Quantum photonic transistor controlled by an atom in a Floquet cavity-QED system

Haozhen Li; Han Cai; Jingping Xu; V. V. Yakovlev; Yaping Yang; Da-Wei Wang

doi:10.1364/OE.27.006946

1. Introduction

Photons are widely used in quantum computing [1,2] and quantum communication [3,4]. Many of the applications are implemented through all-optical devices, which can perform quantum logic operations [5–9] and route information. Among these devices, optical diodes and transistors have been developed in various atom-photon coupled systems [10–17]. Analogous to electronic transistors [18], an optical transistor is a device that can gate the propagation of light [19]. The realization of such an optical transistor has been demonstrated by using Rydberg atoms [20–22], quantum dots [23,24], and cavity-QED systems [25–31]. For example, efficient optical transistors can be realized with quantum dot [23,24] or atom [25–28] embedded in an optical cavity based on the strong light-matter interaction, and a passive optical transistor can be realized by coupling a driven qutrit to two superconducting cavities [31]. The circuit-QED systems [29–31] are particularly advantageous because of their flexibility, tunability, stability and scalability [32–36].

Most of the above-mentioned proposals for all-optical transistors require a single photon to work as the signal and/or gating field, e.g., the propagation of a signal pulse can be completely controlled by a single gate photon [19,27]. As a consequence, a perfect single photon source is required for accomplishing the quantum logic operation. Although various schemes for generating single photon have been proposed [37–40], a reliable single photon source with high fidelity is still challenging to practically realized for high demanding applications of quantum computing and quantum communication [41,42]. Fortunately, this problem can be circumvented by exploiting multiphoton fields, such as coherent states [43,44]. It has been proved that [45] the single-photon pulse can be replaced by a weak coherent pulse in realizing quantum logic gates without essential scaling problem.

In this report, we propose a quantum photonic transistor with coherent light. Our scheme is composed of three single-mode cavities and a two-level atom. This system can be achieved in superconducting circuits [32], where the two-level atom is a superconducting qubit, and the three single-mode cavities are superconducting resonators. The frequencies of the three superconducting resonators are dynamically modulated [46–50]. We investigate the input-output relations of the system and analyze the effect of the atomic states on the photon transmission. We reveal that the light from the input resonator can be guided to different output resonators by the atomic states. A quantum photonic transistor controlled by an atom can be realized by applying an external classical field driving the two-level atom.

Compared to the previous proposals of optical transistors that are based on single photons [19–21,26–31], our present quantum photonic transistor can work with multiple photons, including coherent field. Furthermore, our scheme can route the input signal into different output ports guided by the quantum states of a two-level atom. Such a quantum photonic transistor can be used in constructing quantum logic gates in a network where classical field is guided by a qubit to control other qubits.

This paper is organized as follows. In Section 2, we introduce the Floquet cavity QED system, which consists of three periodically modulated cavities coupled to a two-level atom. We also derive the effective Hamiltonian which will be used to investigate the photon transmission. In Section 3, we explore the input-output relations and the second order correlation functions between the output fields. A particular scheme for achieving a quantum photonic transistor is discussed in detail. Finally, a conclusion is drawn in Section 4.

2. Theoretical model and effective Hamiltonian

The system under investigation consists of three periodically modulated cavities (superconducting resonators), which are coupled to a two-level atom (superconducting qubit) with the vacuum Rabi frequency $g_{ν}$ , as shown in the dashed black box in Fig. 1. Under the rotating-wave approximation, the Hamiltonian of the system is,

Fig. 1 Schematic configuration of the Floquet cavity QED system. Three periodically modulated cavities (superconducting resonators) are coupled to a two-level atom (superconducting qubit). Three transmission lines are connected to the three cavities for input and output signals. The input/output field of transmission line 1, and the output fields of transmission line 2 and 3 are denoted by the red, blue and green arrows, respectively.

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H_{S} = \frac{1}{2} ℏ ω_{a} σ_{z} + \sum_{j = 1}^{3} ℏ ν_{j} (t) a_{j}^{†} a_{j} + \sum_{j = 1}^{3} ℏ g_{ν} (σ^{+} a_{j} + h . c .) .

The first two terms are the Hamiltonians of the atom and the cavities, respectively. The last term is the interaction between the cavities and the atom. $σ^{-} = | g 〉〈 e |$ and $σ_{z} = | e 〉〈 e | - | g 〉〈 g |$ are the atomic lowering operator and the z-component Pauli matrix, respectively. $ω_{a}$ is the atomic transition frequency. $a_{j}$ ( $a_{j}^{†}$ ) is the annihilation (creation) operator of the j-th cavity mode. The frequency of the cavity j is periodically modulated $ν_{j} (t) = ν + Δ \sin (ν_{d} t + ϕ_{j})$ with $Δ$ being the modulating amplitude, $ν_{d}$ the modulating frequency, $ϕ_{j}$ the modulating phase and $ν$ the cavity idle frequency. We transform the Hamiltonian to the interaction picture, where the Hamiltonian has the following form,

H_{I} = \frac{1}{2} ℏ δ σ_{z} + \sum_{j = 1}^{3} ℏ g_{ν} (σ^{+} a_{j} e^{i f \cos (ν_{d} t + ϕ_{j})} + h . c .) .

Here

δ = ω_{a} - ν

is the detuning between the atom and cavity idle frequency,

f = Δ / ν_{d}

is the cavity modulating parameter. Using the relationship

e^{i f \cos (ν_{d} t + ϕ_{j})} = \sum_{n = - \infty}^{\infty} i^{n} J_{n} (f) e^{i n (ν_{d} t + ϕ_{j})}

, where

J_{n} (f)

is the n-th order Bessel’s function of the first kind, we expand the interaction Hamiltonian into a Fourier series,

H_{I} = H_{0} + \sum_{n \neq 0}^{\pm \infty} H_{n} e^{i n ν_{d} t},

with

\begin{array}{l} H_{0} = ℏ δ σ_{z} / 2 + ℏ g_{ν} J_{0} (f) \sum_{j = 1}^{3} (σ^{+} a_{j} + h . c .) . \\ H_{n} = ℏ g_{ν} i^{n} J_{n} (f) \sum_{j = 1}^{3} [σ^{+} a_{j} + {(- 1)}^{n} a_{j}^{†} σ^{-}] e^{i n ϕ_{j}} \end{array}

_{$H_{n}$} is the Hermitian conjugate of

H_{- n}

. Here

n

should be integers running from negative infinity to positive infinity and

n \neq 0

. Under the condition

ν_{d} > > \sqrt{N} g_{ν}

and

δ

, with

N

being the total excitation number of the atom and photons, the second order perturbation is valid. Thus, given

ϕ_{j} = - 2 π (j - 1) / 3

,

δ = 0

and

J_{0} (f) = 0

(with

f = 2.4

), we obtain the effective Hamiltonian [51,52],

H_{e f f} = i ℏ κ σ_{z} \sum_{j = 1}^{3} a_{j + 1}^{†} a_{j} + H . c ..

where

κ = g_{ν}^{2} β (f) / ν_{d}

,

β (f) = \sum_{n = 1}^{\infty} 2 J_{n}^{2} (f) \sin (2 n π / 3) / n

. We obtain

β (f) \approx 0.309

with

f = 2.40

.

From the effective Hamiltonian in Eq. (5), we find that the periodic modulation of the cavity frequency results in the direct coupling between the cavities with a complex coupling coefficient $i κ$ . This complex coupling coefficient introduces an effective magnetic field [53,54], which breaks the time-reversal symmetry [55] and can be used to create an optical circulator in the three cavities [51]. Furthermore, due to the factor $σ_{z}$ , the photons in one cavity can be transferred to the other two cavities in two opposite directions depending on the atomic states $| e 〉$ and $| g 〉$ . If the photons transfer follow the direction from cavities $1 \to 2 \to 3 \to 1$ , we call it clockwise, and otherwise counterclockwise. In the following, we explore the photon transmission of the Floquet cavity QED system with this effective Hamiltonian.

3. Photon transmission of the transmission lines

In this section, we study the effects of the atomic states on the photon transmission features. First, we explore the input-output relations of the three transmission lines under the effective Hamiltonian. Second, we further apply an external classical field to drive the two-level atom and manipulate the photon transmission. To study the input-output relations, the interaction between the atom-cavity system and the outside transmission lines has to be considered, as shown in Fig. 1. The new Hamiltonian of the whole system can be expressed as follows:

H_{1} = H_{S - e f f} + H_{R} + H_{C R},

where

H_{S - e f f} = \frac{1}{2} ℏ ω_{a} σ_{z} + ℏ ν \sum_{j = 1}^{3} a_{j}^{†} a_{j} + i ℏ κ σ_{z} \sum_{j = 1}^{3} (a_{j + 1}^{†} a_{j} - a_{j}^{†} a_{j + 1}),

is the effective Hamiltonian of the atom-cavity system,

H_{R} = ℏ \sum_{j = 1}^{3} \sum_{k} ω_{j k} c_{j k}^{†} c_{j k},

is the Hamiltonian of the transmission lines and,

H_{C R} = \sum_{j = 1}^{3} \sum_{k} ℏ g_{j} (c_{j k}^{†} a_{j} + a_{j}^{†} c_{j k}) .

is the interaction between the cavity mode and the transmission lines. In Eqs. (8) and (9),

c_{j k}

is the annihilation operators of the modes of the transmission line

j

.

ω_{j k}

are their corresponding circular frequencies.

g_{j}

are the coupling strengths between cavity

j

and transmission line

j

. According to the Heisenberg equation

\partial \hat{O} / \partial t = [\hat{O}, \hat{H}] / i ℏ

, the equations of relevant operators can be written as,

{\dot{σ}}_{z} = 0,

{\dot{a}}_{j} = - i ν a_{j} - κ σ_{z} a_{j + 1} + κ σ_{z} a_{j + 2} - i g_{j} \sum_{k} c_{j k},

{\dot{c}}_{j k} = - i ω_{j k} c_{j k} - i g_{j} a_{j} .

By integrating Eq. (12), we obtain the input–output equations for the transmission lines [56,57],

c_{o u t, j} = c_{i n, j} - i \sqrt{γ_{j}} a_{j} .

where

γ_{j} = {| g_{j} |}^{2} τ_{j}

are the leakage rates of the photons from cavity

j

to the transmission line

j

.

τ_{j}

have the dimension of a time and depend on the mode density, which are defined by

\sum_{k} e^{- i ω_{j k} t} = δ (t) τ_{j}

[57]. The input and output operators are introduced as,

\begin{array}{l} c_{i n, j} (t) = \frac{1}{\sqrt{τ_{j}}} \sum_{k} c_{j k} (t_{0}) e^{- i ω_{j k} (t - t_{0})}, t > t_{0} . \\ c_{o u t, j} (t) = \frac{1}{\sqrt{τ_{j}}} \sum_{k} c_{j k} ({t^{'}}_{0}) e^{- i ω_{j k} (t - {t^{'}}_{0})}, t < {t^{'}}_{0} \end{array}

In a frame rotating with the input frequency $ω$ , the Heisenberg equation of motion of the cavity mode $a_{j}$ becomes,

{\dot{a}}_{j} = - [i Δ ω_{c} + \frac{1}{2} (γ_{j} + γ_{c a v, j})] a_{j} - κ σ_{z} a_{j + 1} + κ σ_{z} a_{j + 2} - i \sqrt{γ_{j}} c_{i n, j} .

Here,

Δ ω_{c} = ν - ω

is the detuning between the cavity idle frequency and the input field.

γ_{c a v, j}

are the cavity decay rates due to the modes other than those in the transmission lines. They are much smaller than

γ_{j}

and we set them to be zero (

γ_{c a v, j} = 0

) [58].

In the following, we only consider a coherent field input of the first transmission line. By setting $γ_{1} = γ_{2} = γ_{3} = γ$ , the cavity operators $a_{j}$ are obtained directly from Eq. (15) in the steady state ( ${\dot{a}}_{j} = 0$ ),

a_{1} = - \frac{i \sqrt{γ} [{(i Δ ω_{c} + γ / 2)}^{2} + κ^{2}] c_{i n, 1}}{(i Δ ω_{c} + γ / 2) [{(i Δ ω_{c} + γ / 2)}^{2} + 3 κ^{2}]},

a_{2} = - \frac{i κ \sqrt{γ} [κ + (i Δ ω_{c} + γ / 2) σ_{z}] c_{i n, 1}}{(i Δ ω_{c} + γ / 2) [{(i Δ ω_{c} + γ / 2)}^{2} + 3 κ^{2}]},

a_{3} = - \frac{i κ \sqrt{γ} [κ - (i Δ ω_{c} + γ / 2) σ_{z}] c_{i n, 1}}{(i Δ ω_{c} + γ / 2) [{(i Δ ω_{c} + γ / 2)}^{2} + 3 κ^{2}]} .

To study the properties of the output field, we need to calculate the expectation values of the input and output operators. By substituting Eqs. (16)–(18) into Eq. (13), we obtain the output field,

〈 c_{o u t, 1}^{†} c_{o u t, 1} 〉 = {| \frac{(3 i Δ ω_{c} + γ / 2) κ^{2} + {(i Δ ω_{c} + γ / 2)}^{2} (i Δ ω_{c} - γ / 2)}{(i Δ ω_{c} + γ / 2) [{(i Δ ω_{c} + γ / 2)}^{2} + 3 κ^{2}]} |}^{2} 〈 c_{i n, 1}^{†} c_{i n, 1} 〉,

〈 c_{o u t, 2}^{†} c_{o u t, 2} 〉 = \frac{κ^{2} γ^{2} [(Δ ω_{c}^{2} + γ^{2} / 4) + κ γ 〈 σ_{z} 〉 + κ^{2}]}{{| (i Δ ω_{c} + γ / 2) [{(i Δ ω_{c} + γ / 2)}^{2} + 3 κ^{2}] |}^{2}} 〈 c_{i n, 1}^{†} c_{i n, 1} 〉,

〈 c_{o u t, 3}^{†} c_{o u t, 3} 〉 = \frac{κ^{2} γ^{2} [(Δ ω_{c}^{2} + γ^{2} / 4) - κ γ 〈 σ_{z} 〉 + κ^{2}]}{{| (i Δ ω_{c} + γ / 2) [{(i Δ ω_{c} + γ / 2)}^{2} + 3 κ^{2}] |}^{2}} 〈 c_{i n, 1}^{†} c_{i n, 1} 〉 .

with

〈 c_{i n, 1}^{†} c_{i n, 1} 〉 = {| α |}^{2}

. According to the definition

g_{2 - 3}^{(2)} (0) = \frac{〈 c_{o u t, 2}^{†} c_{o u t, 3}^{†} c_{o u t, 2} c_{o u t, 3} 〉}{〈 c_{o u t, 2}^{†} c_{o u t, 2} 〉 〈 c_{o u t, 3}^{†} c_{o u t, 3} 〉}

, the second-order correlation function between the output fields of transmission line 2 and 3 is,

g_{2 - 3}^{(2)} (0) = \frac{[{(γ / 2 - i Δ ω_{c})}^{2} - κ^{2}] [{(γ / 2 + i Δ ω_{c})}^{2} - κ^{2}]}{{[(Δ ω_{c}^{2} + γ^{2} / 4) + κ^{2}]}^{2} - κ^{2} γ^{2} {〈 σ_{z} 〉}^{2}} \frac{〈 c_{i n, 1}^{†} c_{i n, 1}^{†} c_{i n, 1} c_{i n, 1} 〉}{{〈 c_{i n, 1}^{†} c_{i n, 1} 〉}^{2}} .

Here,

〈 c_{i n, 1}^{†} c_{i n, 1}^{†} c_{i n, 1} c_{i n, 1} 〉 = {| α |}^{4}

is the normally ordered correlation functions of the input field. In deriving Eqs. (19)–(22), we assume

〈 X_{f} σ_{a} 〉 \approx 〈 X_{f} 〉 〈 σ_{a} 〉

for a weak coherent input, where

X_{f}

and

σ_{a}

are the field operators and atomic operators, respectively. This approximation has been widely used in the study of atom-photon coupled systems [13,15,57].

First, we study how the input-output relations vary with the detuning parameter $Δ ω_{c} = ν - ω$ and show how the photon transmission is affected by the atomic states of the two-level atom. In Figs. 2(a)–2(e), the output of each transmission line is plotted. In order to discuss the relations between the outputs of transmission line 2 and 3, we set $κ = γ / 2$ to achieve a $100 %$ transmission and zero reflection at the input transmission line, i.e., $〈 c_{o u t, 1}^{†} c_{o u t, 1} 〉 = 0$ is always satisfied when $Δ ω_{c} = 0$ . Here, the values of $κ$ and $γ$ are experimentally feasible. In one of our recent experiments, $κ$ is about $1 M H z$ [46]. Furthermore, the value of $γ$ can be controlled by a tunable coupler, which has been realized in a circuit QED experiment [59]. It was demonstrated that $γ$ can be adjusted from zero to a photon emission rate about $0.2 G H z$ .

Fig. 2 The input-output relations of the three transmission lines versus the detuning $Δ ω_{c}$ with different atomic states. The pink dashed-dot lines in (a)-(e) are their corresponding second-order correlation $g_{2 - 3}^{(2)} (0)$ of the output field between transmission line 2 and 3.

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It can be seen from Fig. 2 that the transmission is strongly related to the atomic states. In the resonant regime $Δ ω_{c} = 0$ , if the atom is in the excited state ( $〈 σ_{z} 〉 = 1$ ), the input light of transmission line 1 will be outputted from transmission line 2, as shown in Fig. 2(a). If the atom is in the ground state ( $〈 σ_{z} 〉 = - 1$ ), the light will be outputted from transmission line 3, as shown in Fig. 2(b). If the atom is in an arbitrary superposition state $| ψ 〉 = A_{e} | e 〉 + A_{g} | g 〉$ , the probabilities of the light output from transmission line 2 and 3 are proportional to the populations of the atom in the states $| e 〉$ and $| g 〉$ , as shown in Figs. 2(c)–2(e).

The physical mechanism of this atomic states-dependent photon transmission property can be explained as follows. By periodically modulating the resonant frequencies of three identical cavities, the photons feel opposite effective magnetic fields for the two states of the atom such that the photons rotate in opposite directions in the three cavities when the atom is in different states [51]. This can be understood as a spin-orbit coupling [60]. The orbital motion of the photons is coupled with the atom.

A remarkable feature of the output fields is that it is in an entangled state of different channels (transmission line 2 and 3) and the atom when the atom-cavity detuning is zero. In order to further illuminate this feature, we calculated the second-order correlation functions of the two output fields from transmission line 2 and 3, as shown by the pink dashed-dot lines in Figs. 2(a)–2(e). The second-order correlation function always equals to zero ( $g_{2 - 3}^{(2)} (0) = 0$ ) when $Δ ω_{c} = 0$ , which means that the probability of a simultaneous output from these two transmission lines is zero. Namely, two detectors for output fields from transmission line 2 and 3 do not click at the same time. This remarkable feature can be used in quantum communication and quantum computing with coherent fields.

This system is also a beam splitter for entangling coherent states, which has the same effect of an ordinary beam splitter for entangling single photons. A single photon input at an ordinary beam splitter will be either reflected or transmitted. The output state is a single photon entangled state ( ${| 1 〉}_{i n} \to {| ψ 〉}_{o u t} = (i | 10 〉 + | 01 〉) / \sqrt{2}$ ). However, a coherent state input is transformed to a separable state by an ordinary beam splitter ( ${| α 〉}_{i n} \to {| ψ 〉}_{o u t} = | i α / \sqrt{2} 〉 | α / \sqrt{2} 〉$ ). In contrast, the current scheme can transform the incident coherent state to an entangled state of the two output transmission lines and the two-level atom ( ${| α 〉}_{i n} \to {| ψ 〉}_{o u t} = (| α 0 〉 | e 〉 \pm | 0 α 〉 | g 〉) / \sqrt{2}$ ) [51], which is similar to the single photon at an ordinary beam splitter. Here, we have neglected the losses of the cavities to modes other than those in the transmission lines because these losses are small in superconducting circuits [58]. As a result, in the resonant regime, the input and output photon numbers satisfy $〈 c_{i n, 1}^{†} c_{i n, 1} 〉 = 〈 c_{o u t, 2}^{†} c_{o u t, 2} 〉 + 〈 c_{o u t, 3}^{†} c_{o u t, 3} 〉$ , which means that all the input energy is transferred to the other two output transmission lines. In addition, such a transformation is under an ideal condition that the atom does not suffer decay and decoherence.

From the above discussion, we find that the atomic states play a crucial role in the photon transmission. In order to realize an all-optical device, i.e., optical transistor, it is necessary to control the light propagation with another light field. Therefore, we further apply an external classical laser field with frequency $ω_{d}$ to drive the $| e 〉 \leftrightarrow | g 〉$ transition with the Rabi frequency $Ω$ . The Hamiltonian that describes the whole system is given by

H_{2} = H_{1} + ℏ Ω (σ^{+} e^{- i ω_{d} t} + h . c .) .

where

H_{1}

is shown in Eq. (6). By considering the atomic dissipation, the Heisenberg equations of motion for the relevant operators can be written in the rotating frame at the driving fields’ frequencies as,

{\dot{σ}}^{-} = - (i Δ ω_{a d} + Γ / 2) σ^{-} + i Ω σ_{z} + 2 σ^{-} \sum_{j = 1}^{3} κ (a_{j + 1}^{†} a_{j} - a_{j}^{†} a_{j + 1}),

{\dot{σ}}_{z} = - 2 i Ω (σ^{+} - σ^{-}) - Γ (σ_{z} + 1),

{\dot{a}}_{j} = - (i Δ ω_{c} + \frac{1}{2} γ_{j}) a_{j} - σ_{z} κ a_{j + 1} + σ_{z} κ a_{j + 2} - i \sqrt{γ_{j}} c_{i n, j} .

Here

Δ ω_{a d} = ω_{a} - ω_{d}

is the detuning between the atom and the driving laser.

Γ

is the decay rate of the atom. From Eq. (26), we find that the dynamic equations of the cavity operators

a_{j}

are only related to the atomic operator

σ_{z}

, which is the same as in Eq. (15). Thus, the cavity operators

a_{j}

, the input-output relations and the second-order correlation functions between the output fields of transmission line 2 and 3 can be obtained along the same line as shown in Eqs. (16)–(21) and Eq. (22).

However, in the present case, the expectation value of $σ_{z}$ is related to the external driving field. Inserting Eqs. (16)–(18) into Eq. (24), we obtain,

\begin{matrix} {\dot{σ}}^{-} = - (i Δ ω_{a d} + Γ / 2) σ^{-} + i Ω σ_{z} \\ + \frac{8 i κ^{2} Δ ω_{c} γ {| i Δ ω_{c} + γ / 2 |}^{2} σ^{-}}{{| (i Δ ω_{c} + γ / 2) [{(i Δ ω_{c} + γ / 2)}^{2} + 3 κ^{2}] |}^{2}} c_{i n, 1}^{†} c_{i n, 1} . \end{matrix}

Then, by taking the expectation values of Eqs. (25) and (27), we can obtain the steady state expectation value of $σ_{z}$ as,

〈 σ_{z} 〉 = - \frac{1}{1 + x},

where

x

is a saturation parameter,

x = \frac{2 Ω^{2}}{{[Δ ω_{a d} + 8 κ^{2} γ Δ ω_{c} 〈 c_{i n, 1}^{†} c_{i n, 1} 〉 / {| {(i Δ ω_{c} + γ / 2)}^{2} + 3 κ^{2} |}^{2}]}^{2} + {(Γ / 2)}^{2}} .

Here, we analyze the photon transmission in the resonant regime $Δ ω_{a d} = Δ ω_{c} = 0$ . The results show that the propagation of the incident field is effectively controlled by the external driving field. In the absence of the driving field ( $Ω = 0$ ), the atom stays in the ground state $| g 〉$ . The incident light will be guided to transmission line 3, as shown by the green dotted line in Fig. 3. This is consistent with the result shown in Fig. 2(b). When we turn on the external field, the probability of the incident light output from transmission line 2 and 3 are related to the strength of the driving field. By increasing the driving strength, the population of the ground state will be transferred into the excited state $| e 〉$ , which leads to the probability of the light output from transmission line 2 increases and that output from transmission line 3 decreases, as shown by the blue dashed and green dotted lines in Fig. 3. By further increasing the strength of the driving field, the atom becomes saturated and the probability of the light output from transmission line 2 and 3 are eventually equal, which is consistent with the result shown in Fig. 2(e). We have a flexibility in choosing real parameters in this simulation, e.g., $γ = 1 M H z$ , $Γ = 100 k H z$ , and $κ = 500 k H z$ are all within the experimentally achievable region.

Fig. 3 The output of the three transmission lines as a function of the driving strength $Ω$ . The purple dashed-dot line is the expectation value of $σ_{z}$ .

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The scheme proposed in this paper can be used to realize a quantum photonic transistor. The output channel is controlled by a quantum two-level system. When the atom is in the ground (excited) state, the input light is outputted from transmission line 3 (2), as shown in Figs. 2(a) and 2(b). On the other hand, we can use an external light field to drive the two-level atom and continuously tune the probability of the field outputted by transmission line 2 or 3 from 0 to 1/2 or 1/2 to 1, as shown in Fig. 3. It is worth noting that, such a photonic transistor has a quantum nature. Namely, when the atom is in a quantum superposition state, the output state is an entangled state of the photons in two output channels and the atom. This counterintuitive property can be characterized more clearly by the second-order correlation functions between the output fields of transmission line 2 and 3, as shown in Fig. 4. It is found that there exists a dip near the resonant regime, the second-order correlation $g_{2 - 3}^{(2)} (0)$ equals to zero when $Δ ω_{c} = 0$ , which demonstrates that the incident light cannot be outputted from these two channels at the same time. Such a quantum photonic transistor could be more powerful than conventional classical optical transistors in quantum information processing.

Fig. 4 The second-order correlation $g_{2 - 3}^{(2)} (0)$ of the output fields between transmission line 2 and 3 as a function of the detuning $Δ ω_{c}$ with different driving strengths.

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4. Conclusion

In this report, we have investigated for the first time the input-output relations of a Floquet cavity quantum electrodynamics system. The output channel of this system is governed by the quantum states of the atom. Thus, the photon transport can be controlled by manipulating the atomic state. Furthermore, by applying another external classical field to drive the two-level atom, our scheme can be used to realize a quantum photonic transistor. Compared with the previous works, our quantum photonic transistor has a significant advantage. Namely, a quantum logic operation can be implemented with coherent state, which relaxes the requirement of a single photon source in realizing quantum information processing.

Funding

National Natural Science Foundation of China (NSFC) (11574229, 11874287,11504272, 11274242, 11574068, 11874322); Shanghai Science and Technology Committee (18JC1410900); Shanghai Education Commission Foundation; The International Exchange Program for Graduate Students from Tongji University; Air Force Office of Scientific Research (AFOSR) (FA9550-18-1-0141); The National Science Foundation (NSF) (ECCS-1509268, CMMI-1826078); The National Key Research and Development Program of China (2018YFA0307200); The Strategic Priority Research Program of Chinese Academy of Sciences (XDB28000000).

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Quantum photonic transistor controlled by an atom in a Floquet cavity-QED system

Abstract

1. Introduction

2. Theoretical model and effective Hamiltonian

3. Photon transmission of the transmission lines

4. Conclusion

Funding

References

Cited By

Figures (4)

Equations (29)

Optics Express