Adiabatic critical quantum metrology cannot reach the Heisenberg limit even when shortcuts to adiabaticity are applied

Quantum metrology is based on the quantum theory of estimation [43] which states that when estimating an unknown parameter $\theta$, the sensitivity $\mathrm{Var}[\theta]$ is limited by the quantum Cramér-Rao bound [44]

$\displaystyle\mathrm{Var}[\theta]\geq\frac{1}{\sqrt{\mathcal{I}_{\theta}}},$

(1)

where $\mathcal{I}_{\theta}$ is the quantum Fisher information (QFI) which for pure states $|\psi\rangle$ is defined as [44]

$\displaystyle\mathcal{I}_{\theta}\equiv 4\left(\langle\partial_{\theta}\psi|\partial_{\theta}\psi\rangle-\langle\partial_{\theta}\psi|\psi\rangle^{2}\right),$

(2)

with $\partial_{\theta}\equiv\partial/\partial\theta$. If the unknown parameter is a quantum phase imprinted via a coherent dynamical process then $\theta=\omega T$ with $T$ being the total time of the process and $\omega$ being the unknown Hamiltonian parameter. In this case, the QFI gains time dependence ($\mathcal{I}_{\omega}=T^{2}\mathcal{I}_{\theta}$) which means that by performing a long-enough measurement under idealistic conditions, one can obtain an arbitrary precision of estimation. Therefore, the time is considered a resource in quantum metrology.

We now assume that the mechanism imprinting the information about an unknown parameter $\omega$ is known and can be described via a Hamiltonian of the form $\hat{H}=\omega\hat{H}_{\omega}+\hat{H}_{t}(t)$, where $\hat{H}_{t}(t)$ represents a general unknown-parameter-independent Hamiltonian while $\omega\hat{H}_{\omega}$ is the term imprinting the information about $\omega$ ($\hat{H}_{\omega}$ itself does not depend on $\omega$). Starting from an initial state $\left|\psi_{0}\right\rangle$, the time-evolved state after a time $T$ is given by (we set $\hbar=1$ throughout the entire manuscript)

$\displaystyle|\psi_{f}\rangle=\hat{U}|\psi_{0}\rangle=\mathcal{T}\exp\left(-i\int_{0}^{T}\hat{H}\mathrm{d}t\right)|\psi_{0}\rangle,$

(3)

with $\mathcal{T}$ being the time ordering operator. This allows us to rewrite the expression for the QFI in the following way $\mathcal{I}_{\omega}\equiv 4\left(\langle\psi_{0}|\hat{h}^{2}|\psi_{0}\rangle-\langle\psi_{0}|\hat{h}|\psi_{0}\rangle^{2}\right)$, where $\hat{h}=i\hat{U}^{\dagger}\partial_{\omega}\hat{U}$. Moreover, it can be shown that the QFI is upper bounded by [45]

$\displaystyle\mathcal{I}_{\omega}\equiv 4\left(\langle\psi_{0}|\hat{h}^{2}|\psi_{0}\rangle-\langle\psi_{0}|\hat{h}|\psi_{0}\rangle^{2}\right)\leq 4T^{2}\left(\langle\psi_{0}|\hat{H}_{\omega}^{2}|\psi_{0}\rangle-\langle\psi_{0}|\hat{H}_{\omega}|\psi_{0}\rangle^{2}\right).$

(4)

For example, for two-mode systems [46] with a fixed total number of particles $N$, the right hand side of the above inequality can be maximized by using maximally entangled initial states which are in a superposition of eigenstates of the operator $\hat{h}$ with smallest and largest eigenvalue leading to the celebrated Heisenberg limit (HL) $\mathrm{Var}[\omega]=1/NT$ [44]. However, for non-entangled initial states, the above inequality gives rise maximally to the standard quantum limit (SQL) $\mathrm{Var}[\theta]=1/\sqrt{N}T$. Therefore, the quantum-enhanced sensitivity can be used as an entanglement witness [47, 48, 49]. For clarity, throughout this manuscript, we will refer to the above scenario as regular quantum metrology and compare this scheme to the case of critical quantum metrology by carefully taking into account the time resources of either of the two approaches.

Critical quantum metrology takes advantage of criticality associated with continuous quantum phase transitions, for which in the thermodynamic limit the energy gap above the ground state closes [50]. The effect of a vanishing energy gap can be explicitly seen if we calculate the QFI for the ground state $|\psi_{0}(\omega)\rangle$ of the Hamiltonian $\hat{H}(\omega)=\sum_{n=0}E_{n}(\omega)|\psi_{n}(\omega)\rangle\langle\psi_{n}(\omega)|$ [51]

$$\mathcal{I}_{\omega}=4\sum_{n\neq 0}\frac{|\langle\psi_{n}(\omega)|\partial_{\omega}\hat{H}(\omega)|\psi_{0}(\omega)\rangle|^{2}}{[E_{n}(\omega)-E_{0}(\omega)]^{2}},$$

(5)

which coincides with the real part of the quantum geometric tensor [52]. From the expression above, it is clear that if the energy gap above the ground state closes, the QFI diverges due to the vanishing denominator. This property may lead, in principle, to an arbitrarily high estimation precision in the thermodynamic limit. However, the critical unknown-parameter-dependent ground state has to be prepared beforehand. A standard approach to ground state preparation uses adiabatic time evolution in which an easy-to-prepare ground state is adiabatically evolved into the desired target ground state through a change in the Hamiltonian parameters. The adiabatic theorem states that a quantum system will remain in its ground state as long as it is changed slowly enough such that no excitations occur. However, this implies that near a quantum phase transition, where the energy gap vanishes, any change in the system parameters has to be infinitely slow—a behaviour known as critical slowing down. Therefore, a critical quantum metrology protocol which relies on preparing a ground state far away from the critical point and subsequently driving it close to the critical point [20] will inevitably require long preparation times which is often overlooked when calculating the achievable sensitivities and when comparing the approach to the regular quantum metrology scheme. In fact, the QFI in critical quantum metrology is constrained to a tighter limit than the QFI in the regular quantum metrology scheme, which we briefly outline in the following.

Note that adiabatic state preparation is still a unitary dynamical process and hence the final (critical) ground state is obtained via

$\displaystyle\left|\psi_{f}\right\rangle\equiv\left|\psi_{f}(\omega,g_{f})\right\rangle=\hat{U}(T,\omega,g_{0},g_{f})\left|\psi_{0}\right\rangle,$

(6)

where $\left|\psi_{0}\right\rangle\equiv\left|\psi_{0}(\omega,g_{0})\right\rangle$ is the initial state that can in general depend on the unknown parameter $\omega$ and $g_{0}$, $g_{f}$ are the initial and final values of a control parameter $g$ which is adiabatically changed from an uncritical to a critical point of the Hamiltonian $\hat{H}=\omega\hat{H}_{\omega}+\hat{H}_{t}[g(t)]$. The QFI is calculated with respect to the final critical states

$\displaystyle\mathcal{I}_{\omega}=4\left(\left\langle\partial_{\omega}\psi_{f}\vphantom{\partial_{\omega}\psi_{f}}\right|\left.\partial_{\omega}\psi_{f}\vphantom{\partial_{\omega}\psi_{f}}\right\rangle-\left\langle\partial_{\omega}\psi_{f}\vphantom{\psi_{f}}\right|\left.\psi_{f}\vphantom{\partial_{\omega}\psi_{f}}\right\rangle^{2}\right).$

(7)

If we substitute equation (6) into equation (7) it is straightforward to show that the expression for the QFI consists of three terms (see Appendix A)

$\displaystyle\mathcal{I}_{\omega}=\mathcal{I}_{\omega}(\partial_{\omega}\hat{U})+\mathcal{I}_{\omega}(\left|\partial_{\omega}\psi_{0}\right\rangle)+\mathcal{I}_{\omega}(\partial_{\omega}\hat{U},\left|\partial_{\omega}\psi_{0}\right\rangle),$

(8)

where the first term depends only on $\partial_{\omega}\hat{U}$, the second term depends only on $\left|\partial_{\omega}\psi_{0}\right\rangle$ and the third term depends both on $\partial_{\omega}\hat{U}$ and $\left|\partial_{\omega}\psi_{0}\right\rangle$. However, if the initial state is not critical itself, as in the protocols studied in this work, the dependence of the initial ground state on the unknown parameter can be neglected and the QFI becomes

$\displaystyle\mathcal{I}_{\omega}=4\left(\langle\psi_{0}|\hat{h}^{2}|\psi_{0}\rangle-\langle\psi_{0}|\hat{h}|\psi_{0}\rangle^{2}\right),$

(9)

with $\hat{h}=i\hat{U}^{\dagger}\partial_{\omega}\hat{U}$ which coincides with the expression for the QFI in equation (4) and is therefore upper bounded by the same limits, i.e. the SQL for non-entangled states and the HL for maximally entangled states. However, as was previously shown, saturating the upper bound in equation (4) requires different states than the instantaneous ground states of the Hamiltonian $\hat{H}=\omega\hat{H}_{\omega}+\hat{H}_{t}[g(t)]$ [37] and therefore critical quantum metrology is always inferior to the optimal regular quantum metrology scheme when adiabatic ground state preparation is taken into account as well. Hence, the maximal attainable QFI $\mathcal{I}_{\omega}^{c}$ for a critical metrology protocol, i.e., when the time evolved state is the instantaneous ground state of the Hamiltonian $\hat{H}=\omega\hat{H}_{\omega}+\hat{H}_{t}[g(t)]$ obeys

$\displaystyle\mathcal{I}^{c}_{\omega}<\max_{\{|\psi_{0}\rangle\}}4\tau_{c}^{2}\left(\langle\psi_{0}|\hat{H}^{2}_{\omega}|\psi_{0}\rangle-\langle\psi_{0}|\hat{H}_{\omega}|\psi_{0}\rangle^{2}\right),$

(10)

where $\tau_{c}$ is the duration of the adiabatic protocol.

Shortcuts to an effective adiabatic dynamics can be realized with several different techniques [34]. Here, we use counter-diabatic (CD) quantum driving as it ensures transition-less dynamics [36]. This technique relies on adding an extra term to the original Hamiltonian $\hat{H}$ which guarantees time evolution corresponding to the adiabatic dynamics but within an arbitrary time. The CD term can be chosen as [53]

$\displaystyle\hat{H}_{\mathrm{CD}}=i\sum_{n}|\partial_{t}\psi_{n}(t)\rangle\langle\psi_{n}(t)|,$

(11)

where $|\psi_{n}(t)\rangle$ are the instantaneous eigenstates of the original Hamiltonian. In general, the CD term might be non-local [54, 55] and thus often impractical for experimental realizations. Here, however, we are interested in the fundamental limitations under idealistic conditions, therefore a possible non-locality of the CD term does not constitute any issue.

Note that with the addition of the CD term the unitary evolution according to equation (3) is altered to

$\displaystyle\begin{split}|\psi_{f}(T,\omega,\tilde{\omega},g_{f})\rangle&=\hat{U}_{\mathrm{CD}}(T,\omega,\tilde{\omega},g_{0},g_{f})|\psi_{0}\rangle\\ &=\mathcal{T}\exp\left(-i\int_{0}^{T}\left(\omega\hat{H}_{\omega}+\hat{H}_{t}[g(t)]+\hat{H}_{\mathrm{CD}}[\tilde{\omega},g(t)]\right)\mathrm{d}t\right)|\psi_{0}\rangle.\end{split}$

(12)

The appropriate CD term ensuring transitionless driving will in general depend on the parameters of the bare Hamiltonian including the unknown parameter $\omega$ either explicitly or implicitly through the chosen ramp function $g(t,\omega)$. However, the CD term constitutes a control term that has to be engineered and applied externally to the system of interest. Therefore, in any experimentally realistic setting the CD term can only depend on an initial estimate $\tilde{\omega}$ of the unknown parameter rather than on the unknown parameter $\omega$ itself. The estimate $\tilde{\omega}$ has to be updated after each measurement yielding better control terms and hence giving rise to an adaptive metrology scheme [37]. The fact that the CD term does not depend on the true unknown parameter but its estimate $\tilde{\omega}$ allows us to include the CD term into the general time dependent term $\hat{H}_{t}(t)$ of the Hamiltonian. Therefore, the arguments of the previous section still hold and the limitation from equation (10) is still valid. Thus, even if the estimate $\tilde{\omega}$ matches the true value of the unknown parameter to arbitrary precision, i.e. $\tilde{\omega}=\omega$, and therefore the critical ground state is reached with perfect fidelity in an arbitrary short time using CD driving, the QFI is still bounded by the HL. Note that this statement holds for any form of applied optimal control in which the additional control term only depends on an estimate of the unknown parameter rather than on the unknown parameter itself. While these arguments already show that adiabatic critical quantum metrology with and without CD driving cannot beat or even reach the HL, they do not yet quantify the extent to how much better or worse one performs over the other. Therefore, we will explore two different examples in Section 3 comparing the achieved QFI of the regular and the critical quantum metrology approach with and without CD driving.

In order to quantify whether a desired target (ground) state $|\psi_{t}\rangle$ is reached, we will use the fidelity defined as

$\displaystyle\mathcal{F}=|\langle\psi_{f}|\psi_{t}\rangle|^{2},$

(13)

where $|\psi_{f}\rangle$ is the final state obtained after time evolution with the Hamiltonian $\hat{H}+\hat{H}_{\mathrm{CD}}$. We will also refer to the quantum speed limit (QSL) time [56] which for a general Hamiltonian $\hat{H}$ is given by [57]

$\displaystyle\tau_{\mathrm{QSL}}=\mathrm{max}\Bigg{\{}\frac{\hbar}{\mathrm{Var}[\hat{H}]}\arccos|\langle{\psi_{0}|\psi_{t}}\rangle|,\frac{2\hbar}{\pi\langle\hat{H}\rangle}\arccos|\langle{\psi_{0}|\psi_{t}}\rangle|^{2}\Bigg{\}}$

(14)

and constitutes the fundamental maximum rate for quantum time evolution.

The Landau-Zener (LZ) model describes a two-level system in a time-dependent or controlled field $g(t)$ [58]

$\displaystyle\hat{H}_{\mathrm{LZ}}=\frac{\Delta}{2}\hat{\sigma}_{x}+\frac{g(t)}{2}\hat{\sigma}_{z}.$

(15)

where $\Delta$ is the level splitting for $g=0$, and $\hat{\sigma}_{i}$ is the $i$th Pauli matrix. Being a single-particle system, the LZ model does not exhibit a quantum phase transition, however, it features an avoided crossing at $g=0$ and can therefore be considered as a toy model for criticality [59, 60]. The instantaneous ground state of the LZ model is given by

$\displaystyle|\mathrm{\psi_{0}}\rangle=\frac{g-\sqrt{\Delta^{2}+g^{2}}}{\sqrt{2g\left(g-\sqrt{\Delta^{2}+g^{2}}\right)+2\Delta^{2}}}|\!\downarrow\,\rangle+\frac{\Delta}{\sqrt{2g\left(g-\sqrt{\Delta^{2}+g^{2}}\right)+2\Delta^{2}}}|\!\uparrow\,\rangle,$

(16)

and the QFI with respect to an unknown parameter, which we set to $\Delta$, is calculated to

$\displaystyle\mathcal{I}_{\Delta}=\frac{g^{2}}{\left(\Delta^{2}+g^{2}\right)^{2}},$

(17)

by using the definition from equation (2). We plot the dependence of the QFI on the control parameter $g$ in figure 1(a), where we have set $\Delta=0.05$ for the unknown parameter. The QFI reaches its maximum value $\mathcal{I}_{\Delta}=1/(4\Delta^{2})$ close to the point of the avoided crossing at $g=\Delta$ and diverges for $(\Delta,g)\ll 1$. The divergent behavior of the QFI seems promising and suggests that arbitrary high sensitivities can be realized. However, achieving these high sensitivities requires the preparation of the critical ground state in the first place, which will inevitably suffer from the critical slowing down. Therefore, when comparing the critical quantum metrology scheme to the regular scheme we also have to take into account the amount of time it takes to prepare the critical state. In what follows we assume that the initial state of the system is the ground state with $g\gg\Delta$, i.e. the spin-down state $|\psi_{0}\rangle=|\!\downarrow\,\rangle$, and subsequently the control field $g(t)$ is adiabatically decreased to reach the ground state at $g=\Delta$. The required time for adiabatic state preparation with the LZ Hamiltonian of equation (15) can be lower bounded by the QSL time which for the LZ can be calculated according to [61, 62]

$\displaystyle\cos\left(\frac{\Delta}{2}\tau_{\mathrm{QSL}}\right)=|\langle\psi_{0}|\psi_{t}(g)\rangle|,$

(18)

where $|\psi_{0}\rangle$ and $|\psi_{t}\rangle$ is the initial and target ground state, respectively. Figure 1(b) shows the QSL as a function of $g$ and illustrates the critical slowing down close to the avoided crossing.

The QFI for a single-particle system in the regular quantum metrology setting under optimal conditions is given by the SQL (for the LZ model the SQL is also the HL since $N=N^{2}=1$) which for the LZ model takes the simple form $\mathcal{I}_{\Delta}^{\mathrm{SQL}}=T^{2}$ with $T$ being the total evolution time [40]. Figure 1(c) compares the QFI attained in the critical quantum metrology framework [equation (17)] to the SQL (black-dashed line) when using the same time resources, i.e. $\mathcal{I}_{\Delta}^{\mathrm{SQL}}=\tau_{\mathrm{QSL}}(g)^{2}$. The performance of critical quantum metrology in comparison to the SQL becomes worse as $g$ is decreased towards the point of the avoided crossing. Hence, given the time it would take to prepare a critical ground state, one always achieves higher sensitivities by performing regular quantum metrology (under ideal conditions) in the same amount of time.

Note that the QFI from equation (17) asymptotically reaches the SQL for $g\gg\Delta$. The SQL (the maximum QFI) is obtained when the evolved state is in a superposition of eigenstates with highest and lowest eigenvalue of the operator $\hat{h}=i\hat{U}^{\dagger}\partial_{\Delta}\hat{U}$ at all times [37]. In the limit $g\gg\Delta$, i.e. far away from the avoided crossing, the required time for adiabatic state preparation and therefore the QSL time become arbitrarily small. Hence, the operator $\hat{h}$ can be approximated as $\hat{h}=i\exp(i\Delta\hat{\sigma}_{x}\tau_{\mathrm{QSL}}/2)\partial_{\Delta}\exp(-i\Delta\hat{\sigma}_{x}\tau_{\mathrm{QSL}}/2)+\mathcal{O}(\tau_{\mathrm{QSL}}^{2})\sim\hat{\sigma}_{x}\tau_{\mathrm{QSL}}/2$. Therefore, the state maximizing the QFI is the spin-down state which happens to also be the ground state of the LZ model for $g\gg\Delta$. For that reason, optimal regular quantum metrology becomes equivalent to our critical quantum metrology approach in the limit of small evolution times $T\ll 1/\Delta$ and $g\gg\Delta$.

The results above suggest that the critical slowing down near the avoided crossing prevents the QFI to reach or overcome the SQL. Therefore, we now consider a shortcut to adiabaticity, specifically CD driving, which allows to prepare a (critical) ground state in arbitrary short times. We add the CD term $\hat{H}_{\mathrm{CD}}$ to the LZ Hamiltonian ensuring effective adiabatic dynamics for any control field $g(t)$ [28]

$\displaystyle\hat{H}_{\mathrm{CD}}=-\frac{\dot{g}(t)\tilde{\Delta}}{2[\tilde{\Delta}^{2}+g(t)^{2}]}\hat{\sigma}_{y},$

(19)

where the dot notation $\dot{g}(t)$ indicates a time derivative. As mentioned previously the CD term involves the parameter $\Delta$ and is therefore not exactly realizable when the parameter is unknown. Therefore, the parameter $\Delta$ in the CD term is replaced by an initial estimate $\tilde{\Delta}$ which is then updated adaptively after each measurement [37]. The ramp function $g(t)$ can be chosen arbitrarily and therefore we set $g(t)=g_{0}-(g_{0}-g_{f})(t/T)^{1/5}$ which is fast when the energy gap is large and becomes slower close to the avoided crossing. In the following we focus on preparing the ground state at $g_{f}=\Delta$ for which the QFI is maximal and set $\tilde{\Delta}=\Delta=0.05$ which corresponds to the ideal case where the estimate of the unknown parameter coincides with the true value.

As shown in figure 2(c) with the addition of the CD term the target ground state is indeed reached with unit fidelity in arbitrary short times (blue line) while driving without the CD term (dashed-red line) gives rise to excited states for short ramps away from the adiabatic limit. The QFI on the other hand now explicitly depends on time and is plotted in figure 2(a),(b) for both cases of driving with and without the CD term. The critical quantum metrology scheme performs again considerably worse than the SQL (black dashed-dotted line) even though the critical ground state can be prepared in much shorter times [see figure 2(a)]. Moreover, the achieved QFI quickly goes to zero for small evolution times and is able to surpass the previously calculated value of the QFI in the adiabatic limit (black solid line) only for intermediate times [see figure 2(b)]. Therefore, regular quantum metrology operated close to the SQL outperforms critical quantum metrology also in this case when using CD driving and hence when avoiding the problem of critical slowing down. Before discussing these results in Section 4, we first analyze a more complex system that, unlike the LZ model, exhibits a quantum phase transition.

The quantum Rabi model is a finite-component system composed of a single two-level atom interacting with a single bosonic mode

$\displaystyle\hat{H}_{\mathrm{QRM}}=\Delta\hat{a}^{\dagger}\hat{a}+\frac{\Omega}{2}\hat{\sigma_{z}}+\frac{g}{2}\left(\hat{a}^{\dagger}+\hat{a}\right)\hat{\sigma}_{x}.$

(20)

where $\Delta$ is the frequency of the bosonic field represented by its creation and annihilation operators $\hat{a}^{\dagger}$ and $\hat{a}$, $\Omega$ is the energy splitting of a two-level atom represented by Pauli matrices $\hat{\sigma}_{i}$, and $g$ is the coupling parameter between the bosonic field and the two-level atom. In the limit of $\Delta/\Omega\rightarrow 0$, which can be considered as the thermodynamic limit [63], the quantum Rabi model exhibits a superradiant phase transition at $g_{c}\equiv\sqrt{\Delta\Omega}$ that can be harnessed in critical quantum metrology [20].

To obtain an analytical expression for the QFI we need to compute the ground state of the system in an analytical form which is difficult for the general case [64, 65]. However, in the suitable limit of $\Delta/\Omega\rightarrow 0$, the system can be diagonalized with the help of the Schrieffer-Wolff transformation $\hat{U}_{\mathrm{SW}}=\exp\{i(g/2\Omega)(\hat{a}^{\dagger}+\hat{a})\hat{\sigma}_{y}\}$ which up to $\mathcal{O}(\Delta\sqrt{{\Delta}/{\Omega}})$ terms leads to

$\displaystyle\hat{U}_{\mathrm{SW}}\hat{H}_{\mathrm{QRM}}\hat{U}_{\mathrm{SW}}^{\dagger}\equiv\hat{H}_{\mathrm{SW}}\simeq\Delta\hat{a}^{\dagger}\hat{a}+\frac{\Omega}{2}\hat{\sigma}_{z}+\frac{g^{2}}{4\Omega}\left(\hat{a}^{\dagger}+\hat{a}\right)^{2}\hat{\sigma}_{z}.$

(21)

This effective model (for clarity we will keep referring to it as the quantum Rabi model) can be diagonalized and the ground state is given by

$\displaystyle|\psi_{0}\rangle=\hat{S}(\xi)|0\rangle\otimes|\!\downarrow\,\rangle,$

(22)

where $\hat{S}(\xi)\equiv\exp\{(\xi/2)(\hat{a}^{\dagger})^{2}-(\xi^{*}/2)\hat{a}^{2}\}$ is the squeeze operator with $\xi=-\frac{1}{4}\ln\{1-(g/g_{c})^{2}\}$ being the squeezing parameter which is real only for $g<g_{c}$. The latter condition restricts the validity of this ground state to the normal phase. Although an effective model for the superradiant phase can be derived [20], in what follows we will focus on the normal phase only. Given the analytical expression for the ground state we can compute the QFI with respect to an unknown parameter which we assume to be $\Delta$. Near the critical point the QFI becomes [20]

$\displaystyle\mathcal{I}_{\Delta}\simeq\frac{1}{32\Delta^{2}(1-g/g_{c})^{2}}.$

(23)

The QFI diverges at the critical point $g=g_{c}$, where an arbitrarily large precision can be achieved [see red curve in figure 3(a)]. Similarly to the LZ case, the QFI above is attained when the (critical) ground state is prepared adiabatically. To include the required time resources for reaching a specific target ground state, we again use the QSL as a lower bound to the adiabatic evolution time. In the following we assume that the initial state is always given by the ground state of the Hamiltonian at $g=0$, i.e. the vacuum state of the field and the spin-down state $\left|0\right\rangle\otimes\left|\downarrow\right\rangle$. The QSL time for achieving the ground state of the Hamiltonian for a different value of $g$ can be calculated according to the bang-off protocol (see Appendix C for a detailed derivation) in which first a quantum kick is applied such that $g_{\mathrm{bang}}^{2}=2\Delta\Omega$ and $g_{\mathrm{bang}}^{2}T_{\mathrm{bang}}/\Omega=-\ln(1-(g/g_{c})^{2})/2$, and subsequently one waits for a time $T_{\mathrm{off}}=\pi/(4\Delta)$ with $g_{\mathrm{off}}=0$, where $T_{\mathrm{bang}}$ and $T_{\mathrm{off}}$ is the duration of the bang and off time, respectively. The QSL is then given by $\tau_{\mathrm{QSL}}=T_{\mathrm{bang}}+T_{\mathrm{off}}$ and plotted in figure 3(b).

To compare the QFI in the critical quantum metrology scheme of equation (23) to the HL of regular quantum metrology, we first need an expression for the latter, which requires some extra considerations. The quantum Rabi model does not conserve the number of photons and the ground state itself exhibits a different number of photons depending on the coupling parameter $g$. The HL can be computed for every $g$ by calculating the maximal sensitivity that can be achieved in single mode phase estimation [66] for a state with a fixed average number of photons $\langle n\rangle$, which turns out to be the squeezed vacuum state, i.e. the ground state of the quantum Rabi model. The HL is then given by $\mathcal{I}_{\Delta}=8T^{2}(\langle n\rangle^{2}+\langle n\rangle)$ where $T$ is the total evolution time. The HL after setting the time $T$ to the previously computed QSL time $\tau_{\mathrm{QSL}}(g)$ required for ground state preparation in critical quantum metrology is shown as a black dashed-dotted line in figure 3(a),(c). In contrast to the LZ model, if we could adiabatically evolve the quantum Rabi model within the QSL time, it would be possible to overcome the HL for states close to the critical point at $g\sim g_{c}$. We have to keep in mind though that the QSL time is a very optimistic lower bound on the actual required adiabatic evolution time. Hence, the achievable HL (black dashed-dotted line) when considering the true time resources of adiabatic state preparation will likely be larger than the QFI (red solid line) computed from equation (23). However, these general considerations serve as a benchmark result and naively suggest that reducing the time of reaching the critical ground state might indeed lead to a potential advantage of critical quantum metrology for the quantum Rabi model.

Therefore, we next consider adding the appropriate CD term to the Hamiltonian (see Appendix B for a derivation) given by

$\displaystyle\hat{H}_{\mathrm{CD}}=i\frac{g(t)\dot{g}(t)}{4\left(g_{c}^{2}-g^{2}(t)\right)}\left[\left(\hat{a}^{\dagger}\right)^{2}-\hat{a}^{2}\right].$

(24)

In contrast to the LZ model, the CD term for the quantum Rabi model does not explicitly depend on the unknown parameter $\Delta$ (or an estimate $\tilde{\Delta}$ thereof). The dependence on the unknown parameter enters only through the expression for the critical coupling $\tilde{g}_{c}=\sqrt{\tilde{\Delta}\Omega}$ that we have to insert into the ramp function which we choose to be $g(t)=\sqrt{t/T}\tilde{g}_{c}$. We again consider the ideal case where the estimate matches the value of the unknown parameter, i.e. $\tilde{\Delta}=\Delta=0.01$.

Figure 4(b) shows the achieved target ground state fidelities when driving the system with and without the CD term and indicates that critical ground state preparation can be reduced to arbitrary short times with the addition of the CD term as expected. However, the computed QFI for either of the driving protocols [plotted in figure 4(a)] is not able to overcome the HL (black dashed-dotted line) and reaches the adiabatic limit (solid black line) only for very large driving times. Together with the example of the LZ model these results confirm that shortcuts to adiabaticity cannot be used to saturate or beat the HL and therefore inevitably lead to lower sensitivities than performing optimal regular quantum metrology using the same amount of time resources.