the equation for Σ (t) in superoperator formalism as 3. Validity of Eqs. (4), (5), and (6) Let be the initial state of the system associated with the initial matrix density ρ(0). Under adiabatic evolution, the state at a later time t will be given by i (0) } denote the instantaneous Jordan-preserving left and right bases of L(t), respectively 11 . Therefore, from Eq. (A11), we can write the work δW op for an adiabatic dynamics as On the other hand, when no work is realized, we can obtain the heat dQ op for an adiabatic dynamics as so that dQ ad represents the exchanged heat if no work is performed during such dynamics. Morevoer, from Eq. (A20), we can write the entropy production variation as so that we conclude that , with ρ ad log (t)| standing for the adiabatic evolved state associated with ρ log (t)|.
The Hamiltonian H is taken as a constant operator so that no work is realized by/on the system. Assume that the heat exchange between S and its reservoir during the quantum evolution is given by ∆Q. Then, any unitarily related adiabatic dynamics driven Proof. Let us consider that ρ(t) is solution oḟ so, by multiplying both sides of the above equation by U (on the left-hand-side) and U † (on the right-hand-side), we get where Γ (t) = UΓ n (t)U † . In conclusion, we get that ρ (t) = Uρ(t)U † is a solution oḟ where Now, by taking into account that the Hamiltonian H is a constant operator, we have that no work is realized by/on the system. Then, by computing the amount of heat extracted from the system in the prime dynamics during an interval t ∈ [0, τ], we obtain where, by definition, we can use ρ (t) = Uρ(t)U † , ∀t ∈ [0, τ]. Hence where we have used the cyclical property of the trace and that ∆Q = Tr{Hρ(τ)} − Tr{Hρ(0)}. 5. Proof of Eqs. (7)- (10) Consider the Hamiltonian H x = ωσ x , where the system is initialized in the thermal of H x at inverse temperature β. In this case, the initial state can be written as If we rewrite the above state in superoperator formalism as the state |ρ x (0) , we can compute the components ρ x where we define the basis |k = [δ k1 δ kx δ ky δ kz ] t . Moreover, it is possible to show that the set {|1 , |x } satisfies the eigenvalue equation for L(t) as It can be shown that this eigenstates are nondegenerate. Therefore, if the dynamics is adiabatic, we can write the evolved state as |ρ x (t) = c 1 (t)|1 + c x (t)|x , where c y (t) = c y (0) = 0 and c z (t) = c z (0) = 0 because the coefficients evolve independently form each other. Thus, from the adiabatic solution in open quantum system given in Eq. (A21), we obtain c 1 (t) = 1 and c x (0) = − tanh[β ω], so that we can useλ 1 = 0 andλ x = −2γ(t) to obtain Without driving noise (noise amplitude is zero), the dephasing rate of the qubit is fitted as 3.03 Hz, which is caused by the magnetic fluctuation in the laboratory.
In order to certify that the decoherence channel is indeed a σ z channel (dephasing channel) in our experiment, we employed quantum process tomography. A general quantum evolution can be typically described by the operator-sum representation associated to a trace-preserving map ε. For an arbi-trary input state ρ, the output state ε(ρ) can be written as 2 ε(ρ) = m,n χ mn A m ρA † n , where A m are basis elements (usually a fixed reference basis) that span the state space associated with ρ and χ mn is the matrix element of the so-called process matrix χ, which can be measured by quantum state tomography. In a single qubit system, we take A 0 = I, A 1 = σ x , A 2 = σ y , A 3 = σ z . The quantum process tomography is carried out for the quantum process described by the Lindblad equation given by Eq. (B6), where H(t) = ωσ x , with ω = 5.0 × 2π KHz and γ = 2.5 KHz. We fixed the total evolution time as 0.24 ms (here, the noise amplitude is 1.62 V and the modulation depth is 96.00 KHz). The resulting estimated process matrix is shown in Fig. 5. We can calculate the fidelity between the experimental process matrix χ exp and the theoretical process matrix χ id F (χ exp , χ id ) = Tr √ χ exp χ id √ χ exp 2 (B8) We measured several process with different evolution times. For example, when the amplitude of the noise is set to 1.54V, the process fidelities are measured as F t 1 = 99.27%, F t 2 = 99.50%, F t 3 = 99.72%, F t 4 = 99.86% and F t 5 = 99.87%, at times t 1 = 0.08 ms, t 2 = 0.16 ms, t 3 = 0.24 ms, t 4 = 0.32 ms and t 5 = 0.40 ms, respectively. Thus, the dephasing channel can be precisely controlled as desired and it can support the scheme to implement the time-dependent dephsing in experiment.
The function η(t) depends on an amplitude parameter A, which is used to control γ(t). As shown in Fig. 6, we experimentally measured the relation between A and γ(t) for a situation where γ(t) is a time-independent value γ 0 . As result, we find a linear relation between √ γ 0 and A, which reads √ γ 0 = 29.81A + 1.74 .
For the case A = 0, we get the natural dephasing rate γ nd = 1.74 2 Hz of the physical system. Thus, we can see that, if we change the parameter A, which we can do with high controllability, the quantity √ γ 0 can be efficiently controlled. On the other hand, if we need a time-dependent rate γ(t), we just need to consider a way to vary A as a function A(t). To this end, we use a second channel (CH2) of the AWG to perform amplitude modulation (AM) of the Gaussian noise. The temporal dependence of A(t) is achieved by programming the channel (CH2) to change during the evolution time.