Proposal for reversing the weak measurement with arbitrary maximum photon number

We present a proposal for reversing the weak (partial-collapse) quantum measurement on a cavity field with arbitrary maximum photon number. We start by putting forth a protocol to realize quantum phase gates between the cavity field and an ancilla qubit. Afterward, adopting these phase gates and some other quantum gates, we can determine the reversal of the cavity state just by observing the ancilla qubit. Compared to previous proposals, our proposal does not need any other weak measurement and can save reversal time, which is significantly important in quantum informatics and quantum computation.

Figure 1

(a) Schematic diagram of the proposal. Initially, the cavity state $|\phi \rangle$ shown in Eq. (2) is input to process ${\mathbf{S}}_1$ and the process ${\mathbf{S}}_1$ can reverse the cavity state to the state shown in Eq. (6) with some probability. Then the cavity state $|\phi \rangle$ shown in Eq. (6) is input to process ${\mathbf{S}}_2$ and the process ${\mathbf{S}}_2$ can reverse the cavity state to the state shown in Eq. (11) with some probability. After processes ${\mathbf{S}}_1\to {\mathbf{S}}_{n_{\max }}$, the cavity state can be reversed to the original state shown in Eq. (1). Here $H_{\theta j}$ ($j=1,2,...,R$) can prepare the ancilla qubit from ${|D\rangle }_q$ to ${cos\theta |D\rangle }_q+sin\theta {|U\rangle }_q$ with $tan\theta =e^{2^{j-1}\gamma t}$. The cnot${}_i$ ($i=1,2,...,n_{\max }$) are cnot gates of Eq. (4) with the bases $\left({\sum }_{n=0}^{i-1}{\alpha }_n{e}^{-n\gamma t}\right|n\rangle {\left)\right|D\rangle }_q, \left({\sum }_{n=0}^{i-1}{\alpha }_n{e}^{-n\gamma t}\right|n\rangle {\left)\right|U\rangle }_q, \left({\sum }_{n=i}^{n_{\max }}{\alpha }_n{e}^{-n\gamma t}\right|n\rangle {\left)\right|D\rangle }_q$, and $\left({\sum }_{n=i}^{n_{\max }}{\alpha }_n{e}^{-n\gamma t}\right|n\rangle {\left)\right|U\rangle }_q$. (b) The cnot gates consist of two Hadamard gates and one phase gate. The phase gate $Q_{\pi }^i$ denotes that only when the photon number in the cavity is bigger than or equal to $i$ and the ancilla qubit is in the excited state ${|U\rangle }_q$ can there be a $\pi$ phase shift.

Figure 2

Success probability for $R=1,2,3$ compared with the curve $e^{-2n_{\max }\gamma t}$ (from bottom to top). Here we set $n_{\max }=2$.

Figure 3

Protocol diagram showing the three cavities I, II, and III in Figs. 5 and 7. The wide blue arrows are classical laser beams. Atom $\mathrm{\Xi }$ can pass through cavities I and II, while atoms $\mathrm{\Lambda }$ can pass through cavities I and III. The dotted box of the right part is the same apparatus as that in Refs. [25, 31]. Here $L_1, L_2$, and $L_3$ are three Ramsey laser fields: $L_1$ can prepare the initial state of the ancilla qubit $[H_{\theta j}$ in Fig. 1], while $L_2$ and $L_3$ can do Hadamard gates on the ancilla qubit $[H_{\pi /4}$ and $H_{-\pi /4}$ in Fig. 1]. The detector can measure the state of the ancilla qubit.

Figure 4

(a) Structure of atom $\mathrm{\Xi }$. The total angular momenta of degenerate levels $|e\rangle , |a\rangle$, and $|g\rangle$ are $F_g=2, F_a=1$, and $F_g=2$, respectively. The $\pi$-polarized transitions $|g_K{\rangle }_{\mathrm{\Xi }}\to {|e_K\rangle }_{\mathrm{\Xi }}$ ($K=\pm 1,\pm 2$) are resonant with cavity field ${\omega }_e$, the $\pi$-polarized transitions $|g_K{\rangle }_{\mathrm{\Xi }}\to {|a_K\rangle }_{\mathrm{\Xi }}$ ($K=0,\pm 1$) are resonant with cavity field ${\omega }_a$, the ${\sigma }^{+}$-polarized transitions $|g_{K-1}{\rangle }_{\mathrm{\Xi }}\to {|e_K\rangle }_{\mathrm{\Xi }}$ ($K=0,\pm 1,2$) are resonant with the classical beams ${\sigma }_e^{+}$, and the ${\sigma }^{+}$-polarized transitions $|g_{K-1}{\rangle }_{\mathrm{\Xi }}\to {|a_K\rangle }_{\mathrm{\Xi }}$ ($K=0,\pm 1$) are resonant with the classical beam ${\sigma }_a^{+}$. (b) Structure of atom $\mathrm{\Lambda }$. The $\pi$-polarized transition ${|d\rangle }_{\mathrm{\Lambda }}\to {|b\rangle }_{\mathrm{\Lambda }}$ is resonant with cavity field ${\omega }_e$ and the ${\sigma }^{+}$-polarized transition ${|c\rangle }_{\mathrm{\Xi }}\to {|b\rangle }_{\mathrm{\Xi }}$ is resonant with the classical beams ${\sigma }_{bc}^{+}$.

Figure 5

Atom $\mathrm{\Xi }$ passes through cavities I and II. Cavity I has two resonant frequencies ${\omega }_e$ and ${\omega }_a$, while cavity II has resonant frequency ${\omega }_e$. There are classical laser beams propagating through the cavities.

Figure 6

Electrons moving in atom $\mathrm{\Xi }$ according to the adiabatic passages. Here ${\sigma }_e^{+}$ and ${\sigma }_a^{+}$ are classical fields, while ${\pi }_e$ and ${\pi }_a$ are quantum fields.

Figure 7

Atoms $\mathrm{\Lambda }$ pass through cavities I and III. Cavity III has resonant frequency ${\omega }_e$. There are classical laser beams propagating through the cavities.