Abstract
A superconducting circuit with a qubit and a resonator coupled via a twophoton interaction is considered. When the resonator is initially in a superposition of coherent states, optical tomography and quantum coherence dynamics are examined in the context of intrinsic decoherence. The results reveal that optical tomography is a good quantifier of the quantum coherence produced by the qubitresonator interaction. The effects of qubitresonator detuning and intrinsic decoherence on the dynamics of optical tomography distributions for coherent and even coherent states are investigated. The dynamics of optical tomography distributions are highly dependent on detuning and intrinsic decoherence. Our numerical simulations reveal that there is a relation between the optical tomography and the generated quantum coherence. When the qubitresonator detuning and intrinsic decoherence are augmented, the amplitude and intensity, as well as the structure of the optical tomography, change substantially.
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Introduction
A tomographic quantifier is a ubiquitous tool for estimating the physical properties of quantum states, and its procedure depends on the density matrix of the quantum state^{1}. The optical homodyne tomography is a monotonic relation between the density operator and the probability distributions of the coherent field states^{2}, which enables all conceivable information about the quantum states to be extracted^{3,4,5}. Therefore, many nonclassical light states can be characterized using the optical homodyne tomography. Optical tomography is a critical method for testing any quantum information processing implementation^{6,7}. It has been investigated theoretically and experimentally for neuralnetwork quantum states^{8}, maximally entangled states of beam splitter modes^{9}, qdeformed coherent states^{10}, and a single electron spin state associated with a nitrogenvacancy center^{11}. The optical homodyne tomography is used to analyze revivals in a Kerr medium^{12}.
More attention has been paid to realize artificial lightmatter interactions^{13} such as: superconducting circuits^{14,15} quantum dot^{16} trapped ions^{17}, to generate useful nonclassical effects such as coherence, squeezing, and quantum correlations. Superconducting systems, in particular, enable a significant flexibility in the design of superconducting artificial qubits interacting with resonators^{18}. Such systems have opened the door for a number of theoretical studies on theses artificial qubits and the nonlinear interactions^{19}. Twophoton transition has been predicted by GoeppertMayer^{20}. This effect has been widely utilized in fluorescence microscopy^{21}, and dichromatic laser pulses^{22}. Twophoton quantum Rabi model was implement based on proposed superconducting circuit in a solidstate device^{15}.
The potential of establishing quantum computing and information in the context of superconducting circuits has been investigated using Wigner tomography of coherent state superpositions and a fixedfrequency superconducting transmon qubit coupled to a waveguide cavity resonator^{23,24,25}. Recently, superconducting transmon qubits and two photonic qubits were utilized in experiments using quantum tomography to: realize an architecture of two coupled logical qubits^{26}, and to demonstrate controlledphase gates between two Errorcorrectable photonic qubits^{27}, as well as to reveal quantum correlations in bimode cavity^{28}.
Optical homodyne tomography distributions of a quantum state are highly vulnerable to decoherence and dissipation resources^{12} due to the strong relation between the nonclassically loss and the decoherence effect. In this paper, we investigate the dynamics of the proposed qubitresonator interactions using the Milburn motion equation, which governs the dynamics of the system in the presence of intrinsic decoherence^{29}. The intrinsic decoherence model has been used to investigate the dynamics of quantum information resources in numerous systems^{30,31,32,33}. The intrinsic decoherence was theoretically investigated^{29} and implemented in a linear array of coupled trappedion qubits^{34} without any explicit coupling to the surrounding environment. This form of decoherence comes into play in closed qubitresonator interactions. The intrinsic decoherence results in a quantum coherence loss without loss of energy. The dynamics of the system is described in this case by the Milburn equation. Others approaches are governed by Master equations, describing systemreservoir interactions (open quantum systems)^{35,36}, which lead naturally to decoherence as the information is immediately lost after evolution. The intrinsic decoherence differs from the dissipation which occurs in open quantum systems^{37,38,39} that interacts with the surrounding environment.
In various superconducting circuits^{40} and NitrogenVacancy Centers^{41}, the intrinsic decoherence processes have been realistically controlled. The intrinsic decoherence time in isolated quantum systems was investigated, and it was discovered that it is dependent on both the system size and the disorder strength^{34}. Quantum error detection reduces the influence of intrinsic decoherence in a superconducting circuit^{42}. The phase coherence in microcavity polariton condensate^{43} and Josephson qubits^{44} is controlled by intrinsic decoherence processes.
Previously, the dynamics of optical homodyne tomography were restricted to specific models^{9,12}, that ignoring the presence of decoherence and dissipation effects. Optical tomography and coherence dynamics in the context of a superconducting circuit are investigated in this paper. The resonant twophoton interaction of a qubit coupled to a resonator, whose state is initialized in a coherent state or a superposition of coherent states, models the relevant dynamics. In addition, the influence of qubitresonator detuning and intrinsic decoherence on the dynamics of the optical tomography distributions for the selected system’s states is the focus of this study. As compared to the optical tomography considered previously, here the optical tomography distribution is explored with the qubitresonator interaction in the presence of the intrinsic decoherence.
The paper is organized as follows: In the “Qubitresonator system with intrinsic decoherence model” section, the qubitresonator system with intrinsic decoherence model is presented with its analytical solutions. In “Optical tomography distribution, Optical tomography dynamics” sections, the dynamics of the optical tomography distribution for the generated timedependent resonator state is illustrated. “Conclusions” section is dedicated to the conclusion.
Qubitresonator system with intrinsic decoherence model
We consider a twophoton qubitresonator model that describes a qubit and a resonator coupled via twophoton transition. The model can be realized as a superconducting circuit consisting of a dcSQUID with two identical junctions inductively coupled to a superconducting flux qubit. The Hamiltonian of the twophoton qubitresonator with the quadratic coupling is given by (\(\hbar =1\))^{45}
\(\Phi _{DC}\) represents an externally applied static flux. \(\omega _{q}\), M, and \(I_{p}\) represent respectively the frequency, the mutual inductance, and the persistent current states of the qubitSQUID. \(\omega _{SQ}\) is SQUID mode frequency with the annihilation operator \({\hat{\psi }}\). \(0\rangle\) and \(1\rangle\) are the qubit ground and excited states. \({\hat{\sigma }}_{x}\) and \({{\hat{\sigma }}}_{z}\) design the Pauli qubit density matrices. Here, we focus on the SC regime, where the coupling strength is small in comparison to the qubit and cavity eigenfrequencies, but greater than all dissipation rates. In the SC regime, the counterrotating terms \({\hat{\sigma }}_{} {\hat{\psi }}^{2} + H.c\). and \({\hat{\sigma }}_{x}{\hat{\psi }}^{\dagger } {\hat{\psi }}\), rotating at frequencies \(2\omega _{q} + \omega _{SQ}\) and \(\omega _{q}\), respectively, can then be neglected. The Hamiltonian can be written as^{45}
\(\lambda =\frac{\pi }{4} \tan (\frac{\pi \Phi _{DC}}{\Phi _{0}}) \frac{M I_{p}}{\Phi _{0}}\omega _{SQ}\). \(\delta =\omega _{q}2\omega _{SQ}\) represents the qubitresonator detuning, for the offresonance case \(\delta \ne 0\). In the spanned space of qubitresonator states \(\{\varpi _{1}\rangle =1, n \rangle , \varpi _{2}\rangle =0, n+2\rangle \}\), the dressedstates of the Hamiltonian of Eq. (2) are given by
where
The energy eigenvalues have the following expressions
For the ground state energy \(0,0\rangle\), the energy eigenvalue is \(V_{0}=\frac{\delta }{2}\).
Here, the time evolution of the qubitresonator interactions and the decoherence intrinsic effect on the dynamics of the qubitresonator is explored by the Milburn’s equation^{29},
where \(\gamma\) denotes intrinsic decoherence, in which quantum coherence is lost naturally as the system evolves. This intrinsic decoherence process has been introduced as propose a modification of the standard quantum mechanics evolution, built on the hypothesis that the qubitresonator system dynamics under a stochastic succession of identical unitary transformations for sufficiently short time steps rather than a continuous unitary evolution^{29}. In the intrinsicdecoherence models, the offdiagonal elements of the density matrix are intrinsically suppressed in the energy eigenstate basis, consequently, intrinsic decoherence is realized without the dissipation.
To find a particular solution for the differential Milburn’s equation, we assumed that the initial qubit state is in the excited state \(1\rangle \langle 1\). While the initial resonator state is a superposition of the two “opposite” coherent states \(\pm \alpha \rangle\) (these two superposed states are \(\pi\) phased), the initial resonator reduced density matrix is given by
A designs the normalization factor, and the photon distribution function is given by
The values \(r= 0, 1, 1\) correspond to a coherent state, an even coherent state, and an odd coherent state, respectively.
To solve the Eq. (6), the initial qubitresonator state is reexpressed by using the dressedstates and the energy eigenvalues of the Hamiltonian of Eq. (2) that satisfies the condition of the eigenvalueproblem: \({\hat{H}}S_{k}^{n}\rangle =V^{n}_{k}S_{k}^{n}\rangle (k=1,2)\), \(V^{n}_{k}\) correspond to the eigenvalues. Therefore, using the Eq. (6), the density matrix dynamics of the dressed states, \(S^{mn}_{kl}(t)=S_{k}^{m}\rangle \langle S_{k}^{n}\), which have the following expression
where \(D^{mn}(t)=e^{\frac{\gamma }{2}(V^{m}_{k}V^{n}_{l})^{2}t}\) represents the intrinsic decoherence term. Then, using the initial state and the Hamiltonian (2), the particular solution of Eq. (6) is then
where \(l=C\) for the initial coherent state and \(l=Ec\) for the initial even coherent state with the abbreviations:
The study of the dynamics of the optical tomography is mainly built on the resonator reduced density matrix,
with
The optical tomography dynamic of the produced field states will be explored in the next section based on the reduced cavity density matrix \({\hat{\rho }}^{f}(t)\).
Optical tomography distribution
The optical tomography for the timedependent resonator state is calculated by the resonator reduced density matrix \({\hat{\rho }}_{R}(t)\) as^{46}:
where \({\hat{\Lambda }}_{\theta }\rangle\) designs the eigenstate of the homodyne quadrature operator
with eigenvalue \(\Lambda _{\theta }\) and the local oscillator phase \(\theta\). \({\hat{\psi }}\) and \({\hat{\psi }}^{\dagger }\) are the singlemode field operators. Equation (14) verifies the normalization relation
After writing the singlemode cavity field density matrix \({\hat{\rho }}_{R}(t)\) of Eq. (13) in number state representation as: \({\hat{\rho }}_{R}(t)=\sum \limits _{m,n =0}^{\infty }\rho ^{mn}_{R}m\rangle \langle n\), we use this representation into (14). Therefore, the expression of the optical tomography is then
with
where \(H_{n}(.)\) indicates the order n Hermite polynomial. Equation (17) describes the time dynamics of optical tomography in a system including the qubitresonator interaction described by the Eq. (2). In the sections that follow, we will examine the temporal evolution of optical tomography for various types of initial coherent states in the presence of intrinsic decoherence.
Optical tomography dynamics
In Fig. 1, the optical tomography contour \(T(\Lambda _{\theta }, \theta , t=0)\) of the coherent states (CSs), the even coherent state (ECS) and the odd coherent (OCS) are illustrated. Figure 1a shows that the optical tomography of the two “opposite” coherent states \(\pm \alpha \rangle\). The optical CStomography of \(\alpha \rangle\) behaves similar to sinecurve function with \(2\pi\)periodic and constant amplitudes in the \(\theta \Lambda _{\theta }\)plane. In the case, for the increase of the local oscillator phase angle (\(\theta \rightarrow 2\pi\)), the optical CStomography starts from the point \((\Lambda _{\theta }, \theta )= (\alpha \sqrt{2}, 0)\simeq (4.2426,0)\) increasing towards the point \((4.2426, \pi )\), and then it decreases towards the point \((4.2426, 2\pi )\). While the optical CStomography of \(\alpha \rangle\) behaves as cosinecurve function. The optical tomographies of \(\pm \alpha \rangle\) have two opposite distributions which are depending on the coherent intensity \(\alpha \).
Figure 1b illustrates that the optical ECStomography contour is appeared as intersected regular cosinecurve paths. The ECStomography is also depicted for the superposition of the two “opposite” coherent states \(\pm \alpha \rangle\). The InsertColobar shows that the tomography distribution has constant amplitudes in the \(\Lambda _{\theta }\theta\)plane except in the interference regions. The distribution of the amplitudes is different and they are enhanced as the ECStomography tends to its center points \((\theta , \Lambda _{\theta })=(\frac{1}{2}(2n+1),0) (n=0, 1)\). The results of the Fig. 1b,c show that the optical tomographies of the ECS and OCS differ only at interference regions. This difference depends on the initial coherent field intensity. It was proven that, for large field intensities, the nonclassical properties of the even and odd coherent states are approximatively the same^{9,47}. Therefore, our investigations are focused on the coherent states and the even coherent state.
To investigate the relation between the optical tomography and the resonator quantum coherence dynamics, the time evolution of the resonator field entropy under the decoherence and detuning effects are shown in Fig. 1d. The resonator entropy is calculated by:
where \(\pi ^{i}_{n}\) represent the eigenvalues of the resonator reduceddensity matrix \({\hat{\rho }}_{R}(t)\) of Eq. (13). Solid curves of Fig. 1d indicates that the qubitresonator interactions annihilate the coherence of the resonator field (i.e., the generated mixedness of the resonator increases). The subfigure of Fig. 1d illustrates the oscillatory behaviour of the even coherent state during the first \(\pi\)period. Note that at \(\lambda t= \frac{1}{2}\pi\), the resonator can be in a mixed state if its initial state was in the coherent state, and it can be in a pure state if it is initially in the even coherent state. The regularity of the generated resonator mixedness is disappeared after considering the qubitresonator detuning effect. The mixedness is enhanced with more oscillations. Dot and dashdot curves of Fig. 1d show that the resonator entropy grows due to the intrinsic decoherence, in other words, the generated resonator mixedness is enhanced.
Figure 2 illustrates the dynamics of the optical tomography of the coherent state \(\alpha \rangle\) under the qubitresonator interactions in the resonant case \(\delta =0\). At \(t =\frac{1}{4}\pi\) (the resonator state has partial coherence), the optical CStomography evolves to two symmetrical distribution branches around the axis \(\theta =\pi\), see Fig. 2a. The maximum values of the interference regions are larger than those of the initial distribution of Fig. 1a. For \(t =\frac{1}{2}\pi\), the generated optical CStomography has two irregular sinusoidal paths, which are similar to the shape of the initial distribution of the even coherent states. The optical tomography evolution can be used as a good indicator of the field resonator mixedness generated by the qubitresonator interactions.
This result is confirmed for the optical CStomography displayed at \(\lambda t =\pi\) (the resonator state is in a pure state, see Fig. 1d). The generated resonator state presents optical tomography with cosinecurve function which is similar to of the coherent state \(\alpha \rangle\), see Fig. 1a. This means that the generated pure resonator state is the coherent state \(\alpha \rangle\).
Figure 2d shows the dependence of the optical tomography of the coherent state \(\alpha \rangle\), generated at \(\lambda t =\pi\), with the detuning \(\delta =10\lambda\). The amplitudes and the intensity of the optical tomography are enhanced due to the increase of the qubitresonator detuning.
Figure 3, displays the intrinsic decoherence effect \(\gamma =0.05\lambda\) on the generated optical CStomography at the different times \(\lambda t= \frac{1}{2}\pi\) and \(\lambda t=\pi\) of Fig. 2b,c. The intrinsic decoherence effect leads to: (1) the amplitudes and the contour intensity of the CSoptical tomography are reduced at different times. At \(\lambda t= \frac{1}{2}\pi\) and \(\lambda t=\pi\), the sizes of the distribution shape change due to the intrinsic decoherence effect.
In the case of resonance, we observe that optical tomography for a periodic regular dynamics which could be considered as an indicator of the degree of the pureness of the fieldresonator states. The regularity of the generated optical tomography disappears as the qubit resonator detuning increases, i.e., the probability generation another pure coherent resonator states, due to the qubitresonator interactions is very low. The intrinsic decoherence increases the mixedness of the generated filedresonator states, leading the interference optical tomography regions to vanish.
In the Fig. 4, for the qubitresonator interaction, the dynamics of the optical tomography contour of the initial even coherent state at different times \(\lambda t= \frac{1}{4}\pi\), \(\lambda t= \frac{1}{2}\pi\) and \(\lambda t=\pi\) is plotted. For the resonant case \(\delta =0\) and \(\lambda t= \frac{1}{4}\pi\), the contour plot illustrates that the optical tomography has more picks (which appear at high intensity) and more intersected symmetric as well as regular path. The amplitudes and the contour intensity of the generated optical ECStomography are smaller than those of the initial even coherent state, see Fig. 1b. At \(\lambda t= \frac{1}{2}\pi\) (see Fig. 4b), the resonator reduced matrix presents a new shape for the the optical ECStomography with two regular sinecurve paths. It is similar to the initial even coherent state of Fig. 1b as it is expected from the dynamics of the resonator entropy, see the subfigure of Fig. 1d. To prove the relation between the optical ECStomography and the resonator field entropy, the ECStomography is displayed at the time \(\lambda t=\pi\) (in which \(S_{R}(t)=0\)). It should be noted that the produced ECStomography can be roughly restored to the initial distribution.
Figure 4d confirms that the qubitresonator detuning modifies the amplitudes, frequency, and shapes of the optical ECStomography.
Figure 5 shows the intrinsic decoherence effect on the tomography dynamics of the coherent states at different times \(\lambda t= \frac{1}{2}\pi\) and \(\lambda t=\pi\) (Fig. 4b,c). The amplitudes and the intensity of the ECStomography oscillations are reduced as well as the shape of the distributions is modified due to the intrinsic decoherence effect.
From Figs. 1 and 2, we note that the amplitudes, frequency, and shapes of the optical tomography of \(\alpha \rangle\) are timedependent. Therefore, qubitresonator interaction allows to prepare particularly different interesting states. The optical tomography can not be protected by using the coherent state as an initial state. The generated ECStomography may be restored to the initial distribution of the even/odd coherent state, as shown in Figs. 1 and 4. As a result, we deduce that the optical tomography protected information is reliant on the initial coherent/evencoherent resonator state.
Conclusions
We have explored the dynamics of a quantum system formed by a qubit and a resonator coupled by a twophoton interaction. We have analyzed the optical tomography and quantum coherence dynamics of the resonator state when it is initially in a superposition of coherent states in the presence of intrinsic decoherence. The results demonstrate that owing to qubitresonator interactions, there is a connection between the optical tomography and the generated resonator quantum coherence. We have investigated the effects of qubitresonator detuning and intrinsic decoherence on the dynamics of optical tomography distributions for initial coherent and even coherent resonator states. When the qubitresonator detuning and intrinsic decoherence are augmented, the amplitude and intensity, as well as the structure of the optical tomography, change substantially.
Data availability
The datasets used and/or analyzed during the current study available from the corresponding author (A.B.A.M.) on request.
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Acknowledgements
The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number (IFPSAU2021/01/17712).
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A.B. A.M. and H.E. prepared all Figures and performed the mathematical calculations. A.B.A.M. wrote the original draft. H.E. reviewed and edited the draft. All authors have read and agreed to the published version of the manuscript.
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Mohamed, A.B.A., Eleuch, H. Optical tomography dynamics induced by qubitresonator interaction under intrinsic decoherence. Sci Rep 12, 17162 (2022). https://doi.org/10.1038/s41598022213484
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DOI: https://doi.org/10.1038/s41598022213484
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