Shortcuts to adiabatic cat-state generation in bosonic Josephson junctions

Counter-diabatic driving in non-degenerate systems was developed by Demirplak and Rice [66–68] and by Berry [69], independently. We consider a time-dependent Hamiltonian with the non-degenerate eigenstates

$$\begin{eqnarray}&&{{ \mathcal H }}_{0}(t)=\displaystyle \sum _{n}{E}_{n}(t)| n(t)\rangle \langle n(t)| ,\end{eqnarray} \tag{ 9 }$$

where ${E}_{n}(t)$ is the eigen-energy and $| n(t)\rangle$ is the eigenstate of the Hamiltonian ${{ \mathcal H }}_{0}(t)$. If the Hamiltonian varies slowly in time, the adiabatic approximation holds and thus each energy eigenstate evolves as

$$\begin{eqnarray}&&| {\psi }_{n}(t)\rangle ={{\rm{e}}}^{{\rm{i}}{\alpha }_{n}(t)}| n(t)\rangle ,\end{eqnarray} \tag{ 10 }$$

where ${\alpha }_{n}(t)$ is usually expressed by the dynamical and the Berry phases [100, 101]

$$\begin{eqnarray}&&{\alpha }_{n}(t)=-{\int }_{0}^{t}{{\rm{d}}{t}}^{{\prime} }{E}_{n}({t}^{{\prime} })+{\rm{i}}{\int }_{0}^{t}{{\rm{d}}{t}}^{{\prime} }\langle n({t}^{{\prime} })| {\partial }_{{t}^{{\prime} }}n({t}^{{\prime} })\rangle .\end{eqnarray} \tag{ 11 }$$

The Hamiltonian which mimics such adiabatic dynamics is given by

$$\begin{eqnarray}&&{ \mathcal H }(t)={{ \mathcal H }}_{0}(t)+{{ \mathcal H }}_{\mathrm{cd}}(t),\end{eqnarray} \tag{ 12 }$$

where ${{ \mathcal H }}_{\mathrm{cd}}(t)$ is the counter-diabatic Hamiltonian

$$\begin{eqnarray}&&{{ \mathcal H }}_{\mathrm{cd}}(t)={\rm{i}}\displaystyle \sum _{n}(1-| n(t)\rangle \langle n(t)| )| {\partial }_{t}n(t)\rangle \langle n(t)| \,\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&=\,{\rm{i}}\displaystyle \sum _{n\ne m}\displaystyle \frac{\langle m(t)| ({\partial }_{t}{{ \mathcal H }}_{0}(t))| n(t)\rangle }{{E}_{n}(t)-{E}_{m}(t)}| m(t)\rangle \langle n(t)| .\end{eqnarray} \tag{ 14 }$$

In degenerate systems, counter-diabatic driving was developed by Takahashi [78]. We consider a time-dependent Hamiltonian with the degenerate eigenstates

$$\begin{eqnarray}&&{{ \mathcal H }}_{0}(t)=\displaystyle \sum _{n,\mu }{E}_{n}(t)| n,\mu ;t\rangle \langle n,\mu ;t| ,\end{eqnarray} \tag{ 15 }$$

where $| n,\mu ;t\rangle$ is the eigenstate of the Hamiltonian ${{ \mathcal H }}_{0}(t)$ and μ is an additional index due to degeneracies. We assume that there is no level-crossing during time-evolution. In this case, adiabatic dynamics of each energy eigenstate is given by

$$\begin{eqnarray}&&| {\psi }_{n}(t)\rangle =\exp \left(-{\rm{i}}{\int }_{0}^{t}{{\rm{d}}{t}}^{{\prime} }{E}_{n}({t}^{{\prime} })\right)\displaystyle \sum _{\mu }{c}_{\mu }^{(n)}(t)| n,\mu ;t\rangle ,\end{eqnarray} \tag{ 16 }$$

where ${c}_{\mu }^{(n)}(t)$ is given by

$$\begin{eqnarray}&&{c}_{\mu }^{(n)}(t)=\displaystyle \sum _{{\mu }^{{\prime} }}{U}_{\mu {\mu }^{{\prime} }}^{(n)}(t){c}_{{\mu }^{{\prime} }}^{(n)}(0),\end{eqnarray} \tag{ 17 }$$

and

$$\begin{eqnarray}&&{U}^{(n)}(t)={ \mathcal T }\,\exp \left(-{\rm{i}}{\int }_{0}^{t}{{\rm{d}}{t}}^{{\prime} }{A}^{(n)}({t}^{{\prime} })\right),\end{eqnarray} \tag{ 18 }$$

as discussed by Wilczek and Zee [102]. Here, ${A}^{(n)}(t)$ is the gauge potential

$$\begin{eqnarray}&&{{\rm{i}}{A}}_{\mu {\mu }^{{\prime} }}^{(n)}(t)=\langle n,\mu ;t| {\partial }_{t}n,{\mu }^{{\prime} };t\rangle .\end{eqnarray} \tag{ 19 }$$

The Hamiltonian which mimics such dynamics is given by equation (12) with the counter-diabatic Hamiltonian

$$\begin{eqnarray}&&{{ \mathcal H }}_{\mathrm{cd}}(t)={\rm{i}}\displaystyle \sum _{n,\mu }\left(1-\displaystyle \sum _{\nu }| n,\nu ;t\rangle \langle n,\nu ;t| \right)| {\partial }_{t}n,\mu ;t\rangle \langle n,\mu ;t| \,\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&=\,{\rm{i}}\displaystyle \sum _{n\ne m,\mu ,\nu }\displaystyle \frac{\langle m,\nu ;t| ({\partial }_{t}{{ \mathcal H }}_{0}(t))| n,\mu ;t\rangle }{{E}_{n}(t)-{E}_{m}(t)}| m,\nu ;t\rangle \langle n,\mu ;t| .\end{eqnarray} \tag{ 21 }$$

In this section, we review the counter-diabatic driving in the Lipkin–Meshkov–Glick model using the Holstein–Primakoff transformation [78]. We remark that in the disordered phase, ${\rm{\Gamma }}(t)\gt J$, $(2S+1)$ eigenstates are separated from each other, and thus the counter-diabatic Hamiltonian is constructed by equation (13) or (14). In contrast, in the ordered phase, ${\rm{\Gamma }}(t)\lt J$, eigenstates have 2-fold degeneracies except for the highest energy eigenstate, and thus the counter-diabatic Hamiltonian is constructed by equation (20) or (21). Originally, doubly degenerate ground states are the counterparts of the singlet and the triplet states. However, using the Holstein–Primakoff transformation and the harmonic approximation means neglecting inter-well transitions, and thus the degenerate ground states become the ground states of each well. Therefore, in this approximation, the counter-diabatic Hamiltonian should be

$$\begin{eqnarray}&&{{ \mathcal H }}_{\mathrm{cd}}(t)=\displaystyle \sum _{\mu }{{ \mathcal H }}_{\mathrm{cd}}^{(\mu )}(t)\,\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&\,=\,{\rm{i}}\displaystyle \sum _{n\ne m,\mu }\displaystyle \frac{\langle m,\mu ;t| ({\partial }_{t}{{ \mathcal H }}_{0}(t))| n,\mu ;t\rangle }{{E}_{n}(t)-{E}_{m}(t)}| m,\mu ;t\rangle \langle n,\mu ;t| ,\end{eqnarray} \tag{ 23 }$$

that is, the summation of two counter-diabatic Hamiltonian for each well, which are formally independent of each other. As mentioned later, however, the counter-diabatic terms for a chosen well, say the well with positive (negative) magnetization, affect the state of the other well with negative (positive) magnetization. Therefore, the correctness of our approximation must be confirmed by numerical simulations.

In the disordered phase, the spin vector is directed to the x-axis due to the strong external field ${\rm{\Gamma }}(t)$. In contrast, in the ordered phase, the spin vector rotates, pointing at a minimum of the potential, by angles $\pm \phi$, where $\cos \phi ={\rm{\Gamma }}(t)/J$. By applying the Holstein–Primakoff transformation expanded up to the second order of the boson operators, which becomes exact in the thermodynamic limit, and applying the Bogoliubov transformation, the Hamiltonian (1) is mapped to the harmonic oscillator

$$\begin{eqnarray}&&{{ \mathcal H }}_{0}(t)=\omega (t)\left({b}^{\dagger }b+\displaystyle \frac{1}{2}\right),\end{eqnarray} \tag{ 24 }$$

where b and ${b}^{\dagger }$ are the annihilation and the creation operators obtained by the Bogoliubov transformation. Here, we neglect constants, which differ depending on the phases. The frequencies are given by

$$\begin{eqnarray}&&\omega (t)=2\sqrt{{\rm{\Gamma }}(t)({\rm{\Gamma }}(t)-J)},\end{eqnarray} \tag{ 25 }$$

in the disordered phase, and

$$\begin{eqnarray}&&\omega (t)=2\sqrt{{J}^{2}-{{\rm{\Gamma }}}^{2}(t)},\end{eqnarray} \tag{ 26 }$$

for each minimum of the potential in the ordered phase. Note that the sign of the rotation ϕ does not affect the transformed Hamiltonian (24) in this approximation.

The counter-diabatic Hamiltonian of the harmonic oscillator is already known [103] and given by

$$\begin{eqnarray}&&{{ \mathcal H }}_{\mathrm{cd}}(t)=\displaystyle \frac{i({\partial }_{t}\omega (t))}{4\omega (t)}({b}^{2}-{b}^{\dagger 2}).\end{eqnarray} \tag{ 27 }$$

In the representation of the total spin operators, the counter-diabatic Hamiltonian (27) is rewritten as

$$\begin{eqnarray}&&{{ \mathcal H }}_{\mathrm{cd}}(t)=\displaystyle \frac{f(t)}{N}({S}_{y}{S}_{z}+{S}_{z}{S}_{y}).\end{eqnarray} \tag{ 28 }$$

Here, using equations (25) and (26), the schedules of the counter-diabatic driving are given by

$$\begin{eqnarray}&&f(t)=-\displaystyle \frac{(2{\rm{\Gamma }}(t)-J)({\partial }_{t}{\rm{\Gamma }}(t))}{4{\rm{\Gamma }}(t)({\rm{\Gamma }}(t)-J)},\end{eqnarray} \tag{ 29 }$$

in the disordered phase, and

$$\begin{eqnarray}&&f(t)=\displaystyle \frac{{\rm{\Gamma }}(t)({\partial }_{t}{\rm{\Gamma }}(t))}{({J}^{2}-{{\rm{\Gamma }}}^{2}(t))}\end{eqnarray} \tag{ 30 }$$

in the ordered phase. Note that the schedule of the counter-diabatic driving in the ordered phase differs by the factor two from the previous works (see e.g. equation (S-17) in supplemental material of [80]) because we use equation (23) instead of equation (14), which is supported by numerical simulations.

It is obvious that the counter-diabatic Hamiltonian diverges at the critical point ${\rm{\Gamma }}(t)=J$. It is confirmed as follows. The expansion of the transverse field ${\rm{\Gamma }}(t)$ around the critical point ${\rm{\Gamma }}(t)=J$ is given by

$$\begin{eqnarray}&&{\rm{\Gamma }}(t)\approx J+({\partial }_{t}{\rm{\Gamma }}(t)){| }_{{\rm{\Gamma }}(t)=J}\delta {\rm{\Gamma }}(t)+{ \mathcal O }(\delta {{\rm{\Gamma }}}^{2}),\end{eqnarray} \tag{ 31 }$$

and thus it leads to the divergence at the critical point

$$\begin{eqnarray}&&f(t)\propto \displaystyle \frac{{\partial }_{t}{\rm{\Gamma }}(t)}{({\partial }_{t}{\rm{\Gamma }}(t)){| }_{{\rm{\Gamma }}(t)=J}\delta {\rm{\Gamma }}(t)},\end{eqnarray} \tag{ 32 }$$

which inevitably diverges regardless of the choice of schedules of the transverse field ${\rm{\Gamma }}(t)$.

In order to deal with the counter-diabatic Hamiltonian at the critical point, we consider finite-size corrections. The Holstein–Primakoff transformation up to the third order of the annihilation and the creation operators is given by

$$\begin{eqnarray}&&{S}_{x}=\displaystyle \frac{N}{2}-{a}^{\dagger }a,\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&{S}_{y}=\displaystyle \frac{\sqrt{N}}{2}(a+{a}^{\dagger })-\displaystyle \frac{1}{4\sqrt{N}}({a}^{\dagger }{aa}+{a}^{\dagger }{a}^{\dagger }a),\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&{S}_{z}=\displaystyle \frac{\sqrt{N}}{2i}(a-{a}^{\dagger })-\displaystyle \frac{1}{4i\sqrt{N}}({a}^{\dagger }{aa}-{a}^{\dagger }{a}^{\dagger }a),\end{eqnarray} \tag{ 35 }$$

in the disordered phase. In the ordered phase, introducing the rotation operator ${U}^{\phi }=\exp [-{\rm{i}}\phi {S}_{y}]$, the spin operators are rotated as

$$\begin{eqnarray}&&{S}_{x}^{\phi }={U}^{\phi }{S}_{x}{U}^{\phi \dagger }={S}_{x}\cos \phi -{S}_{z}\sin \phi ,\end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray}&&{S}_{y}^{\phi }={U}^{\phi }{S}_{y}{U}^{\phi \dagger }={S}_{y},\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}&&{S}_{z}^{\phi }={U}^{\phi }{S}_{z}{U}^{\phi \dagger }={S}_{z}\cos \phi +{S}_{x}\sin \phi .\end{eqnarray} \tag{ 38 }$$

By substituting equations (33)–(35) for equation (1), performing the harmonic approximation after the normal ordering, and neglecting a constant energy, we obtain

$$\begin{eqnarray}&&{{ \mathcal H }}_{0}(t)=\displaystyle \frac{J}{2}\left(1-\displaystyle \frac{1}{2N}\right)({a}^{2}+{a}^{\dagger 2})+\left[2{\rm{\Gamma }}(t)-J\left(1-\displaystyle \frac{1}{N}\right)\right]{a}^{\dagger }a,\end{eqnarray} \tag{ 39 }$$

in the disordered phase. In contrast, in the ordered phase, we perform the same procedure after replacing the spin operators in equation (1) with the rotated spin operators (36)–(38), leading to the Hamiltonian

$$\begin{eqnarray}{{ \mathcal H }}_{0}^{\pm \phi }(t) & = & \displaystyle \frac{{{\rm{\Gamma }}}^{2}(t)}{2J}\left(1-\displaystyle \frac{1}{2N}\right)({a}^{2}+{a}^{\dagger 2})\\ & & +\left[2J\left(1-\displaystyle \frac{1}{N}\right)-\displaystyle \frac{{{\rm{\Gamma }}}^{2}(t)}{J}\left(1-\displaystyle \frac{3}{N}\right)\right]{a}^{\dagger }a\\ & & \pm \displaystyle \frac{1}{i\sqrt{N}}\displaystyle \frac{{\rm{\Gamma }}(t)}{J}\sqrt{{J}^{2}-{{\rm{\Gamma }}}^{2}(t)}(a-{a}^{\dagger }),\end{eqnarray} \tag{ 40 }$$

for each well of the potential.

In order to eliminate the first order terms in the ordered phase, we consider the displaced annihilation and creation operators

$$\begin{eqnarray}&&a=\tilde{a}+{A}^{\pm },\end{eqnarray} \tag{ 41 }$$

$$\begin{eqnarray}&&{a}^{\dagger }={\tilde{a}}^{\dagger }-{A}^{\pm },\end{eqnarray} \tag{ 42 }$$

where ${A}^{\pm }$ is a pure imaginary and given by

$$\begin{eqnarray}&&{A}^{\pm }=\pm \displaystyle \frac{1}{{{iN}}^{1/2}}\displaystyle \frac{{\rm{\Gamma }}(t)\sqrt{{J}^{2}-{{\rm{\Gamma }}}^{2}(t)}}{2{J}^{2}\left(1-\tfrac{1}{N}\right)-2{{\rm{\Gamma }}}^{2}(t)\left(1-\tfrac{7}{4N}\right)}.\end{eqnarray} \tag{ 43 }$$

Then, the Hamiltonian (40) is rewritten as

$$\begin{eqnarray}{{ \mathcal H }}_{0}^{\pm \phi }(t) & = & \displaystyle \frac{{{\rm{\Gamma }}}^{2}(t)}{2J}\left(1-\displaystyle \frac{1}{2N}\right)({\tilde{a}}^{2}+{\tilde{a}}^{\dagger 2})+\left[2J\left(1-\displaystyle \frac{1}{N}\right)-\displaystyle \frac{{{\rm{\Gamma }}}^{2}(t)}{J}\left(1-\displaystyle \frac{3}{N}\right)\right]{\tilde{a}}^{\dagger }\tilde{a},\end{eqnarray} \tag{ 44 }$$

where we again neglected a constant.

Using the Bogoliubov transformation, we can diagonalize the Hamiltonian (39) and (44) in the same way as obtaining equation (24). The frequencies are given by

$$\begin{eqnarray}&&\omega (t)=2\left\{{\left[{\rm{\Gamma }}(t)-\displaystyle \frac{J}{2}\left(1-\displaystyle \frac{1}{N}\right)\right]}^{2}\right.{\left.-{\left(\displaystyle \frac{J}{2}\right)}^{2}{\left(1-\displaystyle \frac{1}{2N}\right)}^{2}\right\}}^{1/2},\end{eqnarray} \tag{ 45 }$$

in the disordered phase, and given by

$$\begin{eqnarray}\omega (t) & = & 2\left\{{\left[J\left(1-\displaystyle \frac{1}{N}\right)-\displaystyle \frac{{{\rm{\Gamma }}}^{2}(t)}{2J}\left(1-\displaystyle \frac{3}{N}\right)\right]}^{2}\right.{\left.-{\left(\displaystyle \frac{{{\rm{\Gamma }}}^{2}(t)}{2J}\right)}^{2}{\left(1-\displaystyle \frac{1}{2N}\right)}^{2}\right\}}^{1/2},\end{eqnarray} \tag{ 46 }$$

for each rotation in the ordered phase.

For the disordered phase, we can easily go back to the spin operator representation using the relation ${b}^{2}-{b}^{\dagger 2}={a}^{2}-{a}^{\dagger 2}$ and the Holstein–Primakoff transformation up to the second order. The counter-diabatic Hamiltonian for the disordered phase is given by equation (28) with

$$\begin{eqnarray}&&f(t)=-\displaystyle \frac{1}{4}\displaystyle \frac{\left[2{\rm{\Gamma }}(t)-J\left(1-\tfrac{1}{N}\right)\right]({\partial }_{t}{\rm{\Gamma }}(t))}{{\left[{\rm{\Gamma }}(t)-\tfrac{J}{2}\left(1-\tfrac{1}{N}\right)\right]}^{2}-{\left(\tfrac{J}{2}\right)}^{2}{\left(1-\tfrac{1}{2N}\right)}^{2}}.\end{eqnarray} \tag{ 47 }$$

In contrast, in the ordered phase, we have other terms depending on the directions of rotation due to the displacement (41) and (42), leading to the relation

$$\begin{eqnarray}&&{b}^{2}-{b}^{\dagger 2}={\tilde{a}}^{2}-{\tilde{a}}^{\dagger 2}=\displaystyle \frac{2i}{N}({S}_{y}{S}_{z}+{S}_{z}{S}_{y})-\displaystyle \frac{4}{\sqrt{N}}{A}^{\pm }{S}_{y}.\end{eqnarray} \tag{ 48 }$$

However, we sum up both directions of rotation as equation (23), and thus the second term proportional to S_y disappears. Therefore, the counter-diabatic Hamiltonian for the ordered phase is again given by equation (28) with

$$\begin{eqnarray}&&f(t)=\displaystyle \frac{\left[{\rm{\Gamma }}(t)\left(1-\tfrac{1}{N}\right)\left(1-\tfrac{3}{N}\right)+\tfrac{5{{\rm{\Gamma }}}^{3}(t)}{2{J}^{2}}\left(\tfrac{1}{N}-\tfrac{7}{4{N}^{2}}\right)\right]({\partial }_{t}{\rm{\Gamma }}(t))}{{\left[J\left(1-\tfrac{1}{N}\right)-\tfrac{{{\rm{\Gamma }}}^{2}(t)}{2J}\left(1-\tfrac{3}{N}\right)\right]}^{2}-{\left(\tfrac{{{\rm{\Gamma }}}^{2}(t)}{2J}\right)}^{2}{\left(1-\tfrac{1}{2N}\right)}^{2}}.\end{eqnarray} \tag{ 49 }$$

Note that we can easily confirm that the schedule of the counter-diabatic driving does not diverge and is continuous with appropriate choices of schedules of the transverse field when $N\lt \infty$. Of course, both schedules of the counter-diabatic driving (47) and (49) converge to equations (29) and (30) in the thermodynamic limit $N\to \infty$, respectively.

We confirm that our method can improve fidelity to the cat state. For this purpose, we compare the distributions of the eigenstate populations at the final time, i.e., the distributions of $| \langle m| {\rm{\Psi }}({t}_{f})\rangle {| }^{2}$, where $| m\rangle$ is the eigenstate of S_z, with and without the counter-diabatic Hamiltonian. From the Hamiltonian (1), it is obvious that the low-energy states are given by large $| m|$ and the high-energy states have small $| m|$. If the perfect adiabatic dynamics is attained, the distribution consists of two peaks at $m=\pm N/2$ with the populations 0.5, respectively. We plot the distributions for N = 1000 in figure 3. With the counter-diabatic Hamiltonian, the state is distributed over low-energy states and forms a cat state. In contrast, without the counter-diabatic Hamiltonian, the state is distributed over high-energy states. Note that even with the counter-diabatic Hamiltonian, there is a small amount of populations in unfavorable high-energy states. This unfavorable excitations are due to the approximation and fast operation. Slower operation can suppress this unfavorable excitations but deviations due to the approximation are accumulated. Therefore, we have to optimize operation time.

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As mentioned in section 1, automatically freezing dynamics after creation of cat states is one of the advantages of adiabatic generation. We demonstrate generation of the cat state within time s = 1 and dynamics with the final Hamiltonian ${ \mathcal H }({t}_{f})$ up to time s = 2, and plot in figure 4. It is obvious that the cat state is maintained after the creation process and thus we do not have to care about timing unless decoherence becomes problematic.

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Now, we consider the fidelity to the ground state subspace ${ \mathcal F }(t)=| \langle {{\rm{\Psi }}}_{\mathrm{GS}}(t)| {\rm{\Psi }}(t)\rangle {| }^{2}$, where $| {{\rm{\Psi }}}_{\mathrm{GS}}(t)\rangle$ is the ground state for the disordered phase, ${\rm{\Gamma }}(t)\gt J$, and is the sum of the ground state and the first excited state for the ordered phase, ${\rm{\Gamma }}(t)\lt J$. As discussed in [83], this criterion of adiabaticity is rather severe especially for large systems $N\to \infty$. This is because not only deviations are enhanced but also the gap closes in the limit $N\to \infty$. We plot the fidelity ${ \mathcal F }(t)$ of both cases, with and without the counter-diabatic Hamiltonian, in figure 5. Although it is the severe criterion, the fidelity to the ground state subspace remains finite for N = 100.

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It should be noted that the counter-diabatic Hamiltonian results in bad fidelities to the ground state subspace around the critical point (compare figures 5(a) and (b) around $s=1/2$), which was also observed in the previous work (see supplemental material of [80]). This should be attributable to the breakdown of the harmonic approximation as discussed in section 2. One might expect that we can improve adiabaticity by turning off the counter-diabatic driving up to the critical point and turning on it at there. However, at last, results are almost the same.

It is also of interest how the counter-diabatic driving can accelerate adiabatic dynamics. For this purpose, we calculate the residual energy and the incomplete magnetization.

The residual energy is defined as the deviation of the energy from the ground state energy at the final time, $t={t}_{f}$,

$$\begin{eqnarray}&&{E}_{\mathrm{res}}=E({t}_{f})-{E}_{\mathrm{gs}},\end{eqnarray} \tag{ 53 }$$

where $E({t}_{f})$ is the energy given by the dynamical state, $E({t}_{f})=\langle {\rm{\Psi }}({t}_{f})| { \mathcal H }({t}_{f})| {\rm{\Psi }}({t}_{f})\rangle$, and E_gs is the ground state energy, ${E}_{\mathrm{gs}}=-{JN}/2$. The incomplete magnetization is given by the deviation of the order parameter, which is given by

$$\begin{eqnarray}&&m({t}_{f})=\sqrt{\displaystyle \frac{1}{{S}^{2}}\langle {\rm{\Psi }}({t}_{f})| {S}_{{z}^{2}}| {\rm{\Psi }}({t}_{f})\rangle },\end{eqnarray} \tag{ 54 }$$

from that of the ground state

$$\begin{eqnarray}&&{m}_{\mathrm{inc}}={m}_{\mathrm{gs}}-m({t}_{f}),\end{eqnarray} \tag{ 55 }$$

where m_gs is the order parameter of the ground state given by ${m}_{\mathrm{gs}}=1$.

We compare the residual energy and the incomplete magnetization in the cases with and without the counter-diabatic Hamiltonian plotted in figure 6. For fast operations, where bare adiabatic tracking results in failure, we can suppress diabatic transitions using the counter-diabatic driving. Roughly speaking, the counter-diabatic driving accelerates dynamics by ten times or more compared with adiabatic tracking. Note that this approximated counter-diabatic driving leads to rather bad results when bare adiabatic tracking gives enough adiabaticity. However, this does not lower worth of our results because acceleration of adiabatic dynamics is not necessary if the dynamics is already adiabatic.

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Now, we estimate the quantum Fisher information F_Q of the state driven by the counter-diabatic Hamiltonian. If the quantum Fisher information satisfies ${F}_{Q}\gt N$, then we can conclude that the state is entangled and has phase sensitivity beyond the standard quantum limit, i.e., potentially useful applying to quantum metrology. The upper bound of the quantum Fisher information is given by ${F}_{Q}={N}^{2}$, and then phase sensitivity reaches the Heisenberg limit. Note that, for examples, the quantum Fisher information of the GHZ state and the NOON state is given by ${F}_{Q}={N}^{2}$, which reaches the Heisenberg limit, and that of the maximally spin-squeezed Dicke state is given by ${F}_{Q}={N}^{2}/2+N$, which is beyond the standard quantum limit but does not reach the Heisenberg limit.

In our scheme, the state is the pure state and the relative Hermitian operator is S_z, and thus the appropriate quantum Fisher information is given by

$$\begin{eqnarray}&&{F}_{Q}[| {\rm{\Psi }}({t}_{f})\rangle ,{S}_{z}]=4(\langle {\rm{\Psi }}({t}_{f})| {S}_{z}^{2}| {\rm{\Psi }}({t}_{f})\rangle -\langle {\rm{\Psi }}({t}_{f})| {S}_{z}| {\rm{\Psi }}({t}_{f}){\rangle }^{2}).\end{eqnarray} \tag{ 56 }$$

We plot the quantum Fisher information for N = 100, 500, and 1000 with ${t}_{f}=1$ in figure 7. The results show that the quantum Fisher information is far beyond the standard quantum limit, slightly above that of the maximally spin-squeezed Dicke state, and of course below the Heisenberg limit. It is evident that the generated cat states are truly macroscopic entangled states. It is also remarkable that the quantum Fisher information of the generated cat states is larger than that of the maximally spin-squeezed Dicke state, which is the ground state of the anti-ferromagnetic Lipkin–Meshkov–Glick model, and thus our scheme of quantum speedup is significantly useful.

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