1 Introduction

In quantum physical experiments, the sensitivity of signal detections and the precision measurement of observables are greatly restricted by quantum uncertainty of quantum systems. How to break through the restriction of quantum uncertainty (quantum noise) and realize quantum physical experiments with ultra-low quantum noise is not only a dream state but also a strong expectation of physical scientists. The suppression of quantum noise in quantum systems via squeezing effects is one of the most prominent achievements in modern quantum optics and quantum informatics over the last few years. The squeezing effects of fields and atoms have attracted considerable attention due to their potential application in optical communications [1], detections of weak signal like a gravitational wave [2], high-resolution spectroscopy [3], the high-precision atomic fountain clock [4, 5], high-precision spin polarization measurements [6], and quantum information processing. Recently, the mechanical squeezing has been considered as a quantum resource for optimal quantum estimation in a pulsed quantum optomechanical system [7]. The squeezing effects have been studied for many years, and the main research achievements include the following aspects. On one hand, many schemes for generating squeezing effects have been proposed, and a few squeezed states of fields and atoms have been prepared experimentally. On the other hand, some researchers have proposed different types of measures for squeezing, such as the definition of entropy squeezing expressed via the information entropy of systemic observables, which is proposed by one of our authors [8] and overcomes the defect of the measure expressed via the variance of the systemic observables. Although the studies of squeezing have made great progress, there are still problems to be solved. Firstly, in many previous works, the available squeezing effects at most have either optimal squeezing degree or long squeezing time, but not both. It is difficult to prepare the squeezing resource which is not only optimal but also long-lasting. Secondly, the influence of environmental noise on the squeezing of systemic observables is rarely considered. In practice, any real quantum system is inevitably in contact with its environments, and this leads to quantum decoherence which will degrade the squeezing of systemic observables. Because an optimal squeezing state corresponds to a quantum state with minimum quantum noise, and steady squeezing can provide enough time for people to use the squeezing resource to allow performances of quantum information processing, quantum computation, and quantum communication, how to prepare steady and optimal squeezing becomes of great interest in practical quantum physical experiments.

Nowadays, much attention has been focused on the quantum feedback control, which is a typical method to steer the dynamics of quantum systems by means of external controls. The direct (Markovian) feedback introduced by Wiseman and Milburn [9, 10] has been widely applied by feeding back the measurement results to systems to modify the future dynamics of the system, such as suppressing decoherence [11–14], improving the creation of steady-state entanglement in open quantum systems [15–18], manipulating the geometric phase [19] and the stationary state [20] of a dissipative two-level system, increasing quantum discord of two atoms by the inefficiency of detection [21], and enhancing parameter-estimation precision [22]. According to different measurement strategies, the direct feedback can be classified as homodyne feedback and quantum-jump-based feedback. Reference [18] shows that the latter outperforms the prior in protecting entangled states against decoherence. The quantum-jump-based feedback is also used to prepare three-atom singlet states [23] and entanglement state [24, 25]. However, to the best of our knowledge, no detail investigations concerning the influence of feedback strategy on entropy squeezing effects are available at present. In this paper, we research, for the first time, the effect of the quantum-jump-based feedback itself or the quantum-jump-based feedback together with classical driving on the entropy squeezing of a two-level atom coupled to a dissipative cavity. We find that quantum-jump-based feedback can be employed to produce and preserve entropy squeezing of the two-level atom. Moreover, unlike previous studies, we obtain steady and optimal entropy squeezing by choosing an appropriate feedback control and setting the strength of classical driving in a dissipative environment.

This paper is structured as follows. In Sect. 2, the physical model is presented. In Sect. 3, we introduce the calculation of the entropy squeezing for a two-level atom. By numerical calculations, we analyze the properties of the entropy squeezing for the two-level atom coupled to a dissipative cavity under two different controls. And we present a scheme to generate and protect steady and optimal entropy squeezing of the two-level atom. Finally, we provide a conclusion in Sect. 4.

2 The physical model

We consider a two-level atom with excited state \(|1\rangle \) and ground state \(|0\rangle \). The atom is coupled resonantly to a single-mode cavity with coupling strength g and simultaneously driven by an external laser field (classical driving) with strength \(\varOmega \). The cavity mode a is damped with decay rate \(\kappa \), and the atom can spontaneously decay with rate \(\gamma _{0}\). We apply the feedback which is based upon the measurement of the cavity to the atom. According to the output of the leaky cavity, a feedback Hamiltonian \(H_{fb}\) is triggered and added to the atom, completing the feedback scheme. In the absence of the feedback, the master equation for the system consisted of atom and cavity field is of the form \((\hbar =1)\)

$$\begin{aligned} \frac{\text {d}\rho }{\text {d}t}=-i[H,\rho ]+\kappa {{\mathcal {D}}}[a]\rho +\gamma _{0}{{\mathcal {D}}}[\sigma _{-}]\rho , \end{aligned}$$
(1)

where

$$\begin{aligned} H=\frac{\varOmega }{2}(\sigma _{+}+\sigma _{-})+g(\sigma _{+}a+\sigma _{-}a^{\dag }),\end{aligned}$$
(2)
$$\begin{aligned} {{\mathcal {D}}}[c]\rho \equiv c\rho c^{\dag }-\frac{1}{2}(c^{\dag }c\rho +\rho c^{\dag } c). \end{aligned}$$
(3)

Here H is the Hamiltonian of the system in the interaction picture; \({{\mathcal {D}}}[a]\) and \({{\mathcal {D}}}[\sigma _{-}]\) are superoperators, representing the damping of the field by cavity decay and of the atom by spontaneous emission; a and \(a^{\dag }\) are the cavity-mode annihilation and creation operators; \(\sigma _{-}=|0\rangle \langle 1|\) and \(\sigma _{+}=|1\rangle \langle 0|\) are the lowering and raising operators of the atom, respectively.

In the limit that the cavity decay rate \(\kappa \) is much larger than the other relevant frequencies of the system, the cavity mode can be adiabatically eliminated, resulting in a master equation only including the atomic degrees of freedom [22]:

$$\begin{aligned} \frac{\text {d}\rho }{\text {d}t}=-i\frac{\varOmega }{2}[\sigma _{+}+\sigma _{-},\rho ]+\gamma {{\mathcal {D}}}[\sigma _{-}]\rho +\gamma _{0}{{\mathcal {D}}}[\sigma _{-}]\rho . \end{aligned}$$
(4)

where \(\gamma =g^2/\kappa \) is the effective decay rate. Moreover, under the assumption that the effective decay rate is much larger than the spontaneous emission rate of the atom, we can obtain

$$\begin{aligned} \frac{\text {d}\rho }{\text {d}t}=-i\frac{\varOmega }{2}[\sigma _{+}+\sigma _{-},\rho ]+\gamma {{\mathcal {D}}}[\sigma _{-}]\rho . \end{aligned}$$
(5)

For the moment, taking the quantum-jump-based feedback into account. The dynamics of the system with the feedback reads [10, 22]

$$\begin{aligned} \frac{\text {d}\rho }{\text {d}t}=-i\frac{\varOmega }{2}[\sigma _{+}+\sigma _{-},\rho ]+\gamma \left[ F\sigma _{-}\rho \sigma _{+}F^{\dag }+\frac{1}{2}(\sigma _{+}\sigma _{-}\rho +\rho \sigma _{+}\sigma _{-})\right] . \end{aligned}$$
(6)

Here \(F=\exp (-iH_{fb}\delta t)\) represents the finite amount of evolution imposed by the control Hamiltonian on the system. Once a quantum-jump event \(\sigma _{-}\rho \sigma _{+}\) occurs, the unitary operator F is applied, resulting in \(F\sigma _{-}\rho \sigma _{+}F^{\dag }\). There are various choices for the feedback operator F, even considering the limitations imposed by experimental constraints. Under the condition \(FF^{\dag }=1\), we here choose

$$\begin{aligned} F=e^{i\mathbf {\sigma }\cdot \mathbf {A}}=\cos |\mathbf {A}|+i\frac{\sin |\mathbf {A}|}{|\mathbf {A}|}\mathbf {A}\cdot \mathbf {\sigma }. \end{aligned}$$
(7)

where \(\mathbf {\sigma }=(\sigma _{x},\sigma _{y},\sigma _{z})\) which is decomposed by Pauli matrices and \(\mathbf {A}=(A_{x},A_{y},A_{z})\), representing the amplitude of \(\sigma _{x}\), \(\sigma _{y}\), and \(\sigma _{z}\) controls. When we set \(A_{y}=0,A_{z}=0\), the operator F represents the \(\sigma _{x}\) control, which means rotating the Bloch vector with a certain amount of angle around the x axis of Bloch sphere; so do the \(\sigma _{y}\) and \(\sigma _{z}\) controls [26]. It can be seen that the \(\sigma _{z}\) control has no effect on the evolution of the system because of \(e^{iA_{z}\sigma _{z}}\sigma _{-}\rho \sigma _{+}e^{-iA_{z}\sigma _{z}}=\sigma _{-}\rho \sigma _{+}\). So in this paper, we only consider the \(\sigma _{x}\) control and \(\sigma _{y}\) control:

$$\begin{aligned} F_{\sigma _{x}}= & {} \cos A_{x}+i\sin (A_{x})\sigma _{x},\nonumber \\ F_{\sigma _{y}}= & {} \cos A_{y}+i\sin (A_{y})\sigma _{y}. \end{aligned}$$
(8)

The two controls both influence the evolution of the system with a period of \(\pi \), which comes from that \(e^{iA_{x}\sigma _{x}}\sigma _{-}\rho \sigma _{+}e^{-iA_{x}\sigma _{x}}=e^{i(A_{x}+\pi )\sigma _{x}}\sigma _{-}\rho \sigma _{+}e^{-i(A_{x}+\pi )\sigma _{x}}\) and \(e^{iA_{y}\sigma _{y}}\sigma _{-}\rho \sigma _{+}e^{-iA_{y}\sigma _{y}}=e^{i(A_{y}+\pi )\sigma _{y}}\sigma _{-}\rho \sigma _{+}e^{-i(A_{y}+\pi )\sigma _{y}}\) for any \(A_{x}\) and \(A_{y}\).

3 Generation and protection of entropy squeezing

Before discussing the entropy squeezing properties of the atom in our system, we firstly review the definition given by Ref. [8] of it. As we know, the information entropy of the operator \(S_{\alpha } (\alpha =x,y,z)\) for a two-level atom is

$$\begin{aligned} H(S_{\alpha })=-\sum _{i=1}^{2}P_{i}(S_{\alpha })\ln P_{i}(S_{\alpha }). \end{aligned}$$
(9)

where \(S_{\alpha }=\frac{1}{2}\sigma _{\alpha }~(\alpha =x,y,z)\) and \(P_{i}(S_{\alpha })=\langle \psi _{\alpha i}|\rho |\psi _{\alpha i}\rangle ~(i=1,2)\), which is the probability distribution for two possible outcomes of measurements of an operator \(S_{\alpha }\). The information entropies \(H(S_{x}), H(S_{y})\), and \(H(S_{z})\) satisfy the entropy uncertainty relation

$$\begin{aligned} \delta H(S_{x})\delta H(S_{y})\ge \frac{4}{\delta H(S_{z})}, \end{aligned}$$
(10)

here \(\delta H(S_{\alpha })\equiv \exp [H(S_{\alpha })]\). The fluctuations in the component \(S_{\alpha }\) \((\alpha \equiv x \) or y) of the atomic dipole are said to be squeezed in entropy if the information entropy \(H(S_{\alpha })\) of \(S_{\alpha }\) satisfies the condition

$$\begin{aligned} E(S_{\alpha })=\delta H(S_{\alpha })-\frac{2}{[\delta H(S_{z})]^{1/2}}<0,~\alpha \equiv x~\hbox {or}~y. \end{aligned}$$
(11)

It is worth noting from Eq. (11) that the value of optimal entropy squeezing is \(1-\sqrt{2}\approx -0.414\). Using the atomic density operator \(\rho (t)\), we calculate the information entropies of the atomic operator \(S_{\alpha }(\alpha =x,y,z)\) directly by Eq. (9) as

$$\begin{aligned} H(S_{x})= & {} -\left[ \frac{1}{2}+\hbox {Re}\rho _{21}(t)\right] \ln \left[ \frac{1}{2}+\hbox {Re}\rho _{21}(t)\right] \nonumber \\&-\,\left[ \frac{1}{2}-\hbox {Re}\rho _{21}(t)\right] \ln \left[ \frac{1}{2}-\hbox {Re}\rho _{21}(t)\right] , \nonumber \\ H(S_{y})= & {} -\left[ \frac{1}{2}+\hbox {Im}\rho _{21}(t)\right] \ln \left[ \frac{1}{2}+\hbox {Im}\rho _{21}(t)\right] \nonumber \\&-\,\left[ \frac{1}{2}-\hbox {Im}\rho _{21}(t)\right] \ln \left[ \frac{1}{2}-\hbox {Im}\rho _{21}(t)\right] ,\nonumber \\ H(S_{z})= & {} -\rho _{22}(t)\ln \rho _{22}(t)-\rho _{11}(t)\ln \rho _{11}(t). \end{aligned}$$
(12)

Next we will investigate the effect of the quantum-jump-based feedback control with or without classical driving on the entropy squeezing for the component \(S_{\alpha }\) \((\alpha \equiv x \) or y) in our system.

Fig. 1
figure 1

Entropy squeezing factor \(E(S_{x})\) as a function of \(\gamma t\) and the feedback parameter with a the \(\sigma _{x}\) control and b the \(\sigma _{y}\) control. \(\theta =0\), the atom is initially in the excited state

Full size image

3.1 Entropy squeezing with only quantum-jump-based feedback

Now we focus on the case that the system is controlled by quantum-jump-based feedback in the absence of classical driving, namely \(\varOmega =0\) in Eq. (6). Assuming that the atom is initially in the state \(\psi =\cos \theta |1\rangle +\exp (i\phi )\sin \theta |0\rangle \), the elements of the atomic density matrix at time t can be obtained by solving Eq. (6) as

$$\begin{aligned} \rho _{11}(t)= & {} e^{-\gamma t(\cos A)^2}(\cos \theta )^2 ,\nonumber \\ \rho _{12}(t)= & {} e^{-\frac{\gamma t}{2}}\cos \theta \left[ e^{-i\phi }\sin \theta + e^{i\beta -\alpha }(e^{\alpha }-1)\cos \theta \tan (2A)\right] , \nonumber \\ \rho _{21}(t)= & {} \rho _{12}^{*}(t) ,\nonumber \\ \rho _{22}(t)= & {} 1-\rho _{11}(t). \end{aligned}$$
(13)

where \(\alpha = \frac{\gamma t\cos (2A)}{2}\), \(\beta =\pi /2\) and 0 correspond to the \(\sigma _{x}\) control and \(\sigma _{y}\) control, respectively. Substituting Eqs. (12), (13) in Eq. (11), \(E(S_{x})\) and \(E(S_{y})\) can be derived.

We first explore the case that the atom is initially in the excited state (i.e., \(\theta =0\)) in which the entropy squeezing factors \(E(S_{x})\) and \(E(S_{y})\) are both equal to zero at the beginning. The evolution of the entropy squeezing factor \(E(S_{x})\) with the \(\sigma _{x}\) control and \(\sigma _{y}\) control are respectively shown in Fig. 1a, b. Figure 1a indicates that there is no entropy squeezing effect for the component \(S_{x}\) under the \(\sigma _{x}\) control as \(E(S_{x})\ge 0\) all the time, whereas we can see from Fig. 1b that the entropy squeezing for \(S_{x}\) can be generated by the \(\sigma _{y}\) control with an appropriate feedback amplitude \(A_{y}\) as \(E(S_{x})<0\) in some regions, which is explicitly shown in Fig. 2 by plotting the evolution of the entropy squeezing factor \(E(S_{x})\) with a selected feedback amplitude \(A_{y}=1.1\) or 2.04 (the red solid curve) compared with the uncontrolled case (\(A_{y}=0\), the blue dashed curve). For the other component \(S_{y}\), we find that the \(\sigma _{y}\) control can not generate entropy squeezing for it, while a proper \(\sigma _{x}\) control can generate entropy squeezing for this component, and the evolution of \(E(S_{y})\) with the \(\sigma _{x}\) control is the same as that of \(E(S_{x})\) with the \(\sigma _{y}\) control shown in Fig. 1b. These results show that when we choose suitable feedback parameters, the \(\sigma _{x}\) control and \(\sigma _{y}\) control can be respectively used to prepare entropy squeezing for \(S_{y}\) and \(S_{x}\).

In order to examine whether the quantum-jump-based feedback can protect the atomic entropy squeezing, we prepare a initial state \(\psi _{1}=\frac{1}{\sqrt{2}}(|1\rangle +|0\rangle )\) with optimal entropy squeezing by setting \(\theta =\pi /4\), \(\phi =0\), which is an eigenstate of \(S_{x}\). Figure 3a, b displays the evolution of the entropy squeezing factor \(E(S_{x})\) with the \(\sigma _{x}\) control and \(\sigma _{y}\) control, respectively. It can be seen from Fig. 3a the entropy squeezing factor \(E(S_{x})\) quickly increases to zero whatever the feedback parameter \(A_{x}\) is. This indicates that the \(\sigma _{x}\) control can not protect the entropy squeezing for the atomic polarization component \(S_{x}\). However, the entropy squeezing for \(S_{x}\) can last for relatively longer time under an appropriate \(\sigma _{y}\) control as illustrated in Fig. 3b, and we find that \(A_{y}\approx \pi /4\) is an optimal feedback amplitude for this case. In Fig. 4, the evolution of \(E(S_{x})\) with feedback amplitude \(A_{y}=\pi /4\) (the red solid curve) compared with the uncontrolled case (\(A_{y}=0\), the blue dashed curve) is plotted. We can clearly observe that the time of existence of entropy squeezing for \(S_{x}\) is remarkably prolonged under the \(\sigma _{y}\) control with the feedback amplitude \(A_{y}=\pi /4\), which implies that the decoherence is partially suppressed. For this initial state, there is no entropy squeezing for \(S_{y}\) under either the \(\sigma _{x}\) control or the \(\sigma _{y}\) control as \(E(S_{y})\ge 0\) all the time. Similarly, if the atom is initially in \(\psi _{2}=\frac{1}{\sqrt{2}}(|1\rangle +i|0\rangle )\) that is an eigenstate of \(S_{y}\) (i.e., the entropy squeezing factor \(E(S_{y})\) initially exhibits optimal entropy squeezing), we can protect the entropy squeezing for \(S_{y}\) by a proper \(\sigma _{x}\) control, but the \(\sigma _{y}\) control can not protect it. The numerical results show that the \(\sigma _{x}\) control with the feedback amplitude \(A_{x}\approx 3\pi /4\) is a optimal choice for protecting the entropy squeezing for \(S_{y}\). Thus, it is obvious that the \(\sigma _{x}\) control and \(\sigma _{y}\) control with proper feedback parameters also can be utilized to protect entropy squeezing for \(S_{y}\) and \(S_{x}\), respectively.

Fig. 2
figure 2

Entropy squeezing factor \(E(S_{x})\) as a function of \(\gamma t\) with the \(\sigma _{y}\) control for \(A_{y}=1.1\) or 2.04 (red solid curve) versus the uncontrolled case for \(A_{y}=0\) (blue dashed curve). \(\theta =0\), the atom is initially in the excited state (Color figure online)

Full size image
Fig. 3
figure 3

Entropy squeezing factor \(E(S_{x})\) as a function of \(\gamma t\) and the feedback parameter with a the \(\sigma _{x}\) control and b the \(\sigma _{y}\) control. \(\theta =\frac{\pi }{4}\), \(\phi =0\), the atom is initially in an eigenstate of \(S_{x}\)

Full size image
Fig. 4
figure 4

Entropy squeezing factor \(E(S_{x})\) as a function of \(\gamma t\) with the \(\sigma _{y}\) control for \(A_{y}=\frac{\pi }{4}\) (red solid curve) versus the uncontrolled case for \(A_{y}=0\) (blue dashed curve). \(\theta =\frac{\pi }{4}, \phi =0\), the atom is initially in an eigenstate of \(S_{x}\) (Color figure online)

Full size image

In this subsection, we have studied the evolution of the entropy squeezing factors \(E(S_{x})\) and \(E(S_{y})\) with the \(\sigma _{x}\) control and \(\sigma _{y}\) control. By numerical analysis, we can conclude that the \(\sigma _{x}\) control is good for the entropy squeezing for the component \(S_{y}\), while the \(\sigma _{y}\) control is in favor of the entropy squeezing for the component \(S_{x}\). When suitable feedback parameters are chosen, the entropy squeezing for \(S_{x}\) and \(S_{y}\) can be generated and protected. Nevertheless, we can note from Figs. 1, 2, 3, and 4 that in these cases the entropy squeezing decreases with time and finally vanishes because of the dissipative effect; that is to say, the enhancement of squeezing degree and squeezing time is still rather small. So we further study the dynamics of the entropy squeezing factors under the control of both quantum-jump-based feedback and classical driving to seek for the entropy squeezing with high squeezing degree and long squeezing time.

3.2 Steady and optimal entropy squeezing with both quantum-jump-based feedback and classical driving

In this part, the system with both quantum-jump-based feedback and classical driving will be studied. Unlike the case without classical driving, the density matrix of the atom can not be analytically presented, so we discuss the entropy squeezing by numerical calculations. Figure 5 presents the evolution of the entropy squeezing factor \(E(S_{x})\) for different values of the driving strength \(\varOmega \) and different initial states of the atom under the \(\sigma _{y}\) control with \(A_{y}=\pi /4\). From the investigation of the preceding subsection, we have found that the entropy squeezing in those cases will disappear after long-time evolution when there is no classical driving. However, when the classical driving is taken into account, the phenomenon is very different. In Fig. 5a, we plot the entropy squeezing factor \(E(S_{x})\) against the scale time \(\gamma t\) under the condition that the atom is initially in the excited state, and \(\varOmega \) takes different values. It is observed that, when \(\varOmega =0.2\gamma \), \(E(S_{x})\) keeps in a positive value after long-time evolution, which indicates that there is no entropy squeezing for the component \(S_{x}\) in the long-time limit. While for \(\varOmega =2\gamma \) and \(20\gamma \), we can see that the blue dashed curve and the black solid curve finally reach steady negative values. This predicts steady entropy squeezing for the component \(S_{x}\). What is more, for \(\varOmega =20\gamma \), \(E(S_{x})\) attains optimal entropy squeezing \(-0.414\), which means that a initial atomic state without squeezing can be evolved into a state with steady and optimal entropy squeezing via proper quantum-jump-based feedback and classical driving. In Fig. 5b, we choose the initial state \(\psi _{1}=\frac{1}{\sqrt{2}}(|1\rangle +|0\rangle )\), and other parameters are the same as those of Fig. 5a. After long-time evolution, the effect of quantum-jump-based feedback combined with classical driving on the entropy squeezing factor \(E(S_{x})\) is similar to that in Fig. 5a in which the atom is initially in the excited state. The most significant result is that when \(\varOmega =20\gamma \), the initial optimal entropy squeezing of the atom can be kept all the time. This result shows that a initial atomic state with optimal entropy squeezing can be protected from environmental noise during its evolution via quantum-jump-based feedback and classical driving. These results can be interpreted with the help of the steady-state solution of the two-level atom.

Fig. 5
figure 5

Entropy squeezing factor \(E(S_{x})\) as a function of \(\gamma t\) with the \(\sigma _{y}\) control for \(A_{y}=\frac{\pi }{4}\). a \(\theta =0\), the atom is initially in the excited state. b \(\theta =\frac{\pi }{4}, \phi =0\), the atom is initially in an eigenstate of \(S_{x}\). The classical driving strength: \(\varOmega =0.2\gamma \) (green dotted curve), \(\varOmega =2\gamma \) (blue dashed curve) and \(\varOmega =20\gamma \) (black solid curve) (Color figure online)

Full size image

The steady-state solution of the atom with quantum-jump-based feedback and classical driving can be easily obtained by setting \(\dot{\rho }=0\) and the normalization condition. Under the \(\sigma _{x}\) control and \(\sigma _{y}\) control, the density matrix elements of the steady state respectively have the forms

$$\begin{aligned} \rho _{x,11}= & {} \frac{\varOmega ^{2}}{2\varOmega ^{2}+\gamma ^{2}(\cos A_{x})^2+\gamma \varOmega \sin (2A_{x})},\nonumber \\ \rho _{x,12}= & {} \frac{-i\gamma \varOmega (\cos A_{x})^2}{2\varOmega ^{2}+\gamma ^{2}(\cos A_{x})^2+\gamma \varOmega \sin (2A_{x})} , \nonumber \\ \rho _{x,21}= & {} \rho _{x,12}^{*} ,\nonumber \\ \rho _{x,22}= & {} 1-\rho _{x,11}, \end{aligned}$$
(14)

and

$$\begin{aligned} \rho _{y,11}= & {} \frac{\varOmega ^{2}}{2\varOmega ^{2}+\gamma ^{2}(\cos A_{y})^2},\nonumber \\ \rho _{y,12}= & {} \frac{-i\gamma \varOmega (\cos A_{y})^2+\varOmega ^2\sin (2A_{y})}{2\varOmega ^{2}+\gamma ^{2}(\cos A_{y})^2} , \nonumber \\ \rho _{y,21}= & {} \rho _{y,12}^{*} ,\nonumber \\ \rho _{y,22}= & {} 1-\rho _{y,11}. \end{aligned}$$
(15)

Substituting Eq. (15) in Eq. (12), the entropy squeezing factor of the steady state under the \(\sigma _{y}\) control can be obtained. In Fig. 6, we show the variation of entropy squeezing factor \(E(S_{x})\) of the steady state Eq. (15) with respect to \(\varOmega /\gamma \) by setting \(A_{y}=\pi /4\). The figure shows that with the increase of \(\varOmega /\gamma \), \(E(S_{x})\) increases at the initial stage and then decreases to the optimal value \(-0.414\). We also can see from Eq. (15) that when \(\varOmega \) is much larger than the other parameters, the elements of the state reduce to \(\rho _{y,11}=1/2\) and \(\rho _{y,12}=\sin (2A_{y})/2\). So, if we set \(A_{y}=\pi /4\) or \(3\pi /4\), the steady states will become \(\frac{1}{\sqrt{2}}(|1\rangle \pm |0\rangle )\), which are two eigenstates of the component \(S_{x}\) (i.e., \(E(S_{x})\) exhibits optimal entropy squeezing \(-0.414\)). In addition, through numerical analysis, we find that in the steady state Eq. (15), \(E(S_{y})\) only can exhibit very small entropy squeezing about \(-0.02\) by adjusting \(\varOmega \) and \(A_{y}\). These indicate that using the \(\sigma _{y}\) control and classical driving, we can obtain steady and optimal entropy squeezing for \(S_{x}\), but not for \(S_{y}\).

The steady-state entropy squeezing under the \(\sigma _{x}\) control is also analyzed by inserting Eq. (14) into Eq. (12). In this case, there is no entropy squeezing effect for \(S_{x}\) as \(E(S_{x})\ge 0\), no matter what the values of \(\varOmega \) and \(A_{x}\) are, whereas when \(A_{x}=3\pi /4\) and \(\varOmega =0.5\gamma \), \(E(S_{y})\) exhibits optimal entropy squeezing \(-0.414\). This result stems from the fact that the atomic steady state reduces to \(\frac{1}{\sqrt{2}}(|1\rangle +i|0\rangle )\), an eigenstate of \(S_{y}\). Obviously, different from the \(\sigma _{y}\) control, a suitable \(\sigma _{x}\) control combined with proper classical driving can be used to obtain steady and optimal entropy squeezing for \(S_{y}\) rather than \(S_{x}\).

Fig. 6
figure 6

Steady-state entropy squeezing factor \(E(S_{x})\) with the \(\sigma _{y}\) control for \(A_{y}=\frac{\pi }{4}\) as a function of \(\varOmega /\gamma \)

Full size image

4 Conclusion

In summary, we have studied the entropy squeezing of a two-level atom coupled to a dissipative cavity under the control of quantum-jump-based feedback itself or the combination of quantum-jump-based feedback and classical driving. Our results show that the entropy squeezing of the atomic polarization components greatly depends on the control of quantum-jump-based feedback and classical driving. We find that the \(\sigma _{x}\) control and \(\sigma _{y}\) control with proper parameters respectively have an active influence on entropy squeezing for \(S_{y}\) and \(S_{x}\). When there is only the quantum-jump-based feedback, we can generate and protect the entropy squeezing for \(S_{x}\) and \(S_{y}\) by a properly designed \(\sigma _{y}\) control and \(\sigma _{x}\) control, respectively, but we cannot prepare a state with steady and optimal entropy squeezing. Furthermore, when there are both quantum-jump-based feedback and classical driving, an initial atomic state without squeezing can be evolved into a state with steady and optimal entropy squeezing, and one with optimal entropy squeezing can be kept all the time via appropriate feedback controls and classical driving. Our results can be interpreted with the help of the steady-state solution of the two-level atom and are meaningful for the practical application of squeezing as well as the realization of quantum information processing with ultra-low quantum noise.

It is worth noting that if we use variance squeezing to quantify atomic squeezing in our system, some information about squeezing cannot be captured owing to the defect of the definition of atomic variance squeezing [8]. For a two-level atom, the definition of variance squeezing is based on the Heisenberg uncertainty relation \(\varDelta S_{x}\varDelta S_{y}\ge \frac{1}{2}|\langle S_{z}\rangle |\) (\(\varDelta S_{\alpha }=[\langle S^{2}_{\alpha }\rangle -\langle S_{\alpha }\rangle ^{2}]^{1/2}, \alpha =x,y\)), and fluctuations in the component \(S_{\alpha }\) of the atomic dipole are said to be squeezed if \(S_{\alpha }\) satisfied the condition \(V(S_{\alpha })=\varDelta S_{\alpha }-(\frac{|\langle S_{z}\rangle |}{2})^{1/2}<0\). As we know, when the expectation value of the operator \(S_{z}\) is zero, the Heisenberg uncertainty relation is trivial, and \(V(S_{\alpha })\) cannot provide any useful information about squeezing, whereas the definition of entropy squeezing based on the entropy uncertainty relation which is never trivial can always provide full information about atomic squeezing. In this paper, we can obtain steady and optimal entropy squeezing, which stems from the fact that the steady state of the atom becomes the eigenstate of \(S_{\alpha }\) under appropriate feedback controls and classical driving. However, in these states, the expectation value of the operator \(S_{z}\) exactly equals zero, which indicates that the variance squeezing is invalid and fails to provide information about the squeezing. Therefore, in order to capture all information about squeezing, especially the information about optimal squeezing, we use entropy squeezing rather than variance squeezing to quantify the atomic squeezing in this paper.