Criticality-based quantum metrology in the presence of decoherence

Abstract

Because quantum critical systems are very sensitive to the variation of parameters around the quantum phase transition (QPT), quantum criticality has been presented as an efficient resource for metrology. In this paper, we address the issue whether the divergent feature of the inverted variance is realizable in the presence of noise when approaching the QPT. Taking the quantum Rabi model (QRM) as an example, we obtain the analytical result for the inverted variance with single-photon relaxation. We show that the inverted variance may be convergent in time due to the noise. Since the precision of the metrology is very sensitive to the noise, as a remedy, we propose squeezing the initial state to improve the precision under decoherence. In addition, we also investigate the criticality-based metrology under the influence of the two-photon relaxation. Strikingly, although the maximum inverted variance still manifests a power-law dependence on the energy gap, the exponent is positive and depends on the dimensionless coupling strength. This observation implies that the criticality may not enhance but weaken the precision in the presence of two-photon relaxation, due to the non-linearity introduced by the two-photon relaxation.

Graphical abstract

Keywords

criticality / quantum / metrology / decoherence

1 Introduction

Quantum criticality has been shown to provide significant advantages for quantum sensing and metrology [1-11]. In quantum phase transition (QPT), a small variation in the order parameter may cause a huge change in the system’s properties when approaching the quantum critical point [12-14]. Thus, quantum criticality can be viewed as a valuable resource to acquire an ultrahigh-precision estimation. In the last few years, criticality-based quantum metrological schemes have gained increasing attention among both theoretical and experimental researchers. Many criticality-based quantum metrology protocols in QPT systems have been proposed, which can be classified into two types [4,5]. One of them focuses on the critical behaviors of the QPT at equilibrium, and utilizes the ground state of the Hamiltonian near the critical point. Unfortunately, it is very inefficient to prepare the ground-state around the critical point by the state-of-art technology. The other focuses on the dynamical behaviors, which evolve under a Hamiltonian close to the critical point. It is quite similar to the interferometric paradigm in the typical quantum metrology. Previous work [3] has revealed that both two kinds of approaches yield the same scaling for a broad class of quantum many-body systems. Recently, a dynamic framework for criticality-based quantum metrology, which preliminarily focuses on the dynamical approach, has been proposed [2,15,16]. The fundamental idea as well as the details of experimental implementation using quantum Rabi model (QRM) have been illustrated. The QRM is one of the most fundamental models describing quantum light-matter interactions [17]. Ultrastrong light-matter interactions may give rise to a superradiant QPT in the QRM [5,18-27]. The thermodynamic limit can be achieved when the reduced Rabi frequency approaches infinity in some quantum systems with criticality, such as the Dicke model, the Lipkin−Meshkov−Glick model and the QRM [5].

On the other hand, any quantum system inevitably suffers from the interaction with the surrounding environment and thus forms an open quantum system [28]. Quantum metrology makes use of quantum techniques such as entanglement and squeezing to improve measurement accuracy [29-33]. It has been known over two decades that the quantum metrology with the maximum-entangled states shows no advantage over the classical counterpart when the decoherence is present [29]. Interestingly, when placed in a non-Markovian environment, the entangled probes indeed manifest their superiority by the quantum Zeno effect [30,34-38], and recently it was experimentally demonstrated by an exact and efficient quantum simulation approach [39-45].

In this work, inspired by the above discoveries, we analyze the achievable precision in the QRM when considering the thermal relaxation. The upper bound on the achievable precision of the metrology will be dramatically reduced owing to the noise. This result indicates that the dissipation poses severe limitations and thus hinders the metrological advantages in this framework. Furthermore, we find that the inverted variance may converge in time no matter as a result of the noise. We further explore optimization of initial state and find that the precision can be improved by a squeezing operation on the initial state. On the other hand, recently two-photon relaxation has been experimentally realized [46]. It was theoretically shown that the two-photon relaxation preserves the Z_2 symmetry of the QRM. The initial states with even−odd parity manifest qualitatively distinct transient and steady state behaviors, which is present both at ultrastrong and weak couplings [47]. It is interesting to consider the two-photon relaxation’s impact, which has never been considered on the criticality-based quantum metrology in the QRM. Motivated by these observations, we theoretically investigate the quantum dynamics of the metrology in the presence of two-photon relaxation.

This paper is organized as follows. In Section 2, we outline some basic concepts in the quantum metrology based on the criticality and introduce the QRM with noise. In Section 3, we analyze the results by the semiclassical equation of motion and discuss the effects of various physical parameters on the measurement precision. In Section 4, we investigate the effects of the two-photon relaxation on the metrology. The main conclusions of this paper are drawn in Section 5.

2 Model

In this paper, we consider the QRM for criticality-based quantum metrology, which consists of a two-level system coupled to a single cavity mode. It is the most simplified version of the Dicke model [48]. The Hamiltonian of the QRM is

(1)\begin{array}{r}{H}_{\text{Rabi}}=\omega a^{†}a+{\displaystyle \frac{{\omega }_0}{2}}{\sigma }_z-\lambda (a+a^{†}){\sigma }_x,\end{array}

where \omega is the frequency of the bosonic field, a^{†} and a are the creation and annihilation operators of the field, {\omega }_0 is the transition frequency of the two-level atom, {\sigma }_{\alpha } (\alpha =x,y,z) are the Pauli operators of the atom, \lambda is the coupling strength. Let g=2\lambda /\sqrt{\omega {\omega }_0} be the dimensionless coupling constant. When the ratio of two transition frequencies diverges, i.e., \eta ={\omega }_0/\omega \to \mathrm{\infty }, the energy gap closes and there exhibits a second-order normal-to-superradiant QPT at the critical point g=1 [18]. In the limit of \eta \to \mathrm{\infty }, through the Schrieffer−Wolff transformation U_{\text{np}}=\exp [g\sqrt{{\eta }^{-1}}(a+a^{†})({\sigma }_{+}-{\sigma }_{-})/2] [49], we can obtain an effective low-energy normal-phase Hamiltonian in the spin-down subspace as [2]

(2)\begin{array}{r}{H}_{\text{np}}=\omega [P^2+(1-g^2)X^2]/2,\end{array}

where X=(a+a^{†})/\sqrt{2} and P=\mathrm{i}(a^{†}-a)/\sqrt{2} are the quadrature operators. Let H_0=\omega P^2/2 and H_1=\omega X^2/2. In the limit of \eta \to \mathrm{\infty }, the transition frequency of the atom is much larger than the frequency of the bosonic field. Thus, the QRM is in the large detuning regime. In this case, there is no transition between the spin states of the atom and the coupling between the atom and the field exerts an effective potential on the bosonic field. Owing to the divergence of {\mathrm{\Delta }}^{-1} at the critical point the quantum Fisher information (QFI) for the estimation of the parameter g around the critical point can be expressed as [2]

(3)\begin{array}{r}{\mathcal{F}}_g\simeq 16g^2\frac{{\left[\sin (\sqrt{{\mathrm{\Delta }}_g}\omega t)-\sqrt{{\mathrm{\Delta }}_g}\omega t\right]}^2}{{\mathrm{\Delta }}_g^3}\text{Var}(P^2{)}_{|\psi ⟩},\end{array}

where {\mathrm{\Delta }}_g=4(1-g^2) characterizes the energy gap, \text{Var}(P^2{)}_{|\psi ⟩}=⟨\psi |P^2|\psi ⟩-⟨\psi |P|\psi {⟩}^2 is the variance of the momentum in the state |\psi ⟩. In experiments, the precision of measurement is characterized by the inverted variance as F_g={\chi }_g^2/(\mathrm{\Delta }X{)}^2 with {\chi }_g={\mathrm{\partial }}_g⟨X⟩ and (\mathrm{\Delta }X{)}^2=⟨X^2⟩- ⟨X{⟩}^2. Here, ⟨X⟩ is the expectation of X. Assume an initial state |\psi (0)⟩=|\downarrow ⟩\otimes |\psi {⟩}_b with the spin in the spin-down state |\downarrow ⟩ and the bosonic field state |\psi {⟩}_b= (|0⟩+\mathrm{i}|1⟩)/\sqrt{2}, after the Schrieffer-Wolff transformation. The inverted variance will achieve its local maxima F_g(\tau )={\chi }_g^2/(\mathrm{\Delta }X{)}^2{|}_{t=\tau }=8g^2{\omega }^2{\mathrm{\Delta }}_g^{-2}{\tau }^2 at \tau =2m\pi /(\sqrt{{\mathrm{\Delta }}_g}\omega ) with m\in \mathbb{Z}. Notice that the inverted variance diverges in the long-time limit. The QFI at the same time is {\mathcal{F}}_g(\tau )\simeq 16g^2{\omega }^2{\mathrm{\Delta }}_g^{-2}{\tau }^2\text{Var}(P^2{)}_{|\psi {⟩}_b}. The local maximum of the inverted variance F_g(\tau ) is of the same order of the QFI {\mathcal{F}}_g(\tau ).

3 Single-photon relaxation

First of all, we consider the QRM with the single-photon relaxation. The inverted variance can be obtained solving the Lindblad-form master equation numerically by using QuTip [50,51]. In the \eta \to \mathrm{\infty } limit, all the corrections which have an order higher than {\eta }^{-1/2} become zero. Thus, we have

(4)\begin{array}{rl}{U}_{\text{np}}^{†}a{U}_{\text{np}}& \approx a,\end{array}

where we have dropped the terms higher than {\eta }^{-1/2}. Upon the projection onto the spin-down subspace, we obtain the effective master equation \dot{\rho }=-\mathrm{i}[H_{\text{np}},\rho ]+ {\gamma }_{a}D[a]+{\gamma }_{h}D[a^{†}], where D[a]=a\rho a^{†}-a^{†}a\rho /2-\rho a^{†}a/2, {\gamma }_a is the decay rate, {\gamma }_h is the heating rate [52]. Applying a semiclassical equation of motion for the open QRM [53,54], we have

(5)\begin{array}{rl}& \frac{\mathrm{d}}{\mathrm{d}t}⟨X⟩=-\frac{{\gamma }_a-{\gamma }_h}{2}⟨X⟩+\omega ⟨P⟩,\\ & \frac{\mathrm{d}}{\mathrm{d}t}⟨P⟩=-\frac{{\gamma }_a-{\gamma }_h}{2}⟨P⟩-\frac{{\mathrm{\Delta }}_g\omega }{4}⟨X⟩,\\ & \frac{\mathrm{d}}{\mathrm{d}t}⟨X^2⟩=-({\gamma }_a-{\gamma }_h)⟨X^2⟩+\omega ⟨G⟩+\frac{{\gamma }_a+{\gamma }_h}{2},\\ & \frac{\mathrm{d}}{\mathrm{d}t}⟨P^2⟩=-({\gamma }_a-{\gamma }_h)⟨P^2⟩-\frac{{\mathrm{\Delta }}_g\omega }{4}⟨G⟩+\frac{{\gamma }_a+{\gamma }_h}{2},\\ & \frac{\mathrm{d}}{\mathrm{d}t}⟨G⟩=-({\gamma }_a-{\gamma }_h)⟨G⟩+2\omega ⟨P^2⟩-\frac{{\mathrm{\Delta }}_g\omega }{2}⟨X^2⟩,\end{array}

where ⟨\cdot ⟩=\text{Tr}(\cdot {\rho }_b) with {\rho }_b being the density matrix of the bosonic field and G=XP+PX. In general, {\gamma }_a=\kappa (\overline{n}+1) and {\gamma }_h=\kappa \overline{n} describe the coupling of the field to a thermal reservoir at temperature T with mean number

(6)\begin{array}{r}\overline{n}=\frac{1}{{\mathrm{e}}^{\omega /(k_{B}T)}-1}\end{array}

of thermal photons, k_B the Boltzmann constant, \kappa the decay rate at zero temperature [52]. By solving Eq. (5), the inverted variance F_g(t) is analytically given as

(7)\begin{array}{r}{F}_g(t)=\frac{4{\mathrm{\Delta }}_g^{-2}(4-{\mathrm{\Delta }}_g)A^2(t)}{B(t)+\frac{2n+1}{{\mathrm{\Delta }}_g{\omega }^2+{\kappa }^2}C(t)},\end{array}

where the three time-dependent factors are respectively

(8)\begin{array}{rl}A(t)=& [{\mathrm{\Delta }}_g\omega t⟨X(0)⟩+4⟨P(0)⟩]\sin \left(\frac{1}{2}\sqrt{{\mathrm{\Delta }}_g}\omega t\right)\\ & -2\sqrt{{\mathrm{\Delta }}_g}\omega t⟨P(0)⟩\cos \left(\frac{1}{2}\sqrt{{\mathrm{\Delta }}_g}\omega t\right),\end{array}

(9)\begin{array}{rl}B(t)=& 2{\mathrm{\Delta }}_g[1+\cos (\sqrt{{\mathrm{\Delta }}_g}\omega t)]\text{Var}(X^2{)}_{|\psi (0)⟩}\\ & +8[1-\cos (\sqrt{{\mathrm{\Delta }}_g}\omega t)]\text{Var}(P^2{)}_{|\psi (0)⟩}\\ & +8\sqrt{{\mathrm{\Delta }}_g}\sin (\sqrt{{\mathrm{\Delta }}_g}\omega t)\text{Re}[\text{Cov}(X,P)],\end{array}

(10)\begin{array}{rl}C(t)=& {\mathrm{\Delta }}_g({\mathrm{\Delta }}_g{\omega }^2+4{\omega }^2+2{\kappa }^2)({\mathrm{e}}^{\kappa t}-1)-(4-{\mathrm{\Delta }}_g)\\ & \times \{{\kappa }^2[1-\cos (\sqrt{{\mathrm{\Delta }}_g}\omega t)]+\sqrt{{\mathrm{\Delta }}_g}\omega \kappa \sin (\sqrt{{\mathrm{\Delta }}_g}\omega t)\}.\end{array}

Here, ⟨X(0)⟩ and ⟨P(0)⟩ are the expectation values of the position and momentum operators over the initial state of the bosonic field |\psi (0)⟩, \text{Cov}(X,P)=⟨\psi (0)|XP|\psi (0)⟩- ⟨\psi (0)|X|\psi (0)⟩⟨\psi (0)|P|\psi (0)⟩. When the noise is absent, i.e., \kappa =0, it can be proven that the local maximum of the inverted variance at the evolution time \tau =2m\pi /(\sqrt{{\mathrm{\Delta }}_g}\omega ) is equal to 16g^2{\omega }^2{\mathrm{\Delta }}_g^{-2}{\tau }^2⟨P(0){⟩}^2/{\left(\mathrm{\Delta }X(0)\right)}^2 [2]. It scales quadratically with \tau, and shows a divergent feature when approaching the critical point, i.e., F_g(\tau )\propto {\mathrm{\Delta }}_g^{-3}\to \mathrm{\infty }.

At zero temperature, the average photon number \overline{n} equals zero. In this case, we assume the same initial state \left(|0⟩+\mathrm{i}|1⟩\right)/\sqrt{2} as Ref. [4]. Fig.1 shows the inverted variance of the QRM close to the critical point when coupling with a thermal reservoir at zero temperature. As shown in Fig.1(a), the dynamics of F_g(t) manifest abundant phenomena. At first, the envelope increases along with the increase of the time. Then, after passing the maximum, it decreases and converges due to the noise. It is in sharp contrast to the prediction when the noise is absent. Furthermore, we also investigate the maximum values of F_g(t) for different \kappa’s in Fig.1(b). As shown by the numerical fitting, the dependence of the maximum inverted variance F_g(t){|}_{\text{max}} on \kappa can be well described by a{\kappa }^b. For different g’s, although a’s are distinct, all b’s are non-exceptionally equal to 2.

In Fig.2(a), we plot the time dependence of F_g(t) near the critical point for different g’s. Notice that the time is rescaled by {\mathrm{\Delta }}_g\omega /(2\pi ), which is approximately the oscillating frequency of F_g(t) for small \kappa \ll \omega. Similarly as in Fig.1(a), we can see that F_g(t) gradually increase from their initial values F_g(0) = 0 to the maximum values. Then, F_g(t) begins to decrease and eventually vanishes in the long-time limit. By theoretically obtaining F_g(t) under different \kappa’s, we plot the relation of the maximum of F_g(t) with respect to {\mathrm{\Delta }}_g in Fig.2(b). We can see that F_g(t) decays as {\mathrm{\Delta }}_g increases for a given \kappa. It indicates that the noise hinders the metrological advantages in this framework. Again, we numerically fit the maximum of F_g(t) as a function of {\mathrm{\Delta }}_g, i.e., a{\mathrm{\Delta }}_g^b. Despite of different a’s, all curves consistently decays with a nearly-identical b\simeq -1.18>-2, which is the noise-free case. This implies that in the noisy QRM, the inverted variance still diverges when approaching the critical point, i.e., {\mathrm{\Delta }}_g\to 0. In other words, by the scaling behavior, we theoretically prove that the criticality-based metrology is robust in the presence of single-photon relaxation.

We now investigate the influence of average photon number. As it can be seen from Fig.3(a), the time-rescaled F_g(t)’s for different g’s reach their local maxima at the same period while maximum values of F_g(t) decay as \overline{n} increases. Again, we plot the maximum F_g(t) vs the temperature T in Fig.3(b). It is shown that the maximum F_g(t) drops dramatically as T rises. This observation reminds us that although the criticality-based metrology seems immune to the noise, the precision is very sensitive to the temperature. Hereafter, we will propose some method to relieve this disadvantage.

According to Eq. (7), the measurement accuracy highly depends on the initial state. This observation inspires us to perform a squeezing operation, defined by a squeezing operator S(\xi )=\exp [({\xi }^{\ast }{a}^2-\xi a^{†2})/2], on the initial bosonic field state, where \xi is the squeezing parameter. As seen in Fig.4, the maximum of F_g(t) is significantly increased as \xi is enlarged. Lines in Fig.4(b) are numerically fitted by a\exp (b\xi ). We further explore the effects of squeezing parameter. We first consider the thermal reservoir at the zero temperature and plot the maxima of F_g vs the squeezing parameter \xi in Fig.5(a). The numerically-fitted results are similar to Fig.1(b), with all b’s non-exceptionally equal to 2, although a increases along with \xi. This implies that the squeezing operation does not change the scaling behavior. Then, we take the temperature into account. We plot the relation between the maximum of F_g and \xi at different temperatures in Fig.5(b). We can also observe that the maximum of F_g(t) increases as \xi increases. Although this gain can not completely offset the influence of decoherence, it offers an alternative way to improve the precision under decoherence.

4 Two-photon relaxation

In the above investigations, we explore the quantum metrology in the presence of single-photon relaxation. However, the quantum dynamics of the QRM with single-photon and two-photon relaxation are totally different [47]. Inspired by this discovery, we study the quantum metrology in the QRM with two-photon relaxation, which can be described by the effective master equation \dot{\rho }=-\mathrm{i}[H_{\text{np}},\rho ]+{\gamma }_{a}D[a^2]+{\gamma }_{h}D[a^{†2}] [47].

In Fig.6(a), we investigate the dynamics of the inverted variance for different g’s under the influence of the two-photon relaxation. In contrast to the case with the single-photon relaxation, the quantum dynamics for different g’s does not oscillate with the same frequency for the rescaled time. In addition, as g increases, the time to achieve the maximum becomes shorter and shorter. Moreover, we plot the maximum inverted variance as a function of {\mathrm{\Delta }}_g in Fig.6(b). Interestingly, the behavior is quite different from that for the single-photon relaxation in Fig.1(b). The maximum inverted variance increases monotonically as {\mathrm{\Delta }}_g is enlarged. We again numerically fit the data with the function a{x}^b and thus obtain a=9.03\times {10}^3 and b=0.661 for \kappa =0.01\omega. As \kappa increases, both a and b reduce dramatically. In other words, when we strengthen the noise, the inverted variance becomes smaller and shows weaker dependence on {\mathrm{\Delta }}_g. This remarkable difference can be well described by the Schrödinger equation with a non-Hermitian Hamiltonian H_{\text{tot}} [55,56], in the case of weak relaxation. When there is only single-photon relaxation, the non-Hermitian Hamiltonian H_{\text{tot}} is the original Hamiltonian H_{\text{np}}, which can be diagonalized by a linear combination of a and a^{†}, plus -\mathrm{i}({\gamma }_a{a}^{†}a+{\gamma }_{h}a{a}^{†})/2. Although the Hamiltonian H_{\text{tot}} is non-Hermitian, the energy spectrum is still equally-spaced, which is fundamentally required by the criticality-based metrology [2]. However, for the case with two-photon relaxation, the non-Hermitian Hamiltonian H_{\text{tot}} is the original Hamiltonian H_{\text{np}} plus -\mathrm{i}({\gamma }_a{a}^{†2}{a}^2+{\gamma }_h{a}^2{a}^{†2})/2. Due to the non-linearity introduced by the two-photon relaxation, the energy spectrum is not equally-spaced and thus breaks down the underlying physical mechanism of the criticality-based metrology.

5 Conclusion

We have investigated the impact of decoherence on the criticality-based quantum metrology in the QRM, showing that the achieved precision still diverges when approaching the criticality. In particular, we consider the single-photon relaxation described by the Lindblad-form master equation. By the semiclassical equation of motion, we obtain the analytical solution for the precision characterized by the maximum inverted variance. We have shown that the precision is very sensitive to the temperature. As a remedy, we propose to improve the precision by performing the squeezing operation on the initial state. Furthermore, we also investigate the performance of the metrology in the presence of two-photon relaxation. In contrast to the single-photon relaxation, the inverted variance do not oscillate against the rescaled time at the same frequency for different g's. More strikingly, the maximum inverted variance shows a completely-different tendency when closing to the quantum phase transition.

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