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Tunable asymmetric Einstein–Podolsky–Rosen steering of microwave photons in superconducting circuits

Kun Wu, Guangling Cheng, and Aixi Chen

Open Access Open Access

Abstract

We present an alternative scheme for generating the asymmetric steering of microwave photons via using a superconducting circuit system, where a single $ \Delta $-type three-level fluxoninum qubit interacts dispersively with three superconducting resonators. The nondegenerate parametric down-conversion occurs among three microwave modes by adiabatically eliminating the atomic variables of the artificial atom, which is responsible for the existence of quantum correlation. Furthermore, the asymmetric steering is easily established with the help of coherent driving of the resonators, and its directionality can be controlled by adjusting the driving strengths to two modes among three modes without additional noise. The scheme we present is based on general quantum operations under conditions of decoherence and nonideal coupling efficiency, and the asymmetric steering of microwave photons is a useful resource for the construction of long-distance quantum communication networks in solid-state systems.

© 2020 Optical Society of America

1. INTRODUCTION

Quantum steering is a form of quantum correlation intermediating between entanglement [1] and Bell nonlocality [2], which was first introduced by Schrödinger in 1935 [3] responding to the Einstein–Podolsky–Rosen (EPR) paradox [4]. In recent years, EPR steering with continuous variables has been an important and interesting topic in the field of quantum theory, which not only provides novel insight into quantum nonlocality, but also plays a key role in some information processing [5–18]. Of much interest is the inherent asymmetry of EPR steering, which results in one-way steering [6,19]. For the two observers Alice and Bob, the roles played by them are not exchangeable, because observer Alice can steer Bob’s state, but the reverse Bob-to-Alice steering is impossible even though they are entangled. This is because of the interpretation of asymmetric quantum nonlocality, and the potential applications in future asymmetric quantum information processing, one-way steering, has been thoroughly studied both theoretically [20–28] and experimentally [29–35]. We note that most theoretical schemes for generating asymmetric EPR steering focus on physical systems, including nonlinear optical systems [20–22], optomechanical systems [23–26], and Bose–Einstein condensates [27,28].

 figure: Fig. 1.

Fig. 1. (a) Transition frequencies of energy levels versus the flux bias $ {\Phi _{{\rm ext}}}/{\Phi _0} $ for the fluxonium qubit. (b) Charge matrix elements as a function of the flux bias $ {\Phi _{{\rm ext}}}/{\Phi _0} $.

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Recently, a superconducting quantum circuit named after an artificial atom has been a promising platform for the investigation of typical and interesting quantum optical phenomena such as resonance fluorescence [36], electromagnetically induced transparency (EIT) [37], coherent population trapping [38], Autler–Townes splitting [36,39], and nonlinear optical effects [40]. It is worth mentioning that there are some striking features in such a system. First, just one single artificial atom is involved in the present system, but the general atomic system usually includes many atoms. Second, the microwave mode is coupled to the superconducting qubit, which has a longer wavelength than the optical field and is more useful for long-distance communication. Last but not least, the superconducting circuit can be artificially structured with flexible designs and easily controlled for the experimental parameters. As the research moves along, the superconducting system is found to be an important and remarkable candidate for generating quantum squeezing and entanglement by coupling a circuit qubit serving as a two-level or three-level atom to the superconducting resonators [41–45]. In particular, the superconducting fluxonium qubit with electrical-dipole-transition-based cyclic three-level (i.e., $ \Delta $-type) structure, absent in the natural atomic system, has attracted a lot of attention in recent years [46–49]. For instance, the phenomenon of EIT with amplification can occur via the external couplings in such a system [46]. Realizations of microwave amplification, attenuation, and frequency conversion are feasible via using different configurations of coupling [47]. The efficient single-photon frequency up- and down-conversion in the microwave domain has been presented in superconducting quantum circuits based on coherent control [48]. The squeezing and entanglement of two microwave photons have been proposed via the degenerated parametric down-conversion process induced by the dispersive interaction in the cyclic three-level circuit system [49].

Here we present a feasible scheme for generating the tunable asymmetric steering of microwave photons in the circuit system consisting of a single cyclic three-level artificial atom and three superconducting resonators. Under the conditions of large detunings and weak couplings, the nonlinear interaction occurs among three separated resonators, which is helpful for the presence of nonlocal asymmetric correlations. Striking features are included in the present scheme. (i) The direction of the steady-state asymmetric steering can be easily controlled by virtue of the asymmetric coherent driving strengths on the resonators. (ii) The obtainable asymmetric correlations are immune to system decoherence due to the dispersive interactions between the artificial atom and three resonators. (iii) The steering occurs among three photons with the different frequencies in the system, where the resonators are respectively coupled to the different transitions of the circuit qubit with flexible energy levels. (iv) The nonlocal correlation properties of microwave photons are under consideration, which may expand the range of possibilities for quantum state preparation and be available for long-distance quantum communication.

The remaining parts of this paper are organized as follows. In Section 2, we describe the superconducting fluxonium qubit and its interaction with three resonators, where the nonlinear parametric process is deduced under the condition of large detunings. In Section 3, we introduce the EPR steering criteria and discuss the asymmetric EPR steering of microwave modes with the different coherent drivings. A summary is given in Section 4.

2. MODEL AND EQUATION

The single superconducting fluxonium qubit is under consideration, and its Hamiltonian can be written as in Refs. [50–52]:

$${H_f} = 4{E_C}{N^2} - {E_J}\cos ( {\varphi - 2\pi {\Phi _{{\rm ext}}}/{\Phi _0}} ) + \frac{1}{2}{E_L}{\varphi ^2},$$
where the charge is denoted by the operator $ N = Q/( {2e} ) $, in units of Cooper-pair charge $ 2e $, and the operator $ \varphi = 2\pi \Phi /{\Phi _0} $, with $ {\Phi _0} = h/( {2e} ) $ being the flux quantum describing the loop flux. The other coefficients are given by $ {E_C} = {e^2}/( {2{C_J}} ) $, $ {E_J} = {( {{\Phi _0}/2\pi } )^2}/{L_J}, $ and $ {E_L} = {( {{\Phi _0}/2\pi } )^2}/{L_A} $. As shown in Ref. [51], the generalized coordinate $ \varphi $ and momentum $ N $ with the commutation relation $ [ {\varphi ,N} ] = i $ are chosen, and the harmonic-oscillator-like model is presented with $ {E_J} = 0 $. In the new number basis, the fluxonium Hamiltonian can be expressed as $ {H_f} = \sum\nolimits_{n,{n^\prime }} {H_{n{n^\prime }}}| n \rangle \langle {{n^\prime }} | $. Fortunately, we employ a calculating program to obtain the numerical diagonalization Hamiltonian with $ {H_f} = \sum\nolimits_l \hbar {\omega _l}| l \rangle \langle l | $ and the spectroscopy frequencies $ {\omega _l} $. In particular, the energy transition frequencies and the charge matrix elements as a function of the flux bias $ {\Phi _{{\rm ext}}}/{\Phi _0} $ are shown in Fig. 1, where the parameters make reference to experimental data. It is obvious in Fig. 1(b) that all three matrix elements with respect to the lowest three energy levels have comparable values for the appropriate flux bias, and thus the $ \Delta $-type three-level structure is present in such a system, which recently has become a trend in quantum optics and quantum information.

Here we use the interactions of such a system with three superconducting resonators to obtain asymmetric quantum correlations. For the three-level fluxonium, the level states are respectively denoted by $ | 1 \rangle $, $ | 2 \rangle $, and $ | 3 \rangle $, and three microwave modes are respectively coupled to the different transitions of the qubit, as shown in Fig. 2. In the rotating-wave approximation, the total Hamiltonian is expressed as $ H = {H_0} + V(\hbar = 1) $, with

$${H_0} = \sum\limits_{j = 1 - 3} {\omega _j}{\sigma _\textit{jj}} + {v_j}a_j^\dagger {a_j},$$
$$V = {g_1}{a_1}{\sigma _{21}} + {g_2}{a_2}{\sigma _{32}} + {g_3}{a_3}{\sigma _{31}} + {\rm H.c.},$$
where $ {H_0} $ represents the free Hamiltonian for the superconducting qubit and superconducting resonators, and $ V $ describes the interaction between the microwave modes and the qubit. $ {\omega _j} $ and $ {v_j} $ are, respectively, the eigenfrequencies of qubit’s level states $ | j \rangle $ and resonator modes $ j $. $ {\sigma _\textit{jk}} = | j \rangle \langle k| $ are the populations for $ j = k $ and the flip operators when $ j \ne k $. $ a_j^\dagger ( {{a_j}} ) $ is creation (annihilation) operators of modes $ j $, and $ {g_j} $ are the complex coupling strengths. Here we focus on the far-off-resonant interactions between the qubit and microwave modes. For the large detunings $ | {{\Delta _1}} | \gg | {{g_1}} |, $ $ | {\Delta - {\Delta _1}} | \gg | {{g_2}} |, $ and $ | \Delta | \gg | {{g_3}} | $, where the detunings are defined as $ {\Delta _1} = {\omega _{21}} - {v_1}, \Delta = {\omega _{31}} - {v_3} $, and $ \Delta - {\Delta _1} = {\omega _{32}} - {v_2} $, with $ {\omega _\textit{jk}} = {\omega _j} - {\omega _k} $ $ ( {j \gt k;j,k = 1 - 3} ) $ being the transition frequencies, the effective couplings among the three modes can be obtained via eliminating the variables of the qubit. In order to do so, the method of Frölich–Nakajima transformation is used, which has recently been applied in a quantum Zeno-type effect [53], asymmetric spontaneous emission in circuit quantum electrodynamics (QED) systems [54], and coupling of spin ensembles via superconducting flux qubits [55]. First, an anti-Hermitian operator $ S $ is introduced with the form
$$ S = \frac{{g_1^ * }}{{{\Delta _1}}}a_1^\dagger {\sigma _{12}} + \frac{{g_2^ * }}{{\Delta - {\Delta _1}}}a_2^\dagger {\sigma _{23}} + \frac{{g_3^ * }}{\Delta }a_3^\dagger {\sigma _{13}} - {\rm H.c.},$$
which is proved to be satisfied as $ V + [ {{H_0},S} ] = 0 $. Then a unitary transformation can be made by $ {\cal H} = \exp ( { - \lambda S} ) H \exp ( {\lambda S} ) $, where the parameter $ \lambda $ is present in order to effectively use the perturbation theory for the interaction Hamiltonian, and it can be set to 1 after calculations. In terms of the Baker–Hausdorff formula, one has
$$\begin{split}{\cal H} & = {H_0} + \lambda V + \lambda \left[ {{H_0},S} \right] + {\lambda ^2}\left[ {V,S} \right] + \frac{{{\lambda ^2}}}{2}\left[ {S,\left[ {S,{H_0}} \right]} \right]\\ & \quad+ \frac{{{\lambda ^3}}}{2}\left[ {S,\left[ {S,V} \right]} \right] - \frac{{{\lambda ^3}}}{6}\left[ {S,\left[ {S,\left[ {S,{H_0}} \right]} \right]} \right] + O\left( {{\lambda ^4}} \right).\end{split}$$
 figure: Fig. 2.

Fig. 2. (a) Model for the coupling of a single fluxonium qubit with three superconducting resonators. (b) Corresponding transitions coupled to the different modes.

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As is well known, the qubit is hardly excited and populates mainly in the ground state $ | 1 \rangle $ for the case of large detunings. At the same time, the effect of virtual photon-induced modification can be omitted, which is not important for the energy of a qubit in excited state under such circumstances. As a consequence, we can obtain the expression of Hamiltonian ${\cal H}= H_{\rm eff} \otimes | 1 \rangle \langle 1 | $, and the effective Hamiltonian $ {H_{\rm eff}} $ is written as

$$\begin{split}{H_{\rm eff}} & = {H_0} + \frac{1}{2}\left[ {V,S} \right] + \frac{1}{3}\left[ {\left[ {V,S} \right],S} \right] \\ & = {H_{0 {\rm eff}}} + \chi ( {{e^{i\phi }}{a_1}{a_2}a_3^\dagger + {e^{ - i\phi }}a_1^\dagger a_2^\dagger {a_3}} ),\end{split}$$
with the effective free Hamiltonian $ {H_{0 {\rm eff}}} = {v_{1e}}a_1^\dagger {a_1} + {v_{2e}}a_2^\dagger {a_2} + {v_{3e}}a_3^\dagger {a_3} $. The effective frequencies are derived as $ {v_{1e}} = {v_1} - \frac{{{{| {{g_1}} |}^2}}}{{{\Delta _1}}},{v_{2e}} = {v_2} $, and $ {v_{3e}} = {v_3} - \frac{{{{| {{g_3}} |}^2}}}{\Delta }, $ and they satisfy the condition of $ {v_{3e}} = {v_{1e}} + {v_{2e}} $. Here the parameter $ \chi = \frac{{| {{g_1}{g_2}g_3^ * } |}}{{\Delta {\Delta _1}}} $ represents the effective coupling strength among three microwave modes, and $ \phi $ is the global phase contributed from the three couplings between the resonators and the qubit. It is obvious that the parametric down-conversion process occurs among three microwave modes in the present system, where the artificial atom will evolve along the path with $ | 1 \rangle \to | 3 \rangle \to | 2 \rangle \to | 1 \rangle $ by annihilating one photon of the $ {a_3} $ mode and creating two photons of modes $ {a_1} $ and $ {a_2}, $ respectively.

At the same time, the resonator modes are driven by the external fields, and the pumping Hamiltonian is given by

$${H_p} = i\sum\limits_{j = 1 - 3} { \epsilon _j} ( {a_j^\dagger {e^{ - i{v_\textit{jeff}}t}} - {a_j}{e^{i{v_\textit{jeff}}t}}} ),$$
where $ { \epsilon _j} $ describe the strengths of the driving fields and are assumed to be positive. Note that we can obtain the controllable asymmetric steering via choosing the proper driving strengths. For instance, we can consider the following two cases, where the first case is that the modes $ {a_1} $ and $ {a_2} $ are driven and no driving is acted on mode $ {a_3} $, i.e., $ { \epsilon _3} = 0 $, and the other case is the presence of $ { \epsilon _1} $ and $ { \epsilon _3} $ and the absence of $ { \epsilon _2} $. In the rotating frame with respect to $ U( t ) = \exp [ {i{H_{0{\rm eff}}}t} ] $ and with the choice of $ \phi = \pi /2 $, the total Hamiltonian is rewritten as
$$\tilde H = i\sum\limits_{j = 1 - 3} { \epsilon _j} ( {a_j^\dagger - {a_j}} ) + i\chi ( {a_1^\dagger a_2^\dagger {a_3} - {a_1}{a_2}a_3^\dagger } ).$$
Taking into account the decoherence effects, the corresponding master equation for the whole system is given by
$$\dot \rho = - i [ {\tilde H,\rho } ] + \sum\limits_{j = 1 - 3} {\kappa _j} ( {2{a_j}\rho a_j^\dagger - a_j^\dagger {a_j}\rho - \rho a_j^\dagger {a_j}} ),$$
with $ {\kappa _j} $ being the cavity loss rates. We can use the positive-$ P $ pseudoprobability distribution [56], to derive the fully quantum equations of motion. Based on the Fokker–Planck equation with positive-$ P $ representation, the stochastic differential equations are derived as
$$\begin{split}\frac{{d{\alpha _1}}}{{dt}}& = { \epsilon _1} - {\kappa _1}{\alpha _1} + \chi \alpha _2^\dagger {\alpha _3} + {F_{{\alpha _1}}}, \\[-2pt] \frac{{d\alpha _1^\dagger }}{{dt}} &= \epsilon _1^ * - {\kappa _1}\alpha _1^\dagger + \chi {\alpha _2}\alpha _3^\dagger + {F_{\alpha _1^\dagger }}, \\[-2pt] \frac{{d{\alpha _2}}}{{dt}}&= { \epsilon _2} - {\kappa _2}{\alpha _2} + \chi \alpha _1^\dagger {\alpha _3} + {F_{{\alpha _2}}}, \\[-2pt] \frac{{d\alpha _2^\dagger }}{{dt}} &= \epsilon _2^ * - {\kappa _2}\alpha _2^\dagger + \chi {\alpha _1}\alpha _3^\dagger + {F_{\alpha _2^\dagger }}, \\[-2pt] \frac{{d{\alpha _3}}}{{dt}} &= { \epsilon _3} - {\kappa _3}{\alpha _3} - \chi {\alpha _1}{\alpha _2} + {F_{{\alpha _3}}}, \\[-2pt] \frac{{d\alpha _3^\dagger }}{{dt}}& = \epsilon _3^ * - {\kappa _3}\alpha _3^\dagger - \chi \alpha _1^\dagger \alpha _2^\dagger + {F_{\alpha _3^\dagger }},\end{split}$$
where the noise operators $ {F_{{\alpha _{j}}}} $ satisfy $ \langle {{F_{{\alpha _{j}}}}} \rangle = 0 $. The stochastic averages of products of these variables are then equal to the normally ordered expectation values of the corresponding operators. This entails the replacement of conjugate variables $ \alpha _j^ * $ by $ \alpha _j^\dagger $, which are the complex conjugates of the uncrossed variables only in the mean. In terms of the Ornstein–Uhlenbeck process [57], the noise terms may be dropped, and the linearized semiclassical steady-state equation is obtained, allowing for easy calculations of the output spectra. The validity of this linearized fluctuation analysis is usually found by calculating the eigenvalues of the resulting drift matrix for the fluctuations and requires the knowledge of classical steady-state solutions. We expand the positive-$ P $ variables as $ {\alpha _j} = {\bar \alpha _j} + \delta {\alpha _j} $, where $ {\bar \alpha _j} $ are the steady-state classical amplitudes, and $ \delta {\alpha _j} $ correspond to the quantum fluctuation terms. It is well known that the stationary solutions $ {\bar \alpha _j} $ can be solved via setting $ d{\alpha _j}/dt = 0 $ and omitting the noise terms [58]. When we set $ { \epsilon _2} \ne 0 $ and $ { \epsilon _3} = 0 $, the corresponding equation related to $ {\bar \alpha _1} $ has the form
$$ ( {{ \epsilon _1} - {\kappa _1}{{\bar \alpha }_1}} ){ ( {{\kappa _2}{\kappa _3} + {\chi ^2}\bar \alpha _1^2} )^2} = {\kappa _3}{\chi ^2} \epsilon _2^2{\bar \alpha _1},$$
and the steady-state values $ {\bar \alpha _2} $ and $ {\bar \alpha _3} $ are given by
$${\bar \alpha _2} = \frac{{{ \epsilon _2}}}{{{\kappa _2}}} + \frac{{{{\bar \alpha }_1} ( {{\kappa _1}{{\bar \alpha }_1} - { \epsilon _1}} ) ( {{\kappa _2}{\kappa _3} + {\chi ^2}\bar \alpha _1^2} )}}{{{\kappa _2}{\kappa _3}{ \epsilon _2}}},$$
$${\bar \alpha _3} = \frac{{ ( {{\kappa _1}{{\bar \alpha }_1} - { \epsilon _1}} ) ( {{\kappa _2}{\kappa _3} + {\chi ^2}\bar \alpha _1^2} )}}{{{\kappa _3}\chi { \epsilon _2}}}.$$
When we assume $ { \epsilon _2} = 0 $ and $ { \epsilon _3} \ne 0 $, we have the equation with respect to steady-state value $ {\bar \alpha _1} $:
$$\left( {{\kappa _1}{{\bar \alpha }_1} - { \epsilon _1}} \right){\left( {{\kappa _2}{\kappa _3} + {\chi ^2}\bar \alpha _1^2} \right)^2} = {\kappa _2}{\chi ^2} \epsilon _3^2{\bar \alpha _1},$$
and the steady-state values $ {\bar \alpha _2} $ and $ {\bar \alpha _3} $ associated with $ {\bar \alpha _1} $ are determined by
$${\bar \alpha _2} = \frac{{\left( {{\kappa _1}{{\bar \alpha }_1} - { \epsilon _1}} \right)\left( {{\kappa _2}{\kappa _3} + {\chi ^2}\bar \alpha _1^2} \right)}}{{{\kappa _2}{\chi _3}}},$$
$${\bar \alpha _3} = \frac{{{ \epsilon _3}}}{{{\kappa _3}}} - \frac{{{{\bar \alpha }_1}\left( {{\kappa _1}{{\bar \alpha }_1} - { \epsilon _1}} \right)\left( {{\kappa _2}{\kappa _3} + {\chi ^2}\bar \alpha _1^2} \right)}}{{{\kappa _2}{\kappa _3}{ \epsilon _3}}}.$$
We also note that it can be easy to solve analytically the steady-state solutions and threshold conditions for a single coherent driving, as shown in Refs. [22,49]. However, it is found from the above complicated expressions that Eqs. (12) and (13) are fifth-order equations, and so it is very difficult to obtain analytical solutions to the threshold and the specific expressions for spectral quantities such as the EPR steering as follows. We will therefore concentrate on the numerical results. In terms of the above linearized fluctuation analysis, it gives us the set of equations for the fluctuating terms, which can be expressed in the compact form
$$\frac{d}{{dt}}\delta \vec R ( t ) = - A\delta \vec R ( t ) + Bd \vec F,$$
where $ \delta \vec R ( t ) = {( {\delta {\alpha _1},\delta \alpha _1^ * ,\delta {\alpha _2},\delta \alpha _2^ * ,\delta {\alpha _3},\delta \alpha _3^ * ,} )^T} $. $ A $ is the drift matrix, $ B $ contains the steady-state coefficients of the noise terms, and $ d \vec F $ is a vector of Wiener increments. The stability condition of the fluctuations is that the eigenvalues of $ A $ have no negative real parts, which has been numerically tested to be positive with the choice of the following parameters. When this condition is fulfilled as shown in Refs. [20–22], we may calculate the intracavity spectral matrix via using the Fourier transformation $ \delta \vec R( \omega ) = \frac{1}{{\sqrt {2\pi } }}\int \delta \vec R( t ) $ $ {e^{ - i\omega t}}dt $, and have the expression
$$S ( \omega ) = { ( {A + i\omega I} )^{ - 1}}D{ ( {{A^T} - i \omega I} )^{ - 1}},$$
where $ I $ is a unit matrix and $ D = B{B^T} $ is a $ 6 \times 6 $ matrix with the nonzero terms
$$\begin{split}D ( {1,3} ) = D ( {3,1}) = \chi {\bar \alpha _3}, \\ D ( {2,4} ) = D ( {4,2} ) = \chi \bar \alpha _3^ * .\end{split}$$
The matrix $ A $ is given by
$$A = \left[ {\begin{array}{*{20}{c}}{{\kappa _1}}&0&0&{ - \chi {{\bar \alpha }_3}}&{ - \chi \bar \alpha _2^ * }&0\\0&{{\kappa _1}}&{ - \chi \bar \alpha _3^ * }&0&0&{ - \chi {{\bar \alpha }_2}}\\0&{ - \chi {{\bar \alpha }_3}}&{{\kappa _2}}&0&{ - \chi \bar \alpha _1^ * }&0\\{ - \chi \bar \alpha _3^ * }&0&0&{{\kappa _2}}&0&{ - \chi {{\bar \alpha }_1}}\\{\chi {{\bar \alpha }_2}}&0&{\chi {{\bar \alpha }_1}}&0&{{\kappa _3}}&0\\0&{\chi \bar \alpha _2^ * }&0&{\chi \bar \alpha _1^ * }&0&{{\kappa _3}}\end{array}} \right].$$
This intracavity spectra matrix shown by Eq. (18) allows us to obtain useful and measurable output spectra by using the well-known input–output relations, which are accessible based on the homodyne-measurement techniques [59]. Now we concentrate on the quantum steerable correlation properties of the output modes. According to the input–output theory $ a_l^{{\rm in}} + a_l^{{\rm out}} = \sqrt {{\kappa _l}} {a_l} $ and with the choice of a vacuum input, we can easily obtain the output spectra of the resonator modes. Defining the correlation spectra $ \langle {\delta {A_1}\delta {A_2}} \rangle \delta ( {\omega + {\omega ^\prime }} ) = \langle \delta {A_1}( \omega )\delta {A_2}( {{\omega ^\prime }} ) \rangle $ for the arbitrary two operators $ {A_{1,2}} $, and introducing the canonical quadrature position and momentum operators $ {X_l} = {a_l} + a_l^\dagger $, $ {Y_l} = - i ( {{a_l} - a_l^\dagger } ), ( {l = 1 - 3} ) $, we obtain the output correlation spectra with the form $ \langle \delta X_l^{\rm o}\delta X_m^{\rm o} \rangle (\omega) = {{( { - 1} )^{l - m}} \langle \delta Y_l^{\rm o}\delta Y_m^{\rm o} \rangle (\omega) = {\delta _\textit{lm}} + \sqrt {{\kappa _l}{\kappa _m}} \langle \delta {X_l}\delta {X_m}\, +\, \delta {X_m}\delta {X_l} \rangle}$$(\omega) $, with $ {\delta _\textit{lm}} $ being the Kronecker delta, determined by the intracavity correlation spectra. In the following section, we utilize the above contents to discuss the quantum steering features of the present system.
 figure: Fig. 3.

Fig. 3. Spectra of EPR steering as a function of the ratio of driving strengths, with the parameters $ \chi = 0.01,{ \epsilon _2} = 100,{ \epsilon _3} = 0,{\kappa _1} = {\kappa _2} = {\kappa _3} = 1 $ (a) and $ {\kappa _1} = {\kappa _2} = 0.75,{\kappa _3} = 1 $ (b).

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3. EPR STEERING PROPERTIES FOR THE OUTPUT MICROWAVE MODES

In the following, we concentrate on the correlations of the output microwave modes. As is well known, quantum steering is inherently asymmetric and intermediates between entanglement and nonlocality, which is becoming a more useful resource for remote and secure quantum communication. Let us briefly discuss the criteria for bipartite steering. It is normally identified by criteria that are a natural generalization of those for entanglement. Here we focus on position-momentum measurements, as shown in [60]. Defining the two quadratures of each output mode as $ X_j^o( \omega ) = a_j^o( \omega ) + a_j^{o\dagger }( { - \omega } ),{Y_j} = - i[ {a_j^o( \omega ) - a_j^{o\dagger }( { - \omega } )} ] $, it is found that the criterion of EPR steering is expressed as

$${{\rm EPR}_\textit{jk}}\left( \omega \right) = {V_{\inf }}\left( {X_\textit{jk}^o} \right){V_{\inf }}\left( {Y_\textit{jk}^o} \right) \lt 1,\quad ( {k \Rightarrow j} ),$$
where the inferred variances are written as $ {V_{\inf }}\,({{O_\textit{jk}}}) = V( {{O_j}} ) - {V^2}( {{O_j},{O_k}} )/V( {{O_k}} ) $, with $ V( A ) = \langle {A^2}\rangle - {\langle A\rangle ^2} $, $ V( {A,B} ) = \langle AB\rangle - \langle A\rangle \langle B\rangle $. The condition $ {{\rm EPR}_\textit{jk}}\,(\omega ) \lt 1 $ means the steerability from output mode $ k $ to mode $ j $, and $ {{\rm EPR}_\textit{kj}}\,( \omega ) \lt 1 $ indicates the $ j \to k $ steering. It is worth mentioning that the above criteria can be necessary and sufficient for Gaussian states and measurements [61]. The smaller the values of $ {{\rm EPR}_\textit{jk}}\,( \omega ) $ spectra, the stronger the degrees of EPR correlations. Actually, the result of $ {{\rm EPR}_\textit{jk}}\,( \omega ) $ is not always equal to that of $ {{\rm EPR}_\textit{kj}}\,(\omega ) $. The great interest is in one-way EPR steering as the important part of quantum communication and quantum networks, where one of these is less than 1, while the other is greater than 1.

A. Asymmetric Steering of the Microwave Modes in the Presence of Inputs $ { \epsilon _{1,2}} $

Now we discuss the asymmetric quantum correlations of three output microwave modes by the method of alternative coherent inputs. In order to do so, we first focus on the zero-frequency $ {{\rm EPR}_\textit{jk}} $ spectra as a function of the ratio of the pump driving strengths $ { \epsilon _{1,2}} $, shown in Fig. 3, where the parameters are chosen in units of the resonator decay rate $ {\kappa _3} $, given by $ \chi = 0.01,{ \epsilon _2} = 100,{ \epsilon _3} = 0,{\kappa _1} = {\kappa _2} = {\kappa _3} = 1 $ (a) and $ {\kappa _1} = {\kappa _2} = 0.75,{\kappa _3} = 1 $ (b). Obviously, the system exhibits asymmetric steering over the range of the coherent driving strengths. Under the condition of balanced decay rates, as shown in Fig. 3(a), the value of the $ {{\rm EPR}_{32}} $ spectrum (orange short dotted line) is less than one, and the $ {{\rm EPR}_{23}} $ spectrum (magenta short dashed line) has a value greater than one for the case of $ { \epsilon _1}/{ \epsilon _2} \lt 1 $, which means that one-way steering from resonator 2 to resonator 3 is possible. When the driving condition is satisfied $ 1 \lt { \epsilon _1}/{ \epsilon _2} \lt 1.457 $, it is found that the $ {{\rm EPR}_{31}} $ spectrum (green dashed-dotted line) has a value below one, and the result of the $ {{\rm EPR}_{13}} $ spectrum (blue dotted line) is larger than one. Thus, one-way $ 1 \to 3 $ steering can occur. It is interesting that there are remarkable changes in the steerability properties between resonators 1 and 2 at $ { \epsilon _1} = 1.457{ \epsilon _2} $. For $ { \epsilon _1}/{ \epsilon _2} \lt 1.457 $, the results of $ {{\rm EPR}_{12}} $ (black solid line) and $ {{\rm EPR}_{21}} $ (red dashed line) are both less than one, and so two-way steering occurs between resonators 1 and 2. With an increase in the ratio $ { \epsilon _1}/{ \epsilon _2} $, i.e., $ { \epsilon _1}/{ \epsilon _2} \gt 1.457 $, there is asymmetric steering from resonator 1 to resonator 2. Therefore, the switching from symmetric to asymmetric steering of microwave photons from the two resonators can be achieved by adjusting the driving inputs $ { \epsilon _{1,2}} $.

 figure: Fig. 4.

Fig. 4. Positive-frequency $ {{\rm EPR}_\textit{ij}} $ spectra versus the frequency $ \omega $ under the condition of $ { \epsilon _1} = 50 $; other parameters the same as those in Fig. 3(a).

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It is meaningful to observe the effect of unbalanced loss rates on the steering properties. As shown in Fig. 3(b), the degrees of the EPR correlations can be enhanced in the system according to Reid’s inequalities. Meanwhile, the unbalanced loss rates also play an important role in turning the direction of the asymmetric EPR steering. For the case of $ { \epsilon _1}/{ \epsilon _2} \lt 0.727 $, it is found that there are $ {{\rm EPR}_{12}} \lt 1 $ and $ {{\rm EPR}_{21}} \gt 1 $, and resonator 1 can be steered by the measurement made at resonator 2. For the case of $ 0.727 \lt { \epsilon _1}/{ \epsilon _2} \lt 1 $, the value of $ {{\rm EPR}_{23}} $ is greater than one, and the value of $ {{\rm EPR}_{32}} $ is smaller than one, and the one-way steering occurs from resonator 2 to resonator 3. When $ 1 \lt { \epsilon _1}/{ \epsilon _2} \lt 1.17 $, the asymmetric steerability appears between microwave modes 1 and 3. In a word, the simpler method is used to obtain asymmetric steering as we expected, where the steering direction is indeed tunable by properly changing the driving strengths $ { \epsilon _{1,2}} $.

Moreover, we plot the positive-frequency spectra of the EPR correlations with the fixed pump inputs in Fig. 4, where the parameters are $ { \epsilon _1} = 50,{ \epsilon _2} = 100 $, and the other parameters are the same as those in Fig. 3(a). Obviously, asymmetric EPR steering occurs between resonators 2 and 3 due to $ {{\rm EPR}_{23}} $ being larger than one and $ {{\rm EPR}_{32}} $ less than one. The values of $ {{\rm EPR}_{12}} $ and $ {{\rm EPR}_{21}} $ are simultaneously less than one over a broad range, and two-way steering can be observed for resonators 1 and 2. On the contrary, no steering correlation exists between resonators 1 and 3 with the choice of the present parameters. Remarkably, it is found that the degrees of EPR correlations are the largest at zero frequency.

 figure: Fig. 5.

Fig. 5. EPR correlations versus the driving ratio $ { \epsilon _1}/{ \epsilon _3} $ in the absence of $ { \epsilon _2} $. The parameters are chosen as $ {\kappa _1} = {\kappa _2} = {\kappa _3} = 1 $ and $ \chi = 0.01,{ \epsilon _3} = 100 $.

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 figure: Fig. 6.

Fig. 6. Positive-frequency spectra of the $ {{\rm EPR}_\textit{ij}} $ correlations as a function of the frequency $ \omega $, with the parameters $ {\kappa _1} = {\kappa _2} = 0.75,{\kappa _3} = 1,\chi = 0.01, $ $ { \epsilon _2} = 0,{ \epsilon _1} = 50,{ \epsilon _3} = 100 $ (a) and $ { \epsilon _1} = 100,{ \epsilon _3} = 50 $ (b).

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B. Asymmetric Steering of the Microwave Modes for Inputs $ { \epsilon _{1,3}} $

Next, we detail the dependence of asymmetric correlations on different coherent inputs via considering the driven modes $ {a_1} $ and $ {a_3} $. The evolution of $ {{\rm EPR}_\textit{jk}} $ as a function of the ratio $ { \epsilon _1}/{ \epsilon _3} $ is shown in Fig. 5, where the parameters are chosen as $ {\kappa _1} = {\kappa _2} = {\kappa _3} = 1,\chi = 0.01,{ \epsilon _2} = 0,{ \epsilon _3} = 100 $. It is interesting that there is a switching (at $ { \epsilon _1} = 1.3{ \epsilon _3} $) with respect to the direction of the EPR steering between resonators 1 and 2 and between resonators 1 and 3. For the case of $ { \epsilon _1}/{ \epsilon _3} \lt 1.3 $, the pair of resonators 1 and 3 exhibits asymmetric steering from resonator 1 to 3. At the same time, two-way steering occurs between resonators 1 and 2. However, the $ 1 \to 2 $ steerability exists for the other case of $ { \epsilon _1}/{ \epsilon _3} \gt 1.3 $, where there is symmetric steering for the pair of resonators 1 and 3, whereas there is no steering phenomenon found between resonators 2 and 3 in the system.

Now we concentrate on the case of unbalanced loss rates and plot the positive-frequency spectra of the EPR correlations in Fig. 6, where the parameters are chosen as $ {\kappa _1} = {\kappa _2} = 0.75,{\kappa _3} = 1,\chi = 0.01, $ $ { \epsilon _2} = 0,{ \epsilon _1} = 50,{ \epsilon _3} = 100 $ (a), and $ { \epsilon _1} = 100,{ \epsilon _3} = 50 $ (b). In Fig. 6(a), it can be seen that resonators 1 and 2 exhibit two-way steering in the whole frequency range. At the same time, asymmetric steering occurs between resonators 1 and 3 in the low-frequency range, i.e., the value of $ {{\rm EPR}_{31}} $ is less than one and that of $ {{\rm EPR}_{13}} $ is greater than one. In Fig. 6(b), we can observe that there are more dynamical features with the larger pump inputs $ { \epsilon _1} $. Obviously, there is a great change of the subsystem involving resonators 1 and 2. The $ 1 \to 2 $ one-way steering exists for the low frequency. With increasing frequency, the quantum steering exhibits oscillation from vanishing to appearance. Furthermore, one-way steering occurs again, and for high frequency, two-way steering also appears. In addition, there is a change from symmetric steering to asymmetric steering between resonator 1 and resonator 3. Here a simple and promising way is applied to obtain one-way EPR steering via combining the coherent inputs and dispersive interactions between the circuit qubit and resonators, which may be helpful for practical applications.

Before concluding, we compare asymmetric EPR steering with recent work on quantum steering in a pulsed hybrid opto–electro–mechanical system [26]. The superconducting system including a fluxonium qubit and three separately driven resonators is under consideration, and the one-way steering of output microwave photons is studied in view of the asymmetric coherent driving strengths. Moreover, our scheme is based on dispersive electrical-dipole interactions, and nondegenerate parametric down-conversion occurs among three resonators, which is responsible for the generation of quantum steering. In comparison, the optomechanical and electromechanical couplings via radiation pressure in the scheme [26] are applied to establish both beam splitter and parametric types of interaction among three modes with the help of red-detuned and blue-detuned laser pulses. In addition, the directionality of steady-state one-way steering can be easily controlled via adjusting the ratio of driving intensities to two modes among three modes in the present scheme. The dynamical behavior of quantum steering and entanglement has been investigated, and the influence of the squeezing parameter on the amount of tripartite steering has also been examined [26].

4. CONCLUSION

In conclusion, the asymmetric steering of microwave photons from separated superconducting resonators has been discussed in the circuit system, where a single fluxoninum qubit and three superconducting resonators are involved. Based on the dispersive interactions between an artificial atom and three resonators with respect to the different transitions, nondegenerate parametric down-conversion occurs among three microwaves by adiabatically eliminating the atomic variables of the artificial atom. With the help of asymmetric pumping inputs, one-way steering can be achieved between two modes of three resonators. Moreover, the direction of steering can be effectively controlled via simply adjusting the driving strengths. The present scheme may expand the range of possibilities for microwave steering preparation in solid-state systems and provide a feasible way for remote secure quantum communication.

Funding

National Natural Science Foundation of China (11565013, 11704287, 11775190, 11905064).

Disclosures

The authors declare no conflicts of interest.

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Figures (6)
Fig. 1.
Fig. 1. (a) Transition frequencies of energy levels versus the flux bias $ {\Phi _{{\rm ext}}}/{\Phi _0} $ for the fluxonium qubit. (b) Charge matrix elements as a function of the flux bias $ {\Phi _{{\rm ext}}}/{\Phi _0} $.

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Fig. 2.
Fig. 2. (a) Model for the coupling of a single fluxonium qubit with three superconducting resonators. (b) Corresponding transitions coupled to the different modes.

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Fig. 3.
Fig. 3. Spectra of EPR steering as a function of the ratio of driving strengths, with the parameters $ \chi = 0.01,{ \epsilon _2} = 100,{ \epsilon _3} = 0,{\kappa _1} = {\kappa _2} = {\kappa _3} = 1 $ (a) and $ {\kappa _1} = {\kappa _2} = 0.75,{\kappa _3} = 1 $ (b).

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Fig. 4.
Fig. 4. Positive-frequency $ {{\rm EPR}_\textit{ij}} $ spectra versus the frequency $ \omega $ under the condition of $ { \epsilon _1} = 50 $; other parameters the same as those in Fig. 3(a).

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Fig. 5.
Fig. 5. EPR correlations versus the driving ratio $ { \epsilon _1}/{ \epsilon _3} $ in the absence of $ { \epsilon _2} $. The parameters are chosen as $ {\kappa _1} = {\kappa _2} = {\kappa _3} = 1 $ and $ \chi = 0.01,{ \epsilon _3} = 100 $.

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Fig. 6.
Fig. 6. Positive-frequency spectra of the $ {{\rm EPR}_\textit{ij}} $ correlations as a function of the frequency $ \omega $, with the parameters $ {\kappa _1} = {\kappa _2} = 0.75,{\kappa _3} = 1,\chi = 0.01, $ $ { \epsilon _2} = 0,{ \epsilon _1} = 50,{ \epsilon _3} = 100 $ (a) and $ { \epsilon _1} = 100,{ \epsilon _3} = 50 $ (b).

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Equations (21)

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H f = 4 E C N 2 − E J cos ⁡ ( φ − 2 π Φ e x t / Φ 0 ) + 1 2 E L φ 2 ,
H 0 = ∑ j = 1 − 3 ω j σ jj + v j a j † a j ,
V = g 1 a 1 σ 21 + g 2 a 2 σ 32 + g 3 a 3 σ 31 + H . c . ,
S = g 1 ∗ Δ 1 a 1 † σ 12 + g 2 ∗ Δ − Δ 1 a 2 † σ 23 + g 3 ∗ Δ a 3 † σ 13 − H . c . ,
H = H 0 + λ V + λ [ H 0 , S ] + λ 2 [ V , S ] + λ 2 2 [ S , [ S , H 0 ] ] + λ 3 2 [ S , [ S , V ] ] − λ 3 6 [ S , [ S , [ S , H 0 ] ] ] + O ( λ 4 ) .
H e f f = H 0 + 1 2 [ V , S ] + 1 3 [ [ V , S ] , S ] = H 0 e f f + χ ( e i ϕ a 1 a 2 a 3 † + e − i ϕ a 1 † a 2 † a 3 ) ,
H p = i ∑ j = 1 − 3 ϵ j ( a j † e − i v jeff t − a j e i v jeff t ) ,
H ~ = i ∑ j = 1 − 3 ϵ j ( a j † − a j ) + i χ ( a 1 † a 2 † a 3 − a 1 a 2 a 3 † ) .
ρ ˙ = − i [ H ~ , ρ ] + ∑ j = 1 − 3 κ j ( 2 a j ρ a j † − a j † a j ρ − ρ a j † a j ) ,
d α 1 d t = ϵ 1 − κ 1 α 1 + χ α 2 † α 3 + F α 1 , d α 1 † d t = ϵ 1 ∗ − κ 1 α 1 † + χ α 2 α 3 † + F α 1 † , d α 2 d t = ϵ 2 − κ 2 α 2 + χ α 1 † α 3 + F α 2 , d α 2 † d t = ϵ 2 ∗ − κ 2 α 2 † + χ α 1 α 3 † + F α 2 † , d α 3 d t = ϵ 3 − κ 3 α 3 − χ α 1 α 2 + F α 3 , d α 3 † d t = ϵ 3 ∗ − κ 3 α 3 † − χ α 1 † α 2 † + F α 3 † ,
( ϵ 1 − κ 1 α ¯ 1 ) ( κ 2 κ 3 + χ 2 α ¯ 1 2 ) 2 = κ 3 χ 2 ϵ 2 2 α ¯ 1 ,
α ¯ 2 = ϵ 2 κ 2 + α ¯ 1 ( κ 1 α ¯ 1 − ϵ 1 ) ( κ 2 κ 3 + χ 2 α ¯ 1 2 ) κ 2 κ 3 ϵ 2 ,
α ¯ 3 = ( κ 1 α ¯ 1 − ϵ 1 ) ( κ 2 κ 3 + χ 2 α ¯ 1 2 ) κ 3 χ ϵ 2 .
( κ 1 α ¯ 1 − ϵ 1 ) ( κ 2 κ 3 + χ 2 α ¯ 1 2 ) 2 = κ 2 χ 2 ϵ 3 2 α ¯ 1 ,
α ¯ 2 = ( κ 1 α ¯ 1 − ϵ 1 ) ( κ 2 κ 3 + χ 2 α ¯ 1 2 ) κ 2 χ 3 ,
α ¯ 3 = ϵ 3 κ 3 − α ¯ 1 ( κ 1 α ¯ 1 − ϵ 1 ) ( κ 2 κ 3 + χ 2 α ¯ 1 2 ) κ 2 κ 3 ϵ 3 .
d d t δ R → ( t ) = − A δ R → ( t ) + B d F → ,
S ( ω ) = ( A + i ω I ) − 1 D ( A T − i ω I ) − 1 ,
D ( 1 , 3 ) = D ( 3 , 1 ) = χ α ¯ 3 , D ( 2 , 4 ) = D ( 4 , 2 ) = χ α ¯ 3 ∗ .
A = [ κ 1 0 0 − χ α ¯ 3 − χ α ¯ 2 ∗ 0 0 κ 1 − χ α ¯ 3 ∗ 0 0 − χ α ¯ 2 0 − χ α ¯ 3 κ 2 0 − χ α ¯ 1 ∗ 0 − χ α ¯ 3 ∗ 0 0 κ 2 0 − χ α ¯ 1 χ α ¯ 2 0 χ α ¯ 1 0 κ 3 0 0 χ α ¯ 2 ∗ 0 χ α ¯ 1 ∗ 0 κ 3 ] .
E P R jk ( ω ) = V inf ( X jk o ) V inf ( Y jk o ) < 1 , ( k ⇒ j ) ,
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