Experimental demonstration of work fluctuations along a shortcut to adiabaticity with a superconducting Xmon qubit

For an arbitrary non-degenerate quantum system driven by a time-dependent Hamiltonian H₀(t), we introduce its instantaneous eigenbasis $\{| n(t)\rangle \}$ satisfying ${H}_{0}(t)| n(t)\rangle ={\varepsilon }_{n}(t)| n(t)\rangle$, where ε_n(t) is the nth instantaneous eigenenergy and $| n(t)\rangle$ is the associated eigenstate. The Hamiltonian is thus expanded in this instantaneous eigenbasis, becoming

$$\begin{eqnarray}&&{H}_{0}(t)=\displaystyle \sum _{n}{\varepsilon }_{n}(t)| n(t)\rangle \langle n(t)| .\end{eqnarray} \tag{ 1 }$$

In the scenario of ${d}_{t}{H}_{0}(t)\to 0$, the system follows an adiabatic trajectory, $| n(0)\rangle \to | n(t)\rangle$, if it is prepared at $| n(0)\rangle$ initially. The quantum state at time t is given by $| {\rm{\Psi }}(t)\rangle ={U}_{\mathrm{ad}}(t)| n(0)\rangle ={c}_{n}(t)| n(t)\rangle$, where ${U}_{\mathrm{ad}}(t)={T}_{+}\exp [-({\rm{i}}/{\rm{\hslash }}){\int }_{0}^{t}{H}_{0}(\tau ){\rm{d}}\tau ]$ is the adiabatic time evolution operator and the coefficient c_n(t) includes the accumulation of both dynamic and geometric phases [52]. Here T₊ denotes the time ordering operator and ℏ = h/2π is the reduced Planck constant.

In practice, we often hope that the quantum system can follow an adiabatic trajectory but in a short time scale. In the STA protocol [5–13], an additional counter-diabatic Hamiltonian ${H}_{\mathrm{cd}}(t)$ is constructed to cancel the non-adiabatic transition. In a formally exact way, ${H}_{\mathrm{cd}}(t)$ is written as [6]

$$\begin{eqnarray}&&{H}_{\mathrm{cd}}(t)={\rm{i}}{\rm{\hslash }}\displaystyle \sum _{n}{{ \mathcal P }}_{n}^{\perp }(t)| {\partial }_{t}n(t)\rangle \langle n(t)| ,\end{eqnarray} \tag{ 2 }$$

where ${{ \mathcal P }}_{n}^{\perp }(t)=1-| n(t)\rangle \langle n(t)|$ is the projection operator onto the subspace perpendicular to $| n(t)\rangle$. The quantum system driven by $H(t)={H}_{0}(t)+{H}_{\mathrm{cd}}(t)$ evolves rigorously along the adiabatic trajectory of H₀(t), i.e., $| {\rm{\Psi }}(t)\rangle ={U}_{\mathrm{STA}}(t)| n(0)\rangle ={c}_{n}(t)| n(t)\rangle$ with the STA time evolution operator ${U}_{\mathrm{STA}}(t)={T}_{+}\exp [-({\rm{i}}/{\rm{\hslash }}){\int }_{0}^{t}H(\tau ){\rm{d}}\tau ]$.

In general, the reference Hamiltonian H₀(t) may be expressed as a function of time-dependent control parameters, λ(t) = {λ₁(t), λ₂(t), ⋯} [52]. A formal representation, H₀(t) ≡ H₀(λ), is applied, where the t-dependence of λ is implicitly assumed. The same representation can be applied to the reference instantaneous eigenstates, i.e. $| n(t)\rangle \equiv | n(\lambda )\rangle$. Based on its definition in equation (2), the counter-diabatic Hamiltonian is formulated as ${H}_{\mathrm{cd}}(t)\equiv {H}_{\mathrm{cd}}(\lambda ,\dot{\lambda })$ with

$$\begin{eqnarray}&&{H}_{\mathrm{cd}}(\lambda ,\dot{\lambda })={\rm{i}}{\rm{\hslash }}\displaystyle \sum _{n}\displaystyle \sum _{\mu }{{ \mathcal P }}_{n}^{\perp }(\lambda )| {\partial }_{\mu }n(\lambda )\rangle \langle n(\lambda )| {\dot{\lambda }}_{\mu }.\end{eqnarray} \tag{ 3 }$$

The symbol ∂_μ stands for ∂/∂ λ_μ and the time derivatives, $\dot{\lambda }(t)=\{{\dot{\lambda }}_{1}(t),{\dot{\lambda }}_{2}(t),\cdots \}$, specify a designed STA protocol.

In our experiment, we study a single superconducting Xmon qubit which can be mapped onto a spin-1/2 particle driven by an external field [53]. For consistency, we will mostly take the notation of the up $(| \uparrow \rangle )$ and down $(| \downarrow \rangle )$ states rather than the equivalent ground $(| 0\rangle )$ and excited $(| 1\rangle )$ states. In the rotating frame, the time-dependent reference Hamiltonian follows a general form

$$\begin{eqnarray}&&{H}_{0}(t)={\rm{\hslash }}{{\boldsymbol{B}}}_{0}(t)\cdot {\boldsymbol{\sigma }}/2,\end{eqnarray} \tag{ 4 }$$

where ${{\boldsymbol{B}}}_{0}(t)={\rm{\Omega }}(t)(\sin \theta (t)\cos \phi (t),\sin \theta (t)\sin \phi (t),\cos \theta (t))$ is the vector of an external field and ${\boldsymbol{\sigma }}=({\sigma }_{x},{\sigma }_{y},{\sigma }_{z})$ is the vector of Pauli matrices. The time-dependent control parameters are the amplitude Ω(t), the polar angle θ(t) and the azimuthal angle ϕ(t), i.e., λ(t) = {Ω(t), θ(t), ϕ(t)}. By calculating the reference instantaneous eigenstates

$$\begin{eqnarray}&&\left\{\begin{array}{l}| {s}_{\uparrow }(t)\rangle =\cos [\theta (t)/2]| \uparrow \rangle +\sin [\theta (t)/2]{{\rm{e}}}^{{\rm{i}}\phi (t)}| \downarrow \rangle \\ | {s}_{\downarrow }(t)\rangle =-\sin [\theta (t)/2]{{\rm{e}}}^{-{\rm{i}}\phi (t)}| \uparrow \rangle +\cos [\theta (t)/2]| \downarrow \rangle \end{array}\right.\end{eqnarray} \tag{ 5 }$$

and substituting them into equation (2), we obtain an analytical expression of the counter-diabatic Hamiltonian, which reads

$$\begin{eqnarray}&&{H}_{\mathrm{cd}}(t)={\rm{\hslash }}{{\boldsymbol{B}}}_{\mathrm{cd}}(t)\cdot {\boldsymbol{\sigma }}/2.\end{eqnarray} \tag{ 6 }$$

The three elements of the counter-diabatic field ${{\boldsymbol{B}}}_{\mathrm{cd}}(t)=({B}_{\mathrm{cd};x}(t),{B}_{\mathrm{cd};y}(t),{B}_{\mathrm{cd};z}(t))$ are explicitly given by [5, 6, 19–21]

$$\begin{eqnarray}&&\left\{\begin{array}{l}{B}_{\mathrm{cd};x}(t)=-\dot{\theta }(t)\sin \phi (t)-\dot{\phi }(t)\sin \theta (t)\cos \theta (t)\cos \phi (t)\\ {B}_{\mathrm{cd};y}(t)=\dot{\theta }(t)\cos \phi (t)-\dot{\phi }(t)\sin \theta (t)\cos \theta (t)\sin \phi (t)\\ {B}_{\mathrm{cd};z}(t)=\dot{\phi }(t){\sin }^{2}\theta (t)\end{array}\right..\end{eqnarray} \tag{ 7 }$$

The total STA field is a sum of the reference and counter-diabatic fields, i.e., ${\boldsymbol{B}}(t)={{\boldsymbol{B}}}_{0}(t)+{{\boldsymbol{B}}}_{\mathrm{cd}}(t)$.

Although the non-adiabatic transition is fully suppressed in the STA protocol, the introduction of the counter-diabatic Hamiltonian ${H}_{\mathrm{cd}}(t)$ is not cost-free. For the total Hamiltonian H(t), we introduce the STA instantaneous eigenbasis $\{| {\psi }_{k}(t)\rangle \}$, which leads to

$$\begin{eqnarray}&&H(t)=\displaystyle \sum _{k}{E}_{k}(t)| {\psi }_{k}(t)\rangle \langle {\psi }_{k}(t)| \end{eqnarray} \tag{ 8 }$$

with E_k(t) the kth instantaneous eigenenergy and $| {\psi }_{k}(t)\rangle$ the associated eigenstate. For a given initial state of $| {\rm{\Psi }}(0)\rangle =| n(0)\rangle$, the probability of observing the quantum state $| {\psi }_{k}(t)\rangle$ at time t is given by

$$\begin{eqnarray}&&{P}_{k| n}(t)=| \langle {\psi }_{k}(t)| {U}_{\mathrm{STA}}(t)| n(0)\rangle {| }^{2}=| \langle {\psi }_{k}(t)| n(t)\rangle {| }^{2}.\end{eqnarray} \tag{ 9 }$$

The corresponding joint probability is written as ${P}_{{kn}}(t)={P}_{k| n}(t){P}_{n}(0)$ with P_n(0) an initial probability at the state $| n(0)\rangle$.

Next we introduce the concept of an STA work in the framework of quantum thermodynamics. The initial system is assumed to follow a canonical distribution, $\rho (0)={\sum }_{n}{P}_{n}(0)| n(0)\rangle \langle n(0)|$ with ${P}_{n}(0)\propto \exp [-\beta {\varepsilon }_{n}(0)]$. Based on the two-point measurement scheme [25, 39–41], the probability distribution function of a quantum work at time t is written as

$$\begin{eqnarray}&&P(W,t)=\displaystyle \sum _{{nk}}\delta (W-\delta {E}_{{kn}}(t)){P}_{{kn}}(t)\end{eqnarray} \tag{ 10 }$$

with $\delta {E}_{{kn}}(t)={E}_{k}(t)-{\varepsilon }_{n}(0)$. In equation (10), we implicitly assume that the initial counter-diabatic Hamiltonian is zero, i.e., ${H}_{\mathrm{cd}}(0)=0$ [32]. The general mth moment of the quantum work is defined as

$$\begin{eqnarray}&&\langle {W}^{m}(t)\rangle =\int {W}^{m}P(W,t){\rm{d}}{W},\end{eqnarray} \tag{ 11 }$$

which can be rewritten as

$$\begin{eqnarray}&&\langle {W}^{m}(t)\rangle =\displaystyle \sum _{n}\overline{{W}_{n}^{m}(t)}{P}_{n}(0).\end{eqnarray} \tag{ 12 }$$

The quantity, $\overline{{W}_{n}^{m}(t)}={\sum }_{k}{[{E}_{k}(t)-{\varepsilon }_{n}(0)]}^{m}{P}_{k| n}(t)$, is the mth moment of the quantum work for the initial eigenstate $| n(0)\rangle$. This term is independent of the initial distribution but fully determined by the designed STA protocol [32]. In this paper, we will present an experimental investigation of $\overline{{W}_{n}^{m}(t)}$ instead of the statistics of the total work $\langle {W}^{m}(t)\rangle$.

In an adiabatic process, the conditional probabilities satisfy ${P}_{k| n}(t)={\delta }_{k,n}$ and the mth moment is simplified to be $\overline{{W}_{\mathrm{ad};n}^{m}(t)}={[{\varepsilon }_{n}(t)-{\varepsilon }_{n}(0)]}^{m}$. In the STA protocol, this quantity is changed due to an quantum uncertainty induced by ${H}_{\mathrm{cd}}(t)$. Here we focus on the first and second moments, which are related to the average work and its variance. Through a straightforward derivation (see the appendix), we obtain the first equality [32]

$$\begin{eqnarray}&&\overline{{W}_{n}(t)}=\overline{{W}_{\mathrm{ad};n}(t)}+{\rm{i}}{\rm{\hslash }}\langle n(t)| {{ \mathcal P }}_{n}^{\perp }| {\partial }_{t}n(t)\rangle .\end{eqnarray} \tag{ 13 }$$

The second term on the right-hand side (rhs) of equation (13) vanishes due to an orthogonal projection, $\langle n(t)| {{ \mathcal P }}_{n}^{\perp }=0$. In the STA protocol, the preservation of the adiabatic trajectory, $| n(0)\rangle \to | n(t)\rangle$, is represented alternatively by the conservation of the average work, i.e.

$$\begin{eqnarray}&&\overline{{W}_{n}(t)}=\overline{{W}_{\mathrm{ad};n}(t)}.\end{eqnarray} \tag{ 14 }$$

However, the quantum uncertainty created in the STA instantaneous eigenbasis $\{| {\psi }_{k}(t)\rangle \}$ cannot be cancelled in the second order moment of the STA work. Through another straightforward derivation (see the appendix), we obtain the second equality [32]

$$\begin{eqnarray}&&\overline{{W}_{n}^{2}(t)}=\overline{{W}_{\mathrm{ad};n}^{2}(t)}+{{\rm{\hslash }}}^{2}\langle {\partial }_{t}n(t)| {{ \mathcal P }}_{n}^{\perp }| {\partial }_{t}n(t)\rangle .\end{eqnarray} \tag{ 15 }$$

With the introduction of the control parameter set λ(t), equation (15) is rewritten as

$$\begin{eqnarray}&&\delta \overline{{W}_{n}^{2}(t)}={{\rm{\hslash }}}^{2}\displaystyle \sum _{\mu ,\nu }{g}_{\mu \nu }^{(n)}{\dot{\lambda }}_{\mu }{\dot{\lambda }}_{\nu },\end{eqnarray} \tag{ 16 }$$

where $\delta \overline{{W}_{n}^{2}(t)}=\overline{{W}_{n}^{2}(t)}-\overline{{W}_{\mathrm{ad};n}^{2}(t)}$ denotes the STA excess of work fluctuations. The term, ${g}_{\mu \nu }^{(n)}=\mathrm{Re}\,{Q}_{\mu \nu }^{(n)}=\mathrm{Re}\,\langle {\partial }_{\mu }n(\lambda )| {{ \mathcal P }}_{n}^{\perp }| {\partial }_{\nu }n(\lambda )\rangle$, is the real part of the quantum geometric tensor of the $| n(t)\rangle$-state manifold defined in the parameter space of λ [32, 54]. The time dependence on the rhs of equation (16) arises from the varying speed of the control parameters. Since the eigenvalues of the ${g}_{\mu \nu }^{(n)}$-tensor are always equal or greater than zero, the STA work variance is always equal or greater than that under the adiabatic condition, i.e.,

$$\begin{eqnarray}&&\overline{{W}_{n}^{2}(t)}\geqslant \overline{{W}_{\mathrm{ad};n}^{2}(t)}.\end{eqnarray} \tag{ 17 }$$

For our spin-1/2 particle, the instantaneous eigenenergies of the STA Hamiltonian H(t) are ${E}_{\pm }(t)=\pm {\rm{\hslash }}| {\boldsymbol{B}}(t)| /2$ and the associated eigenstates are denoted as $| {\psi }_{\pm }(t)\rangle$. Accordingly, the four conditional probabilities involved are ${P}_{\pm | \uparrow }(t)=| \langle {\psi }_{\pm }(t)| {U}_{\mathrm{STA}}(t)| {s}_{\uparrow }(0){\rangle }^{2}$ and ${P}_{\pm | \downarrow }(t)=\langle {\psi }_{\pm }(t)| {U}_{\mathrm{STA}}(t)| {s}_{\downarrow }(0){\rangle }^{2}$. Since the reference instantaneous eigenstates are independent of the amplitude Ω(t), a two-element set, λ(t) ={θ(t), ϕ(t)}, is used to calculate the geometric tensor [32, 54]

$$\begin{eqnarray}{g}^{\uparrow }(\lambda )={g}^{\downarrow }(\lambda )=\displaystyle \frac{1}{4}\left(\begin{array}{cc}1 & 0\\ 0 & {\sin }^{2}\theta \end{array}\right).\end{eqnarray} \tag{ 18 }$$

In our experiment, the STA protocol is designed to compress a target adiabatic process evenly through time. With an operation time T, the reference field is defined by the same form of ${{\boldsymbol{B}}}_{0}(\tilde{t}=t/T)$. Consequently, we obtain the STA excess of work fluctuations in the single-qubit system as

$$\begin{eqnarray}&&\delta \overline{{W}_{n}^{2}(t)}=\displaystyle \frac{{{\rm{\hslash }}}^{2}}{4{T}^{2}}\left[{\left(\displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}\tilde{t}}\right)}^{2}+{\sin }^{2}\theta {\left(\displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}\tilde{t}}\right)}^{2}\right],\end{eqnarray} \tag{ 19 }$$

which shows an inverse square dependence of the STA operation time T.

The experimental verification of STA work fluctuations requires a measurement scheme in the instantaneous eigenbasis of the total Hamiltonian H(t). Here we design two different sequences to detect the eigenenergies and the population of eigenstates separately.

The eigenenergy measurement scheme is shown in figure 3(a). A π/2-pulse is initially applied to the up-state (equivalent to the ground state) to create a superposition state $(| \uparrow \rangle +| \downarrow \rangle )/\sqrt{2}$, which is then driven by the STA field ${\boldsymbol{B}}(t)$. At an intermediate time τ_m ($0\leqslant {\tau }_{m}\leqslant T$), the original external field ${\boldsymbol{B}}(t)$ is interrupted and becomes fixed at ${\boldsymbol{B}}({\tau }_{m})$. The constant field is sustained for an extra time duration of τ_d, leading to ${\boldsymbol{B}}({\tau }_{m}\leqslant t\leqslant {\tau }_{m}+{\tau }_{d})\,={\boldsymbol{B}}({\tau }_{m})$ and $H({\tau }_{m}\leqslant t\leqslant {\tau }_{m}+{\tau }_{d})={\rm{\hslash }}{\boldsymbol{B}}({\tau }_{m})\cdot {\boldsymbol{\sigma }}/2$. This pulse design is named as the frozen-Hamiltonian scheme accordingly. Since the quantum state at time τ_m is a linear combination of two instantaneous eigenstates, $| {\rm{\Psi }}({\tau }_{m})\rangle ={c}_{+}({\tau }_{m})| {\psi }_{+}({\tau }_{m})\rangle +{c}_{-}({\tau }_{m})| {\psi }_{-}({\tau }_{m})\rangle$, a quantum oscillation is expected in the frozen period, where the oscillation frequency is determined by ${E}_{+}({\tau }_{m})-{E}_{-}({\tau }_{m})=2{E}_{+}({\tau }_{m})$. This frozen-Hamiltonian measurement can experimentally detect the instantaneous eigenenergies, E₊(τ_m) and E₋(τ_m) [57]. In figure 3(a), a Ramsey-type sequence is used to enhance the amplitude of quantum oscillations despite the fact that an arbitrary initial state is allowed. For T = 25 ns, figure 3(b) presents the experimental measurement of the up-state population ${P}_{\uparrow }({\tau }_{m}+{\tau }_{d})$ with the change of the interruption time τ_m and the duration time τ_d. The time interval of τ_m is 1 ns and a total time range of 1 μs is swept for τ_d. For each value of τ_m, the measured population ${P}_{\uparrow }({\tau }_{m}+{\tau }_{d})$ periodically oscillates with τ_d and the oscillation frequency varies with τ_m. The experimentally extracted results of E_±(τ_m) are plotted in figure 3(c). An excellent agreement is observed between the experimental measurement and the theoretical design. Compared to the adiabatic increase of ${\varepsilon }_{\uparrow }({\tau }_{m})/h$ from 5 MHz to 10 MHz, the STA protocol with T = 25 ns induces a bump of 8 MHz around τ_m ≈ 16 ns for the instantaneous eigenenergy ${E}_{+}({\tau }_{m})/h$, where h is the Planck constant.

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Figure 3(d) demonstrates the scheme of measuring the population of the two STA instantaneous eigenstates. The qubit is initialized at the up-state that is an initial eigenstate, i.e., $| {s}_{\uparrow }(0)\rangle =| \uparrow \rangle$, and then driven by the STA field ${\boldsymbol{B}}(t)$. At each intermediate time τ_m, the external field ${\boldsymbol{B}}(t)$ is suddenly interrupted and replaced by a different slowly varying field ${{\boldsymbol{B}}}_{0}^{{\prime} }(t)$, which satisfies ${{\boldsymbol{B}}}_{0}^{{\prime} }(t\lt {\tau }_{m})=0$ and ${{\boldsymbol{B}}}_{0}^{{\prime} }({\tau }_{m})={\boldsymbol{B}}({\tau }_{m})$. At the starting point (t = τ_m) of this extra adiabatic trajectory, $| {\psi }_{+}({\tau }_{m})\rangle$ and $| {\psi }_{-}({\tau }_{m})\rangle$ are the eigenstates, and their populations are given by ${P}_{+| \uparrow }({\tau }_{m})=| {c}_{+}({\tau }_{m}){| }^{2}$ and ${P}_{-| \uparrow }({\tau }_{m})=| {c}_{-}({\tau }_{m}){| }^{2}$. Accordingly, the populations of two instantaneous eigenstates are fixed at ${P}_{\pm | \uparrow }({\tau }_{m})$ over time. At the end point ($t={\tau }_{m}+{T}^{{\prime} }$) of the adiabatic trajectory, the final external field is ramped back to the z-direction and the eigenstates become $| {\psi }_{+}({\tau }_{m}+{T}^{{\prime} })\rangle =| \uparrow \rangle$ and $| {\psi }_{-}({\tau }_{m}+{T}^{{\prime} })\rangle =| \downarrow \rangle$. The final population measured at the up and down states thus provides an experimental determination of ${P}_{\pm | \uparrow }({\tau }_{m})$ [37]. This pulse design is named as the frozen-population measurement scheme. To avoid a non-adiabatic error, the STA protocol is used practically with ${{\boldsymbol{B}}}^{{\prime} }(t)={{\boldsymbol{B}}}_{0}^{{\prime} }(t)+{{\boldsymbol{B}}}_{\mathrm{cd}}^{{\prime} }(t)$. A great flexibility is allowed in the functional form of ${{\boldsymbol{B}}}_{0}^{{\prime} }(t)$. In our experiment, we choose a geodesic line for the reference adiabatic trajectory of ${{\boldsymbol{B}}}_{0}^{{\prime} }(t)/| {{\boldsymbol{B}}}_{0}^{{\prime} }(t)|$ and this second STA operation time is set at ${T}^{{\prime} }=100\,\mathrm{ns}$. The measured result of ${P}_{\pm | \uparrow }({\tau }_{m})$ for T = 25 ns is plotted in figure 3(e), showing the same excellent agreement with the theoretical prediction. The quantum uncertainty, ${P}_{+| \uparrow }({\tau }_{m})\lt 1$ and ${P}_{-| \uparrow }({\tau }_{m})\gt 0$, in the STA operation is observed through time. The maximum uncertainty of ∼20% appears around τ_m ≈ 16 ns. We perform the experimental measurement of ${P}_{\pm | \downarrow }({\tau }_{m})$ by applying a π-pulse to the up-state qubit followed by the same frozen-population approach (see figure 3(d)). Since the down state is equivalent to the excited state of the qubit, an error is accumulated in the experimental measurement of ${P}_{\pm | \downarrow }({\tau }_{m})$ due to inevitable quantum dissipation of T₁ and ${T}_{2}={[{(2T)}_{1}^{-1}+{({T}_{2}^{* })}^{-1}]}^{-1}$ (see figure 3(f)). This dissipation-induced error is gradually increased with the increase of the operation time T.

The measurement of the STA instantaneous eigenenergies, E_±(τ_m), and the conditional probabilities, ${P}_{\pm | \uparrow }({\tau }_{m})$ and ${P}_{\pm | \downarrow }({\tau }_{m})$, allows us to obtain experimental statistics of the STA work. In our experiment, the STA operation time is set at T = 25, 50, 100, 200 and 500 ns. The STA evolution with T = 500 ns is very close to the adiabatic passage since ${{\boldsymbol{B}}}_{\mathrm{cd}}(t)$ is nearly negligible.

The first moment of the STA work at each measurement time τ_m is experimentally determined by $\overline{{W}_{n}({\tau }_{m})}\,={\sum }_{k}[{E}_{k}({\tau }_{m})-{\varepsilon }_{n}(0)]{P}_{k| n}({\tau }_{m})$ for both initial conditions, $| n(0)\rangle =| {s}_{\uparrow }(0)\rangle$ and $| {s}_{\downarrow }(0)\rangle$. For a given operation time T, we rescale the measurement time by ${\tilde{t}}_{m}={\tau }_{m}/T$ and present the result of $\overline{{W}_{n}({\tilde{t}}_{m})}$ as a function of the reduced time ${\tilde{t}}_{m}$. Since the reference external field with different values of T follows the same form of ${{\boldsymbol{B}}}_{0}(\tilde{t}=t/T)$, the rescaled time provides a transparent comparison on the influence of the STA operation time. As shown in figures 4(a) and (b), the data of $\overline{{W}_{n}({\tilde{t}}_{m})}$ with different T collapse to a single curve for each initial condition, which is consistent with the theoretical prediction in equation (14) [32]. With the designed reference field in equation (20), the averaged STA work follows the adiabatic behavior of $\overline{{W}_{n}({\tilde{t}}_{m})}=\overline{{W}_{\mathrm{ad};n}({\tilde{t}}_{m})}=\pm [{\rm{\Omega }}({\tilde{t}}_{m})-{\rm{\Omega }}(0)]/2$. A systematic error is found for $\overline{{W}_{\downarrow }({\tilde{t}}_{m})}$, especially with T = 500 ns, which is due to the quantum dissipation of the qubit (see section 4.1).

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The second moment of the STA work is experimentally determined by $\overline{{W}_{n}^{2}({\tau }_{m})}={\sum }_{k}{[{E}_{k}({\tau }_{m})-{\varepsilon }_{n}(0)]}^{2}{P}_{k| n}({\tau }_{m})$. The experimental results of $\overline{{W}_{\uparrow }^{2}({\tilde{t}}_{m})}$ and $\overline{{W}_{\downarrow }^{2}({\tilde{t}}_{m})}$ are shown in figures 4(c) and (d), respectively. Different from the behavior of the averaged STA work, the work variance increases monotonically with the decrease of the STA operation time. Since the STA operation with T = 500 ns can mimic the adiabatic process, we thus experimentally verifies the inequality of the STA work fluctuations, $\overline{{W}_{n}^{2}({\tilde{t}}_{m})}\geqslant \overline{{W}_{\mathrm{ad};n}^{2}({\tilde{t}}_{m})}$ [32]. The equality is obtained at the initial and final moments where the counter-diabatic field vanishes, given by ${{\boldsymbol{B}}}_{\mathrm{cd}}(0)={{\boldsymbol{B}}}_{\mathrm{cd}}(T)=0$. As a comparison, both $\overline{{W}_{\uparrow }^{2}({\tilde{t}}_{m})}$ and $\overline{{W}_{\downarrow }^{2}({\tilde{t}}_{m})}$ agree excellently with the theoretical prediction for T = 25, 50 and 100 ns. As the operation time is further increased to T = 200 and 500 ns, the dissipation-induced error deviates the experimental result of $\overline{{W}_{\downarrow }^{2}({\tilde{t}}_{m})}$ away from the theoretical prediction.

Based on the theoretical design, the STA excess of work fluctuations, $\delta \overline{{W}_{n}^{2}(t)}=\overline{{W}_{n}^{2}(t)}-\overline{{W}_{\mathrm{ad};n}^{2}(t)}$, is fully determined by the functional form of ${H}_{\mathrm{cd}}(t)$. From an alternative perspective of differential geometry, ${H}_{\mathrm{cd}}(t)$ guides a parallel transport of the reference instantaneous eigenstate $| n(t)\rangle$ by satisfying $\langle n(t)| {H}_{\mathrm{cd}}(t)| n(t)\rangle =0$ [32, 58]. With a geometric tensor, ${g}_{\mu \nu }^{(n)}=\mathrm{Re}\langle {\partial }_{\mu }n(\lambda )| {P}_{n}^{\perp }| {\partial }_{\nu }n(\lambda )\rangle$, defined in the space of control parameters λ, the counter-diabatic Hamiltonian is characterized by $\dot{\lambda }$. Therefore, it is straightforward to expect a connection between the STA excess of work fluctuations and a quantum geometric quantity. As shown in section 2.2, this connection is described by the equality in equation (16) [32], which is further simplified to be equation (19) for a single-qubit system.

To verify equation (19), we first experimentally determine the STA excess of work fluctuations with the operation time T. Since the second moments for both initial eigenstates are theoretically identical, only $\overline{{W}_{\uparrow }^{2}({\tilde{t}}_{m})}$ are inspected to reduce the dissipation-induced error. For each operation time (T = 25, 50, 100 and 200 ns), the adiabatic reference is taken from the result of T = 500 ns, giving

$$\begin{eqnarray}&&\delta \overline{{W}_{\uparrow }^{2}({\tilde{t}}_{m})}=\overline{{W}_{\uparrow }^{2}({\tilde{t}}_{m};T)}-\overline{{W}_{\uparrow }^{2}({\tilde{t}}_{m};T=500\,\mathrm{ns})}.\end{eqnarray} \tag{ 22 }$$

Equation (19) shows that $\delta \overline{{W}_{\uparrow }^{2}({\tilde{t}}_{m})}$ is inversely proportional to the square of the operation time. Accordingly, we plot the rescaled value, ${T}^{2}\overline{\delta {W}_{\uparrow }^{(2)}({\tilde{t}}_{m})}$, in figure 5(a). Consistent with the theoretical prediction, all the data of ${T}^{2}\overline{\delta {W}_{\uparrow }^{(2)}({\tilde{t}}_{m})}$ fall into a single curve. The amplified error with T = 200 ns is due to a very small difference between $\overline{{W}_{\uparrow }^{2}({\tilde{t}}_{m};T=200\,\mathrm{ns})}$ and $\overline{{W}_{\uparrow }^{2}({\tilde{t}}_{m};T=500\,\mathrm{ns})}$.

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Next we experimentally probe the quantum geometric quantity on the rhs of equation (19). For each operation time, the STA evolution is interrupted frequently with an time interval of $\delta {\tilde{t}}_{m}T=0.02T$ and the qubit state is measured in the fixed frame of $\{| \uparrow \rangle ,| \downarrow \rangle \}$ by the quantum state tomography. The extracted parameters, $\{{r}_{q}({\tilde{t}}_{m}),{\theta }_{q}({\tilde{t}}_{m}),{\phi }_{q}({\tilde{t}}_{m})\}$ of the qubit vector describe the qubit evolution in response to the STA field. In an ideal scenario, the qubit initialized at the up-state $| {s}_{\uparrow }(0)\rangle$ follows the same evolution of ${{\boldsymbol{B}}}_{0}(t)/| {{\boldsymbol{B}}}_{0}(t)|$, giving ${\theta }_{q}({\tilde{t}}_{m})=\theta ({\tilde{t}}_{m})$ and ${\phi }_{q}({\tilde{t}}_{m})=\phi ({\tilde{t}}_{m})$. For each operation time, these two equalities are confirmed in figures 5(b) and (c). Notice that a small measurement error in the off-diagonal element ${\rho }_{\uparrow \downarrow }({\tilde{t}}_{m})$ can lead to an amplified error in the phase determination if ${\theta }_{q}({\tilde{t}}_{m})$ is around zero. The trajectory of the qubit vector allows us to experimentally estimate the time derivatives,

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{x}({\tilde{t}}_{m})}{{\rm{d}}\tilde{t}}=\displaystyle \frac{x({\tilde{t}}_{m}+\delta {\tilde{t}}_{m})-x({\tilde{t}}_{m}-\delta {\tilde{t}}_{m})}{2\delta {\tilde{t}}_{m}}\end{eqnarray} \tag{ 23 }$$

with $x({\tilde{t}}_{m})={\theta }_{q}({\tilde{t}}_{m})$ and ${\phi }_{q}({\tilde{t}}_{m})$. The geometric quantity in equation (19) is calculated as

$$\begin{eqnarray}&&{\left[\displaystyle \frac{{\rm{d}}{l}({\tilde{t}}_{m})}{{\rm{d}}{\tilde{t}}_{m}}\right]}^{2}=\displaystyle \frac{1}{4}\left[{\left(\displaystyle \frac{{\rm{d}}{\theta }_{q}({\tilde{t}}_{m})}{{\rm{d}}\tilde{t}}\right)}^{2}+{\sin }^{2}{\theta }_{q}({\tilde{t}}_{m}){\left(\displaystyle \frac{{\rm{d}}{\phi }_{q}({\tilde{t}}_{m})}{{\rm{d}}\tilde{t}}\right)}^{2}\right].\end{eqnarray} \tag{ 24 }$$

As shown in figure 5(d), the data of ${[{\rm{d}}{l}({\tilde{t}}_{m})/{\rm{d}}{\tilde{t}}_{m}]}^{2}$ with different operation times also fall into a single curve from the theoretical prediction, although experimental errors are found. Through the combination of the above two measurements, we experimentally verify the general equality connecting the STA excess of the work fluctuations and the quantum geometric quantity in a simple quantum system of a single-qubit. For an initial Boltzmann distribution, ${P}_{\uparrow }(0)\propto \exp (-\beta {\varepsilon }_{\uparrow })$ and ${P}_{\downarrow }(0)\propto \exp (-\beta {\varepsilon }_{\downarrow })$, the equalities of STA work fluctuations lead to an energy-time uncertainty relation [32]. The verification requires an initial preparation of a mixed state, which can be realized in a future multi-qubit experiment.