Violations of the Leggett-Garg inequality for coherent and cat states

A violation of Bell’s inequality illustrates the fact that quantum mechanics is essentially incompatible with classical mechanics [1]. To compute Bell’s inequality, first we prepare states of two qubits spatially separated, second observe them independently, and third evaluate expectation values by taking averages of the set of the measurements. The violation of Bell’s inequality tells us that quantum mechanics is able to exhibit strong correlations which cannot be explained by the hidden variable theory.

The Leggett-Garg inequality (LGI) was proposed by Leggett and Garg in 1985 for testing whether or not macroscopic coherence could be realized in the laboratory [2, 3, 4]. When we consider Bell’s inequality, we pay attention to the correlation between the two spatially separated qubits. By contrast, although the LGI is an analogue of Bell’s one, it examines a correlation of measurements at two different times for a single particle, and thus we can regard it as temporal Bell’s inequality.

To derive this inequality, Leggett and Garg introduced the following two assumptions. The first one is “macroscopic realism”. This assumption requires that results of observation are determined by hidden variables that are attributes of the observed system. Moreover, we postulate that the hidden variables are independent of the observation itself. The second one is “non-invasive measurability”. This premise implies that a result of measurement at time $t_{2}$ is independent of a result of previous measurement at time $t_{1}(<t_{2})$. Thus, we can specify the state of the system uniquely without disturbing the system.

In general, classical mechanics satisfies these two assumptions, so that dynamics of all systems that evolves according to the classical theory does not violate the LGI. However, the two assumptions do not always hold in the quantum mechanics. Thus, it is possible that the dynamics of a quantum system violates the LGI. In fact, concrete examples of quantum systems which violate the inequality have been already found theoretically [5, 6, 7, 8, 9, 10, 11]. Recently, experimental demonstrations of the violation of the LGI have been reported [12, 13, 14].

In the current paper, we investigate the LGI for a cavity mode weakly coupled to a zero-temperature environment. We make the boson system undergo dissipation due to the interaction with the environment but we assume that it does not receive dephasing. Because there is no thermal fluctuation of the external reservoir, we can solve the master equation of the boson system exactly in a closed-form expression [15, 16].

In the present paper, we focus on cases where initial states of the system are given by the coherent state $|\alpha\rangle$ and the cat state $(|\alpha\rangle+|-\alpha\rangle)$ and compare their violations of the LGI. The cat state is also called the even coherent state and we can consider it to be a special one of the Yurke-Stoler coherent states [17, 18, 19, 20].

In order to evaluate the LGI, selection of operators for dichotomic observation is important. In the derivation of the inequality, we performs measurements of the dichotomic variable at three equally spaced times, $t_{1}=0$, $t_{2}=\tau$, and $t_{3}=2\tau$, where $\tau>0$. In the present paper, we choose the displaced parity operators to measure the state of the boson system [21, 22, 23, 24, 25]. These operators are characterized by an arbitrary complex number $\beta=re^{i\theta}$. Thus, adjusting the two real parameters, $r$ and $\theta$, we can vary the violation of the LGI.

In the current paper, we concentrate on a maximization problem, where we look for the optimum values of the parameters, $\theta$ and $r$, for maximizing the violation. Because the LGI reflects the macroscopic coherence of the system of interest, we can regard its violation as a quantity of the quantum feature of the system. One of the motivations of the present paper is to clarify which state exhibits a characteristic of quantum nature more strongly, the coherent state or the cat state.

We obtain the following result: the violation of the inequality for the coherent state is larger than that for the cat state if we let the time difference $\tau$ of the LGI be a value belonging to a specific range. This result is unexpected because the cat state is a superposition of two different coherent states, $|\alpha\rangle$ and $|-\alpha\rangle$, and we can consider intuitively that the cat state $(|\alpha\rangle+|-\alpha\rangle)$ exhibits the characteristic of quantum nature more intensely than the ordinary coherent state $|\alpha\rangle$. This counter intuitive fact is novel and interesting.

Moreover, if we let the time difference $\tau$ approach zero, the parameter $\beta$ diverges to infinity as $|\beta|=r\to\infty$ for the optimized displaced parity operators and the maximized violation of the inequality converges to $3/2$ but not unity, so that the LGI shows strong singularity. Because such behaviour of the violation does not occur in the inequality for a two-level system (i.e. using projection operators as observables, the upper bound of the inequality for the two-level system gets closer to unity as the time difference $\tau$ approaches zero), we can guess that the origin of the singularity is the infiniteness of the dimension of the Hilbert space for the boson system.

Here, we mention previous works. In the current paper, we study the LGI for a cavity mode interacting with a zero-temperature environment. In Ref. [7], Chen et al. investigated the LGI of a qubit coupled to a zero-temperature environment. In Refs. [8, 9], the LGI of a qubit interacting with a thermal environment was studied. In Ref. [11], Thenabadu and Reid discussed the violation of the LGI of the cat state of the boson system but they did not consider the interaction with an environment.

In Ref. [26], two inequalities inspired by the LGI were derived to distinguish quantum from classical transport through nanostructures. In Refs. [27, 28], the extended LGI proposed in Ref. [26] was utilized on a double quantum dot and a multiple-quantum-well structure. In Ref. [29], new practical quantum witnesses to verify quantum coherence for complex systems were proposed and they were regarded as refined ones compared with the LGI. In Ref. [30], the violation of the LGI in electronic Mach-Zehnder interferometers was investigated.

The current paper is organized as follows. In Sect. 2, we review the master equation of the boson system, the LGI, and the displaced parity operators. In Sect. 3, we derive the explicit form of the LGI for the system prepared initially in the coherent state. In Sect. 4, we derive the rigorous form of the inequality with assuming that the system is initially in the cat state. In Sect. 5, we estimate the violation of the inequality numerically in the case where the system is initialized in the coherent state $|\alpha\rangle$. In Sect. 6, we evaluate the violation numerically in the case where the system is initially put in the cat state $(|\alpha\rangle+|-\alpha\rangle)$. In Sects. 5 and 6, we perform optimization of the displaced parity operators. In Sect. 7, we compare the maximized violations of the inequality for the two initial states, the coherent and cat states. In Sect. 8, we give brief discussion. In Appendices A and B, we give explicit mathematical forms that appear in Sects. 3 and 4, respectively.

In the current section, first of all, we review the master equation for the boson system and its exact solution [15, 16]. Next, we introduce the LGI [2, 3, 4] and the displaced parity operators [21, 22, 23, 24, 25].

The master equation of the boson system weakly coupled to a zero-temperature environment is given by

$$\dot{\rho}(t)=-i[H,\rho(t)]+\Gamma[2a\rho(t)a^{\dagger}-a^{\dagger}a\rho(t)-\rho(t)a^{\dagger}a],$$

(1)

$$H=\omega a^{\dagger}a,$$

(2)

where we put $\hbar=1$, and $\Gamma$ and $\omega$ represent the spontaneous emission rate and the angular velocity, respectively. We assume that $\Gamma$ and $\omega$ are real. Here, we consider how to solve Eq. (1) rigorously.

Taking the interaction picture, we simplify the master equation (1) as

$$\dot{\rho}(t)=\Gamma[2a\rho(t)a^{\dagger}-a^{\dagger}a\rho(t)-\rho(t)a^{\dagger}a].$$

(3)

We introduce superoperators $\hat{J}$ and $\hat{L}$, where we put hats on the symbols to emphasize that they are superoperators, as follows:

$$\hat{J}\rho(t)=a\rho(t)a^{\dagger},$$

(4)

$$\hat{L}\rho(t)=-(1/2)[a^{\dagger}a\rho(t)+\rho(t)a^{\dagger}a].$$

(5)

Then, the master equation (3) can be rewritten in the form,

$$\dot{\rho}(t)=2\Gamma(\hat{J}+\hat{L})\rho(t).$$

(6)

Using a commutation relation $[\hat{J},\hat{L}]=-\hat{J}$, we obtain a formal solution of Eq. (6),

	$\displaystyle\rho(t)$	$\displaystyle=$	$\displaystyle\exp[2\Gamma t(\hat{J}+\hat{L})]\rho(0)$		(7)
		$\displaystyle=$	$\displaystyle\exp(2\Gamma t\hat{L})\exp\{[1-\exp(-2\Gamma t)]\hat{J}\}\rho(0).$

Now, we substitute an operator $v(0)=|\alpha\rangle\langle\beta|$ for $\rho(0)$, where $|\alpha\rangle$ and $|\beta\rangle$ are coherent states and $\alpha$ and $\beta$ are arbitrary complex numbers. Then, $v(t)$ is given by

	$\displaystyle v(t)$	$\displaystyle=$	$\displaystyle\exp\{-(1/2)(|\alpha|^{2}+|\beta|^{2}-2\alpha\beta^{*})[1-\exp(-2\Gamma t)]\}$		(8)
			$\displaystyle\times|\alpha\exp(-\Gamma t)\rangle\langle\beta\exp(-\Gamma t)|.$

Here, we change the interaction picture of Eq. (8) into the Schrödinger picture. Finally, we attain

	$\displaystyle v(t)$	$\displaystyle=$	$\displaystyle\exp\{-(1/2)(|\alpha|^{2}+|\beta|^{2}-2\alpha\beta^{*})[1-\exp(-2\Gamma t)]\}$		(9)
			$\displaystyle\times|\alpha\exp(-i\Omega t)\rangle\langle\beta\exp(-i\Omega t)|,$

where $\Omega=\omega-i\Gamma$.

We pay attention to the following facts. If we put $v(0)=|\alpha\rangle\langle\beta|$ and $\tilde{v}(0)=|\beta\rangle\langle\alpha|$, $\tilde{v}(t)^{\dagger}=v(t)$ holds. Moreover, if we set $\rho(0)=|\alpha\rangle\langle\alpha|$, we obtain
$\rho(t)=|\alpha\exp(-i\Omega t)\rangle\langle\alpha\exp(-i\Omega t)|$.

The LGI is defined as follows. For the sake of simplicity, we assume that the dichotomic observables are given by projection operators. We consider an operator $\hat{O}$ whose eigenvalues are equal to $\pm 1$. To emphasize that it is an operator, we put a hat on the symbol. Next, we introduce equally spaced three times, $t_{1}=0$, $t_{2}=\tau$, and $t_{3}=2\tau$, where $\tau>0$. We describe an observed value of the measurement with $\hat{O}$ at time $t_{1}$ as $O_{1}$. Obviously, $O_{1}=\pm 1$ holds. We write a probability that observed values at times $t_{1}$ and $t_{2}$ are equal to $O_{1}$ and $O_{2}$ respectively as $P_{21}(O_{1},O_{2})$. We define the correlation function $C_{21}$ as

$$C_{21}=\sum_{O_{1},O_{2}\in\{-1,+1\}}O_{2}O_{1}P_{21}(O_{1},O_{2}).$$

(10)

Then, the LGI is given by

$$K_{3}=C_{21}+C_{32}-C_{31},$$

(11)

$$-3\leq K_{3}\leq 1.$$

(12)

We define projection operators which are called the displaced parity operators as

	$\displaystyle\Pi^{(+)}(\beta)$	$\displaystyle=$	$\displaystyle D(\beta)\sum_{n=0}^{\infty}|2n\rangle\langle 2n|D^{\dagger}(\beta),$
	$\displaystyle\Pi^{(-)}(\beta)$	$\displaystyle=$	$\displaystyle D(\beta)\sum_{n=0}^{\infty}|2n+1\rangle\langle 2n+1|D^{\dagger}(\beta),$		(13)

where

$$D(\beta)=\exp(\beta a^{\dagger}-\beta^{*}a).$$

(14)

In the present paper, hereafter, we choose the following operator for the orthogonal measurement of the LGI,

$$\hat{O}=\Pi^{(+)}(\beta)-\Pi^{(-)}(\beta).$$

(15)

In the present section, putting a coherent state $|\alpha\rangle$ as the initial state, we derive a closed-form expression of the LGI.

The probability that we obtain $O_{1}=1$ for the measurement of the initial state $|\alpha\rangle$ at time $t_{1}=0$ is given by

	$\displaystyle P_{1+}$	$\displaystyle=$	$\displaystyle\langle\alpha|\Pi^{(+)}(\beta)|\alpha\rangle$		(16)
		$\displaystyle=$	$\displaystyle\exp(-|\alpha-\beta|^{2})\cosh|\alpha-\beta|^{2}.$

Then, the state collapses from $|\alpha\rangle$ to $w_{1+}(0)/P_{1+}$, where $w_{1+}(0)$ is given by

$$w_{1+}(0)=\Pi^{(+)}(\beta)|\alpha\rangle\langle\alpha|\Pi^{(+)}(\beta),$$

(17)

$$\Pi^{(+)}(\beta)|\alpha\rangle=(1/2)[|\alpha\rangle+\exp(\beta^{*}\alpha-\beta\alpha^{*})|2\beta-\alpha\rangle].$$

(18)

In the above derivation, we use a relation,

$$\sum_{n=0}^{\infty}|2n\rangle\langle 2n|\alpha-\beta\rangle=(1/2)[|\alpha-\beta\rangle+|-(\alpha-\beta)\rangle].$$

(19)

Similarly, the probability that we obtain $O_{1}=-1$ for the measurement of the initial state $|\alpha\rangle$ at time $t_{1}=0$ is written in the form,

	$\displaystyle P_{1-}$	$\displaystyle=$	$\displaystyle\langle\alpha|\Pi^{(-)}(\beta)|\alpha\rangle$		(20)
		$\displaystyle=$	$\displaystyle\exp(-|\alpha-\beta|^{2})\sinh|\alpha-\beta|^{2}.$

Then, the initial state $|\alpha\rangle$ reduces to $w_{1-}(0)/P_{1-}$, where $w_{1-}(0)$ is given by

$$w_{1-}(0)=\Pi^{(-)}(\beta)|\alpha\rangle\langle\alpha|\Pi^{(-)}(\beta),$$

(21)

$$\Pi^{(-)}(\beta)|\alpha\rangle=(1/2)[|\alpha\rangle-\exp(\beta^{*}\alpha-\beta\alpha^{*})|2\beta-\alpha\rangle].$$

(22)

In the above derivation, we use a relation,

$$\sum_{n=0}^{\infty}|2n+1\rangle\langle 2n+1|\alpha-\beta\rangle=(1/2)[|\alpha-\beta\rangle-|-(\alpha-\beta)\rangle].$$

(23)

According to Eq. (9), $w_{1\pm}(0)$, the unnormalized state at time $t_{1}=0$, evolves into the state $w_{1\pm}(\tau)$ at time $t_{2}=\tau$, whose explicit form is given by Eq. (74) in Appendix A. Describing the probability that we detect $O_{2}=1$ with the observation of $w_{1\pm}(\tau)$ at time $t_{2}=\tau$ as $p_{1\pm,2+}$, we obtain it in the form of Eq. (75) in Appendix A. Similarly, writing down the probability that we have $O_{2}=-1$ with the observation of $w_{1\pm}(\tau)$ at time $t_{2}=\tau$ as $p_{1\pm,2-}$, we acquire it in the form of Eq. (76) in Appendix A.

From Eqs. (75) and (76), the correlation function $C_{21}$ is given by

$$C_{21}=p_{1+,2+}-p_{1+,2-}-p_{1-,2+}+p_{1-,2-}.$$

(24)

Here, we regard $C_{21}$ as a function of multiple variables,

$$C_{21}=C(\alpha,\beta,\omega,\Gamma,\tau).$$

(25)

Then, $C_{31}$ and $C_{32}$ are given by

$$C_{31}=C(\alpha,\beta,\omega,\Gamma,2\tau),$$

(26)

$$C_{32}=C(\alpha e^{-i\Omega\tau},\beta,\omega,\Gamma,\tau).$$

(27)

Thus, from Eqs. (24), (25), (26), and (27), we obtain $K_{3}$ eventually.

In this section, preparing the cat state $(|\alpha\rangle+|-\alpha\rangle)$ as the initial state, we derive a mathematically rigorous form of the LGI.

We assume that the initial state of the system at time $t_{1}=0$ is given by the cat state,

$$\rho(t_{1})=|\psi\rangle\langle\psi|,$$

(28)

$$|\psi\rangle=q(\alpha)^{-1/2}(|\alpha\rangle+|-\alpha\rangle),$$

(29)

$$q(\alpha)=2[1+\exp(-2|\alpha|^{2})],$$

(30)

where $|\alpha\rangle$ and $-|\alpha\rangle$ are coherent states. It is convenient to use the following notation for calculations carried out hereafter:

$$\rho(t_{1})=q(\alpha)^{-1}(K+L+L^{\dagger}+M),$$

(31)

	$\displaystyle K$	$\displaystyle=$	$\displaystyle|\alpha\rangle\langle\alpha|,$
	$\displaystyle L$	$\displaystyle=$	$\displaystyle|\alpha\rangle\langle-\alpha|,$
	$\displaystyle M$	$\displaystyle=$	$\displaystyle|-\alpha\rangle\langle-\alpha|.$		(32)

The probability that we obtain $O_{1}=\pm 1$ with the observation of $\rho(t_{1})$ at time $t_{1}=0$ is written down in the form,

$$P_{1\pm}=\mbox{Tr}[\Pi^{(\pm)}(\beta)\rho(t_{1})].$$

(33)

Then, the state collapses from $\rho(t_{1})$ to $w_{1\pm}(0)/P_{1\pm}$, where $w_{1\pm}(0)$ is given by

	$\displaystyle w_{1\pm}(0)$	$\displaystyle=$	$\displaystyle\Pi^{(\pm)}(\beta)\rho(t_{1})\Pi^{(\pm)}(\beta)$		(34)
		$\displaystyle=$	$\displaystyle q(\alpha)^{-1}\Pi^{(\pm)}(\beta)(K+L+L^{\dagger}+M)\Pi^{(\pm)}(\beta).$

Moreover, we divide operators $K$, $L$, and $M$ into operators as

	$\displaystyle\Pi^{(\pm)}(\beta)K\Pi^{(\pm)}(\beta)$	$\displaystyle=$	$\displaystyle(1/4)[K^{(1)}(0)\pm K^{(2)}(0)\pm K^{(3)}(0)+K^{(4)}(0)],$
	$\displaystyle\Pi^{(\pm)}(\beta)L\Pi^{(\pm)}(\beta)$	$\displaystyle=$	$\displaystyle(1/4)[L^{(1)}(0)\pm L^{(2)}(0)\pm L^{(3)}(0)+L^{(4)}(0)],$
	$\displaystyle\Pi^{(\pm)}(\beta)M\Pi^{(\pm)}(\beta)$	$\displaystyle=$	$\displaystyle(1/4)[M^{(1)}(0)\pm M^{(2)}(0)\pm M^{(3)}(0)+M^{(4)}(0)],$		(35)

where $K^{(3)}(0)=K^{(2)}(0)^{\dagger}$ and $M^{(3)}(0)=M^{(2)}(0)^{\dagger}$. We give explicit forms of $\{K^{(j)}(0):j=1,2,4\}$, $\{L^{(j)}(0):j=1,2,3,4\}$, and $\{M^{(j)}(0):j=1,2,4\}$ in Eqs. (77), (78), and (79) in Appendix B. Closed-form expressions of time evolution from time $t_{1}=0$ to time $t_{2}=\tau$ of these operators, $\{K^{(j)}(\tau):j=1,2,4\}$, $\{L^{(j)}(\tau):j=1,2,3,4\}$, and $\{M^{(j)}(\tau):j=1,2,4\}$, are given by Eqs. (80), (81), and (82) in Appendix B explicitly.

We consider that the state $w_{1\pm}(0)$ at time $t_{1}=0$ evolves into $w_{1\pm}(\tau)$ at time $t_{2}=\tau$. Then, we can obtain the state $w_{1\pm}(\tau)$ by replacing $\{K^{(j)}(0):j=1,2,4\}$, $\{L^{(j)}(0):j=1,2,3,4\}$, and $\{M^{(j)}(0):j=1,2,4\}$ in Eqs. (34) and (35) with $\{K^{(j)}(\tau):j=1,2,4\}$, $\{L^{(j)}(\tau):j=1,2,3,4\}$, and $\{M^{(j)}(\tau):j=1,2,4\}$ given by Eqs. (80), (81), and (82). Describing the probability that we detect $O_{2}=1$ with the observation of the state $w_{1\pm}(\tau)$ at time $t_{2}=\tau$ as $p_{1\pm,2+}$, we obtain it in the form of Eq. (LABEL:p1pm2p) in Appendix B.

Letting the probability that we obtain $O_{2}=-1$ with the observation of $w_{1\pm}(\tau)$ at time $t_{2}=\tau$ be equal to $p_{1\pm,2-}$, we can write down it as Eq. (LABEL:p1pm2m) in Appendix B. Then, we can derive the correlation function $C_{21}$ from Eq. (24). We can also obtain $C_{31}$ from Eqs. (25) and (26).

Next, we think about the correlation function $C_{32}$. Here, we remark that we cannot obtain $C_{32}$ from Eq. (27). The reason why is going to be clarified by Eqs.(36) and (37). The initial state at time $t_{1}=0$ is given by Eqs. (28), (29), and (30). The initial state $\rho(t_{1})$ evolves into the following state at time $t_{2}=\tau$:

$\displaystyle\rho(t_{2})$

$\displaystyle=$

$\displaystyle q(\alpha)^{-1}\Bigl{(}\tilde{K}+\exp\{-2|\alpha|^{2}[1-\exp(-2\Gamma\tau)]\}(\tilde{L}+\tilde{L}^{\dagger})+\tilde{M}\Bigr{)},$

(36)

	$\displaystyle\tilde{K}$	$\displaystyle=$	$\displaystyle|\alpha\exp(-i\Omega\tau)\rangle\langle\alpha\exp(-i\Omega\tau)|,$
	$\displaystyle\tilde{L}$	$\displaystyle=$	$\displaystyle|\alpha\exp(-i\Omega\tau)\rangle\langle-\alpha\exp(-i\Omega\tau)|,$
	$\displaystyle\tilde{M}$	$\displaystyle=$	$\displaystyle|-\alpha\exp(-i\Omega\tau)\rangle\langle-\alpha\exp(-i\Omega\tau)|.$		(37)

Looking at Eqs. (36) and (37), we notice that $\rho(t_{2})$ is different from $\rho^{\prime}=|\psi^{\prime}\rangle\langle\psi^{\prime}|$ where $|\psi^{\prime}\rangle=q(\alpha^{\prime})^{-1/2}(|\alpha^{\prime}\rangle+|-\alpha^{\prime}\rangle)$ and $\alpha^{\prime}=\alpha\exp(-i\Omega\tau)$. Thus, we cannot derive $C_{32}$ from Eq. (27).

The probability that we obtain $O_{2}=\pm 1$ with the observation of $\rho(t_{2})$ at time $t_{2}=\tau$ is given by

$$P_{2\pm}=\mbox{Tr}[\Pi^{(\pm)}(\beta)\rho(t_{2})].$$

(38)

Then, the state of the system $\rho(t_{2})$ reduces to $w_{2\pm}(\tau)/P_{2\pm}$, where $w_{2\pm}(\tau)$ is given by

	$\displaystyle w_{2\pm}(\tau)$	$\displaystyle=$	$\displaystyle\Pi^{(\pm)}(\beta)\rho(t_{2})\Pi^{(\pm)}(\beta)$
		$\displaystyle=$	$\displaystyle q(\alpha)^{-1}\Pi^{(\pm)}(\beta)\Bigl{(}\tilde{K}+\exp\{-2|\alpha|^{2}[1-\exp(-2\Gamma\tau)]\}(\tilde{L}+\tilde{L}^{\dagger})+\tilde{M}\Bigr{)}\Pi^{(\pm)}(\beta).$

We can divide $\Pi^{(\pm)}(\beta)\tilde{K}\Pi^{(\pm)}(\beta)$, $\Pi^{(\pm)}(\beta)\tilde{L}\Pi^{(\pm)}(\beta)$, and $\Pi^{(\pm)}(\beta)\tilde{M}\Pi^{(\pm)}(\beta)$ into operators,

$\displaystyle\Pi^{(\pm)}(\beta)\tilde{K}\Pi^{(\pm)}(\beta)$

$\displaystyle=$

$\displaystyle(1/4)[\tilde{K}^{(1)}(0)\pm\tilde{K}^{(2)}(0)\pm\tilde{K}^{(2)}(0)^{\dagger}+\tilde{K}^{(4)}(0)],$

(40)

$$\tilde{K}^{(j)}(0)=\left.K^{(j)}(0)\right|_{\alpha\rightarrow\alpha\exp(-i\Omega\tau)}\quad\mbox{for $j=1,2,3,4$},$$

(41)

$\displaystyle\Pi^{(\pm)}(\beta)\tilde{L}\Pi^{(\pm)}(\beta)$

$\displaystyle=$

$\displaystyle(1/4)[\tilde{L}^{(1)}(0)\pm\tilde{L}^{(2)}(0)\pm\tilde{L}^{(2)}(0)^{\dagger}+\tilde{L}^{(4)}(0)],$

(42)

$$\tilde{L}^{(j)}(0)=\left.L^{(j)}(0)\right|_{\alpha\rightarrow\alpha\exp(-i\Omega\tau)}\quad\mbox{for $j=1,2,3,4$},$$

(43)

$\displaystyle\Pi^{(\pm)}(\beta)\tilde{M}\Pi^{(\pm)}(\beta)$

$\displaystyle=$

$\displaystyle(1/4)[\tilde{M}^{(1)}(0)\pm\tilde{M}^{(2)}(0)\pm\tilde{M}^{(2)}(0)^{\dagger}+\tilde{M}^{(4)}(0)],$

(44)

$$\tilde{M}^{(j)}(0)=\left.M^{(j)}(0)\right|_{\alpha\rightarrow\alpha\exp(-i\Omega\tau)}\quad\mbox{for $j=1,2,3,4$}.$$

(45)

For example, Eq. (41) implies the following. The operator $\tilde{K}^{(j)}(0)$ is equal to the operator $K^{(j)}(0)$ with the replacement of $\alpha$ by $\alpha\exp(-i\Omega\tau)$ in Eq. (77).

Referring to Eqs. (80), (81), and (82), closed-form expressions of time evolution from time $t_{2}=\tau$ to time $t_{3}=2\tau$ of $\{\tilde{K}^{(j)}(0):j=1,2,3,4\}$, $\{\tilde{L}^{(j)}(0):j=1,2,3,4\}$, and $\{\tilde{M}^{(j)}(0):j=1,2,3,4\}$ are given by

	$\displaystyle\tilde{K}^{(j)}(\tau)$	$\displaystyle=$	$\displaystyle\left.K^{(j)}(\tau)\right|_{\alpha\rightarrow\alpha\exp(-i\Omega\tau)},$
	$\displaystyle\tilde{L}^{(j)}(\tau)$	$\displaystyle=$	$\displaystyle\left.L^{(j)}(\tau)\right|_{\alpha\rightarrow\alpha\exp(-i\Omega\tau)},$
	$\displaystyle\tilde{M}^{(j)}(\tau)$	$\displaystyle=$	$\displaystyle\left.M^{(j)}(\tau)\right|_{\alpha\rightarrow\alpha\exp(-i\Omega\tau)}\quad\mbox{for $j=1,2,3,4$}.$		(46)

Thus, the state $w_{2\pm}(2\tau)$, that is to say, the time evolution from time $t_{2}=\tau$ to time $t_{3}=2\tau$ of the state $w_{2\pm}(\tau)$, is obtained by replacing $\{\tilde{K}^{(j)}(0):j=1,2,3,4\}$, $\{\tilde{L}^{(j)}(0):j=1,2,3,4\}$, and $\{\tilde{M}^{(j)}(0):j=1,2,3,4\}$ with Eq. (46) in Eqs. (LABEL:w-2pm-tau-definition), (40), (41), (42), (43), (44), and (45).

We describe the probability that we obtain $O_{3}=1$ with the observation of $w_{2\pm}(2\tau)$ at time $t_{3}=2\tau$ as $p_{2\pm,3+}$. Similarly, we let $p_{2\pm,3-}$ represent the probability that we obtain $O_{3}=-1$ with the observation of $w_{2\pm}(2\tau)$ at time $t_{3}=2\tau$. Then, they are given by Eqs. (LABEL:p2pm2p-formula) and (LABEL:p2pm2m-formula) in Appendix B.

Thus, the correlation function $C_{32}$ is given in the form,

$$C_{32}=p_{2+,3+}-p_{2+,3-}-p_{2-,3+}+p_{2-,3-}.$$

(47)

Because we have just derived $C_{21}$, $C_{31}$, and $C_{32}$, finally we obtain $K_{3}$.

Here, we point out that $K_{3}$ has the following symmetry about the variable $\beta$. The value of $K_{3}$ is invariant under a transformation $\beta\rightarrow-\beta$. This fact is understood from relations,

	$\displaystyle\mbox{Tr}[K^{(1)}(\tau)\Pi^{(+)}(-\beta)]$	$\displaystyle=$	$\displaystyle\mbox{Tr}[M^{(1)}(\tau)\Pi^{(+)}(\beta)],$
	$\displaystyle\mbox{Tr}[K^{(2)}(\tau)\Pi^{(+)}(-\beta)]$	$\displaystyle=$	$\displaystyle\mbox{Tr}[M^{(2)}(\tau)\Pi^{(+)}(\beta)],$
	$\displaystyle\mbox{Tr}[K^{(4)}(\tau)\Pi^{(+)}(-\beta)]$	$\displaystyle=$	$\displaystyle\mbox{Tr}[M^{(4)}(\tau)\Pi^{(+)}(\beta)],$
	$\displaystyle\mbox{Tr}[L^{(1)}(\tau)\Pi^{(+)}(-\beta)]$	$\displaystyle=$	$\displaystyle\mbox{Tr}[L^{(1)}(\tau)\Pi^{(+)}(\beta)]^{*},$
	$\displaystyle\mbox{Tr}[L^{(2)}(\tau)\Pi^{(+)}(-\beta)]$	$\displaystyle=$	$\displaystyle\mbox{Tr}[L^{(3)}(\tau)\Pi^{(+)}(\beta)]^{*},$
	$\displaystyle\mbox{Tr}[L^{(3)}(\tau)\Pi^{(+)}(-\beta)]$	$\displaystyle=$	$\displaystyle\mbox{Tr}[L^{(2)}(\tau)\Pi^{(+)}(\beta)]^{*},$
	$\displaystyle\mbox{Tr}[L^{(4)}(\tau)\Pi^{(+)}(-\beta)]$	$\displaystyle=$	$\displaystyle\mbox{Tr}[L^{(4)}(\tau)\Pi^{(+)}(\beta)]^{*}.$		(48)

Hence, putting $\beta=r\exp(i\theta)$ and examining a value of $K_{3}$ with variation of $\beta$, we only have to try $\theta$ in a range of $0\leq\theta\leq\pi$. In other words, $K_{3}$ is a periodic function of $\theta$ and its period is equal to $\pi$.

In the current section, preparing the system initially in a coherent state $|\alpha\rangle$ for the LGI, we compute $K_{3}$ numerically and examine its properties. Moreover, we go into the optimization problem of the displaced parity operators. In order to let discussion of the current section be simple, we assume

$$\alpha=1/2,\quad\omega=1.$$

(49)

Because we set $\omega=1$, that is we let the angular velocity be equal to unity, we can adopt the following system of units. Because of $\omega=1$, we can rewrite a dimensionless quantity $\omega\tau$ as $\tau$. Thus, we can regard the time variable $\tau$ as dimensionless. Similarly, we can rewrite a dimensionless quantity $\Gamma/\omega$ as $\Gamma$, so that we can consider $\Gamma$ to be dimensionless, as well. Hereafter, we use this system of units.

Here, we define a simple notation as follows. Fixing the variables $\alpha$ and $\omega$ at $\alpha=1/2$ and $\omega=1$ respectively, we describe $K_{3}$ as a function of $\Gamma$, $\beta=re^{i\theta}$, and $\tau$,

$$K_{3}(\Gamma;\theta,r,\tau).$$

(50)

In Fig. 1, we plot $K_{3}(\Gamma;0,1/2,\tau)$ as a function of $\tau$ with $\beta=1/2$, that is $\theta=0$ and $r=1/2$. The thick solid red, thin solid blue, and thin dashed purple curves represent graphs with $\Gamma=0$, $0.05$, and $0.2$, respectively. The graph of the thick solid red curve has a period $2\pi$. As $\Gamma$ becomes larger, amplitudes of the curves shrink.

We can confirm numerically that the curve of $\Gamma=0.2$ converges to $0.367{\,}879...$ as $\tau$ increases. We can also validate this fact in the manner of analytical mathematics. Fixing $\Gamma$ at a finite positive value $\Gamma_{0}(>0)$ and letting $\tau$ approach infinity, $\tau\to\infty$, we obtain

	$\displaystyle\lim_{\tau\rightarrow\infty}K_{3}(\Gamma_{0};0,1/2,\tau)$	$\displaystyle=$	$\displaystyle 1/e$		(51)
		$\displaystyle\simeq$	$\displaystyle 0.367{\,}879....$

From the above result, we understand that $K_{3}$ converges to $1/e$ for $\Gamma_{0}>0$ and $\tau\rightarrow\infty$.