Quantum state protection in finite-temperature environment via quantum gates

Xiaodong Zeng; Xiaodong Zeng; Xiaodong Zeng; Guo-Qin Ge; Guo-Qin Ge; M. Suhail Zubairy; M. Suhail Zubairy; M. Suhail Zubairy

doi:10.1364/OE.27.025789

1. Introduction

Due to the inevitable interaction with the noisy environment, quantum state decoheres and quantum entanglement vanishes, which have been the main barriers that hinder the wide usage of quantum technology. As a consequence, the protection of the quantum states and quantum entanglements for a long time is important. Consequently several schemes, including those based on dynamic decoupling and quantum Zeno effect, have been proposed to overcome the effect of environment on quantum systems [1–5].

Another interesting method to protect the quantum states or entanglements is by using weak measurements and their reversals [6–19]. In these schemes, the quantum state is firstly transferred to more robust states by a weak measurement to resist the decoherence. After the decoherence process, the state is reversed back to the original state by another weak measurement. The behind physics is that the initial state is firstly projected to a more robust state. For example for an initial state $\alpha _1|0\rangle +\alpha _2|1\rangle$, we can project the state to $(\alpha _1|0\rangle +\alpha _2x|1\rangle )/\sqrt {|\alpha _1|^2+|\alpha _2|^2x^2}$ with $x<1$. This state has a larger probability to stay and keep on state $|0\rangle$, which decreases the influence of the decay. After the interaction with the environment, another weak measurement with $x>1$ is applied to reverse the quantum state. Due to the failure rates of the weak measurements, these schemes have limited success probabilities. In our previous works, we have shown that quantum gates can replace weak measurements in reversing the quantum states and protecting the entanglements [20–22]. In those works, ancilla qubits prepared in certain states are used to interact with the target qubits to construct the quantum gates. After the quantum gates, the ancilla qubits are detected. If the ancilla qubits are detected in certain state, such as a ground state, the reversal processes succeed and the states of the target qubits are reversed completely or approximately.

In all the above schemes, the quantum states decohere in a zero temperature environment. In this situation, only the quantum fluctuation is considered and the qubit decays from the excited state exponentially. However, in many quantum systems, such as superconducting circuits and Rydberg atom systems, this approximation is usually not valid and thermal environment has to be taken into account. The qubit not only decays to the ground state but also gets excited to the excited state incoherently by thermal photons. Protecting the quantum states and entanglements in such situation by using weak measurements and their reversals has been investigated [23,24]. In these procedures, the time-consuming weak measurements are required, which makes the whole process slow.

In this article, we present a scheme using quantum gates to protect the quantum states and entanglements of qubits under finite-temperature thermal noise (FTTN), which has in principle almost the same success probability and fidelity as that of weak measurement. We also present a physical implementation in a cavity QED system, where the quantum states that we consider are stored in superconducting cavities. They can be encoded to flying atoms adiabatically and then undergo spontaneous decay and thermal photon excitation. Finally, the atomic states are encoded back to other identical cavities. Ancilla qubits are used to interact with the cavities to construct quantum phase gates and subsequently control not (CNOT) gates. The CNOT gates are applied to the system before and after the cavity states undergo the thermal noise. The quantum state reversal is achieved with increased fidelity by projecting the ancilla qubits to certain states. This proposal does’t require time-consuming weak measurement (such as monitoring the cavities by photon counting devices) and consequently save the reversal time. Additionally, compared to the implementation of previous simulation schemes based on linear optics [11,24], the present implementation allows the qubits to undergo real decay and excitation processes, which can be useful in quantum state protection and quantum information.

The article is organized as follows: In Sec. 2, we present the general theory to protect the quantum states and entanglement via quantum gates. In Sec. 3, we give the physical implementation in a cavity QED system. Finally, in Sec. 4, we present the concluding remarks.

2. Protection theory and mathematical simulations

2.1. Quantum state under finite-temperature and its reversal

In a thermal bath of finite-temperature $T$, a quantum state $\rho _s^k$ of a qubit evolves to [25]

(1)$$\rho_s^n=\sum_{m=1}^{4}E_m\rho_s^k E_m^\dagger,$$

where

(2)$$\begin{aligned}E_1=\sqrt{p}\left(\begin{array}{cc}1 & 0\\0 & \sqrt{1-r}\end{array}\right);& E_2=\left(\begin{array}{cc}0 & \sqrt{pr}\\0 & 0\end{array}\right);\\ E_3=\sqrt{1-p}\left(\begin{array}{cc}\sqrt{1-r} & 0\\0 & 1\end{array}\right);&E_4=\left(\begin{array}{cc}0 & 0\\ \sqrt{r(1-p)} & 0\end{array}\right). \end{aligned}$$

Here, for simplicity, we have assumed the quantum system to be a two level system. The parameters $r$ and $p$ are related to the decay rate and temperature, respectively. For a simple case of an atom in a thermal environment, $r=1-e^{-\gamma _0 t}$ and $p=1/(1+e^{-\hbar \omega /k_B T})$ where $\omega$ is the transition frequency, $\gamma _0$ is the decay rate, and $k_{B}$ is the Boltzman constant.

The scheme for the quantum state reversal is shown in Fig. 1(a). In order to reverse the state, we need an ancilla qubit prepared in the state [20]

(3)$$\rho_{a}=|\varphi\rangle_{a}\langle\varphi|,$$

where

(4)$$|\varphi\rangle_{a}=\cos{\theta_r}|0\rangle+\sin{\theta_r}|1\rangle,$$

to interact with the system. Here $|0\rangle$ and $|1\rangle$ are the two eigen states of the ancilla qubit.

Fig. 1. The schematics for the protocol of the quantum state protection. Ancilla qubits interact with the system qubit to construct CNOT gates. (a) The CNOT gates are applied after the qubit undergoes the FTTN. (b ) Two CNOT gates are applied before and after the qubit undergoes the FTTN.

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After a CNOT gate

(5)$$C=\left(\begin{array}{cccc}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\end{array}\right)$$

applied on the system as shown in Fig. 1(a), the state can be described in the systematic basis ($00,01,10,11$) by [20,26]

(6)$$C[\rho_s^n\otimes \rho_{a}]C^{{\dagger}}=\frac{1}{1+x_r^2}\left(\begin{array}{cccc}\tilde{\rho}_{11} & \tilde{\rho}_{11}x_r & \tilde{\rho}_{12}x_r & \tilde{\rho}_{12}\\ \tilde{\rho}_{11}x_r & \tilde{\rho}_{11}x_r^2 & \tilde{\rho}_{12}x_r^2 & \tilde{\rho}_{12}x_r\\ \tilde{\rho}_{21}x_r & \tilde{\rho}_{21}x_r^2 & \tilde{\rho}_{22}x_r^2 & \tilde{\rho}_{22}x_r\\ \tilde{\rho}_{21} & \tilde{\rho}_{21}x_r & \tilde{\rho}_{22}x_r & \tilde{\rho}_{22}\end{array}\right).$$

Here $\rho _{ij}$ ($i,j=1,2$) are the elements of $\rho _s^i$; $\tilde {\rho }_{11}=\rho _{11}(1-r)+pr$; $\tilde {\rho }_{12}=\tilde {\rho }_{21}^*=\rho _{12}\sqrt {1-r}$; $\tilde {\rho }_{22}=\rho _{22}+\rho _{11}r-pr$ and we have set $\tan \theta _r=x_r$. At this point a measurement is made on the ancilla qubit. If the ancilla qubit is detected in state $|0\rangle$, the target state is projected to

(7)$$\rho_s^{f}=\frac{1}{\tilde{\rho}_{11}+\tilde{\rho}_{22}x_r^2}\left(\begin{array}{cc}\tilde{\rho}_{11} & \tilde{\rho}_{12}x_r\\ \tilde{\rho}_{21}x_r & \tilde{\rho}_{22}x_r^2 \end{array}\right)$$

and the protocol is considered a success. In this process, the success probability to obtain the above state is $\cos ^2\theta _r=1/(1+x_r^2)$. As an example, we assume $\rho _s^k$ to be a pure state $\alpha _1|g\rangle +\alpha _2|e\rangle$. The fidelity of the state after the CNOT can be obtained as [27,28]

(8)$$F(\rho_s^k,\rho_s^{f})=\frac{|\alpha_1|^4(1-r)+|\alpha_1|^2pr+2|\alpha_1\alpha_2|^2\sqrt{1-r}x_r +|\alpha_2|^2(|\alpha_2|^2+|\alpha_1|^2r-pr)x_r^2}{|\alpha_1|^2(1-r)+pr+(|\alpha_2|^2+|\alpha_1|^2r-pr)x_r^2}.$$

In Fig. 2, we plot the increase of the fidelity $F_{increase}=F(\rho _s^k,\rho _s^{f})-F(\rho _s^k,\rho _s^{n})$ of the state with the reversal process compared to the state without the reversal process as functions of $|\alpha _1|$ and $x_r$. It is shown that the fidelity can be enhanced in a large parameter range by choosing appropriate $\theta _r$. For example, if $x_r=1.2$, the fidelity can be enhanced when $0<|\alpha _1|<0.88$. In the rest of the range, the fidelity decreases slightly. The reason is that if $\rho _s^k$ is mainly in the ground state, the spontaneous decay is smaller than the gain from the thermal photon excitation and consequently the population of the excited state increases due to the thermal noise. If we choose $x_r>1$, the reversal process increase the population of the excited state more and $\rho _s^f$ is further to the initial state. This means that the quantum gate decreases the fidelity. However, in this range, the fidelity has always a large value and a small decrease of fidelity can be endured. As a consequence, for given values $r$ and $p$, we can find $x_r$ to keep the quantum state with a relatively large fidelity no matter what is the initial state.

Fig. 2. The increase of the fidelity $F_{increase}$. Here $r=0.5$ and $p=0.9$.

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If the ancilla qubit is detected in the state $|1\rangle$, the target qubit is projected to

(9)$$\rho_s^{f}=\frac{1}{\tilde{\rho}_{11}x_r^2+\tilde{\rho}_{22}}\left(\begin{array}{cc}\tilde{\rho}_{11}x_r^2 & \tilde{\rho}_{12}x_r\\ \tilde{\rho}_{21}x_r & \tilde{\rho}_{22} \end{array}\right)$$

and the protocol is considered a failure. Now we assume the above state to be $\rho _s^n$ as shown in Eq. (1) and do the above operations again by choosing an appropriate parameter $\theta _r$. In our previous paper [22], we showed that the total success probability to obtain the target state is approximately equal to that using weak measurement.

In order to improve the fidelity further, we modify our protocol as shown in Fig. 1(b). Here we use another ancilla qubit to interact with the target system and apply a CNOT gate on them before $\rho _s^k$ undergoes the thermal noise. This operator maps $\rho _s^k$ to be more robust against the thermal noise, which lowers the decoherence of the qubit. The ancilla qubit has a similar initial state as shown in Eq. (3) but different $\theta _l$ with $\tan \theta _l=x_l$. In Fig. 3, we set $x_l=0.65, x_r=2$ and plot the fidelity varying with $|\alpha _1|$. The result shows that the fidelity increases when two CNOT gates are applied. These results demonstrate that, compared to using weak measurements and their reversals used in the previous schemes [23,24], the quantum gates can also reverse the quantum states efficiently.

Fig. 3. The increase of the fidelity $F_{increase}$. The black-solid line is for the case shown in Fig. 1(a) while the red-dashed line is for the case shown in Fig. 1(b). Here $x_l=0.65, x_r=2$. The other parameters are the same as in Fig. 2.

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In FIg. 4, we plot the average fidelity increase as a function of $x_1$ and $x_2$ for $1000$ random generated initial mixed state. The figure shows that by choosing $x_1$ and $x_2$ appropriately, the present protocol can protect the mixed states efficiently with a maximum average fidelity increase about $0.06$.

Fig. 4. The average increase of the fidelity $F_{increase}$ for $1000$ random generated initial mixed states. Here $r=0.5$ and $p=0.9$.

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2.2. Protecting the entanglement from finite-temperature thermal noise

In this section, we show that the quantum gates can also protect the entanglement from thermal noise. We assume that the initial state of the quantum system to be a two-qubit entangled state. As described by Eq. (1), after time $t$, the state evolves to $\rho _s^n$. The corresponding elements $\rho _{ij}$ can be expressed as

(10)$$\begin{aligned} \tilde{\rho}_{11}= & \rho_{11}(1-r+pr)^2 +(\rho_{22}+\rho_{33})(1-r+pr)pr+\rho_{44}p^2r^2;\\ \tilde{\rho}_{12}= & \tilde{\rho}_{21}^*=\rho_{12}(1-r+pr)\sqrt{1-r}+\rho_{34}pr\sqrt{1-r};\\ \tilde{\rho}_{22}= & (1-r+pr)[\rho_{22}(1-pr)+\rho_{11}r(1-p)]+pr[\rho_{44}(1-pr)+\rho_{33}r(1-p)];\\ \tilde{\rho}_{13}= & \tilde{\rho}_{31}^*=\sqrt{1-r}[\rho_{13}(1-r+pr)+ \rho_{24}pr];\\ \tilde{\rho}_{14}= & \tilde{\rho}_{41}^*=\rho_{14}(1-r);\\ \tilde{\rho}_{23}= & \tilde{\rho}_{32}^*=\rho_{23}(1-r);\\ \tilde{\rho}_{24}= & \tilde{\rho}_{42}^*=\sqrt{1-r}[\rho_{24}(1-pr)+\rho_{13}r(1-p)];\\ \tilde{\rho}_{33}= & (1-pr)[\rho_{33}(1-r+pr)+\rho_{44}pr]+[\rho_{11}(1-r+pr)+\rho_{22}pr]r(1-p);\\ \tilde{\rho}_{34}= & \tilde{\rho}_{43}^*=\rho_{34}(1-pr)\sqrt{1-r}+\rho_{12}r(1-p)\sqrt{1-r};\\ \tilde{\rho}_{44}= & (1-pr)[\rho_{44}(1-pr)+\rho_{33}r(1-p)]+[\rho_{22}(1-pr)+\rho_{11}r(1-p)]r(1-p). \end{aligned}$$

As shown in Fig. 5, before the system undergoes the finite-temperature thermal noise, we use two ancilla qubits with initial state as shown in Eq. (3) to interact with the two system qubits separately to construct two CNOT gates. The full state of the system and ancilla qubit has the form

(11)$$\rho^{k}_{s,a}=\rho_{a1}\otimes\rho_s^k\otimes\rho_{a2}.$$

After the two CNOT gates, the whole system can be expressed as

(12)$$\rho_{s,a}^{C_l}=(UC_l\otimes C_l)\cdot \rho^{k}_{s,a}\cdot (UC_l^{{\dagger}}\otimes C_l^{{\dagger}})$$

where

(13)$$UC_l=\left(\begin{array}{cccc}1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\end{array}\right),$$

and $C_l$ is shown in Eq. (5). We detect the ancilla qubits. If both qubits are detected in state $|0\rangle$, the process is successful and we proceed to the next step. Otherwise we repeat the process with an approximate $\theta _l$. After the detections, the system state is projected to

(14)$$\rho_s^{C_l}=Tr_{a1,a2}[T_l\cdot\rho_{s,a}^{C_l}\cdot T_l^{{\dagger}}]$$

where the projection operator

(15)$$T_l=(|0\rangle_{a1}\langle 0|\otimes\left(\begin{array}{cc}1 & 0\\0 & 1\end{array}\right))\otimes(\left(\begin{array}{cc}1 & 0\\0 & 1\end{array}\right)\otimes|0\rangle_{a2}\langle 0|).$$

Here $a1$ and $a2$ denote the first and second ancilla qubits. After the system qubits undergo thermal noise, we carry out these operations on the qubits again with $\tan \theta _r=x_r$. The final state is the same as that after the first CNOT gate shown in Eq. (14) but with the replacement of $\rho _{s,a}^{C_l}$ by $\rho _{s,a}^{n}$.

Fig. 5. The schematics for the protocol to protect the quantum entanglement from FTTN. Two ancilla qubits interact with the system qubits separately before and after the quantum state undergoes the thermal noise to construct the CNOT gates.

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As an example, we assume $\rho _s^k$ to be $\alpha _1|gg\rangle +\alpha _2|ee\rangle$. After the first pair of CNOT gates, the state is projected to

(16)$$\rho_s^{C_l}=\frac{1}{\Lambda_1}\left(\begin{array}{cccc}|\alpha_1|^2 & 0 & 0 & \alpha_1\alpha_2^{*}x_l^2\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\ \alpha_1^{*}\alpha_2x_l^2 & 0 & 0 & |\alpha_2|^2x_l^4\end{array}\right)$$

where $\Lambda _1=|\alpha _1|^2+|\alpha _2|^2x_l^4$. Then the system undergoes the finite temperature thermal noise and the state evolves to $\rho _s^n$ with all matrix elements to be 0 except $\tilde {\rho }_{11}=[|\alpha _1|^2(1-r+pr)^2+|\alpha _2|^2x_l^4p^2r^2]/\Lambda _1$, $\tilde {\rho }_{14}=\tilde {\rho }_{41}=\alpha _1\alpha _2^{*}x_l^2(1-r)/\Lambda _1$, $\tilde {\rho }_{22}=\tilde {\rho }_{33}=[|\alpha _1|^2(1-r+pr)r(1-p)+|\alpha _2|^2x_l^4pr(1-pr)]/\Lambda _1$, $\tilde {\rho }_{44}=[|\alpha _2|^2x_l^4(1-pr)^2+|\alpha _1|^2r^2(1-p)^2]/\Lambda _1$. We perform the CNOT gates on the two system qubits separately again. The elements of the final state $\rho _s^{f}$ are $\rho ^{f}_{11}=\tilde {\rho }_{11}/\Lambda _2$, $\rho ^{f}_{22}=\rho ^{f}_{33}=\tilde {\rho }_{22}x_r^2/\Lambda _2$, $\rho ^{f}_{44}=\tilde {\rho }_{44}x_r^4/\Lambda _2$, and $\rho ^{f}_{14}=\rho ^{f}_{41}=\tilde {\rho }_{14}x_r^2/\Lambda _2$. Here $\Lambda _2=\tilde {\rho }_{11}+2\tilde {\rho }_{22}x_r^2+\tilde {\rho }_{44}x_r^4$. The concurrence of the two system qubits can be described as $E=\max \{0,\Xi \}$, where $\Xi =\sqrt {\lambda _1}-\sqrt {\lambda _2}-\sqrt {\lambda _3}-\sqrt {\lambda _4}$ [29] and $\lambda _i$ ($i=1,2,3,4$) are the eigenvalue, in decreasing order, of the Hermitian matrix $\sqrt {\sqrt {\rho ^f_s}\rho _p\sqrt {\rho ^f_s}}$ with $\rho _p=(\sigma _y\otimes \sigma _y)(\rho ^{f}_s)^*(\sigma _y\otimes \sigma _y)$. For the present case, $E$ can be expressed as

(17)$$E=\max\{0,2(|\rho_{14}|-\rho_{22})\}.$$

In Fig. 6, we plot the increase of the entanglement $E_{increase}=E(\rho _s^f)-E(\rho _s^n)$ of the state with the protection process compared to the state without the protection process. It is shown that the entanglement is indeed improved. In certain region, the entanglement increases strongly. Similar to the state reversal of the quantum state discussed in the previous subsection, the first time of CNOT gates project the quantum system to be a robust quantum state that can resist the decoherence since the thermal noise efficiently. Figure 6(a) shows that for given $r$ and $p$, by adjusting $x_l$ and $x_r$, the entanglement can be enhanced or kept when $\alpha _1$ lies in a large range. When $\alpha _1$ is small, there is no entanglement after the decoherence process and the reversal process also doesnot generate the entanglement. When $\alpha _1$ is large, the protection process can improve the entanglement strongly. However, when $\alpha _1$ approaches 1, the entanglement is a little weakened. In FIg. 6(b), we present the entanglement increase varying with $x_l$ and $x_r$. $x_l<1$ denotes that the system state is projected to a state having large probability to be in the ground state (shown in Eq. (16)), which means that the state varies slower under the FTTN due to we have set the spontaneous decay to be largerr than the incoherent excitation rate, i.e., $1-r>1- p$. The second time of CNOT gates can reverse the system state to approach to $\rho _s^k$. We should note that large values of $x_l$ and $x_r$ denote the success probability of the protection process to be low. Similar to protecting the quantum state of a single qubit, in Fig. 7, we plot the average concurrence increase as a function of $x_1$ and $x_2$ for $1000$ random generated initial mixed state of two qubits. The figure shows that by choosing $x_1$ and $x_2$ appropriately, the present protocol can also protect the mixed states efficiently with a maximum average concurrence increase about $0.07$.

Fig. 6. The increase of the concurrence $E_{increase}$. In (a), $x_l$ is set to 0.65 while in (b), the initial state is set to be $(|ee\rangle +|gg\rangle )/\sqrt {2}$. Here $r=0.5$ and $p=0.9$.

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Fig. 7. The average increase of the concurrence $E_{increase}$ for $1000$ random generated initial mixed states. Here $r=0.5$ and $p=0.9$.

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3. Protecting the quantum states and entanglements of cavity states

In this section we present a physical implementation of the above proposal in the cavity QED system. Our setup is schematically shown in Fig. 8(a). Cavities 1 and 2 are two identical superconducting cavities. we consider an initial state of the cavities, which have 0 or 1 photon in them and can be considered as qubits. As already discussed, in order to reverse the quantum states and entanglements between the two cavity fields, we need to construct CNOT gates between the two cavities and two ancilla qubits separately.

Fig. 8. (a) The physical implementation of the proposal. Two identical atoms fly through two identical cavities. $\Omega _1$ is coupling strength between the classical laser and the transition dipole from $|a\rangle$ to $|c\rangle$ of the atom shown in (b); $g_c$ is the coupling strength between the quantum field and the transition dipole from $|a\rangle$ to $|g\rangle$; and $\Omega _2$ is a $\pi$ pulse that can transfer the atom between state $|c\rangle$ and $|e\rangle$. The atoms on the superposition states of $|e\rangle$ and $|g\rangle$ undergo spontaneous decay and thermal photon excitation in the FTTN area, which can be realized by illuminating this area by thermal light. (b) The energy structure of atoms 1 and 2. (c) The right is the physical diagram of the CNOT gates, where $R_{1,2,3}$ are Ramsey laser beams. The left is the atomic structure of the ancilla qubit.

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3.1. CNOT gates

First, we discuss how to realize the CNOT gates used in the reversal processes. Here we utilize ancilla qubits to interact with the cavities. The ancilla qubit, as a Rydberg atom shown in Fig. 8(c), has a down state $|D\rangle$, an upper state $|U\rangle$, and an additional state $|A\rangle$. The transition dipole from $|U\rangle$ to $|D\rangle$ can interact with the Ramsey fields $R_{1,2,3}$ resonantly. By manipulating the field intensity of $R_1$, the ancilla qubit initially in state $\rho _{a}=|D\rangle \langle D|$ evolves to [20,22,30]

(18)$$\rho^{R_1}_{a}=H_{\theta_l}\cdot\rho_{a}\cdot H_{\theta_l}^{{\dagger}},$$

where $H_{\theta }$ is the Hadamard gate having the form

(19)$$H_{\theta}=\left(\begin{array}{cc}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{array}\right).$$

Similarly, by controlling the intensity of $R_2$, a Hadamard gate $H_{\theta }$ with $\theta =-\pi /4$ can be applied on the ancilla qubit. The ancilla qubit evolves to

(20)$$\rho^{R_2}_{a}=H_{-\pi/4}\cdot\rho^{R_1}_{a}\cdot H_{-\pi/4}^{{\dagger}}.$$

After that, the ancilla qubit enters the cavity. The transition from $|U\rangle$ to $|A\rangle$ can interact with the cavity field dispersively with a frequency detuning $\Delta$. The corresponding coupling strength is $g_a$ and $\Delta \gg g_{a}$. The interaction Hamiltonian has the following effective form [38]

(21)$$\hat{H}_{eff}={-}\frac{\hbar g_{a}^{2}}{\Delta}(a a^{{\dagger}}|A\rangle\langle A| -a^{{\dagger}}a|U\rangle\langle U|).$$

where $a^{\dagger }, a$ are the creation and annihilation operators of cavity 1 or 2. After the qubit passes the cavity with an interaction time $\tau$, the state of the ancilla qubit and the cavity evolves to

(22)$$\rho_{s,a}^{phase}=P_{\tau}\cdot (\rho_s^k\otimes\rho_{a}^{R_2})\cdot P_{\tau}^{{\dagger}},$$

where

(23)$$P_{\tau}= \left(\begin{array}{cccc}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & e^{{-}i\frac{g_{a}^{2}\tau}{\Delta}}\end{array}\right)$$

Then $R_3$ can do a Hadamard gate $H_{\pi /4}$ on the ancilla qubit again. The state evolves to

(24)$$\begin{aligned} \rho^{R_3}_{s,a} &= (I\otimes H_{\pi/4})\cdot P_{\tau}\cdot (I\otimes H_{-\pi/4})\cdot (\rho_s^k\otimes \rho_a^{R_1})\cdot (I\otimes H_{-\pi/4}^{{\dagger}})\cdot P_{\tau}^{{\dagger}}\cdot(I\otimes H_{\pi/4}^{{\dagger}})\\ & =C\cdot\rho_s^k\otimes\rho_{a}^{R_1}\cdot C^{{\dagger}}, \end{aligned}$$

where $C$ is a CNOT gate shown in Eq. (5) by setting $\tau =\pi \Delta /g_a^2$.

3.2. Detailed processes to protect the quantum states and entanglements

In the following steps, we give the detailed processes to protect the quantum entanglement of the two qubits from FTTN.

Step 1: As shown in Fig. 8(a), before the system undergoes the finite temperature thermal noise, we use two ancilla qubits to interact with the two cavities separately to realize the CNOT gates. After the two CNOT gates, we detect the ancilla qubits. If both qubits are in the ground states $|D\rangle$, we move on to the next step. Otherwise we modify the intensity of laser $R_1$ and do the above process again. The cavity state can be expressed as $\rho _s^{C_l}$ in Eq. (14).

Step 2: We pass two identical atoms shown in Fig. 8(b) through cavities 1 and 2 separately. The atoms are both initially prepared in ground state $|g\rangle$. Before the atom enters the cavity, it interacts with a Ramsey field resonantly with $|a\rangle$ to $|c\rangle$ transition and Rabi strength $\Omega _1$. When the atom enters the cavity, the transition dipole from $|a\rangle$ to $|g\rangle$ interacts with the cavity field resonantly with a coupling strength $g_c$. The dark state of the atom interacting with the two kinds of fields has the following form [12,31–35]:

(25)$$(\Omega_1|g\rangle-g_c|c\rangle)/\sqrt{|\Omega_1|^2+|g_c|^2}.$$

Before the atom arrives at the center of the cavity, $\Omega _1\gg g_c$, which means that the atom is trapped in state $|g\rangle$. However, when the atom leaves the center, $\Omega _1$ decreases and $g_c$ increases. The atom absorbs the photon in the cavity and move to state $|c\rangle$ adiabatically if there is one photon in the cavity initially. However, if the cavity is in the vacuum state initially, no interaction $g_c$ happens and consequently the atom stays in state $|g\rangle$. When the atom leaves the cavity, the cavity state will be encoded in the atom and the atom is in the superposition state of $|g\rangle$ and $|c\rangle$. The atoms’ state still can be expressed as $\rho _s^{C_l}$ but has different basis with cavity state $|1\rangle _{cavity 1/2}$ replaced by atom state $|c\rangle$.

Step 3: After the atom flies out of the cavity, it meets a $\pi$ beam. The beam interacts with transition dipole from $|c\rangle$ to $|e\rangle$ with Rabi frequency $\Omega _2$, such as by Raman process. The atom evolves from

(26)$$|c\rangle\longrightarrow|e\rangle.$$

After this beam, the atom stays in the superposition state of $|e\rangle$ and $|g\rangle$. Similarly, the atoms’ state can still be expressed as $\rho _s^{C_l}$ but has different basis with cavity state $|1\rangle _{cavity 1/2}$ replaced by atom state $|e\rangle$.

Step 4: The atom passes through the area with finite-temperature, which can be realized by illuminating this area by thermal light [36,37]. The atom on the superposition state of $|e\rangle$ and $|g\rangle$ interacts with the vacuum modes and the thermal photons incoherently. It can both lose and gain excitations; the dynamic evolution function can be expressed as [38]

(27)$$\begin{aligned} \dot{\rho}(t)= & \frac{\gamma_0}{2}(N+1)[2\sigma_-\rho(t)\sigma_+{-}\sigma_+\sigma_-\rho(t)-\rho(t)\sigma_+\sigma_-]\\ & +\frac{\gamma_0}{2}(N)[2\sigma_+\rho(t)\sigma_-{-}\sigma_-\sigma_+\rho(t)-\rho(t)\sigma_-\sigma_+]. \end{aligned}$$

Here $\sigma _{\pm }$ are the rising and lowering operators of the atom, $\gamma _0$ is the spontaneous decay rate , $N=1/[\exp (-\hbar \omega /k_BT)-1]$ is the average thermal photon number. After time $t$, the state $\rho _s^n$ has the form shown in Eq. (10).

Step 5: The atom passes through another $\pi$ beam and cavity. As the inverse processes of $steps$ 3 and 2, the atom state transfers from $|e\rangle$ to $|c\rangle$ firstly by interacting with the $\pi$ beam. A photon is emitted back to cavity 2 and the atom goes back to level $|g\rangle$ from $|c\rangle$ adiabatically. The state of the cavity 1, influenced by the FTTN, is transferred to cavity 2. Meanwhile, the state of cavity 2, influenced by the FTTN, is transferred to cavity 1.

Step 6: As the same as $step$ 1, we perform two other CNOT gates on the cavities 1 and 2 separately. By detecting the states of the ancilla qubits to the ground states, we can project the cavity state to the final state with increasing fidelity and entanglement compared to that without the protection processes.

By the above processes, we can protect the quantum entanglement of two cavities. Note that the CNOT gates on the two cavity states are separated. They can be different, which is similar to the different weak measurement parameters in [23,24].

To reverse the quantum state, only one atom shown in Fig. 8(b) is required to pass the two cavities. The CNOT gates can be constructed as the same method discussed in the above subsection. After the same steps as the case to protect the entanglement, the cavity state can be reversed and stored in another cavity.

In a real experiment, the time for an atom passing through the cavity to construct a quantum phase gate or absorb and emit a photon adiabatically is about $20\mu s$ [30]. The total required time for reversal process is about hundreds of microseconds. The lifetime of the atom can be millisecond. Before and after the atoms fly in the thermal noise region, they stay in state $|g\rangle$ or $|c\rangle$, which always have long life times and can restrain the influence of the spontaneous decay on the system. The main experimental requirement is the slow decay rate of the cavities, which can hold the cavity states when the CNOT gates are applied. Since the technology of micromachining has improved, the damping time of the superconducting cavity is about 130ms [39,40]. As a consequence, we believe our implementation can be realized experimentally. Meanwhile, the interaction times between the cavities and the atoms are much shorter than the lifetime of the Rydberg atoms as well as the flying time of the atom through the FTTN area. If the entire system is placed in an environment with finite temperature, the effect of the thermal photons on the cavity state can still be expressed as Eq. (1). This means that our protocol is still valid even the entire system is placed in FTTN, which makes the implementation be easily to be realized. These effects can be approximately involved to the $p$ and $r$ in Eq. (1). However, similar to weak measurement requiring to swap the quantum state, such as swapping the cavity state from $\alpha _1|0\rangle +\alpha _2|1\rangle$ to $\alpha _2|0\rangle +\alpha _1|1\rangle$ by interacting with atoms, and having limited succeed probability [19], the quantum gate also has limited fidelity, which will decrease the fidelity of the final state. As the technology advaces, the fidelity of quantum gate, such as CNOT gate, based on Rydberg states are very high, even more than 0.9999 in [41]. If we can realize a CNOT gate with fidelity 0.99, the state after the CNOT gate can be expressed $0.99\rho _s^C+0.01\rho _s^F$ with $\rho _s^C$ being the state we want and $\rho _s^F$ being the state we donot want. Even $\rho _s^F$ is orthogonal to the initial state, it just has about 0.02 fidelity decrease, which is much smaller than the fidelity increase due to the protocol, i.e., on average 0.06 in Fig. 4 for mixed state. As a consequence, the physical implementation is still valid.

4. Conclusions

In this article, we presented a protocol to protect the quantum states and entanglements from FTTN via quantum gates. The fidelity and entanglement can be improved efficiently with a large probability. In the real physical implementation, by using the adiabatic passage of multi-level Rydberg atoms passing through the superconducting cavities, the quantum states stored in the cavities can be transferred to the atoms and subsequently undergo the FTTN effectively. Meanwhile, it is convenient to realize a quantum phase gate and subsequently a CNOT gate by passing an ancilla qubit through the cavity. By detecting the ancilla qubits, we can reverse the quantum states and entanglements with certain success probability. Compared to the protocols based on weak measurements and their reversals, the present protocol can be much faster. Additionally, compared to the previous simulation schemes based on linear optics, the qubits in the implementation of our proposal based on cavity QED system undergoes the real decay and thermal photon excitation processes as described in Eq. (1) and the physical setup is relatively easy to be realized with current experimental technologies. Our investigation can be useful in quantum state protection and quantum information.

Funding

National Natural Science Foundation of China (11804219); King Abdulaziz City for Science and Technology.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Quantum state protection in finite-temperature environment via quantum gates

Abstract

1. Introduction

2. Protection theory and mathematical simulations

2.1. Quantum state under finite-temperature and its reversal

2.2. Protecting the entanglement from finite-temperature thermal noise

3. Protecting the quantum states and entanglements of cavity states

3.1. CNOT gates

3.2. Detailed processes to protect the quantum states and entanglements

4. Conclusions

Funding

Disclosures

References

Cited By

Figures (8)

Equations (27)

Optics Express