Energy efficient quantum machines

Efficiency and power are prominent quantifiers of the performance of a thermal machine. Efficiency is defined as the ratio of energy output to energy input, while power characterizes the rate of energy output per unit time [1]. A key current challenge is to design energy efficient engines that deliver more output for the same input, without sacrificing power [2]. Such devices would reduce both energy consumption and energy costs —they are for these reasons of particular interest. In conventional finite-time thermodynamics, an efficiency increase is usually associated with a power decrease, and vice versa [3,4]. We here develop the general finite-time thermodynamics of thermal machines that are driven by shortcut-to-adiabaticity (STA) protocols [5,6]. We show that higher efficiency and higher power may be achieved simultaneously, even when the energetic cost of the STA driving is included.

Shortcut-to-adiabaticity techniques allow the engineering of adiabatic dynamics in finite time. While truly adiabatic transformations require infinitely slow driving, transitionless protocols may be implemented at finite speed by adding properly designed time-dependent terms H_STA(t) to the Hamiltonian of a system [5,6]. By suppressing nonadiabatic excitations, these fast processes reproduce the same final state as that of adiabatic driving. In that sense, they provide a shortcut to adiabaticity. Over the past few years, there has been remarkable progress, both theoretical [5–13] and experimental [14–22], in developing STA methods for quantum and classical systems (see ref. [23] for a review). Successful applications include high-fidelity driving of a BEC [16], fast transport of trapped ions [17,18], fast adiabatic passage using a single spin in diamond [19] and cold atoms [20], as well as swift equilibration of a Brownian particle [21].

Shortcut-to-adiabaticity protocols have recently been extended to thermal machines as a means to enhance their performance. Classical [24,25] and quantum [26,27] single-particle heat engines as well as multiparticle quantum motors [27–29] have been theoretically investigated. Nonadiabatic transitions are well-known sources of entropy production that reduce the efficiency of thermal machines [3,4]. Successfully suppressing them using STA methods therefore appears a promising strategy to boost both their work and power output.

However, a crucial point that needs to be addressed in order to assess the usefulness of shortcut techniques in thermodynamics is the proper computation of the efficiency of a STA engine. Since the excitation suppressing term H_STA(t) in the Hamiltonian is often assumed to be zero at the beginning and at the end of a transformation [23], its work contribution vanishes. The energetic cost of the additional STA driving is therefore commonly not included in the calculation of the efficiency [24–29]. As a result, the latter quantity reduces to the adiabatic efficiency, even for fast nonadiabatic driving of the machine: the STA driving thus appears to be for free. This situation is somewhat reminiscent of the power of a periodic signal which is zero at the beginning and at the end of one period. While the instantaneous power vanishes at the end of the interval, the actual power of the signal is given by the nonzero time-averaged power [30]. As a matter of fact, the energetic cost of STA protocols has lately been defined in universal quantum computation and adiabatic gate teleportation models as the time-averaged norm of the STA Hamiltonian H_STA(t) [31,32] (see also refs. [33–36]). However, the chosen Hilbert-Schmidt norm is related to the variance of the energy and not to its mean [37]. It is hence of limited relevance to the investigation of the energetics of a heat engine.

In this paper, we evaluate the performance of a STA engine by properly taking the energetic cost of the transitionless driving into account. We consider commonly employed local counterdiabatic (LCD) control techniques [11–13] which have latterly been successfully implemented experimentally [15,16]. We evaluate both efficiency and power for a paradigmatic harmonic quantum Otto engine [38–43]. We explicitly compute the cost of the STA protocol as the time-averaged expectation value of the Hamiltonian H_STA(t) for compression and expansion phases of the engine cycle. We find that the energetic cost of the STA driving exceeds the potential work gain for very slow protocols. STA engines may therefore outperform traditional quantum motors for fast cycles, albeit with an efficiency smaller than the corresponding adiabatic efficiency. We additionally derive generic upper bounds on both STA efficiency and power, valid for any heat engine cycle, based on the concept of quantum speed limit times for driven unitary dynamics [44]. We demonstrate that these quantum bounds are tighter than conventional bounds that follow from the second law of thermodynamics.

We consider a quantum engine whose working medium is a harmonic oscillator with time-dependent frequency ω_t. The corresponding Hamiltonian is of the usual form, $H_0(t) = p^2/(2 m) + m\omega_t^2 x^2/2$, where x and p are the position and momentum operators of an oscillator of mass m. The Otto cycle consists of four consecutive steps as shown in fig. 1 [38–42]: (1) Isentropic compression A → B: the frequency is varied from ω₁ to ω₂ during time τ₁ while the system is isolated. The evolution is unitary and the von Neumann entropy is constant. (2) Hot isochore B → C: the oscillator is weakly coupled to a bath at inverse temperature β₂ at fixed frequency and thermalizes to state C during time τ₂. (3) Isentropic expansion C → D: the frequency is changed back to its initial value during time τ₃ at constant von Neumann entropy. (4) Cold isochore D → A: the system is weakly coupled to a bath at inverse temperature $\beta_1>\beta_2$ and relaxes to state A during τ₄ at fixed frequency. We will assume, as commonly done [38–42], that the thermalization times τ_2,4 are much shorter than the compression/expansion times τ_1,3. The total cycle time is then $\tau_\text{cycle}= \tau_1+ \tau_3=2 \tau$ for equal step duration.

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In order to evaluate the performance of the Otto engine, we need to compute work and heat for each of the above steps. Work is performed during the first and third unitary strokes, whereas heat is exchanged with the baths during the isochoric thermalization phases two and four. The mean work may be calculated by using the exact solution of the Schrödinger equation for the parametric oscillator for any given frequency modulation [45,46]. For the compression/expansion steps, it is given by [42],

$$\begin{eqnarray} \left\langle W_1\right\rangle &= \frac{\hbar}{2} (\omega_2Q^\ast_1 - \omega_1) \coth\left(\frac{\beta_1\hbar\omega_1}{2}\right)\!, \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} \left\langle W_3\right\rangle &= \frac{\hbar}{2} (\omega_1 Q^\ast_3 - \omega_2) \coth\left(\frac{\beta_2\hbar\omega_2}{2}\right)\!, \end{eqnarray} \tag{ 2 }$$

where we have introduced the dimensionless adiabaticity parameter $Q^*_{i}\ (i=1,3)$ [47]. It is defined as the ratio of the mean energy and the corresponding adiabatic mean energy and is thus equal to one for adiabatic processes [46]. Its explicit expression for any frequency modulation ω_t may be found in refs. [45,46]. Furthermore, the mean heat absorbed from the hot bath reads [42]

$$\begin{equation} \left\langle Q_2\right\rangle = \frac{\hbar \omega_2}{2} \left[\coth\left(\frac{\beta_2\hbar\omega_2}{2}\right) - Q^\ast_1 \coth\left(\frac{\beta_1 \hbar \omega_1}{2}\right) \right]\!. \end{equation} \tag{ 3 }$$

For an engine, the produced work is negative, $\langle W_1\rangle +\langle W_3\rangle <0$, and the absorbed heat is positive, $\langle Q_2\rangle >0$.

The compression and expansion phases (1) and (3) may be speeded up, while suppressing unwanted nonadiabatic transitions, by adding a local harmonic potential H_STA to the system Hamiltonian H₀. The local counterdiabatic Hamiltonian may then written in the form $H_\text{LCD}(t) = H_0(t) + H_\text{STA}(t)$ with [11–13]

$$\begin{equation} H_\mathrm{STA} = \frac{m}{2} \left ( \Omega^2_t -\omega^2_t \right ) x^2= \frac{m}{2} \left(- \frac{3 \dot{\omega}_t^2}{4 \omega_t^2} + \frac{\ddot{\omega}_t}{2 \omega_t}\right) x^2. \end{equation} \tag{ 4 }$$

Boundary conditions ensuring that $H_\text{STA}(0,\tau)=0$ at the beginning and at the end of the driving are given by

$$\begin{equation} \begin{array}{rclrclrcl} \omega(0) &=& \omega_i,&{\qquad} \dot{\omega}(0) &=& 0,&{\qquad} \ddot{\omega}(0) &=& 0,\\ \omega(\tau) &=&\omega_f,& \dot{\omega}(\tau) &=& 0,& \ddot{\omega}(\tau) &=& 0, \end{array} \end{equation} \tag{ 5 }$$

where $\omega_{i,f}= \omega_{1,2}$ denote the respective initial and final frequencies of the compression/expansion steps. The conditions (5) are, for instance, satisfied by [11–13]

$$\begin{equation} \omega_t= \omega_i + 10(\omega_f - \omega_i)s^3 - 15(\omega_f-\omega_i)s^4+ 6(\omega_f - \omega_i) s^5, \end{equation} \tag{ 6 }$$

with $s = t/\tau$. Note that $\Omega_t^2 > 0$ to avoid trap inversion. Implementing the STA driving (4) leads to a unit adiabaticity parameter, $Q^*_i(\tau)=1\ (i=1,3)$. As a consequence, the work performed in finite time during the two compression/expansion phases is equal to the adiabatic work, $\langle W_1\rangle_\text{STA} =\langle W_1\rangle_\text{AD}$ and $\langle W_3\rangle_\text{STA} =\langle W_3\rangle_\text{AD}$.

We define the efficiency of the STA motor as

$$\begin{equation} \eta_\text{STA} = \frac{\mathrm{energy \,output}}{\mathrm{energy\, input}} = \frac{-(\left\langle W_1\right\rangle_\text{STA}+\left\langle W_3\right\rangle_\text{STA})}{\left\langle Q_2\right\rangle + \left\langle H^1_\mathrm{STA}\right\rangle_\tau + \left\langle H^3_\mathrm{STA}\right\rangle_\tau}. \end{equation} \tag{ 7 }$$

In the above expression, the energetic cost of the transitionless driving is taken into account by including the time average, $\langle H^i_\text{STA}\rangle_\tau = (1/\tau) \int_0^\tau \text{d}t\,\langle H^i_\text{STA}(t)\rangle\ (i=1,3)$, of the local potential (4) for the compression/expansion steps. Equation (7) reduces to the adiabatic efficiency $\eta_\text{AD}$ in the absence of these two contributions. For further reference, we also introduce the usual nonadiabatic efficiency of the engine, $\eta_\text{NA} = -(\langle W_1\rangle+\langle W_3\rangle)/\langle Q_2\rangle$, based on the formulas (1)–(3) without any shortcut.

The expectation value of the local counterdiabatic potential (4) may be calculated explicitly for an initial thermal state in terms of the initial energy of the system $\langle H_0(0)\rangle$. We find (see appendix)

$$\begin{equation} \left\langle H_\mathrm{STA}(t) \right\rangle = \frac{\omega_t}{\omega_i} \left\langle H_0(0) \right\rangle\left[ -\frac{\dot{\omega}_t^2}{4 \omega_t^4} + \frac{\ddot{\omega}_t}{4 \omega_t^3}\right]. \end{equation} \tag{ 8 }$$

We shall use eq. (8) to numerically compute the time averages $\langle H^i_\text{STA}\rangle_\tau\ (i=1,3)$ for compression/expansion that are needed to evaluate the STA efficiency (7).

Figure 2 shows, as an illustration, the energetic cost of the STA driving $\langle H^1_\text{STA}\rangle_\tau+ \langle H^3_\text{STA}\rangle_\tau$, for the compression and expansion steps (1) and (3), as a function of the driving time τ. We also display, for comparison, the corresponding nonadiabatic work $\langle W_1 \rangle_\text{NA}+ \langle W_3 \rangle_\text{NA}$, defined as the difference between the actual work and the adiabatic work, $\langle W_i \rangle_\text{NA}= \langle W_i \rangle - \langle W_i\rangle_\text{AD}\ (i=1,3)$; this quantity measures the importance of nonadiabatic excitations induced by fast protocols and is often referred to as internal friction [4,26,33,40]. We observe that the time-averaged STA energy (red dot-dashed line) increases significantly with decreasing process time as expected. This increase is much faster than that of the nonadiabatic work $\langle W_1 \rangle_\text{NA}+\langle W_3 \rangle_\text{NA}$ (grey dotted line). Eventually, for very rapid driving, the energetic price of the shortcut will dominate nonadiabatic energy losses.

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Figure 3 exhibits the STA efficiency $\eta_\text{STA}$ (red dot-dashed line), eq. (7), as a function of the driving time τ, together with the adiabatic efficiency $\eta_\text{AD}$ (black dotted line) and the nonadiabatic efficiency $\eta_\text{NA}$ (blue dashed line). Three points are worth emphasizing: i) if the energetic cost of the shortcut is not included, the STA efficiency is equal to the maximum possible value given by the constant adiabatic efficiency $\eta_\text{AD}$, as noted in refs. [24–29]; ii) by contrast, if that energetic cost is taken into account, the STA efficiency $\eta_\text{STA}$ drops for decreasing τ, reflecting the sharp augmentation of the time-averaged STA energy seen in fig. 2; iii) however, we observe that $\eta_\text{STA}<\eta_\text{NA}$ for large time τ, while $\eta_\text{STA}>\eta_\text{NA}$ for small enough τ (inset). We may thus conclude that the STA driving is of advantage for sufficiently short cycle durations, leading to a sizable efficiency increase compared to conventional machines. For long cycles, the energetic cost of the shortcut outweighs the work gained by emulating adiabaticity.

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Another benefit of the STA appears for very small time τ. An examination of eq. (3) reveals that the heat $\langle Q_2\rangle$ becomes negative for very strongly nonadiabatic processes, when $Q^*_1(\tau) > \coth({\beta_2\hbar\omega_2}/{2}) / \coth({\beta_1 \hbar \omega_1}/{2})$. In this regime, heat is pumped into the hot reservoir, instead of being absorbed from it, and the machine stops working as an engine [48]. Since $Q^*_1(\tau) =1$ for all τ for the local counterdiabatic driving, this problem never occurs for the STA motor, even in the limit of very short cycles $\tau \rightarrow 0$.

The power of the STA machine is given by

$$\begin{equation} P_\text{STA} = -\frac{\left\langle W_{1}\right\rangle_\text{STA} + \left\langle W_{3}\right\rangle_\text{STA}}{\tau_\text{cycle}}. \end{equation} \tag{ 9 }$$

Since the STA protocol ensures adiabatic work output, $\langle W_{i}\rangle_\text{STA} = \langle W_{i}\rangle_\text{AD}\ (i=1,3)$, in a shorter cycle duration $\tau_\text{cycle}$, the STA power $P_\text{SA}$ is always greater than the nonadiabatic power $P_\text{NA}=-(\langle W_{1}\rangle + \langle W_{3}\rangle)/{\tau_\text{cycle}}$ (see fig. 4). This ability to considerably enhance the power of a thermal machine is another advantage of the STA approach. However, in view of the discussion above, it is not possible to reach arbitrarily large power at maximum efficiency [24–27], in agreement with recent general proofs that forbid the simultaneous attainability of maximum power and maximum efficiency [49].

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We finally derive generic upper bounds for both the STA efficiency (7) and the STA power (9), based on the concept of quantum speed limits (see refs. [50–52] and references therein). Contrary to classical physics, quantum theory limits the speed of evolution of a system between given initial and final states. In particular, there exists a lower bound, called the quantum speed limit time $\tau_\text{QSL}\leq \tau$, on the time a system needs to evolve between these two states. An important restriction of the STA technique is the time required to successfully implement the counterdiabatic driving (4), which depends on the first two time derivatives of the frequency ω_t (see footnote ¹ ). For this unitary driven dynamics, a Margolus-Levitin–type bound on the evolution time reads [44],

$$\begin{equation} \tau \geq \tau_\text{QSL}= \frac{\hbar \mathcal{L}(\rho_i,\rho_f)}{\left\langle H_\text{STA}\right\rangle_\tau}, \end{equation} \tag{ 10 }$$

where $\mathcal{L}(\rho_i,\rho_f)$ denotes the Bures angle between the initial and final density operators of the system [53] (see appendix) and $\langle H_\text{STA}\rangle_\tau$ the time-averaged STA energy (8). We expect eq. (10) to be a proper bound for the compression/expansion phases, when the engine dynamics is dominated by the STA driving for small τ (see refs. [35,36] for related results connecting speed limits and driving costs).

The efficiency of a classical thermal machine is limited by the second law of thermodynamics [1]. To derive an upper bound on the STA efficiency (7) of the quantum engine, we use inequality (10) to obtain

$$\begin{equation} \eta_\text{STA} \leq \eta_\text{STA}^\text{QSL} = -\frac{\left\langle W_1\right\rangle_\text{AD}+\left\langle W_3\right\rangle_\text{AD}}{\left\langle Q_2\right\rangle +\hbar(\mathcal{L}_1+\mathcal{L}_3)/\tau}, \end{equation} \tag{ 11 }$$

where $\mathcal{L}_i\ (i=1,3)$ are the respective Bures angles for the compression/expansion steps. On the other hand, an upper bound on the STA power (9) is

$$\begin{equation} P_\text{STA}\leq P_\text{STA}^\text{QSL} = -\frac{\left\langle W_{1}\right\rangle_\text{AD} + \left\langle W_{3}\right\rangle_\text{AD}}{\tau^1_\text{QSL}+\tau^3_\text{QSL}}, \end{equation} \tag{ 12 }$$

where $\tau^i_\text{QSL}\ (i=1,3)$ are the respective speed limit bounds (10) for the compression/expansion phases.

The two quantum speed limit bounds (11) and (12) are shown in figs. 3, 4 (green solid line). We first notice that the quantum bound (11) on the efficiency is sharper than the thermodynamic bound given by the constant adiabatic efficiency $\eta_\text{AD}$. Surprisingly, quantum theory further imposes an upper bound on the power, whereas the second law of thermodynamics does not [49]. Quantum thermodynamics hence establishes tighter bounds than classical thermodynamics. The latter result may be understood by noting that thermodynamics does not have the notion of time scale, contrary to quantum mechanics, and does not impose any restrictions on the speed of a process. We also note that the quantum bound (11) (green line) reduces to the usual thermodynamic bound (black line) for slow processes (large τ), see fig. 3, as expected. Finally, we stress that the two speed limit bounds directly follow from the definitions of efficiency and power. As a result, they are independent of the thermodynamic cycle considered and generically apply to any quantum engine, not just to the quantum Otto motor. They may be, e.g., readily adapted to nonunitary compression/expansion steps by using the corresponding expression of the quantum speed limit [37].

We have performed a detailed study of efficiency and power of a STA quantum heat engine. We have explicitly accounted for the energetic cost of the STA driving, defined as the time average of the local counterdiabatic potential. We have shown that STA machines outperform their conventional counterparts for short cycles, when the work gain generated by the counterdiabatic driving outweighs its energetic cost. Remarkably, this leads to an enhancement of both efficiency and power. These findings depart from the usual efficiency-power trade-off expected from traditional finite-time thermodynamics. Shortcuts-to-adiabaticity engines therefore appear as unique energy efficient machines that are able to produce more output from the same input at higher power. This follows from the fact that STA protocols not only speed up the dynamics (thus increasing power), but also ensure that the final target state is an adiabatic instead of a highly excited state (hence reducing entropy production and increasing efficiency). We have additionally derived generic upper bounds on STA efficiency and STA power based on the idea of quantum speed limits. These quantum bounds, valid for general thermal motors, are tighter than the usual bounds based on the second law of thermodynamics. We therefore expect them to be useful for future theoretical and experimental investigations of thermal machines in the quantum regime [54].

This work was partially supported by the EU Collaborative Project TherMiQ (Grant Agreement 618074) and the COST Action MP1209.

Appendix. Local counterdiabatic energy

We here present a derivation of the mean energy of the local counterdiabatic (LCD) Hamiltonian $H_\text{LCD}$ and of the corresponding adiabaticity parameter $Q^*_\text{LCD}$ used during the compression/expansion protocols. We consider a time-dependent harmonic oscillator with Hamiltonian

$$\begin{eqnarray} &&H_0(t) = p^2/(2m) + m \omega_t^2 x^2/2, \end{eqnarray} \tag{ A.1 }$$

where ω_t is the time-dependent angular frequency, m the mass, and (p, x) the respective momentum and position operators. The initial energy eigenstates at t = 0 with $\omega(0) = \omega_0$ in coordinate representation are given by

$$\begin{eqnarray} \psi_n(x,0) &= \frac{1}{\sqrt{2^n n!}} \left(\frac{m\omega_0}{\pi \hbar}\right)^{1/4} \exp\left(-\frac{m\omega_0}{2\hbar}x^2\right) \nonumber\\ & \quad \times\mathcal{H}_n\left(\sqrt{\frac{m\omega_0}{\hbar}}x\right), \end{eqnarray} \tag{ A.2 }$$

where $\mathcal{H}_n$ are Hermite polynomials and $E_n^0 = \hbar \omega_0 (n{\,+\,1/2})$ the corresponding energy eigenvalues. The instantaneous eigenstates and their corresponding eigenvalues are obtained by replacing $\omega_0$ with ω_t.

The STA may be implemented by adding a time-dependent counterdiabatic (CD) term to the system Hamiltonian (A.1) [7,10–12]:

$$\begin{eqnarray} &&H_\mathrm{STA}^\mathrm{CD}(t) = -\frac{\dot{\omega}_t}{4\, \omega_t}({x}{p}+ {p} {x}). \end{eqnarray} \tag{ A.3 }$$

The total Hamiltonian $H_\mathrm{CD}(t) = H_\mathrm{0}(t) + H_\mathrm{STA}^\mathrm{CD}(t)$ is still quadratic in x and p and may thus be considered that of a generalized harmonic oscillator [10,55]. However, since the Hamiltonian (A.3) is a nonlocal operator, it is often convenient to look for a unitarily equivalent Hamiltonian with a local potential [11,12]. Applying the canonical transformation, $U_x = \exp\left({i m \dot{\omega} x^2}/{4 \hbar \omega}\right)$, which cancels the cross-terms xp and px, to the Hamiltonian (A.3) leads to a new local counterdiabatic (LCD) Hamiltonian [11,12]

$$\begin{eqnarray} &&H_\mathrm{LCD}(t) = U_x^\dagger (H_\mathrm{CD}(t) - i\hbar \dot{U}_x U_x^\dagger) U_x =\frac{p^2}{2 m} + \frac{m\Omega_t^2 x^2}{2}, \end{eqnarray} \tag{ A.4 }$$

with the modified time-dependent (squared) frequency

$$\begin{eqnarray} &&\Omega^2(t) = \omega_t^2 -\frac{3\dot{\omega}_t^2}{4\omega_t^2}+\frac{\ddot{\omega}_t}{2\omega_t}. \end{eqnarray} \tag{ A.5 }$$

This resulting Hamiltonian is local and still drives the evolution along the adiabatic trajectory of the system of interest. By demanding that $H_\mathrm{LCD}(t) = H_\mathrm{0}(t)$ at $t = {0,\tau}$ and imposing $\dot{\omega} (\tau) = \ddot{\omega} (\tau) = 0$, the final state is equal for both dynamics, even in phase, and the final vibrational state populations coincide with those of a slow adiabatic process [11]. It can be readily shown that $\Omega^2(t)$ approaches $\omega^2(t)$ for very slow expansion/compression process [12].

Exact solutions of the Schrödinger equation for a time-dependent harmonic oscillator have been extensively investigated [56,57]. Following Lohe [57], a solution based on the invariants of motion is of the form

$$\begin{eqnarray} &&I(t) = \frac{b^2}{2 m} p^2 + \frac{m\dot{b}^2}{2}x^2 - \frac{b\, \dot{b}}{2}(p x + x p) + \frac{m \omega_0^2}{2b^2} x^2, \end{eqnarray} \tag{ A.6 }$$

where $\omega_0$ is an arbitrary constant —a convenient choice is to set $\omega_0 = \omega(0)\ (\omega_0^2 > 0)$. The scaling factor $b = b(t)$ is a solution of the Ermakov differential equation

$$\begin{eqnarray} &&\ddot{b} + \omega_t^2 b = \omega_0^2/b^3. \end{eqnarray} \tag{ A.7 }$$

In the adiabatic limit, $\ddot{b} \simeq 0$ and $b(t) \rightarrow b_{ad} = \sqrt{\omega_0/\omega_t}$. Equation (A.7) is valid for any given ω_t and its general solution can be constructed from the solutions f(t) of the linear equation of motion for the classical time-dependent harmonic oscillator [58],

$$\begin{eqnarray} &&\ddot{f} + \omega_t^2 f = 0, \end{eqnarray} \tag{ A.8 }$$

according to $b^2/\omega_0 = f_1^2 + W^{-2}f_2^2$, where f₁, f₂ are independent solutions of eq. (A.8) and the Wronskian $W[f_1,f_2] = f_1\dot{f}_2 - \dot{f}_1 f_2$ is a nonzero constant. We note that the Wronskian properties of eq. (A.8) can be used to show the equivalence of the adiabaticity parameter derived here and that of Husimi [46,47] (see ref. [28]). The general solution of the time-dependent Schrödinger equation for the Hamiltonian (A.4) is hence

$$\begin{eqnarray} &&\Psi_n(x,t) = \frac{1}{\sqrt{2^n n!}} \left(\frac{m \omega_0}{\pi \hbar b^2}\right)^{1/4} \exp\left[\frac{i m \dot{b}}{2 \hbar b}x^2 - i\int_0^t \frac{\omega_0(n+1/2)}{b(t^\prime)^2}\text{d}t^\prime\right] \exp\left(-\frac{m \omega_0}{2 \hbar b^2}x^2\right) \mathcal{H}_n\left(\sqrt{\frac{m\omega_0}{\hbar}}\frac{x}{b}\right). \end{eqnarray} \tag{ A.9 }$$

The time-dependent energy eigenstates, $H_\text{LCD}|\Psi_n(x,t) \rangle = E |\Psi_n(x,t) \rangle$, are given by

$$\begin{eqnarray} E &= {\langle {\Psi_n(x,t)}|} H_\mathrm{LCD} {|{\Psi_n(x,t)}\rangle} \nonumber \\ &= \frac{E_n^0}{b^2} - \frac{m}{2}\left(b\ddot{b} - \dot{b}^2\right) {\langle {\Psi_n(x,0)}|} x^2 {|{\Psi_n(x,0)}\rangle} \nonumber \\ & \quad + \frac{\dot{b}}{2 b} {\langle {\Psi_n(x,0)}|} x p + px {|{\Psi_n(x,0)}\rangle}. \end{eqnarray} \tag{ A.10 }$$

We next consider a quantum oscillator initially prepared in thermal equilibrium state with density operator

$$\begin{eqnarray} &&\rho_{eq} = \sum_{n =0}^{\infty} p_n^0 {|{\Psi_n(x,0)}\rangle} {\langle {\Psi_n(x,0)}|}, \end{eqnarray} \tag{ A.11 }$$

where $p_n^0 = \exp(-\beta E_n^0)/Z_0$ is the probability that the oscillator is in state ${|{\Psi_n(x,0)}\rangle}$ and Z₀ is the partition function. The initial thermal mean energy at $t = 0$ is thus

$$\begin{eqnarray} &&\left\langle H(0) \right\rangle = m\omega_0^2 \left\langle x^2(0)\right\rangle = \frac{\hbar \omega_0}{2} \coth\left(\frac{\beta \hbar \omega_0}{2}\right). \end{eqnarray} \tag{ A.12 }$$

The expectation value of the local counterdiabatic Hamiltonian $H_\text{LCD}(t)$ at time t follows from eqs. (A.10)–(A.12) as

$$\begin{eqnarray} \left\langle H_\text{LCD}(t) \right\rangle &= \sum_{n=0}^{\infty} p_n^0 {\langle {\Psi_n(x,t)}|} H_\text{LCD} {|{\Psi_n(x,t)}\rangle}\nonumber\\ &= \frac{\left\langle H(0)\right\rangle}{b^2} + \frac{(\dot{b}^2 - b\, \ddot{b})}{2 \omega_0^2} \left\langle H(0)\right\rangle\!, \end{eqnarray} \tag{ A.13 }$$

where we have used the fact that $\langle \{x,p\} (0)\rangle = 0$ for the thermal equilibrium state. Since the squared frequency (A.5) can be rewritten in the adiabatic limit as

$$\begin{eqnarray} &&\Omega_t^2 = \omega_t^2 - \frac{\ddot{b}_{ad}}{b_{ad}}, \end{eqnarray} \tag{ A.14 }$$

we obtain, using eqs. (A.5) and (A.14), the expression

$$\begin{eqnarray} &&b\, \ddot{b} - \dot{b}^2 = b^2_{ad} \left\{-\frac{\ddot{\omega}_t}{2 \omega_t} +\frac{1}{2}\left(\frac{\dot{\omega}_t}{\omega_t}\right)^2\right\}. \end{eqnarray} \tag{ A.15 }$$

Finally, substituting eq. (A.15) into eq. (A.13), the mean energy of the local counterdiabatic driving is found to be

$$\begin{eqnarray} &&\left\langle H_\text{LCD}(t)\right\rangle= \frac{Q^\ast_\text{LCD} (t)}{b_{ad}^2} \left\langle H(0)\right\rangle, \end{eqnarray} \tag{ A.16 }$$

where the adiabaticity parameter $Q^\ast_\text{LCD} (t)$ is given by (see fig. 5)

$$\begin{eqnarray} &&Q^\ast_\text{LCD} (t) = 1 - \frac{\dot{\omega}_t^2}{4 \omega_t^4} + \frac{\ddot{\omega}_t}{4\omega_t^3}. \end{eqnarray} \tag{ A.17 }$$

Note that eq. (A.17) corrects eq. (51) in ref. [27].

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Appendix. Shortcut energy

The expectation value of the STA potential H_STA may be evaluated from eqs. (4) and (A.5). We have

$$\begin{eqnarray} &&H_\mathrm{STA}(t) = H_\mathrm{LCD}(t) - H_\mathrm{0}(t) = \frac{m}{2}\left(- \frac{3 \dot{\omega}_t^2}{4 \omega_t^2} + \frac{\ddot{\omega}_t}{2\omega_t}\right) x^2. \end{eqnarray} \tag{ A.18 }$$

As a consequence, we obtain

$$\begin{eqnarray} &&\left\langle H_\mathrm{STA}(t)\right\rangle = \frac{\omega_t}{\omega_0} \left\langle H(0) \right\rangle\left[- \frac{\dot{\omega}_t^2}{4 \omega_t^4} + \frac{\ddot{\omega}_t}{4 \omega_t^3}\right]. \end{eqnarray} \tag{ A.19 }$$

The properties of the shortcut imply that $\left\langle H_\mathrm{STA}(0,\tau)\right\rangle = 0$ at the beginning and at the end of the protocol.

Appendix. Bures length

The Bures length between initial and final density operators of the system is $\mathcal{L}(\rho_\tau,\rho_0) = \arccos \sqrt{F(\rho_\tau,\rho_0)}$, where the $F(\rho_\tau,\rho_0)$ is the fidelity between the two states [59]. For the considered driven harmonic oscillator, initial and final states are Gaussian and the fidelity is explicitly given by [53]:

$$\begin{eqnarray} &&\mathcal{F}(\rho_\tau,\rho_0) =\frac{2}{\sqrt{ct^2(\beta \epsilon_0/2) + ct^2(\beta \epsilon_1/2) + 2 Q^\ast ct(\beta \epsilon_0/2) ct(\beta \epsilon_1/2) + c^2(\beta \epsilon_0/2)c^2(\beta \epsilon_1/2)} - c(\beta \epsilon_0/2)c(\beta \epsilon_1/2)}, \end{eqnarray} \tag{ A.20 }$$

where $\epsilon_i = \hbar \omega_i$, $ct(x) = \coth(x)$ and $c(x) = \mathrm{csch}(x)$.