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

In a `shortcut-to-adiabaticity' (STA) protocol, the counter-diabatic Hamiltonian, which suppresses the non-adiabatic transition of a reference `adiabatic' trajectory, induces a quantum uncertainty of the work cost in the framework of quantum thermodynamics. Following a theory derived recently [Funo et al 2017 Phys. Rev. Lett. 118 100602], we perform an experimental measurement of the STA work statistics in a high-quality superconducting Xmon qubit. Through the frozen-Hamiltonian and frozen-population techniques, we experimentally realize the two-point measurement of the work distribution for given initial eigenstates. Our experimental statistics verify (i) the conservation of the average STA work and (ii) the equality between the STA excess of work fluctuations and the quantum geometric tensor.


I. INTRODUCTION
The fast growing man-made quantum devices have demonstrated the potential for quantum computation, quantum information processing and quantum simulation [1]. Among many quantum algorithms, the utilization of an adiabatic trajectory has attracted a lot of interest in various problems [2][3][4]. However, the bottleneck of an adiabatic quantum operation is its inevitable dissipation-induced error accumulated in a slow process. The 'shortcut-to-adiabaticity' (STA) is a theoretical protocol to speed-up the adiabatic operation [5][6][7][8][9][10][11][12][13]]. An additional counter-diabatic Hamiltonian is introduced to suppress the non-adiabatic transition and the system undergoes a reference 'adiabatic' trajectory in a short time scale. Soon after the theoretical proposal, experimental implementation has been executed in various quantum devices [14][15][16][17][18][19][20][21]. In our recent studies, we have applied the STA protocol in a superconducting qubit for the realization of a Berry phase measurement [19], quantum state transfer [20] and high-fidelity gates [21]. The STA protocol was also used by us to simulate topological phase transition through an experimental construction of the first Brillouin zone [20].
From an energy perspective, questions about the thermodynamic nature of quantum systems have been raised [22][23][24][25], in parallel with development of fluctuation theorems in small-scale systems [26,27]. The statistical uncertainty of a quantum system can be separated into three categories: an initial distribution from a pre-thermalization, a random force from a surrounding environment, and the intrinsic uncertainty of quantum mechanics. To avoid the distinction between two basic thermodynamic variables, work and heat, many theoretical proposals and experimental implementations have been restricted in closed systems without quantum heat transfer between the system and the bath [28][29][30][31][32][33][34][35][36][37][38]. To account the remaining two sources of uncertainty, the two-point measurement scheme has been proposed for the distribution of quantum work cost [24,25,[39][40][41]. The probability of an accessible work is a product of the probability of a certain initial eigenstate and the conditional probability from this initial eigenstate to an instantaneous eigenstate at the measurement time. Despite a simple theoretical formulation, the experimental determination of the work distribution is a nontrivial task [33][34][35][36][37][38].
In an ideal adiabatic evolution, the intrinsic quantum uncertainty is removed as the system propagates along an adiabatic trajectory of instantaneous eigenstates, and the work distribution is fully determined by the initial distribution. The accelerated STA operation has been proposed to engineer friction-free quantum machines [42][43][44]. However, the exact instantaneous eigenbasis is rotated away from the reference adiabatic basis due to the introduction of the counter-diabatic Hamiltonian. Therefore, each STA protocol is physically characterized by its deviation of the work statistics, which is originated from the quantum uncertainty of the counter-diabatic Hamiltonian. In a recent paper [32], the statistics of the STA work were theoretically resolved. An equality connecting the STA excess of work fluctuations and the quantum geometric tensor was derived. The experimental verification of this STA work theory was proposed for a harmonic oscillator in a trapped ion system. Alternatively, high-quality qubits with sufficiently long quantum coherence have been achieved in a superconducting circuit [45][46][47][48][49][50]. Sophisticated microwave control techniques allow us to precisely manipulate and measure a superconducting qubit system. In this paper, we drive a single superconducting Xmon qubit with an STA field and experimentally determine the statistics of the STA work for a given initial eigenstate. To realize the two-point measurement scheme, we design the frozen-Hamitlonian and frozen-population sequences to extract the instantaneous eigenenergies and the population of instantaneous eigenstates, respectively. Our study thus reports an experimental verification of the STA work theory in Ref. [32].

II. THEORY
In this section, we will briefly review the theory of STA work fluctuations in Ref. [32] and apply it to the quantum system of a single qubit.

A. STA Protocol
For an arbitrary non-degenerate quantum system driven by a time-dependent Hamiltonian H 0 (t), we introduce its instantaneous eigenbasis {|n(t) } satisfying H 0 (t)|n(t) = ε n (t)|n(t) , where ε n (t) is the n-th instantaneous eigenenergy and |n(t) is the associated eigenstate. The Hamiltonian is thus expanded in this instantaneous eigenbasis, becoming In the scenario of d t H 0 (t) → 0, the system follows an adiabatic trajectory, |n(0) → |n(t) , if it is prepared at |n(0) initially. The quantum state at time t is given by t 0 H 0 (τ )dτ ] is the adiabatic time evolution operator and the coefficient c n (t) includes the accumulation of both dynamic and geometric phases [51]. Here T + denotes the time ordering operator and = h/2π is the reduced Planck constant.
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 [52]. For consistency, we will mostly take the notation of the up (|↑ ) and down (|↓ ) states rather than the equivalent ground (|0 ) and excited (|1 ) states. In the rotating frame, the time-dependent reference Hamiltonian follows a general form, where B 0 (t) = Ω(t)(sin θ(t) cos φ(t), sin θ(t) sin φ(t), cos θ(t)) is the vector of an external field and σ = (σ x , σ y , σ 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, and substituting them into Eq. (2), we obtain an analytical expression of the counter-diabatic Hamitlonian, which reads The three elements of the counter-diabatic field B cd (t) = (B cd;x (t), B cd;y (t), B cd;z (t)) are explicitly given by [5,6,[19][20][21]] The total STA field is a sum of the reference and counter-diabatic fields, i.e., B(t) = B 0 (t) + B cd (t).

B. Statistics of STA Work
Although the non-adiabatic transition is fully suppressed in the STA protocol, the introduction of the counter-diabatic Hamiltonian H cd (t) is not cost-free. For the total Hamiltonian H(t), we introduce the STA instantaneous eigenbasis {|ψ k (t) }, which leads to with E k (t) the k-th instantaneous eigenenergy and |ψ k (t) the associated eigenstate. For a given initial state of |Ψ(0) = |n(0) , the probability of observing the quantum state |ψ k (t) at time t is given by 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) .
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, ρ(0) = n P n (0)|n(0) n(0)| with P n (0) ∝ exp[−βε n (0)]. Based on the two-point measurement scheme [25,[39][40][41], the probability distribution function of a quantum work at time t is written as with δE kn (t) = E k (t) − ε n (0). In Eq. (10), we implicitly assume that the initial counterdiabatic Hamiltonian is zero, i.e., H cd (0) = 0 [32]. The general m-th moment of the quantum work is defined as which can be rewritten as The quantity, W m , is the m-th moment of the quantum work for the initial eigenstate |n(0) . 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 W m n (t) instead of the statistics of the total work W m (t) . In an adiabatic process, the conditional probabilities satisfy P k|n (t) = δ k,n and the m-th moment is simplified to be W m ad;n (t) = [ε n (t) − ε n (0)] m . In the STA protocol, this quantity is changed due to an quantum uncertainty induced by H 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 Appendix A), we obtain the first equality [32], The second term on the right-hand side (RHS) of Eq. (13) vanishes due to an orthogonal projection, n(t)|P ⊥ n = 0. In the STA protocol, the preservation of the adiabatic trajectory, |n(0) → |n(t) , is represented alternatively by the conservation of the average work, i.e., However, the quantum uncertainty created in the STA instantaneous eigenbasis {|ψ k (t) } cannot be cancelled in the second order moment of the STA work. Through another straightforward derivation (see Appendix A), we obtain the second equality [32], With the introduction of the control parameter set λ(t), Eq. (15) is rewritten as where δW 2 n (t) = W 2 n (t) − W 2 ad;n (t) denotes the STA excess of work fluctuations. The term, g µν -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., For our spin-1/2 particle, the instantaneous eigenenergies of the STA Hamiltonian H(t) are E ± (t) = ± |B(t)|/2 and the associated eigenstates are denoted as |ψ ± (t) . Accordingly, the four conditional probabilities involved are P ±|↑ (t) = | ψ ± (t)|U STA (t)|s ↑ (0) 2 and 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 [32,53], 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 B 0 (t = t/T ). Consequently, we obtain the STA excess of work fluctuations in the single-qubit system as which shows an inverse square dependence of the STA operation time T .

III. EXPERIMENTAL SETUP
Our study of the STA work is performed in a cross-shaped Xmon qubit [46][47][48], with the same experimental setup as in Ref. [21]. Through the bottom arm of the cross (see for a high fidelity measurement [54,55]. Compared with our previous experiment [21], the qubit coherence at the sweet point is further improved with a relaxation time of T 1 = 22 µs and a pure decoherence time of T * 2 = 64 µs (see Figs. 1(b) and 1(c)). The qubit device is measured in a dilution refrigerator whose base temperature is ∼ 10 mK.
The total STA field supplied in our experiment is given by B(t) = B 0 (t) + B cd (t). Equation (21) shows that the counter-diabatic field vanishes at the initial and final moments, i.e., around t ≈ 16 ns), a large difference is found between B 0 (t) and B(t), due to the acceleration of adiabaticity in a short operation time. With the increase of T , the amplitude of the counter-diabatic field is decreased and the adiabatic limit of B(t) ≈ B 0 (t) is gradually approached.

A. Frozen-Hamiltonian and Frozen-Population Measurements in the Instantaneous Eigenbasis
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 Fig. 3(a). A π/2-pulse is initially applied to the up state (equivalent to the ground state) to create a superposition state  [56]. In Fig. 3(a), a Ramseytype sequence is used to enhance the amplitude of quantum oscillations despite the fact that an arbitrary initial state is allowed. For T = 25 ns, Fig. 3  showing the same excellent agreement with the theoretical prediction. The quantum uncertainty, P +|↑ (τ m ) < 1 and P −|↑ (τ m ) > 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 ±|↓ (τ m ) by applying a π-pulse to the up-state qubit followed by the same frozen-population approach (see Fig. 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 ±|↓ (τ m ) due to inevitable quantum dissipation of T 1 and Fig. 3(f)). This dissipation-induced error is gradually increased with the increase of the operation time T .

B. The First and Second Moments of STA Work
The measurement of the STA instantaneous eigenenergies, E ± (τ m ), and the conditional probabilities, P ±|↑ (τ m ) and P ±|↓ (τ 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 B cd (t) is nearly negligible.
The first moment of the STA work at each measurement time τ m is experimentally deter- and |s ↓ (0) . For a given operation time T , we rescale the measurement time byt m = τ m /T and present the result of W n (t m ) as a function of the reduced timet m . Since the reference external field with different values of T follows the same form of B 0 (t = t/T ), the rescaled time provides a transparent comparison on the influence of the STA operation time. As shown in Figs. 4(a) and 4(b), the data of W n (t m ) with different T collapse to a single curve for each initial condition, which is consistent with the theoretical prediction in Eq. (14) [32].
With the designed reference field in Eq. (20), the averaged STA work follows the adiabatic Based on the theoretical design, the STA excess of work fluctuations, δW 2 n (t) = W 2 n (t) − W 2 ad;n (t), is fully determined by the functional form of H cd (t). From an alternative perspective of differential geometry, H cd (t) guides a parallel transport of the reference instantaneous eigenstate |n(t) by satisfying n(t)|H cd (t)|n(t) = 0 [32,57]. With a geometric tensor, g (n) µν = Re ∂ µ n(λ)|P ⊥ n |∂ ν n(λ) , defined in the space of control parameters λ, the counterdiabatic Hamiltonian is characterized byλ. Therefore, it is straightforward to expect a connection between the STA excess of work fluctuations and a quantum geometric quantity.  As shown in Sec. II B, this connection is described by the equality in Eq. (16) [32], which is further simplified to be Eq. (19) for a single-qubit system.
Equation (19) shows that δW 2 ↑ (t m ) is inversely proportional to the square of the operation time. Accordingly, we plot the rescaled value, T 2 δW with x(t m ) = θ q (t m ) and φ q (t m ). The geometric quantity in Eq. (19) is calculated as As shown in Fig. 5(d), the data of [dl(t m )/dt 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.

V. SUMMARY
In this paper, we apply a superconducting Xmon qubit to experimentally verify statistics of 'shortcut to adiabaticity' (STA) work theoretically proposed in Ref. [32]. The counter- Hamiltonian, i.e., W n (t) = W ad;n (t); (2) the energy cost of H cd (t) is reflected by an increased work fluctuation, i.e., W 2 n (t) ≥ W 2 ad;n (t); (3) the STA excess of work fluctuations, δW 2 n (t) = W 2 n (t) − W 2 ad;n (t), is connected with the quantum geometric tensor of the STA protocol, δW 2 n (t) = 2 µν g (n) µνλµλν . In the case of an evenly time compression, H 0 = H 0 (t = t/T ), the relation of δW 2 n (t) ∝ T −2 is verified for the STA excess of work fluctuations. This paper demonstrates the experimental availability of exploring quantum thermodynamics with a high-quality superconducting Xmon qubit device. of the total Hamiltonian H(t) = H 0 (t) + H cd (t) satisfies m m(t)|ψ k (t) ε m (t)|m(t) + i P ⊥ m (t)|∂ t m(t) = E k (t)|ψ k (t) , and [E k (t) − ε n (t)] n(t)|ψ k (t) = i m n(t)|P ⊥ m (t)|∂ t m(t) .
For the first moment of the STA work, its deviation from the adiabatic result is given by By substituting Eq. (A2) into Eq. (A3), we obtain W n (t) − W ad;n (t) = i k m n(t)|P ⊥ m (t)|∂ t m(t) m(t)|ψ k (t) ψ k (t)|n(t) = i n(t)|P ⊥ n (t)|∂ t n(t) , where the unitary operator, I = k |ψ k (t) ψ k (t)|, and the orthogonality, m(t)|n(t) = δ m,n , are used. With the orthogonal projection, n(t)|P ⊥ n (t) = 0, the conservation of the average STA work in Eq. (14) is derived. For the second moment of the STA work, its deviation from the adiabatic result is given by W 2 n (t) − W 2 ad;n (t) = k [E k (t) − ε n (t)] 2 P k|n (t) = k [E k (t) − ε n (t)] 2 n(t)|ψ k (t) ψ k (t)|n(t) .