Quantum nondestructive determination of qubit states in low-Q cavities via single-photon input-output process

Hao Yuan; Lian-Fang Han

doi:10.1364/OE.24.005487

1. Introduction

The determination of an unknown quantum state is one of the most essential tasks in quantum information processing. An effective method to realize quantum state determination (QSD) is to reconstruct the density matrix of this unknown quantum state [1]. Specifically, one first needs to perform a series of projective measurements on an ensemble of identically prepared quantum states, and then obtain all the elements of the density matrix from measurement outcomes statistics [2–4 ]. In recent decades, QSD has been experimentally demonstrated in various physical systems with photons [5], trapped ions [6], semiconductor quantum dots [7], superconducting circuit [8], etc.

Cavity quantum electrodynamics (QED) system [9] is a promising platform to implement quantum information processing. In cavity-based quantum networks [10] for scalable quantum information processing, atoms trapped in the cavity act as stationary qubits for storing quantum information, and photons as flying qubits for fast and reliable transport of quantum information between spatially separated cavities. Also, the input-output process of the flying photons through the cavity plays an important role in the manipulation of the atomic and photonic states. Based on the input-output process of the photons, Duan and Kimble [11] proposed an interesting scheme to realize quantum controlled phase-flip gate for scalable photonic quantum computation. In their scheme, high-Q cavity with strong coupling between the atom and the photon is required. However, high-Q cavity does not facilitate to carry out the input-output process of the photons. Later, An et al. [12] loosed this requirement to the low-Q cavity with moderate coupling between the atom and the photon, and proposed an efficient scheme to implement some quantum information processing tasks via photonic Faraday rotation. After An et al.’s work [12], a variety of schemes for quantum information processing in different physical systems have been proposed, including entanglement generation [13–16 ], quantum logic gates realization [17–21 ], entanglement concentration [22–26 ] and purification [23, 24, 26], and so on. In almost all the aforementioned schemes [11–26 ], we note that the atom is assumed to be resonantly interacted with the cavity field.

Different from the previous works [11–26 ], we focus our attention on the dispersive interaction between the atom and the low-Q cavity. In the dispersive regime, the atomic-state-dependent Stark shifts of the cavity frequency will be induced [27–29 ]. This allows us to nondestructively determine the atomic states by applying an external driving field at the input port of the cavity and then detecting the information of the photons at the output port of the cavity [30]. In our previous works [27–29 ], we have shown that the quantum nondestructive determination (QNDD) of the atomic states can be realized by measuring the amplitude information of the transmitted photons through the driven cavity. In this paper, we will present another alternative approach to realize the QNDD of atomic states in two types of low-Q cavities, i.e., single-sided and double-sided cavity QED systems. We show that the QNDD of the atomic states can also be accomplished by detecting the phase shifts of the reflected or transmitted photons from the cavity. The distinct feature of our proposal is that our proposal works in the dispersive regime of low-Q cavities and it is insensitive to both cavity decay and atomic spontaneous emission.

The rest of this paper is organized as follows. In Sec. 2, we show the phases of the reflected photons and how to realize the QNDD of single-qubit states in type-I cavity QED system. In Sec. 3, we present the phases of the transmitted photons and the realization of the QNDD of single-qubit states in type-II cavity QED system. The investigations in Sec. 2 and Sec. 3 are extended straightforwardly to the case of multiple-qubit states in Sec. 4. Finally, the discussion about the experimental feasibility of our proposal and the conclusion are given in Sec. 5.

2. QNDD of single-qubit states in type-I cavity QED system

2.1. QSDPSs of the reflected photons

We first consider the type-I cavity QED system schematically shown in Fig. 1(a), wherein a qubit (two-level atom) is coupled to a single-mode and single-sided cavity with one mirror partially reflective and another mirror perfectly reflective. Under the rotating wave approximation, the qubit-cavity system is described by the Hamiltonian [31] (h̄ = 1; hereafter the same)

H = ω_{c} a^{†} a + \frac{ω_{a}}{2} σ_{z} + g (a^{†} σ_{-} + σ_{+} a),

where ω_c is the cavity frequency, a ^†(a) is the creation (annihilation) operator of the cavity field, ω_a is the transition frequency of the qubit with the Pauli operators: σ ₋ = |0〉〈1|, σ ₊ = |1〉〈0|, and σ_z = |1〉〈1| − |0〉〈0|, g is the coupling strength between the qubit and the cavity.

Fig. 1 Schematic of the generic models investigated in this paper. (a) Type-I cavity QED system wherein a qubit is dispersively coupled to a single-mode and single-sided cavity with one mirror partially reflective and another mirror perfectly reflective. (b) Type-II cavity QED system consisting of a qubit dispersively coupled to a single-mode and double-sided cavity with both mirror partially reflective. In (a) and (b), the QNDD of unknown single-qubit states can be realized by detecting the QSDPSs of the reflected or transmitted photons from the cavity. κ denotes the cavity decay rate, and γ is the qubit decay rate.

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In the dispersive regime (Δ = ω_a − ω_c ≫ g), we make the unitary transformation $U = exp [\frac{g}{Δ} (a σ_{+} - a^{†} σ_{-})]$ and expand to second order in g to obtain the effective Hamiltonian

H_{eff} = U H U^{†} \approx (ω_{c} + Γ σ_{z}) a^{†} a + (ω_{a} + Γ) \frac{σ_{z}}{a}

with

Γ = \frac{g^{2}}{Δ}

.

We assume that a single-photon pulse with the frequency ω_d is input to drive the cavity. In a frame rotating at the frequency ω_d for both the qubit and the cavity, i.e., making the unitary transformation $U^{'} = exp [i ω_{d} t (a^{†} a + \frac{σ_{z}}{2})]$ , the Hamiltonian in Eq. (2) becomes

H_{eff}^{RF} = U^{'} H_{eff} U^{' †} - i U^{'} \frac{\partial U^{' †}}{\partial t} = - (Δ_{d c} - Γ σ_{z}) a^{†} a - (Δ_{d a} - Γ) \frac{σ_{z}}{2},

where Δ_dc = ω_d − ω_c is the frequency detuning of the input pulse from the cavity field, and Δ_da = ω_d − ω_a is the frequency detuning of the input pulse from the qubit.

The Heisenberg-Langvian equation of motion for the qubit operator σ_z(t) is derived as

\frac{d σ_{z} (t)}{d t} = - γ [σ_{z} (t) + 1] - 2 \sqrt{γ} [c_{in}^{†} (t) σ_{-} + σ_{+} c_{in} (t)],

where γ is the qubit decay rate and c_in(t) satisfying the commutation relation

[c_{in} (t), c_{in}^{†} (t^{'})] = δ (t - t^{'})

is the vacuum input field operator of the qubit. The input field c_in(t) is assumed to be vacuum always, thus the contribution of this field to the cavity output can be neglected [32]. The detection time is mainly determined by the photon lifetime (t ∼ 1/κ, where κ is the photon decay rate of the cavity). In the low-Q cavity (κ ≫ γ), the expectation value of σ_z(t) is safely supposed to be unchanged during the detection. That is, the expectation value of the operator σ_z(t) is obtained from Eq. (4) as

〈 σ_{z} (t) 〉 = e^{- γ t} [〈 σ_{z} (0) 〉 + 1] - 1 ≃ 〈 σ_{z} (0) 〉 .

Moreover, the Heisenberg-Langvian equation of motion for the cavity field operator a(t) is obtained as

\frac{d a (t)}{d t} = i [Δ_{d c} - Γ σ_{z} (t)] a (t) - \frac{κ}{2} a (t) - \sqrt{κ} b_{in} (t),

where b_in(t) with the commutation relation

[b_{in} (t), b_{in}^{†} (t^{'})] = δ (t - t^{'})

is the input field to the cavity. The input-output relation between the input and output fields to the cavity related to the cavity field operator a(t) is given by [33]

b_{out} (t) = b_{in} (t) + \sqrt{κ} a (t) .

Averaging for both sides of Eqs. (6) and (7) and under the mean-field approximation 〈σ_za〉 = 〈σ_z〉〈a〉 [32], we adiabatically eliminate the cavity field operator a(t) and get the phase of the reflected photons satisfying

e^{i ϕ_{1}} \equiv \frac{b_{out} (t)}{b_{in} (t)} = \frac{i [Δ_{d c} - Γ 〈 σ_{z} (0) 〉] + \frac{κ}{2}}{i [Δ_{d c} - Γ 〈 σ_{z} (0) 〉] - \frac{κ}{2}} .

In the case of g = 0, Eq. (8) could get back to the phase of the reflected photons from an empty cavity (EC) satisfying

e^{i ϕ_{0}} = \frac{i Δ_{d c} + \frac{κ}{2}}{i Δ_{d c} - \frac{κ}{2}} .

From Eqs. (8) and (9), it can be found that the phase ϕ ₁ of the reflected photons is shifted horizontally to the left or right by the quantity Γ〈σ_z(0)〉, which is qubit-state-dependent, compared with the phase ϕ ₀ for the EC. This can be confirmed by numerical simulation for the specific single-qubit states. In Fig. 2, we show the numerically simulated phases ϕ ₁ of the reflected photons with Eq. (8) for the computational basis states |1〉 (green line) and |0〉 (red line), and an arbitrary state ρ with the diagonal elements of its density matrix as diag(0.7, 0.3) (blue line). For comparison, the phase ϕ ₀ of the reflected photons from the EC (black line) is also plotted with Eq. (9). Here, the available experimental parameters are chosen as Γ = −2π × 7.38MHz and κ = 2π × 1.69MHz [34]. From Fig. 2, it can be found that the phase ϕ ₁ of the reflected photons is shifted horizontally to the left (right) by the quantity Γ (−Γ) for the computational basis state |1〉 with 〈σ_z(0)〉 = 1 (|0〉 with 〈σ_z(0)〉 = −1), relative to the phase ϕ ₀ for the EC. For the state ρ with 〈σ_z(0)〉 = 0.4, the phase is shifted horizontally to the left by 0.4Γ. Assume the phase detection is performed at Δ_dc = 0, i.e., ω_d = ω_c. It can be seen that the qubit-state-dependent phase shifts (QSDPSs) ϕ ₁ are −0.073π for the computational basis state |1〉, 0.073π for the computational basis state |0〉, and −0.178π for the state ρ, respectively. This indicates that the detection of the QSDPS ϕ ₁ of the reflected photons at a certain driving frequency (e.g., Δ_dc = 0) can be utilized to directly determine the relevant information of the qubit involved in 〈σ_z(0)〉, i.e., the diagonal elements of its density matrix. On the other hand, it can be seen from Eq. (3) that the interaction Hamiltonian between the qubit and the cavity is H_int = Γσ_za ^† a, which commutes with the qubit operator σ_z, i.e., [H_int, σ_z] = 0. This means that the proposed method to detect the QSDPSs is of the nondestructive property [33], and what we directly detect is the reflected photons from the cavity, rather than the intracavity qubit itself.

Fig. 2 The numerically simulated phases ϕ ₁ of the reflected photons from the cavity as a function of the detuning Δ_dc for the EC (black line), the computational basis states |1〉 (green line) and |0〉 (red line), and an arbitrary state ρ with the diagonal elements of its density matrix as diag(0.7, 0.3) (blue line). The available experimental parameters are chosen as Γ = −2π × 7.38MHz and κ = 2π × 1.69MHz [34].

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2.2. QNDD of single-qubit states by detecting the QSDPSs of the reflected photons

We now demonstrate how to realize the QNDD of unknown single-qubit states by detecting the QSDPSs of the reflected photons proposed in Sec. 2.1 and numerically show the QNDD of an exemplified qubit state.

In general, the density matrix of an arbitrary single-qubit state can be represented as

ρ_{1} = \frac{1}{2} \sum_{j = 0, x, y, z} r_{j} σ_{j} = \frac{1}{2} (\begin{matrix} 1 + r_{z} & r_{x} - i r_{y} \\ r_{x} + i r_{y} & 1 - r_{z} \end{matrix}),

where r_x,y,z are real parameters and r ₀ = 1 due to the normalization condition Trρ ₁ = 1, σ_x,y,z are Pauli operators and σ ₀ is an identity operator. To acquire the complete information of this unknown single-qubit state, we have to identify three parameters (r_x, r_y, r_z). For the state ρ ₁, it is calculated that 〈σ_z(0)〉 = Tr(ρ ₁ σ_z) = r_z. Based on Eq. (8), the parameter r_z can be determined by one kind of QSDPS of the reflected photons. To determine the other two parameters r_x and r_y in the non-diagonal locations, one needs to transfer the non-diagonal elements to the diagonal locations with the assistance of single-qubit unitary operations (UOs), prior to the measurements. To be specific, we perform single-qubit UO

U_{x} (\frac{π}{4}) = e^{i \frac{π}{4} σ_{x}}

on the state ρ ₁. After that, the state ρ ₁ becomes

ρ_{1}^{'} = U_{x} (\frac{π}{4}) ρ_{1} U_{x}^{†} (\frac{π}{4}) = \frac{1}{2} (\begin{matrix} 1 - r_{y} & r_{x} - i r_{z} \\ r_{x} + i r_{z} & 1 + r_{y} \end{matrix})

with 〈σ_z(0)〉 = Tr(ρ′ ₁ σ_z) = −r_y. Then by another kind of QSDPS of the reflected photons, the parameter r_y can be determined. Similarly, after performing single-qubit UO

U_{y} (\frac{π}{4}) = e^{i \frac{π}{4} σ_{y}}

on the state ρ ₁, we obtain

ρ_{1}^{″} = U_{y} (\frac{π}{4}) ρ_{1} U_{y}^{†} (\frac{π}{4}) = \frac{1}{2} (\begin{matrix} 1 + r_{x} & - r_{z} - i r_{y} \\ - r_{z} + i r_{y} & 1 - r_{x} \end{matrix})

with 〈σ_z(0)〉 = Tr(ρ″ ₁ σ_z) = r_x. And the parameter r_x can be determined in the same way.

We numerically demonstrate the above procedure with an example. Without loss of generality, if we choose r_x = 0.5, r_y = 0.6, and r_z = 0.2, the density matrix (10) is specified as

ρ_{1} = (\begin{matrix} 0.6 & 0.25 - 0.3 i \\ 0.25 + 0.3 i & 0.4 \end{matrix})

with 〈σ_z(0)〉 = Tr(ρ ₁ σ_z) = 0.2. After performing single-qubit UOs

U_{x} (\frac{π}{4}) = e^{i \frac{π}{4} σ_{x}}

and

U_{y} (\frac{π}{4}) = e^{i \frac{π}{4} σ_{y}}

, the state ρ ₁ is changed as

ρ_{1}^{'} = (\begin{matrix} 0.2 & 0.25 - 0.1 i \\ 0.25 + 0.1 i & 0.8 \end{matrix})

with 〈σ_z(0)〉 = Tr(ρ′ ₁ σ_z) = −0.6, and

ρ_{1}^{″} = (\begin{matrix} 0.75 & - 0.1 - 0.3 i \\ - 0.1 + 0.3 i & 0.25 \end{matrix})

with 〈σ_z(0)〉 = Tr(ρ″ ₁ σ_z) = 0.5. According to Eq. (8), the numerically simulated QSDPSs of the reflected photons from the cavity as a function of the detuning Δ_dc are shown in Fig. 3. Panels (a)–(c) correspond to the exemplified detected states ρ ₁, ρ′ ₁, and ρ″ ₁, respectively. The available experimental parameters are the same as those in Fig. 2. Here we assume that the detection of QSDPSs is performed at Δ_dc = 0. From Figs. 3(a)–3(c), we can directly read out the QSDPSs ϕ ₁ as −0.331π, 0.12π, and −0.143π. This implies that once these three QSDPSs are obtained, we can identify three unknown parameters r_z, r_y, and r_x, and consequently the unknown single-qubit state ρ ₁ can be determined in a deterministic way.

Fig. 3 The numerically simulated QSDPSs ϕ ₁ of the reflected photons from the cavity as a function of the detuning Δ_dc. Panels (a)–(c) correspond to the exemplified detected states ρ ₁, ρ′ ₁, and ρ″ ₂, respectively. The detection of the QSDPSs is assumed to be performed at Δ_dc = 0. The available experimental parameters are the same as those in Fig. 2.

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3. QNDD of single-qubit states in type-II cavity QED system

3.1. QSDPSs of the transmitted photons

In this section, we investigate how to realize the QNDD of unknown single-qubit states in type-II cavity QED system illustrated in Fig. 1(b). In this system, a qubit interacts with a single-mode and double-sided cavity with both two mirrors partially reflected. In the dispersive regime and in the frame rotating at the frequency ω_d, such a system is also described by the Hamiltonian (3). Further, we can obtain Eqs. (4) and (5) in type-II low-Q cavity.

In this system, the Heisenberg-Langvian equation of motion for the cavity field operator a(t) is derived as

\frac{d a (t)}{d t} = i [Δ_{d c} - Γ σ_{z} (t)] a (t) - \frac{1}{2} (κ_{l} + κ_{r}) a (t) - \sqrt{κ_{l}} b_{in}^{l} (t) - \sqrt{κ_{r}} b_{in}^{r} (t),

where κ_l (κ_r) is the photon decay rate of the left (right) mirror of the cavity,

b_{in}^{l (r)} (t)

with the commutation relation

[b_{in}^{l (r)}, b_{in}^{l (r) †} (t^{'})] = δ (t - t^{'})

is the input field to the left (right) mirror of the cavity. The input-output relations between the input and output fields to the left and the right mirrors of the cavity related to the cavity field operator a(t) are given by [33]

\begin{array}{l} b_{out}^{l} (t) = b_{in}^{l} (t) + \sqrt{κ_{l}} a (t), \\ b_{out}^{r} (t) = b_{in}^{r} (t) + \sqrt{κ_{r}} a (t) . \end{array}

We assume that there is no input field to the right mirror (i.e.,

b_{in}^{r} (t) = 0

). Similar to the derivation of Eq. (8), from Eqs. (16) and (17) together with Eq. (5), we can derive the phase θ ₁ of the transmitted photons through the cavity related to the transmission coefficient

T_{1} e^{i θ_{1}} \equiv \frac{b_{out}^{r} (t)}{b_{in}^{l} (t)} = \frac{\sqrt{κ_{l} κ_{r}}}{i [Δ_{d c} - Γ 〈 σ_{z} (0) 〉] - \frac{1}{2} (κ_{l} + κ_{r})},

where T ₁ is the amplitude of the transmission coefficient. Particularly, if g = 0, Eq. (18) recovers the phase θ ₀ of the transmitted photons through the EC related to the transmission coefficient

T_{0} e^{i θ_{0}} = \frac{\sqrt{κ_{l} κ_{r}}}{i Δ_{d c} - \frac{1}{2} (κ_{l} + κ_{r})},

where T ₀ is the amplitude of the transmission coefficient for the EC.

From Eqs. (18) and (19), we can find that the phase θ ₁ of the transmitted photons is shifted horizontally to the left or right by the qubit-state-dependent quantity Γ〈σ_z(0)〉, compared with the phase θ ₀ for the EC. This can be confirmed by numerical simulation for the specific single-qubit states. In Fig. 4, we present the numerical simulation of the phases θ ₁ of the transmitted photons with Eq. (18) for the computational basis states |1〉 (green line) and |0〉 (red line), and an arbitrary state ρ with the diagonal elements of its density matrix as diag(0.7, 0.3) (blue line). For comparison, we also plot the phase θ ₀ of the transmitted photons from the EC (black line) with Eq. (19). The available experimental parameters are the same as those in Fig. 2 and κ_l = κ_r = κ for simplicity. Through numerical analysis, we can find that the phase θ ₁ of the transmitted photons for the computational basis state |1〉 with 〈σ_z(0)〉 = 1 (|0〉 with 〈σ_z(0)〉 = −1) is shifted horizontally to the left (right) by the quantity Γ (−Γ), relative to the phase θ ₀ for the EC. Additionally, this result can agree well with the experimentally measured results in [35]. Furthermore, we can find that the phase θ ₁ for the state ρ with 〈σ_z(0)〉 = 0.4 is shifted horizontally to the left by 0.4Γ. When the detection of QSDPSs is made at Δ_dc = 0, the QSDPSs θ ₁ of the transmitted photons are 0.857π/2 for the computational basis state |1〉, −0.857π/2 for the computational basis state |0〉, and 0.669π/2 for the state ρ, respectively. This illustrates that the relevant information of single-qubit state involved in 〈σ_z(0)〉, i.e., the diagonal elements of its density matrix, can be directly determined by detecting the QSDPSs θ ₁ of the transmitted photons through the cavity at a certain driving frequency, Δ_dc = 0 for instance.

Fig. 4 The numerically simulated QSDPSs θ ₁ of the transmitted photons through the cavity as a function of the detuning Δ_dc for the EC (black line), the computational basis states |1〉 (green line) and |0〉 (red line), and an arbitrary state ρ with the diagonal elements of its density matrix as diag(0.7, 0.3) (blue line). The available experimental parameters are the same as those in Fig. 2 and κ_l = κ_r = κ for simplicity.

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3.2. QNDD of single-qubit states by detecting the QSDPSs of the transmitted photons

The approach to detect the QSDPSs proposed in Sec. 3.1 can maintain the nondestructive property [33], and can also be utilized to realize the QNDD of unknown single-qubit states. Following the procedure in Sec. 2.2 and according to Eq. (18), we numerically simulate the QSDPSs θ ₁ of the transmitted photons through the cavity as a function of the detuning Δ_dc in Fig. 5. Panels (a)–(c) correspond to the exemplified detected state ρ ₁, ρ′ ₁, and ρ″ ₁, respectively. The available experimental parameters are chosen as same as those in Fig. 2 and κ_l = κ_r = κ for simplicity. The detection of the QSDPSs is also made at Δ_dc = 0. From Figs. 5(a)–5(c), we can directly read out the QSDPSs θ ₁ as 0.457π/2, −0.768π/2, and 0.727π/2. Conversely, once we obtain these QSDPSs, we can determine the three unknown parameters (r_z, r_y, r_x) characterized the unknown single-qubit state, and thus achieve the QNDD of unknown single-qubit state deterministically.

Fig. 5 The numerically simulated QSDPSs θ ₁ of the transmitted photons through the cavity as a function of the detuning Δ_dc. Panels (a)–(c) correspond to the exemplified detected states ρ ₁, ρ′ ₁, and ρ″ ₁, respectively. The detection of the QSDPSs is assumed to be performed at Δ_dc = 0. The available experimental parameters are the same as those in Fig. 2 and κ_l = κ_r = κ for simplicity.

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4. QNDD of multiple-qubit states in two types of cavity QED systems

4.1. QSDPSs of the reflected and transmitted photons for multiple-qubit states

In the preceding two sections, we have derived the QSDPSs of the reflected and transmitted photons in two types of cavity QED systems, and have demonstrated the QNDD of single-qubit states. Here we extend the above investigations to the case of multiple-qubit states in a straightforward way. We consider the system wherein N qubits interact with a single-mode cavity. Similar to the derivations of Eqs. (2) and (3), in the dispersive regime ( $Δ_{j} = ω_{a}^{j} - ω_{c} ≫ g_{j}$ ) and in the frame rotating at the frequency ω_d, the system is governed by the Hamiltonian

ℋ_{eff}^{RF} = - (Δ_{d c} - \sum_{j = 1}^{N} Γ_{j} σ_{z}^{j}) a^{†} a - \sum_{j = 1}^{N} (Δ_{d a}^{j} - Γ_{j}) \frac{σ_{z}^{j}}{2},

where

Γ_{j} = \frac{g_{j}^{2}}{Δ_{j}}

, g_j is the coupling strength between the j-th qubit and the cavity,

Δ_{d a}^{j} = ω_{d} - ω_{a}^{j}

is the frequency detuning of the input pulse from the j-th qubit.

In two types of cavity QED systems, we can obtain the Heisenberg-Langvian equation of motion for each qubit operator which is the same as Eq. (4) and further obtain Eq. (5). In the type-I cavity QED system, the Heisenberg-Langvian equation of motion for the cavity field operator a(t) is given by

\frac{d a (t)}{d t} = i [Δ_{d c} - \sum_{j = 1}^{N} Γ_{j} σ_{z}^{j} (t)] a (t) - \frac{κ}{2} a (t) - \sqrt{κ} b_{in} (t) .

With the input-output relation (7) and Eq. (5), after the adiabatic elimination of the cavity field operator, we can get the QSDPS ϕ_N of the reflected photon from the cavity satisfying

e^{i ϕ_{N}} \equiv \frac{b_{out} (t)}{b_{in} (t)} = \frac{i [Δ_{d c} - \sum_{j = 1}^{N} Γ_{j} 〈 σ_{z}^{j} (0) 〉] + \frac{κ}{2}}{i [Δ_{d c} - \sum_{j = 1}^{N} Γ_{j} 〈 σ_{z}^{j} (0) 〉] - \frac{κ}{2}} .

In the type-II cavity QED system, the Heisenberg-Langvian equation of motion for the cavity field operator a(t) is written as

\frac{d a (t)}{d t} = i i [Δ_{d c} - \sum_{j = 1}^{N} Γ_{j} σ_{z}^{j} (t)] a (t) - \frac{1}{2} (κ_{l} + κ_{r}) a (t) - \sqrt{κ_{l}} b_{in}^{l} (t) - \sqrt{κ_{r}} b_{in}^{r} (t) .

Likewise, with the input-output relations (17) and Eq. (5), we can obtain the QSDPS θ_N of the transmitted photons through the cavity related to the transmission coefficient

T_{N} e^{i θ_{N}} \equiv \frac{b_{out}^{r} (t)}{b_{in}^{l} (t)} = \frac{\sqrt{κ_{l} κ_{r}}}{i [Δ_{d c} - \sum_{j = 1}^{N} Γ_{j} 〈 σ_{z}^{j} (0) 〉] - \frac{κ_{l} + κ_{r}}{2}},

where T_N is the amplitude of the transmission coefficient.

The density matrix of an arbitrary N-qubit state to be determined is generally expressed as

ρ_{N} = \frac{1}{2^{N}} \sum_{j_{1}, j_{2} \dots, j_{N} = 0, x, y, z} r_{j_{1} j_{2} \dots j_{N}} σ_{j_{1}} \otimes σ_{j_{2}} \dots \otimes σ_{j_{N}},

where r _{j₁j₂··· j_N} are real parameters and r _00···0 = 1 due to the normalization condition Trρ _N = 1. To determine the unknown N-qubit state ρ_N, we actually need to identify 4^N − 1 real parameters r _{j₁j₂··· j_N}. As demonstrated in Sec. 2.2 and Sec. 3.2, these parameters r _{j₁j₂··· j_N} can be identified by detecting the QSDPSs in (22) and (24) of the reflected and transmitted photons from the cavity, with the help of the combination of the local and nonlocal UOs [36]. In a similar way, the QNDD of unknown N-qubit states can be achieved deterministically.

4.2. An example for N = 2

To show our idea clearly, we will present an example for N = 2 in the following. Specifically, the density matrix of the two-qubit state ρ ₂ has 15 real parameters r _j₁j₂ to be determined. Prior to the detection of QSDPSs of the reflected or transmitted photons, we first need to perform 15 proper UOs U_k(k = 1, 2, ···, 15). After that, two-qubit state ρ ₂ is transformed as $ρ_{2}^{k} = U_{k} ρ_{2} U_{k}^{†}$ . Further, the quantity $\sum_{j = 1, 2} Γ_{j} 〈 σ_{z}^{j} (0) 〉$ in Eqs. (22) and (24) can be calculated. The correspondence relation between the required proper UOs and the calculated quantity $\sum_{j = 1, 2} Γ_{j} 〈 σ_{z}^{j} (0) 〉$ is shown in Table 1. Therein, I is an identity operator, $U_{x (y, z)}^{1 (2)} (ϑ) = e^{i ϑ σ_{x (y, z)}^{1 (2)}}$ is a local single-qubit UO, and $U^{12} = exp [i \frac{π}{4} (σ_{y}^{1} \otimes σ_{y}^{2} + σ_{z}^{1} \otimes σ_{z}^{2})]$ is a nonlocal two-qubit UO. The superscript “1(2)” of UO is labeled for the operation performed on qubit 1(2).

Table 1. Correspondence relation between the required proper unitary operations (UOs) and the calculated quantity $\sum_{j = 1, 2} Γ_{j} 〈 σ_{z}^{j} (0) 〉$ . See the text for details.

View Table

With the calculated quantity $\sum_{j = 1, 2} Γ_{j} 〈 σ_{z}^{j} (0) 〉$ shown in Table 1, we numerically show the QNDD of unknown two-qubit states in two types of cavity QED systems. Without loss of generality, the 15 real parameters are chosen as r _0x = 0.65, r _0y = 0.2, r _0z = 0.1, r _x0 = 0.5, r_yy = 0.45, r_xy = 0.05, r_xz = 0.15, r _y0 = 0.3, r_yx = 0.25, r_yy = 0.2, r_yz = 0.25, r _z0 = 0.2, r_zx = 0.3, r_zy = 0.15, and r_zz = 0.05. In type-I cavity QED system, the numerically simulated QSDPSs ϕ ₂ of the reflected photons according to Eq. (22) as a function of the detuning Δ_dc are shown in Fig. 6. Panels (a)–(o) correspond to the states $ρ_{2}^{k}$ (k = 1, 2, ···, 15), respectively. The accessible experimental parameters are selected as (Γ₁, Γ₂) = 2π × (13, 4)MHz [37] and κ = 2π × 1.69MHz [34]. At Δ_dc = 0, the detected QSDPSs ϕ ₂ can be read out from Fig. 6(a)–6(o) as 0.175π, −0.234π, −0.151π, 0.078π, 0.279π, 0.103π, 0.273π, 0.633π, 0.167π, −0.75π, −0.463π, 0.75π, −0.888π, 0.517π, and 0.336π, respectively. In type-II cavity QED system, we numerically simulate the QSDPSs θ ₂ of the transmitted photons based on Eq. (24) vs. the detuning Δ_dc in Fig. 7. Panels (a)–(o) correspond to the states $ρ_{2}^{k}$ (k = 1, 2, ···, 15), respectively. The available experimental parameters are the same as those in Fig. 6 and κ_l = κ_r = κ for simplicity. When the detection is made at Δ_dc = 0, we can directly read out the QSDPSs θ ₂ of the transmitted photons from Fig. 7(a)–7(o) as −0.673π/2, 0.583π/2, 0.714π/2, −0.847π/2, −0.52π/2, −0.8π/2, −0.529π/2, −0.2π/2, −0.687π/2, 0.13π/2, 0.326π/2, −0.13π/2, 0.056π/2, −0.282π/2, and −0.451π/2, respectively. Conversely, with these obtained QSDPSs of the reflected or transmitted photons, we can identify the unknown 15 real parameters and thus the QNDD of unknown two-qubit state ρ ₂ can be realized.

Fig. 6 The numerically simulated QSDPSs ϕ ₂ of the reflected photons as a function of the detuning Δ_dc. Panels (a)–(o) correspond to the states $ρ_{2}^{k}$ (k = 1, 2, ···, 15), respectively. The accessible experimental parameters are selected as (Γ₁, Γ₂) = 2π × (13, 4) MHz [37] and κ = 2π × 1.69MHz [34].

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Fig. 7 The numerically simulated QSDPSs θ ₂ of the transmitted photons vs. the detuning Δ_dc. Panels (a)–(o) correspond to the states $ρ_{2}^{k}$ (k = 1, 2, ···, 15), respectively. The accessible experimental parameters are the same as those in Fig. 6 and κ_l = κ_r = κ for simplicity.

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5. Discussion and conclusion

The generic models investigated in this paper are suitable for various physical systems, such as neutral atoms trapped in an optical microcavity [32, 38–41 ], quantum dots coupled to an optical microcavity [42–45 ], nitrogen-vacancy centers in diamond coupled to photonic crystal nanocavity [46], and superconducting qubits [47–49 ] coupled to a resonator [35, 50–54 ].

Now we take circuit QED system [35,50–56 ] as an example to discuss the experimental feasibility of our proposal. Firstly, it is required that our proposal works in the dispersive regime of circuit QED system. This is readily satisfied by adjusting the external flux bias on each superconducting qubit so that its transition frequency is detuned largely from the cavity frequency. Secondly, two types of resonators in circuit QED systems can function as the needed low-Q cavities. That is, reflection-line resonator [57] shown in Fig. 8(a) acts as the single-sided cavity in type-I cavity QED system, and transmission-line resonator [35, 50–54 ] illustrated in Fig. 8(b) functions as the double-sided cavity in type-II cavity QED system. These two types of resonators both have the relatively low quality factor, Q ≈ 10⁴ [35], and κ ≫ γ is ensured for the typical experimental parameters, e.g., κ = 2π × 1.69MHz [34] and γ = 2π × 0.02MHz [58]. Thirdly, the required single-qubit UOs U_x,y,z can be implemented via various methods [59], and two-qubit UO U ¹² can be realized with the so-called “FLICFORQ” protocol [59]. The times needed to realize them are about a few ns, which is significantly less by two orders at least than the relaxation and dephasing times for qubit, e.g., T ₁ = 7.3μs and T ₂ = 500ns [58]. Fourthly, the input driving field should be sufficiently weak so that the average photon number is less than the critical photon number n_c = Δ²/(4g ²) [35] to maintain the nondestructive property of the QNDD. Meanwhile, it also ensures that the mean-field approximation is validated [32] and the populations of the qubit are almost unchanged. Finally, the QSDPS of the reflected or transmitted photons can be measured experimentally with the sophisticated homodyne detection technique shown in Fig. 8(c). The resonator output field is first mixed with the local oscillator (LO). Then the QSDPS (ϕ_N or θ_N) is extracted by the network analyzer in experiment since the resulting signal out of the mixer is proportional to cosϕ_N or cosθ_N. From the above discussion, it is seen that our proposal is feasible with the current experimental technology.

Fig. 8 Schematic of two types of circuit QED systems. (a) Type-I circuit QED system wherein a superconducting qubit is dispersively coupled to a reflection-line resonator. (b) Type-II circuit QED system wherein a superconducting qubit is dispersively coupled to a transmission-line resonator. (c) The resonator output field in (a) and (b) is measured with the homodyne detection technique. The resonator output field is first mixed with the local oscillator (LO) and then the resulting signal out of the mixer is measured by the network analyzer in experiment.

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We next address the robustness of our proposal. On one hand, our proposal is required to work in the low-Q cavities (i.e., a relatively larger cavity decay rate). Our proposal is based on the input-output process of photons. The larger cavity decay rate facilitates the input-output process of photons, and thus the detection time of QSDPSs of the reflected or transmitted photons can be decreased. Hence, cavity decay plays an active role in our proposal. From the numerical simulations in Figs. 2 –7, it can also be found that our proposal is insensitive to the cavity decay. On the other hand, our proposal works in the dispersive regime. This means that the atom inside the cavity is only virtually excited. Also, it can be seen from Eq. (5) that the average of the atomic operator σ_z is almost unchanged during the whole process of the detection since κ ≫ γ. Therefore, our proposal is insensitive to atomic spontaneous emission. In a word, the distinct property of our proposal is that it is insensitive to both cavity decay and atomic spontaneous emission.

In conclusion, we have proposed an efficient approach to achieve the QNDD of unknown qubit states in two types of low-Q cavities. Different from the previous works, our proposal is required to work in the dispersive regime. We have shown that the QNDD of unknown qubit states can be realized by detecting the QSDPSs of the reflected or transmitted photons from the cavity in the single-photon input-output process. Our proposal is straightforwardly extended to the case of multiple-qubit states, and is feasible with the current experimental technology. The manifest property of our proposal is that it works in the dispersive regime of low-Q cavities and it is robust to both cavity decay and atomic spontaneous emission.

Acknowledgments

We thank Prof. Zheng-Wei Zhou for useful discussion. This work was supported in part by the National Basic Research Program of China (Grant Nos. 2011CBA00200 and 2011CB921200), the Strategic Priority Research Program(B) of the Chinese Academy of Sciences (Grant No. XDB01030200), and the Natural Science Foundation of China (Grant Nos. 11405171 and 11574294), the Anhui Provincial Natural Science Foundation (Grant No. 1608085QF139), and the Grant for Scientific Research of BSKY (No. XJ201519) from Anhui Medical University.

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UOs	$\sum_{j = 1, 2} Γ_{j} 〈 σ_{z}^{j} (0) 〉$	UOs	$\sum_{j = 1, 2} Γ_{j} 〈 σ_{z}^{j} (0) 〉$
U ₁ = I	Γ₁ r _z0 + Γ₂ r _0z	$U_{9} = U_{x}^{1} (\frac{π}{4}) U^{12} U_{x}^{1} (\frac{π}{4})$	Γ₁ r_xz + Γ₂ r_zx
$U_{2} = U_{x}^{1} (\frac{π}{2})$	−Γ₁ r _z0 + Γ₂ r _0z	$U_{10} = U_{y}^{2} (\frac{π}{4}) U^{12} U_{x}^{1} (\frac{π}{4})$	Γ₁ r_xy − Γ₂ r_yx
$U_{3} = U_{x}^{1} (\frac{π}{4})$	−Γ₁ r _y0 + Γ₂ r _0z	$U_{11} = U^{12} U_{y}^{1} (\frac{π}{4})$	−Γ₁ r_zy + Γ₂ r_yx
$U_{4} = U_{y}^{1} (\frac{π}{4})$	Γ₁ r _x0 + Γ₂ r _0z	$U_{12} = U_{x}^{1} (\frac{π}{4}) U^{12} U_{y}^{1} (\frac{π}{4})$	−Γ₁ r_zz + Γ₂ r_yx
$U_{5} = U_{x}^{2} (\frac{π}{4})$	Γ₁ r _z0 − Γ₂ r _0y	$U_{13} = U_{x}^{2} (\frac{π}{4}) U^{12} U_{y}^{1} (\frac{π}{4})$	−Γ₁ r_zy + Γ₂ r_xx
$U_{6} = U_{y}^{2} (\frac{π}{4})$	Γ₁ r _z0 + Γ₂ r _0x	$U_{14} = U^{12} U_{z}^{1} (\frac{π}{4})$	Γ₁ r_yy − Γ₂ r_xx
$U_{7} = U^{12} U_{x}^{1} (\frac{π}{4})$	Γ₁ r_xy + Γ₂ r_zx	$U_{15} = U_{x}^{1} (\frac{π}{4}) U^{12} U_{z}^{1} (\frac{π}{4})$	Γ₁ r_yz − Γ₂ r_xx
$U_{8} = U_{x}^{1} (\frac{π}{2}) U^{12} U_{x}^{1} (\frac{π}{4})$	−Γ₁ r_xy + Γ₂ r_zx

Quantum nondestructive determination of qubit states in low-Q cavities via single-photon input-output process

Abstract

1. Introduction

2. QNDD of single-qubit states in type-I cavity QED system

2.1. QSDPSs of the reflected photons

2.2. QNDD of single-qubit states by detecting the QSDPSs of the reflected photons

3. QNDD of single-qubit states in type-II cavity QED system

3.1. QSDPSs of the transmitted photons

3.2. QNDD of single-qubit states by detecting the QSDPSs of the transmitted photons

4. QNDD of multiple-qubit states in two types of cavity QED systems

4.1. QSDPSs of the reflected and transmitted photons for multiple-qubit states

4.2. An example for N = 2

5. Discussion and conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Tables (1)

Equations (25)

Optics Express