Abstract
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 begin by theoretically discussing thermodynamic adiabatic processes in open quantum systems, which evolve nonunitarily under decoherence due to its interaction with its surrounding environment. From a general approach for adiabatic nonunitary 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 opensystem 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 energylevel 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.
Introduction
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 Schrodingereigenspaces, composed by the eigenvectors of the Hamiltonian with distinct noncrossing 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 timelocal master equations, the definition of adiabaticity can be naturally extended to the decomposition of the HilbertSchmidt space into LindbladJordan eigenspaces associated with distinct eigenvalues of the generator of the dynamics^{11,12,13,14,15,16,17}.
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^{18}. 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)〉^{5,19}. From this definition, we obtain the work (dW) and exchanged heat (dQ) between the reservoir and system as
respectively. As originally introduced in Ref. ^{19}, these quantities are defined in the weak coupling limit between system and reservoir (see also Refs. ^{20,21} for recent attempts to examine strongly coupled quantum systems and Refs. ^{22,23} for separation of internal energy variation in terms of entropy changes). Notice also that dW and dQ are exact differential forms when at least one of them vanishes, thus the nonvanishing quantity can be identified with the internal energy variation ΔU(t) during the entire process. For example, for a unitary transformation associated with a closed quantum system, we necessarily have dQ_{closed} = 0, so that any variation ΔU(t) is due some work performed on/by the system^{5,24}. 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^{25,26,27} or to minimize quantum friction in a nonequilibrium setup^{28,29,30}.
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, heat may be exchanged in the case of nonunitary evolution. In particular, we will establish a sufficient condition to ensure that an adiabatic opensystem dynamics (associated with LindbladJordan decoupled eigenspaces) leads to an adiabatic thermodynamical process (associated with no heat exchange). Moreover, for thermodynamically nonadiabatic processes, we evaluate the von Neumann entropy, discussing its relation with heat for arbitrary 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 timedependent amplitude, which is then used to simulate a dephasing channel with a timedependent decoherece rate γ(t).
Results
Work and heat in the adiabatic dynamics of open systems
We start by introducing heat and work in a general formalism for adiabaticity in open quantum systems, namely, the superoperator formalism^{11}. In this work, we will consider a discrete quantum systems \({\mathcal{S}}\) defined over a ddimensional Hilbert space. The system \({\mathcal{S}}\) interacts with its surrounding environment \({\mathcal{A}}\). The dynamics is assumed to be described by a timelocal master equation \(\dot{\rho }(t)={{\mathcal{L}}}_{t}[\rho (t)]\), where ρ(t) is the density operator associated with \({\mathcal{S}}\) and \({{\mathcal{L}}}_{t}[\bullet ]\) is a timedependent Liouville operator. The Liouville operator takes the form \({{\mathcal{L}}}_{t}[\rho (t)]={{\mathcal{H}}}_{t}[\rho (t)]+{{\mathcal{R}}}_{t}[\rho (t)]\), where \({{\mathcal{H}}}_{t}[\bullet ]=(1/i\hslash )[H(t),\bullet ]\) is the unitary part of the dynamics and \({{\mathcal{R}}}_{t}[\bullet ]\) describes the decohering effects of \({\mathcal{A}}\) over \({\mathcal{S}}\).
In the superoperator formalism, the opensystem dynamics can be provided from a Schrödingerlike equation \(\left.\left\dot{\rho }(t)\right\rangle \right\rangle ={\mathbb{L}}(t)\left.\left\rho (t)\right\rangle \right\rangle\), where \({\mathbb{L}}(t)\) is termed the Lindblad superoperator and the density operator \(\left.\left\rho (t)\right\rangle \right\rangle\) is represented by a D^{2}dimensional vector (hence the double ket notation), whose components ϱ_{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 \(\rho (t)=\frac{1}{2}\mathop{\sum }\nolimits_{k = 0}^{3}{\varrho }_{k}(t){\sigma }_{k}\) and \({\varrho }_{k}(t)={\rm{Tr}}\{\rho (t){\sigma }_{k}\}\), with σ_{k} denoting an element of the Pauli basis. Moreover, \({\mathbb{L}}(t)={\mathbb{H}}(t)+{\mathbb{R}}(t)\), where \({\mathbb{H}}(t)\) and \({\mathbb{R}}(t)\) are (D^{2} × D^{2})dimensional supermatrices, whose elements are \({{\mathbb{H}}}_{ki}(t)=(1/D){\rm{Tr}}\{{\sigma }_{k}^{\dagger }{{\mathcal{H}}}_{t}[{\sigma }_{i}]\}\) and \({{\mathbb{R}}}_{ki}(t)=(1/D){\rm{Tr}}\{{\sigma }_{k}^{\dagger }{\mathcal{R}}[{\sigma }_{i}]\}\), respectively. The thermodynamic quantities defined in Eq. (1) are then rewritten as (see Methods section)
with the components h_{k}(t) of \(\left\langle \left\langle h(t)\right\right.\) defined by \({h}_{k}(t)={\rm{Tr}}\{H(t){\sigma }_{k}\}\). In this notation, the inner product of vectors \(\left.\leftu\right\rangle \right\rangle\) and \(\left.\leftv\right\rangle \right\rangle\) associated with operators u and v, respectively, is defined as 〈〈u∣v〉〉 = (1/D)Tr(u^{†}v).
Because \({\mathbb{L}}(t)\) is nonHermitian, 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 \({\mathbb{L}}(t)\)^{11}. More specifically, adiabaticity is associated with a completely positive tracepreserving dynamics that can be decomposed into decoupled LindbladJordan eigenspaces associated with distinct noncrossing instantaneous eigenvalues λ_{i}(t) of \({\mathbb{L}}(t)\). 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 \({\mathbb{L}}(t)\), completing these eigensets in order to compose right \(\{{{\mathcal{D}}}_{i}^{({k}_{i})}(t)\rangle \rangle \}\) and left \(\{\langle \langle {{\mathcal{E}}}_{i}^{({k}_{i})}(t) \}\) vector bases, where \({{\mathcal{D}}}_{i}^{({k}_{i})}(t)\rangle \rangle\) and \(\langle \langle {{\mathcal{E}}}_{i}^{({k}_{i})}(t)\) are the k_{i}th right and left vectors, respectively, associated with the eigenspace with eigenvalue λ_{i}(t) in the Jordan decomposition of \({\mathbb{L}}(t)\). These Jordan left and right bases can always be built such that they satisfy a biorthonormal relationship \(\langle \langle {{\mathcal{E}}}_{i}^{(\alpha )}(t) {{\mathcal{D}}}_{j}^{(\beta )}(t)\rangle \rangle ={\delta }_{ij}{\delta }^{\alpha \beta }\). Assuming an opensystem adiabatic dynamics, we can analytically derive work, heat, and entropy variation. Indeed, by taking the initial density operator as \(\rho (0)\rangle \rangle ={\sum }_{i,{k}_{i}}{c}_{i}^{({k}_{i})}{{\mathcal{D}}}_{i}^{({k}_{i})}(0)\rangle \rangle\), we obtain that work and heat are provided by
with dW^{ad} (dQ^{ad}) being identified to the amount of work (heat) performed on/by the system.
The validity of Eqs. (3) and (4) is shown in the Methods section. As long as we are in the weak coupling regime and the system is driven by a timelocal master equation, Eqs. (3) and (4) provide expressions for work and heat for the adiabatic decohering dynamics. Notice also that the adiabatic dynamics will require a slowly varying Liouville superoperator \({\mathbb{L}}(t)\)^{11}. Starting from Eq. (2), we are allowed to evaluate the density operator \(\left.\left\rho (t)\right\rangle \right\rangle\) through an arbitrary strategy. For instance, we could apply a piecewise deterministic process approach via FeynmanVernon path integral for the corresponding propagator^{31}. Alternatively, we could implement a numerical simulation via a Monte Carlo wave function method (see, e.g., Ref. ^{32} and references therein). In all these cases, from Eqs. (3) and (4), 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 \(\{{{\mathcal{D}}}_{i}^{({k}_{i})}(0)\rangle \rangle \}\) with eigenvalue λ_{i}(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 \({\mathbb{L}}(t)\). This condition is satisfied by a quantum system that adiabatically evolves under a steady state trajectory, since such dynamics can be described by an eigenstate (or a superposition of eigenstates) of \({\mathbb{L}}(t)\) with eigenvalue zero^{14}. As an example, Ref. ^{33} has considered the adiabatic evolution of 2D topological insulators, where the system evolves through its steady state trajectory. For this system, the evolved state \(\left.\left{\rho }_{{\rm{s}}s}(t)\right\rangle \right\rangle\), associated with the steady state of the system ρ_{ss}(t), satisfies \({\mathbb{L}}(t)\left.\left{\rho }_{{\rm{s}}s}(t)\right\rangle \right\rangle =0\), ∀ t. This means that \(\left.\left{\rho }_{{\rm{s}}s}(t)\right\rangle \right\rangle\) is an instantaneous eigenstate of \({\mathbb{L}}(t)\) with eigenvalue λ(t) = 0.
Thermal adiabaticity for a qubit adiabatic dynamics
As a further illustration, let us consider a twolevel 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 \(\{\left{E}_{n}(t)\right\rangle \}\) of H(t). Thus, we describe the coupling between the system and its reservoir through \({{\mathcal{R}}}_{t}^{{\rm{dp}}}[\bullet ]=\gamma (t)[{\Gamma }^{{\rm{dp}}}(t)\bullet {\Gamma }^{{\rm{dp}}}(t)\bullet ]\), where \({\Gamma }^{{\rm{dp}}}(t)=\left{E}_{0}(t)\right\rangle \left\langle {E}_{0}(t)\right\left{E}_{1}(t)\right\rangle \left\langle {E}_{1}(t)\right\). In this case, the set of eigenvectors of \({\mathbb{L}}(t)\) can be obtained from set of operators \({P}_{nm}(t)=\left{E}_{n}(t)\right\rangle \left\langle {E}_{m}(t)\right\), where the components \({{\mathcal{D}}}_{nm}^{(i)}(t)\) of \(\left.\left{{\mathcal{D}}}_{nm}(t)\right\rangle \right\rangle\) are given by \({{\mathcal{D}}}_{nm}^{(i)}(t)={\rm{Tr}}\{{P}_{nm}(t){\sigma }_{i}\}\). Moreover, the eigenvalue equation for \({\mathbb{L}}(t)\) can be written as \({\mathbb{L}}(t)\left.\left{{\mathcal{D}}}_{nm}(t)\right\rangle \right\rangle ={\lambda }_{nm}(t)\left.\left{{\mathcal{D}}}_{nm}(t)\right\rangle \right\rangle\), where λ_{nm}(t) = E_{n}(t) − E_{m}(t) − 2(1 − δ_{nm})γ(t). In the superoperator formalism, the initial state ρ_{th}(0) is written as \(\left.\left{\rho }_{{\rm{t}}h}(0)\right\rangle \right\rangle ={{\mathcal{Z}}}^{1}(0){\sum }_{n}{e}^{\beta {E}_{n}(0)}\left.\left{{\mathcal{D}}}_{nn}(0)\right\rangle \right\rangle\), where \({\mathcal{Z}}(t)={\rm{Tr}}\{{e}^{\beta H(t)}\}\) is the partition function of the system. Therefore, since \(\left.\left{\rho }_{{\rm{t}}h}(0)\right\rangle \right\rangle\) is given by a superposition of eigenvectors of \({\mathbb{L}}(t)\) with eigenvalue λ_{nn}(t) = 0, we obtain from Eq. (4) that dQ^{ad} = 0. Therefore, thermal adiabaticity is achieved for an arbitrary opensystem adiabatic dynamics subject to dephasing in the energy eigenbasis. Hence, any internal energy variation for this situation should be identified as work.
Heat exchange for a qubit adiabatic dynamics
In contrast, we can use a similar qubit system to find a process in which heat can be exchanged, i.e., dQ^{ad} ≠ 0. To this end, let us consider dephasing in the computational basis, with the coupling between the system and its reservoir through \({{\mathcal{R}}}_{t}^{{\rm{z}}}[\bullet ]=\gamma (t)\left[{\sigma }_{z}\bullet {\sigma }_{z}\bullet \right]\). 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 dW^{ad} = 0). Since \({{\mathcal{R}}}_{t}^{{\rm{z}}}[\bullet ]\) must not be written in the eigenbasis of the Hamiltonian, we assume a Hamiltonian H_{x} = ħωσ_{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 some arbitrary temperature β. By letting the system undergo a nonunitary adiabatic dynamics under dephasing, the evolved state is (see Methods section)
From Eq. (4) we then compute the amount of exchanged heat during an infinitesimal time interval dt as \(d{Q}^{{\rm{ad}}}(t)=2\hslash \tanh (\beta \hslash \omega )\omega \gamma (t){e}^{2\mathop{\int}\nolimits_{\!\!{0}}^{t}\gamma (\xi )d\xi }dt\). The negative argument in the exponential shows that the higher the meanvalue of γ(t) the faster the heat exchange ends (see Methods section). Thus, if we define the amount of exchanged heat during the entire evolution as \(\Delta Q({\tau }_{{\rm{dec}}})=\mathop{\int}\nolimits_{\!{0}}^{{\tau }_{{\rm{dec}}}}[d{Q}^{{\rm{ad}}}(t)/dt]dt\), where τ_{dec} is the total evolution time of the nonunitary dynamics, we get
where \(\bar{\gamma }=(1/{\tau }_{{\rm{dec}}})\mathop{\int}\nolimits_{\!{0}}^{{\tau }_{{\rm{dec}}}}\gamma (\xi )d\xi\) is the average dephasing rate during τ_{dec}. Notice that ΔQ(τ_{dec}) > 0 for any value of \(\bar{\gamma }\). Therefore, the dephasing channel considered here works as an artificial thermal reservoir at inverse temperature \(\tilde{\beta }={\beta }_{{\rm{deph}}}\, < \, \beta\), with \({\beta }_{{\rm{deph}}}=(1/\hslash \omega ){\rm{arctanh}}[{e}^{2\bar{\gamma }{\tau }_{{\rm{dec}}}}\tanh (\beta \hslash \omega )]\) (see Methods section). We can further compute the maximum exchanged heat from Eq. (6) as a quantity independent of the environment parameters and given by \(\Delta {Q}_{\max }=\hslash \omega \tanh (\beta \hslash \omega )\). It would be worth to highlight that, for quantum thermal machines weakly coupled to thermal reservoirs at different temperatures^{19}, the maximum heat \(\Delta {Q}_{\max }\) is obtained with hightemperature hot reservoirs^{25,34,35}.
Despite we have provided a specific opensystem adiabatic evolution, we can determine infinite classes of systemenvironment interactions exhibiting the same amount of heat exchange dQ. In particular, there are infinite engineered environments that are able to extract a maximum heat amount \(\Delta {Q}_{\max }\). A detailed proof of this result can be found in Methods section.
Experimental realization
We now discuss an experimental realization to test the thermodynamics of adiabatic processes in an opensystem evolution. This is implemented using the hyperfine energy levels of an Ytterbium ion ^{171}Yb^{+} confined by a sixneedles Paul trap, with a qubit encoded into the ^{2}S_{1/2} ground state, \(\left0\right\rangle \equiv {\left.\right}^{2}{S}_{1/2};\ F=0,{m}_{F}=0\left.\right\rangle\) and \(\left1\right\rangle \equiv \ {\left.\right}^{2}{S}_{1/2};\ F=1,{m}_{F}=0\left.\right\rangle\), as shown in Fig. 1a^{36}. The qubit initialization is obtained from the standard Rabi Oscillation sequence^{36}, where we first implement the Doppler cooling for 1 ms, after we apply a standard optical pumping process for 0.01 ms to initialize the qubit into the \(\left0\right\rangle\) state, and then we use microwave to implement the desired dynamics. The target Hamiltonian H_{x} can be realized using a resonant microwave with Rabi frequency adjusted to ω. To this end, the channel 1 (CH1) waveform of a programmable twochannel arbitrary waveform generator (AWG) is used, which has been programmed to the angular frequency 2π × 200 MHz. As depicted in Fig. 1(b), 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^{37}. Since we need to implement a timedependent 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). The dephasing rates are calibrated by fitting the Rabi oscillation curve with exponential decay. Since the heat flux depend on the nonunitary process induced by the systemreservoir coupling, then by using a different kind of noise (other than the Gaussian form) we may obtain a different heat exchange behavior. See Methods section for a detailed description of the experimental setup, including the implementation of the quantum channel and the quantum process tomography (see Methods section).
As a further development, we analyze in Fig. 2 the experimental results for the heat exchange ΔQ(τ_{dec}) as a function of τ_{dec}, where we have chosen γ(t) = γ_{0}(1 + t/τ_{dec}), where τ_{dec} is experimentally controlled through the time interval associated to the action of our decohering quantum channel. The solid curves in Fig. 2 are computed from Eq. (6), while the experimental points are computed through the variation of internal energy as ΔQ(τ_{dec}) = U_{fin} − U_{ini}, where \({U}_{{\rm{fin(ini)}}}={\rm{Tr}}\{{\rho }_{{\rm{fin(ini)}}}H(\tau )\}\). 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} (See Methods section). Thus, since we have an adiabatic path in open system (see Methods section), the curves in Fig. 2 represent the heat exchanged during the adiabatic dynamics. It is worth highlighting here that we can have different noise sources in the trapped ion system in addition to dephasing. However, the coherence timescale of the Ytterbium hyperfine qubit is around 200 ms^{37}. Therefore, it is much larger than the timescale of the experimental implementation. Indeed, the dephasing rates implemented in our realization are simulated by the experimental setup.
As previously mentioned, since the Hamiltonian is timeindependent, any internal energy variation is identified as heat. In order to provide a more detailed view of this heat exchange, we analyze the von Neumann entropy \(S(\rho )={\rm{t}}r\ (\rho \mathrm{log}\,\rho )\) during the evolution. To this end, by adopting the superoperator formalism as before, the entropy variation for an infinitesimal time interval dt reads \(dS=(1/D)\left\langle \left\langle {\rho }_{\mathrm{log}\,}(t)\right\right.{\mathbb{L}}(t)\left.\left\rho (t)\right\rangle \right\rangle\), where \(\left\langle \left\langle {\rho }_{\mathrm{log}\,}(t)\right\right.\) is a supervector with components given by \({\varrho }_{n}^{\mathrm{log}\,}(t)={\rm{Tr}}\left\{{\sigma }_{n}\mathrm{log}\,\rho (t)\right\}\) (see Methods section). Thus, for an adiabatic evolution in an open system we find that (see Methods section)
where \({\Gamma }_{i,{k}_{i}}(t)=\langle \langle {\rho }_{\mathrm{log}\,}^{{\rm{ad}}}(t) {{\mathcal{D}}}_{i}^{({k}_{i}1)}(t)\rangle \rangle +{\lambda }_{i}(t)\langle \langle {\rho }_{\mathrm{log}\,}^{{\rm{ad}}}(t) {{\mathcal{D}}}_{i}^{({k}_{i})}(t)\rangle \rangle\), with \(\langle \langle {\rho }_{\mathrm{log}\,}^{{\rm{ad}}}(t)\) defined here as a supervector with components \({\varrho }_{\mathrm{log}\,}^{{\rm{ad}}}(t)={\rm{Tr}}\{{\sigma }_{n}\mathrm{log}\,{\rho }_{\mathrm{log}\,}^{{\rm{ad}}}(t)\}\). For the adiabatic dynamics considered in Fig. 2 the infinitesimal von Neumann entropy variation dS in interval dt is given by
where we define \(g(t)={e}^{2\mathop{\int}\nolimits_{\!\!{0}}^{t}\gamma (\xi )d\xi }\tanh (\beta \hslash \omega )\). Notice that the relation between heat and entropy can be obtained by rewriting the exchanged heat dQ in the interval dt as dQ^{ad}(t) = 2\(\hslash\)ωγ(t)g(t)dt. In conclusion, the energy variation can indeed be identified as heat exchanged along the adiabatic dynamics. Indeed, by computing the thermodynamic relation between dS(t) and dQ^{ad}(t) we get dS(t) = β_{deph}dQ^{ad}(t), where β_{deph} is the inverse temperature of the simulated thermal bath.
Discussion
From a general approach for adiabaticity in open quantum systems driven by timelocal master equations, we provided a relationship between adiabaticity in quantum mechanics and in quantum thermodynamics in the weak coupling regime between system and reservoir. 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 opensystem 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 turn, we have experimentally shown that heat exchange can be directly provided along the adiabatic path in terms of the decoherence rates as a function of the total evolution time. In particular, the relationship between heat and entropy is naturally derived in terms of a simulated thermal bath. Our implementation exhibits high controllability, opening perspectives for analyzing thermal machines (or refrigerators) in open quantum systems under adiabatic evolutions. Moreover, a further point to be explored is the speed up of the adiabatic path through the transitionless quantum driving (TQD) method for open systems^{39}. Indeed, TQD can be incorporated in the formalism for adiabatic thermodynamics we introduced in this work. The starting point is the generalization of Eqs. (3) and (4) through the introduction of the superadiabatic Lindbladian superoperator \({{\mathbb{L}}}_{{\rm{TQD}}}(t)\) governing the open system evolution^{39}. Notice that \({{\mathbb{L}}}_{{\rm{TQD}}}(t)\) will include counterdiabatic contributions generally obtained by reservoir engineering. Suppression of heat may be possibly obtained by constraining the evolution inside the Jordan block of \({{\mathbb{L}}}_{{\rm{TQD}}}(t)\) with vanishing eigenvalue. Naturally, the requirements of weak coupling and timelocal master equations are still to be kept. The associated effects of the engineered reservoirs on the thermal efficiencies and TQD dynamics are left for future research.
Methods
Thermodynamics in the superoperator formalism
Let us consider the heat exchange as
where we have used the equation \(\dot{\rho }(t)={\mathcal{L}}[\rho (t)]\). To derive the corresponding expression in the superoperator formalism we first define the basis of operators given by {σ_{i}}, i = 0, ⋯, D^{2} − 1, where \({\rm{Tr}}\{{\sigma }_{i}^{\dagger }{\sigma }_{j}\}=D{\delta }_{ij}\). In this basis, we can write ρ(t) and H(t) generically as
where we have \({h}_{n}(t)={\rm{Tr}}\{H(t){\sigma }_{n}\}\) and \({\varrho }_{n}(t)={\rm{Tr}}\{\rho (t){\sigma }_{n}^{\dagger }\}\). Then, we get
Now, we use the definition of the matrix elements of the superoperator \({\mathbb{L}}(t)\), associated with \({\mathcal{L}}[\bullet ]\), which reads \({{\mathbb{L}}}_{mn}=(1/D){\rm{Tr}}\{{\sigma }_{m}^{\dagger }{\mathcal{L}}[{\sigma }_{n}]\}\), so that we write
In conclusion, by defining the vector elements
we can rewrite Eq. (12), yielding
Equivalently,
where we have used Eq. (10) to write \(\dot{H}(t)=(1/D)\mathop{\sum }\nolimits_{n = 0}^{{D}^{2}1}{\dot{h}}_{n}(t){\sigma }_{n}^{\dagger }\) and, consequently,
so that we use the definition of the coefficients ϱ_{n}(t) to get
By using Eqs. (13) and (14) into Eq. (18), we conclude that
In thermodynamics, heat exchange is accompanied of an entropy variation. Then, in order to provide a complete thermodynamic study from this formalism, we now compute the instantaneous variation of the von Neumann entropy \(S(t)={\rm{Tr}}\{\rho (t)\mathrm{log}\,[\rho (t)]\}\), which reads
By using that \({\rm{Tr}}\{\rho (t)\}=1\), we get \({\rm{Tr}}\{\dot{\rho }(t)\}=0\). Therefore
where we also used that \(\dot{\rho }(t)={{\mathcal{L}}}_{t}[\rho (t)]\). Now, let us to write
so that we can define the vectors \(\left\langle \left\langle {\rho }_{\mathrm{log}\,}(t)\right\right.\) associated to \(\mathrm{log}\,\rho (t)\) with components \({\varrho }_{n}^{\mathrm{log}\,}(t)\) obtained as \({\varrho }_{n}^{\mathrm{log}\,}(t)={\rm{Tr}}\{{\sigma }_{n}\mathrm{log}\,\rho (t)\}\). Thus, we get
In the superoperator formalism, we then have
Alternatively, it is possible to get a similar result for the entropy variation in an interval Δt = t − t_{0} as
where we can use Eq. (10) to write
so that we can identify \({\varrho }_{n}^{\mathrm{log}\,}(t)={\rm{Tr}}\left\{{\sigma }_{n}\mathrm{log}\,\rho (t)\right\}\) and we finally write
Adiabatic quantum thermodynamics
Let us start by briefly reviewing the adiabatic dynamics in the context of open systems. To this end, let us consider the local master equation (in the superoperator formalism)
which describes a general timelocal physical process in open systems. The dynamical generator \({\mathcal{L}}[\bullet ]\) is requested to be a linear operation, namely,
for any complex numbers α_{1,2} and matrices ρ_{1,2}(t), with α_{1} + α_{2} = 1, because we need to satisfy \({\rm{Tr}}\left\{{\alpha }_{1}{\rho }_{1}(t)+{\alpha }_{2}{\rho }_{2}(t)\right\}=1\). Thus, by using this property of the operator \({\mathcal{L}}[\bullet ]\), it is possible to rewrite Eq. (27) as^{11}
where \({\mathbb{L}}(t)\) and \(\left.\left\rho (t)\right\rangle \right\rangle\) have been already previously defined. In general, due to the nonHermiticity of \({\mathbb{L}}(t)\), there are situations in which \({\mathbb{L}}(t)\) cannot be diagonalized, but it is always possible to write a blockdiagonal form for \({\mathbb{L}}(t)\) via the Jordan block diagonalization approach^{40}. Hence, it is possible to define a set of right and left quasieigenstates of \({\mathbb{L}}(t)\), respectively, as
From the above equations, we can write the Jordan form of \({\mathbb{L}}(t)\) as
where N is the sum of the geometric multiplicities of all the distinct eigenvalues λ_{α}(t) and each block J_{α}(t) is given by
In the adiabatic dynamics of closed systems, the decoupled evolution of the set of eigenvectors \(\left{E}_{n}^{{k}_{n}}(t)\right\rangle\) of the Hamiltonian associated with an eigenvalue E_{n}(t), where k_{n} denotes individual eigenstates, characterizes what we call Schrödingerpreserving eigenbasis. In an analogous way, the set of right and left quasieigenstates of \({\mathbb{L}}(t)\) associated with the Jordan block J_{α}(t) characterizes the Jordanpreserving left and right bases. Here, we will restrict our analysis to a particular case where each block J_{α}(t) is onedimensional, so that the set of quasieigenstates given in Eq. (30) becomes a genuine eigenstate equation given by
In this case, we can expand the matrix density \(\left.\left\rho (t)\right\rangle \right\rangle\) in basis \(\left.\left{{\mathcal{D}}}_{\alpha }(t)\right\rangle \right\rangle\) as
with r_{β}(t) being parameters to be determined. By using the Eq. (29), one gets the dynamical equation for each r_{β}(t) as
Now, we can define a new parameter p_{β}(t) as
so that one finds an equation for p_{β}(t) given by
with the first term in righthandside being the responsible for coupling distinct JordanLindblad eigenspaces during the evolution. If we are able to apply some strategy to minimize the effects of such a term in the above equation, we can approximate the dynamics to
Then, the adiabatic solution r_{β}(t) for the dynamics can be immediately obtained from Eq. (36), which reads
where we already used p_{β}(t_{0}) = r_{β}(t_{0}). In conclusion, if the system undergoes an adiabatic dynamics along a nonunitary process, the evolved state can be written as
with \({\tilde{\lambda }}_{\alpha }(t)={\lambda }_{\alpha }(t)\langle \langle {{\mathcal{E}}}_{\alpha }(t) {\dot{{\mathcal{D}}}}_{\alpha }(t)\rangle \rangle\) being the generalized adiabatic phase accompanying the dynamics of the nth eigenvector. The same mathematical procedure can be applied for multidimensional blocks^{11}. In this scenario, let \(\rho (0)\rangle \rangle ={\sum }_{i,{k}_{i}}{c}_{i}^{({k}_{i})}{{\mathcal{D}}}_{i}^{({k}_{i})}(0)\rangle \rangle\) be the initial state of the system associated with the initial matrix density ρ(0). By considering a general adiabatic evolution, the state at a later time t will be given by^{11}
with \({\tilde{\lambda }}_{i,{k}_{i}}(t)={\lambda }_{i}(t)\langle \langle {{\mathcal{E}}}_{i}^{({k}_{i})}(t) {\dot{{\mathcal{D}}}}_{i}^{({k}_{i})}(t)\rangle \rangle\), where \(\{\langle \langle {{\mathcal{E}}}_{i}^{({k}_{i})}(t)\}\) and \(\{{{\mathcal{D}}}_{i}^{({k}_{i})}(t)\rangle \rangle \}\) denote the instantaneous Jordanpreserving left and right bases of \({\mathbb{L}}(t)\), respectively^{11}. Therefore, from Eq. (2), we can write the work dW_{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. Moreover, from Eq. (24), we can write the von Neumann entropy variation as
so that we can use the Eq. (30) to write
where \({\Gamma }_{i,{k}_{i}}(t)=\langle \langle {\rho }_{\mathrm{log}\,}^{{\rm{ad}}}(t) {{\mathcal{D}}}_{i}^{({k}_{i}1)}(t)\rangle \rangle +{\lambda }_{i}(t)\langle \langle {\rho }_{\mathrm{log}\,}^{{\rm{ad}}}(t) {{\mathcal{D}}}_{i}^{({k}_{i})}(t)\rangle \rangle\), with \(\langle \langle {\rho }_{\mathrm{log}\,}^{{\rm{ad}}}(t)\) standing for the adiabatic evolved state associated with \(\left\langle \left\langle {\rho }_{\mathrm{log}\,}(t)\right\right.\).
Heat in adiabatic quantum processes
We will discuss how to determine infinite classes of systems exhibiting the same amount of heat exchange dQ. This is provided in Theorem 1 below.
Theorem 1
Let \({\mathcal{S}}\)be an open quantum system governed by a timelocal master equation in the form \(\dot{\rho }(t)={\mathcal{H}}[\rho (t)]+{{\mathcal{R}}}_{t}[\rho (t)]\), where \({\mathcal{H}}[\bullet ]=(1/i\hslash )[H,\bullet ]\)and \({{\mathcal{R}}}_{t}[\bullet ]={\sum }_{n}{\gamma }_{n}(t)[{\Gamma }_{n}(t)\bullet {\Gamma }_{n}^{\dagger }(t)(1/2)\{{\Gamma }_{n}^{\dagger }(t){\Gamma }_{n}(t),\bullet \}]\). 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 \({\mathcal{S}}\)and its reservoir during the quantum evolution is given by dQ. Then, any unitarily related adiabatic dynamics driven by \({\dot{\rho }}^{\prime}(t)={{\mathcal{H}}}^{\prime}[{\rho }^{\prime}(t)]+{{\mathcal{R}}}_{t}^{\prime}[{\rho }^{\prime}(t)]\), where \({\dot{\rho }}^{\prime}(t)=U\dot{\rho }(t){U}^{\dagger }\), \({{\mathcal{H}}}^{\prime}[\bullet ]=U{\mathcal{H}}[\bullet ]{U}^{\dagger }\) and \({{\mathcal{R}}}_{t}^{\prime}[\bullet ]=U{{\mathcal{R}}}_{t}[\bullet ]{U}^{\dagger }\), for some constant unitary U, implies in an equivalent heat exchange \(d{Q}^{\prime}=dQ\).□
Proof
Let us consider that ρ(t) is solution of
so, by multiplying both sides of the above equation by U (on the lefthandside) and U^{†} (on the righthandside), we get
thus, by using the relations [UAU^{†}, UBU^{†}] = U[A, B]U^{†} and {UAU^{†}, UBU^{†}} = U{A, B}U^{†}, we find
where \({\Gamma }^{\prime}(t)=U{\Gamma }_{n}(t){U}^{\dagger }\). In conclusion, we get that \({\rho }^{\prime}(t)=U\rho (t){U}^{\dagger }\) is a solution of
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 \({\rho }^{\prime}(t)=U\rho (t){U}^{\dagger }\), ∀ t ∈ [0, τ]. Hence
where we have used the cyclical property of the trace and that \(\Delta Q={\rm{Tr}}\{H\rho (\tau )\}{\rm{Tr}}\{H\rho (0)\}\). ■
As an example of application of the above theorem, let us consider a systemreservoir interaction governed by \({{\mathcal{R}}}_{t}^{{\rm{x}}}[\bullet ]=\gamma (t)\left[{\sigma }_{x}\bullet {\sigma }_{x}\bullet \right]\) (bitflip channel). We can then show that the results previously obtained for dephasing can be reproduced if the quantum system is initially prepared in thermal state of \({H}_{{\rm{y}}}^{0}=\omega {\sigma }_{y}\). Such a result is clear if we choose U = R_{x}(π/2)R_{z}(π/2). Then, it follows that \({{\mathcal{R}}}_{t}^{{\rm{x}}}[\bullet ]=U{{\mathcal{R}}}_{t}^{{\rm{z}}}[\bullet ]{U}^{\dagger }\) and \({{\mathcal{H}}}^{\prime}[\bullet ]=U{\mathcal{H}}[\bullet ]{U}^{\dagger }\), where R_{z(x)}(θ) are rotation matrices with angle θ around z(x)axes for the case of a single qubit. Thus, the above theorem assures that the maximum exchanged heat will be \(\Delta {Q}_{\max }=\hslash \tilde{\omega }\tanh [\beta \hslash \omega ]\).
Let us discuss now the adiabatic dynamics under dephasing and heat exchange. Consider the Hamiltonian H_{x} = \(\hslash\)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 \(\left.\left{\rho }^{{\rm{x}}}(0)\right\rangle \right\rangle\), we can compute the components \({\rho }_{n}^{{\rm{x}}}(0)\) of \(\left.\left{\rho }^{{\rm{x}}}(0)\right\rangle \right\rangle\) from \({\rho }_{n}^{{\rm{x}}}(0)={\rm{Tr}}\{\rho (0){\sigma }_{n}\}\), where σ_{n} = {\({\mathbb{1}}\), σ_{x}, σ_{y}, σ_{z}}. Thus we get
where we define the basis \(\left.\leftk\right\rangle \right\rangle ={[\begin{array}{cccc}{\delta }_{k1}&{\delta }_{kx}&{\delta }_{ky}&{\delta }_{kz}\end{array}]}^{t}\). If we drive the system under the master equation
the superoperator \({\mathbb{L}}(t)\) associated with the generator \({\mathcal{L}}[\bullet ]\) reads
Thus, it is possible to show that the set \(\{\left.\left1\right\rangle \right\rangle ,\left.\leftx\right\rangle \right\rangle \}\) satisfies the eigenvalue equation for \({\mathbb{L}}(t)\) as
It can be shown that this eigenstates are nondegenerate. Therefore, if the dynamics is adiabatic, we can write the evolved state as \(\left.\left{\rho }^{{\rm{x}}}(t)\right\rangle \right\rangle ={c}_{1}(t)\left.\left1\right\rangle \right\rangle +{c}_{x}(t)\left.\leftx\right\rangle \right\rangle\), 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. (41), we obtain c_{1}(t) = 1 and \({c}_{x}(0)=\tanh [\beta \hslash \omega ]\), so that we can use \({\tilde{\lambda }}_{1}=0\) and \({\tilde{\lambda }}_{x}=2\gamma (t)\) to obtain
Notice that Eq. (7) in the main text directly follows by rewriting Eq. (58) in the standard operator formalism. Moreover, by using this formalism, it is also possible to show that the dephasing channel can be used as a thermalization process if we suitably choose the parameter γ(t) and the total evolution time τ_{dec}. In fact, we can define a new inverse temperature β_{deph} so that Eq. (58) behaves as thermal state, namely,
where we immediately identify
In particular, by using the mean value theorem, there is a value \(\bar{\gamma }\) so that \(\bar{\gamma }=(1/{\tau }_{{\rm{dec}}})\mathop{\int}\nolimits_{0}^{{\tau }_{{\rm{dec}}}}\gamma (t)dt\). Then, the above equation becomes
In addition, heat can be computed from Eq. (43) as
where we already used c_{i} = 0, for i = y, z. Now, we can use that the vector \(\left\langle \left\langle h(t)\right\right.\) has components h_{n}(t) given by \({h}_{n}(t)={\rm{Tr}}\{\rho (0)H(t)\}\), in which H(t) is the Hamiltonian that acts on the system during the nonunitary dynamics. In conclusion, by using this result and Eq. (57), we get
Now, let us to use the meanvalue theorem for real functions to write \(\bar{\gamma }=(1/\Delta t)\mathop{\int}\nolimits_{\!{0}}^{t}\gamma (\xi )d\xi\) within the interval Δt, so that we get \({e}^{2\mathop{\int}\nolimits_{\!{0}}^{t}\gamma (\xi )d\xi }={e}^{2\bar{\gamma }\Delta t}\). It shows that the higher the meanvalue of γ(t) the faster the heat exchange ends. Now, by integrating the above result
To solve the above equation, we need to solve
where we can note that
Therefore, we can write the Eq. (65) as
where we used the meanvalue theorem in the last step. Therefore, by using this result in Eq. (64), we find
In order to study the the average power for extracting/introducing the amount ∣ΔQ(τ_{dec})∣, we define the quantity \(\bar{{\mathcal{P}}}({\tau }_{{\rm{dec}}})= \Delta Q({\tau }_{{\rm{dec}}}) /{\tau }_{{\rm{dec}}}\), where τ_{dec} is the time interval necessary to extract/introduce the amount of heat ∣Q(τ_{dec})∣. Thus, from the above equation we obtain
with \(\Delta {Q}_{\max }=\hslash \omega \tanh [\beta \hslash \omega ]\) and \(\eta ({\tau }_{{\rm{dec}}},\bar{\gamma })=(1{e}^{2\bar{\gamma }{\tau }_{{\rm{dec}}}})/{\tau }_{{\rm{dec}}}\). This result is illustrated in Fig. 3, where we have plotted \(\bar{{\mathcal{P}}}({\tau }_{{\rm{dec}}})\) during the entire heat exchange (within the interval τ_{dec}) as a function of τ_{dec}. Notice that, as in the case of ΔQ(τ_{dec}), the asymptotic behavior of the average power is independent of γ_{0}.
For our dynamics, the entropy variation is obtained from Eq. (44) for a onedimensional block Jordan decomposition. Thus, by computing \(\langle \langle {\rho }_{\mathrm{log}\,}^{{\rm{Ad}}}(t)\), where we find
with \(g(t)={e}^{2\mathop{\int}\nolimits_{\!{0}}^{t}\gamma (\xi )d\xi }\tanh (\beta \hslash \omega )\). Then, from Eq. (44) we get
where \({\Gamma }_{i}(t)={\lambda }_{i}(t)\langle \langle {\rho }_{\mathrm{log}\,}^{{\rm{ad}}}(t) {{\mathcal{D}}}_{i}(t)\rangle \rangle\). Hence, from the set of adopted values for our parameters and the spectrum of the Lindbladian, we get
Trappedion experimental setup
We encode a qubit into hyperfine energy levels of a trapped Ytterbium ion ^{171}Yb^{+}, denoting its associated states by \(\left0\right\rangle \equiv {\left.\right}^{2}{S}_{1/2};F=0,m=0\left.\right\rangle\) and \(\left1\right\rangle \equiv {\left.\right}^{2}{S}_{1/2};F=1,m=0\left.\right\rangle\). By using an arbitrary waveform generator (AWG) we can drive the qubit through either a unitary or a nonunitary dynamics (via a frequency mixing scheme). The detection of the ion state is obtained from use of a “readout” laser with wavelength 369.526 nm.
Applying a static magnetic field with intensity 6.40 G, we get a frequency transition between the qubit states given by ω_{hf} = 2π × 12.642825 GHz. Therefore, by denoting the states \(\left0\right\rangle\) and \(\left1\right\rangle\) as ground and excited states, respectively, the inner system Hamiltonian is given by
where \({\sigma }_{z}=\left1\right\rangle \left\langle 1\right\left0\right\rangle \left\langle 0\right\). Therefore, to unitarily drive the system through coherent population inversions within the subspace \(\{\left0\right\rangle ,\left1\right\rangle \}\), we use a microwave at frequency ω_{mw} whose magnetic field
interacts with the electron magnetic dipole moment \(\hat{\mu }={\mu }_{M}\hat{S}\), with μ_{M} a constant and \(\hat{S}\) is the electronic spin. Then, the system Hamiltonian reads
Thus, by defining the Rabi frequency \(\hslash {\Omega }_{{\rm{R}}}\equiv {\mu }_{M} {\overrightarrow{B}}_{0} /4\)^{41}, we obtain that the effective Hamiltonian that drives the qubit is (in interaction picture)
where ω = ω_{hf} − ω_{mw} and \({\sigma }_{x}=\left1\right\rangle \left\langle 0\right+\left0\right\rangle \left\langle 1\right\). By using the AWG we can efficiently control the parameters ω and Ω_{R}. In particular, in our experiment to implement the Hamiltonian \({\tilde{H}}_{{\rm{x}}}\), we have used a resonant (ω_{mw} = ω_{hf}) microwave with Rabi frequency \({\Omega }_{{\rm{R}}}=\tilde{\omega }\), while the frequency ω_{hf} has been adjusted around 2π × 12.642 GHz, with \(\tilde{\omega }\) modulated by using the channel 1 (CH1) of the AWG.
After the experimental qubit operation, we use the statedependent florescence detection method to implement the quantum state binary measurement. We can observe on average 13 photons for the bright state \(\left1\right\rangle\) and zero photon for the dark state \(\left0\right\rangle\) in the 500 μs detection time interval, as shown in Fig. 4. These scattered photons at 396.526 nm are collected by an objective lens with numerical aperture NA = 0.4. After the capture of these photons, they go through an optical bandpass filter and a pinhole, after which they are finally detected by a photomultiplier tube (PMT) with 20% quantum efficiency. By using this procedure, the measurement fidelity is measured to be 99.4%.
Due to the long coherence time of the hyperfine qubit, the decoherence effects can be neglected in our experimental timescale. However, since we are interested in a nontrivial nonunitary evolution, we need to perform environment engineering. This task can be achieved by using a Gaussian noise source to mix the carrier microwave \({\overrightarrow{B}}_{{\rm{un}}}(t)\) by a frequency modulation (FM) method. Thus, by considering the noise source encoded in the function η(t) = Ag(t), where A is average amplitude of the noise and g(t) is a random analog voltage signal, the driving magnetic field will be in form
where \( {\overrightarrow{B}}_{0}\) is field intensity and C is the modulation depth supported by the commercial microwave generator E8257D. If C is a fixed parameter (for example, C = 96.00 KHz/V), the dephasing rate γ(t) associated with Lindblad equation
is controlled from the average amplitude of the Gaussian noise function η(t). To see that η(t) is a Gaussian function in the frequency domain, we show its spectrum in Fig. 5.
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 operatorsum representation associated to a tracepreserving map ε. For an arbitrary input state ρ, the output state ε(ρ) can be written as^{42}
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 socalled 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. (78), 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. 6. We can calculate the fidelity between the experimental process matrix χ_{exp} and the theoretical process matrix χ_{id}
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 \({{\mathcal{F}}}_{{t}_{1}}=99.27 \%\), \({{\mathcal{F}}}_{{t}_{2}}=99.50 \%\), \({{\mathcal{F}}}_{{t}_{3}}=99.72 \%\), \({{\mathcal{F}}}_{{t}_{4}}=99.86 \%\) and \({{\mathcal{F}}}_{{t}_{5}}=99.87 \%\), 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 timedependent dephsing in experiment.
The function η(t) depends on an amplitude parameter A, which is used to control γ(t). As shown in Fig. 7, we experimentally measured the relation between A and γ(t) for a situation where γ(t) is a timeindependent value γ_{0}. As result, we find a linear relation between \(\sqrt{{\gamma }_{0}}\) and A, which reads
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 \(\sqrt{{\gamma }_{0}}\) can be efficiently controlled. On the other hand, if we need a timedependent 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.
In order to guarantee that the dynamics of the system is really adiabatic^{11} we compute the fidelity \({\mathcal{F}}({\tau }_{{\rm{dec}}})\) of finding the system in a path given by Eq. (5), where \({\mathcal{F}}(t)={\rm{Tr}}\left\{{\left[{\rho }_{{\rm{exp}}}^{1/2}(t){\rho }_{{\rm{ad}}}(t){\rho }_{{\rm{exp}}}^{1/2}(t)\right]}^{1/2}\right\}\), with ρ_{ad}(t) the density matrix provided Eq. (5) and ρ_{exp}(t) the experimental density matrix obtained from quantum tomography. In Table 1 we show the minimum experimental fidelity \({{\mathcal{F}}}_{{\rm{min}}}=\mathop{\min }\nolimits_{{\tau }_{{\rm{dec}}}}{\mathcal{F}}({\tau }_{{\rm{dec}}})\) 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.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The code that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank YuanYuan Zhao, Zhibo Hou, JunFeng Tang, and Yuexin Huang for valuable discussion. This work was supported by the National Key Research and Development Program of China (No. 2017YFA0304100), National Natural Science Foundation of China (Nos. 11734015, 11774335), Anhui Initiative in Quantum Information Technologies (AHY070000, AHY020100), Anhui Provincial Natural Science Foundation (No. 1608085QA22), Key Research Program of Frontier Sciences, CAS (No. QYZDYSSWSLH003), the Fundamental Research Funds for the Central Universities (WK2470000026, WK2470000027, WK2470000028), and the China Postdoctoral Science Foundation (Grant No. 2020M671861). A.C.S. is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPqBrazil). M.S.S. is supported by CNPqBrazil (No. 303070/20161) and Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) (No. 203036/2016). D.O.S.P is supported by Brazilian funding agencies CNPq (Grants No. 142350/20176 and 305201/20166), and FAPESP (Grant No. 2017/037270). A.C.S., D. O. S.P. and M.S.S. also acknowledge financial support in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior  Brasil (CAPES) (Finance Code 001) and by the Brazilian National Institute for Science and Technology of Quantum Information [CNPq INCTIQ (465469/20140)]. A.C.S., D. O. S.P. and M.S.S. would like to thank F. Brito and J. G. Filgueiras for fruitful comments.
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A.C.S., D.O.S.P. and M.S.S. developed and performed the theoretical analysis. C.K.H., J.M.C., Y.F.H. and C.F.L. designed the experiment. C.K.H., J.M.C. and Y.F.H. performed the experiment. C.K.H., Y.F.H., A.C.S., D.O.S.P. and M.S.S. wrote the manuscript. C.F.L. and G.C.G. supervised the project. All authors discussed and contributed to the analysis of the experimental data.
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Hu, CK., Santos, A.C., Cui, JM. et al. Quantum thermodynamics in adiabatic open systems and its trappedion experimental realization. npj Quantum Inf 6, 73 (2020). https://doi.org/10.1038/s41534020003002
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DOI: https://doi.org/10.1038/s41534020003002
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