Introduction

Dressed atom formalism was developed in 1969 by Cohen-Tannoudji and Haroche1 to explain the behavior of atoms exposed to radio-frequency fields described in terms of photons2. In fact, the Floquet quasienergy diagram is equivalent to the fully quantized dressed-atom picture in the limit of strong fields3. Generalization of the Floquet theory for non-perturbative treatment of infinite-level systems, including both bound and continuum states, was first introduced by Chu and Reinhardt4 in 1977. Dressed superconducting qubits5,6, have been theoretically studied7, and experimentally demonstrated8,9. For strongly driven superconducting qubits, the Floquet formalism can describe the ac Stark level shift and power broadening for multiphoton resonance processes, which appear beyond rotating-wave approximation (RWA)10. Also, the RWA is not applicable in the ultrastrong-coupling (USC) regime11,12. This new regime of cavity quantum electrodynamics (QED), where the coupling rate becomes an appreciable fraction of the unperturbed frequency of the bare systems, has been experimentally reached in a variety of solid state systems13,14,15,16. In RWA the excitation-number-nonconserving processes or virtual transitions have been excluded in calculations. Therefore the Jaynes-Cummings model cannot describe higher order resonant transitions via intermediate states connected by counter-rotating terms17,18.

The main purpose of this paper is to provide a generalized and systematic graph theoretical approach19,20, motivated by dressed states picture of Floquet theory, to analyze a generic circuit QED system consisting of a two-level qubit coupled to a single-mode resonator in any arbitrary coupling strength regime. We will demonstrate the results from the small normalized coupling rate to deep strong coupling (DSC) regime, where the coupling strength is compatible or larger than the oscillator frequency21. By considering the counter-rotating terms, this graph-theoretical method will allow us to represent the virtual transitions, which are proven to be essential when the number of excitations in the cavity-emitter system is no longer conserved13.

Here, the proposed graph theoretical construction of the interacting quantum system should not be connected to a totally different concept of quantum graph22. Quantum graphs are usually introduced either through a differential operator acting on the functions defined on the edges of a graph or through directly specifying the scattering matrices at the vertices of the graphs23. An important relevance of the proposed graph product scheme is its natural connection to the so-called Floquet Hilbert space24. One can assign a natural Hilbert space to each graph, G, by considering all square summable, complex-valued functions on the vertex set VG = V(G) Such Hilbert space is usually denoted by . If G is 2-colored, then the coloring operator CG defines a grading on the Hilbert space , which becomes a graded Hilbert space (also called a super Hilbert space). The Floquet Hilbert space is defined as the Hilbert space tensor product

here is the spin Hilbert space, and is the Fourier Hilbert space. The spin Hilbert space for a two-level system is 2-dimensional with basis , and the Fourier Hilbert space is infinite dimensional with basis 24,25. Consequently, we can naturally identify with and with with and denoting the two node complete graph and infinite path graph, respectively. The dressed states

form a basis for the Floquet Hilbert space, and correspond to the vertices of any type of product between K2 and P ∞. Moreover, when K2 is colored, the spin Hilbert space will be graded by the coloring matrix , hence the resulting Floquet Hilbert space will be graded by considering the graded tensor product,

The picture provided by Floquet states are closely similar to the n-photon, or Fock states. Fock states are of special importance because quantum behavior in an oscillator is most obvious in these states26,27. They are frequently used in theoretical calculations, and form the basis of quantum computation and communications28. To see the resemblance between these pictures, we introduce the Hamiltonian for the quantum system as HQ = Ha + Hf + Hi, where Ha is the Hamiltonian of the atom, Hf is the Hamiltonian of the field, and Hi is the interaction Hamiltonian21,29. There are only two differences. In HQ, the index n runs from zero to infinity, but here it runs from −∞ to ∞. Also the off-diagonal elements of HQ depend on n (e.g. proportional to if Hi is proportional to the annihilation operator), whereas those of our matrices do not ref. 29.

Mathematical Construction and Theoretical Details

Throughout this paper, the graphs are directed in general, but they don’t have loops (i.e., edges that connect a vertex to itself). The reported graphs are all weighted. By a weighted graph , we mean a directed graph G with a countable vertex set V(G) and edge set together with the additional labeling of the vertices and edges by complex numbers (called weights)30.

Decomposition of a Weighted Graph

Let (vi) be a certain enumeration of the vertices of G. Denote the edge weight assigned to the directed edge (vi, vj) by and the vertex weight assigned to vi by . We make the convention that if and only if This defines the weighted adjacency matrix by30,31,

The weighted adjacency matrix is Hermitian if and only if all the vertex weights are real and the edge weights satisfy. We are mainly interested in Hermitian adjacency matrices that correspond to quantum observables, chief among them being the Hamiltonian of our quantum systems. When the adjacency matrix is real and symmetric, we will simply consider an undirected graph with at most one edge between any two vertices weighted by . In particular, for an unweighted graph we can assume that for all vertices, but any nonzero edge weight is equal to 1. Such a weight is said to be trivial, which obviously gives rise to the (unweighted) adjacency matrix of the graph. There is a one-to-one correspondence between weighted graphs and weighted adjacency matrices, so one can define a weighted graph given a symmetric matrix as its adjacency matrix31.

Each weighted adjacency matrix can be decomposed into the sum of a diagonal matrix encoding the vertex weights and an off-diagonal matrix encoding the edge weights. In other words, we write , where . We call this the weight decomposition of . In particular, if is trivial, and .

Colored Weighted Graphs and Their Products

Coloring vertices (or edges) of a graph is another method of labeling graphs. In this paper, we only consider the simplest case of vertex 2-coloring, namely, when all vertices are assigned by either of the two certain colors, and edges remain uncolored. A 2-coloring is said to be proper when every edge in the graph connects vertices with different colors, in which case the graph is called bipartite. Although properness is usually very useful in applications, we do not assume it for our graphs, unless it becomes a necessity. Suppose is a weighted graph with a certain 2-coloring of its vertices, say, blue and red. The information about coloring of vertices can be stored in a diagonal matrix (coloring matrix) denoted by

where ci = 1 if the vertex vi is blue, and ci = −1 if the vertex vi is red. Based on these definitions, a colored weighted graph is a triple If the graph is uncolored, we simply assume that the coloring matrix CG is the identity matrix. Let and be two colored weighted graphs. Our goal is to define the colored weighted version of the well-known products (such as direct, Cartesian, and strong) between the two graphs. Let denote any such weighted product, e.g., for direct, Cartesian, and strong product, respectively30. In all these cases, defines a coloring on the vertex set of the product graph. Here, the notation ⊗ stands for the Kronecker product of matrices31.

We now define the adjacency matrix of the colored weighted direct product as,

where is the identity matrix of the size of graph Gi.

The adjacency matrix of the colored weighted Cartesian product is introduced as,

And finally, we define the adjacency matrix of the colored weighted strong product as,

Eq. (8) implies that

Results and Discussion

Although the method can be applied to any number of multilevel systems, here, we limit our construction to a single two-level case. We model the two-level system as a two-dimensional spin Hilbert space with orthonormal basis of , and , where α and β are real valued energies of the two states. Let K2 be a colored weighted graph with the weighted adjacency matrix

and with a vertex coloring defined as follows: the vertex weighted by α is red (so −1) and the vertex weighted by β is blue (so +1). We consider the interaction of this two-level system with a single mode oscillator. In general, the oscillating interaction connecting the states of the two-level system, ε(t), can be expanded with the Fourier components 10,29,

where and is the field amplitude corresponding to . We model such discretized oscillating external field by a path graph, P∞. The vertices are assigned with the weights of , where .

The significance of our definition of colored-weighted graph products will be demonstrated by constructing the time-invariant Floquet Hamiltonian of a variety of QED systems. Without loss of generality, we consider a system consisting of a two level qubit coupled to a single-mode resonator in any arbitrary coupling strength regime. In the following we will show three case scenarios where the Floquet Hamiltonian HF in an appropriate basis for the different physical models can be obtained as the weighted adjacency matrix of , for either the direct, Cartesian, or strong product:

Transverse coupling: Colored-weighted direct product

Natural atoms couple with electromagnetic fields at transverse mode due to the well-defined inversion symmetry of the potential energy32. Within the Bloch representation33, the time-dependent Hamiltonian of such two-level atom with transverse coupling is given by29,

where , and . , and are Pauli matrices. Here, is the oscillating interaction connecting (through off-diagonal coupling) these states with a matrix element , where b is the real-valued field amplitude, and ω is the main angular frequency. Atomic units are used throughout this paper. We set . Following Eq. (11), for this case, all the Fourier coefficients, , vanish, except for . We assign equal weights of b to all the edges of the graph P∞. As mentioned before, we can now split the adjacency matrix of the oscillating field, which is modeled by the graph P∞, into the diagonal and off-diagonal terms34. In a 2 × 2 representative form,

by definition, Eq. (6), the direct product P∞ × K2 gives the structure appeared in Fig. 1(b). The edges on this new graph product is weighted by b. The adjacency matrix, Fig. 1(c), for Eq. (6) generates the same structure as one obtains by applying the Floquet approach29.

Figure 1
figure 1

Transverse coupling.

(a) Illustration of the P∞(is presented) and K2 graphs before the direct production. P∞ models the discrete monochromatic laser field, and K2 represents the two-level system. (b,c) The schematic of the direct product graph, and its corresponding adjacency matrix are given, respectively. (d) The contour plot of the transition probability with respect to the coupling strength, b, and the energy separation between the two states. (e) Four branches of the quasienergies of the interacting system, and (f ) Transition probability as a function of energy separation for the two-level system driven by a single mode laser field with , that is modeled by graph direct product P∞ × K2.

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After solving the eigenvalue problem for the colored weighted adjacency matrix of the direct product, the time-averaged transition probability between and can be calculated as the probability to go from a single initial vertex on the product graph to a final vertex, summed over all the paths containing the intermediate vertices in the product graph. This can be numerically calculated as ref. 29.

where n (and also j) is the Fourier index that runs over all the integers, γ = α, β, are the dressed states in the Floquet Hilbert space , are the quasienergy eigenvalues of the product adjacency matrix, and are the corresponding normalized eigenvectors. This is associated with the probability of finding the excited state of the qubit in the experiment. Figure 1(d) shows the contour map of the transition probability with respect to the coupling strength due to the external field, b, and the energy separation |α − β| between the two states of K2. Figure 1(e,f) present the qusienergy and the transition probability for the case when . This result is in agreement with the results reported before29. As b increases, each resonance in Fig. 1(f) broadens and shifts toward smaller values of separation energy.

One can easily see the similarity between the schematic diagram of the direct product presented in Fig. 1(b) with the Hilbert space splitting in two unconnected subspaces or parity chains, P = +1, −121. As schematically in Fig. 1(b), the direct product only allows odd-walks transitions in the graph. This is demonstrated, in Fig. 1(f), as the appearance of the transition peaks near the odd values of the separation energy. It can also be easily confirmed that these paths are only the passages responsible for the odd-multiphoton resonances. For example, removing all paths of length 3 from the product graph - by removing the corresponding edge weights from the adjacency matrix and the associated terms in Eq. (15) - eliminates the transition probability peak near the energy separation of 3 in Fig. 1(f).

Longitudinal coupling: Colored-weighted Cartesian product

Since the potential energy for superconducting qubits can be tuned, the inversion symmetry for these artificial atoms can be broken and multiphoton processes can be observed32. The existence of the longitudinal coupling between superconducting qubits and applied magnetic fields were shown35, when the inversion symmetry of the potential energy of the superconducting qubit is broken. When a superconducting flux qubit is driven by a strong ac field, the time dependent Hamiltonian, which describes the longitudinal coupling is given by

where 36. Here, the parameter Δ is the tunnel splitting and ε0 is the detuning energy proportional to the dc flux bias. A is the amplitude of the ac field that is parameterized in the energy unit and is proportional to the ac flux bias37. In this case, the adjacency matrix of the corresponding K2 graph which represents the two-level system is given by

The dynamics of such a system, as formulated by Eq. (16), can be modeled by the Cartesian product, , defined in Eq. (7). By this definition, the new product graph has the structure given in Fig. 2(b). The corresponding adjacency matrix is given in Fig. 2(c). Figure 2(d,e) present the qusienergy and the transition probability for this case as a function of the energy separation between the states of the two-level system with fixed parameters of and , computed from the matrix presented in Fig. 2(c). The peaks in Fig. 2(e) at nω indicate the multiphoton resonance processes. This result is in agreement with the results reported before10. Due to the periodicity of the quasienergy, the quasienergy plot has repeating structure by ω with the avoided crossings between the lower and upper Floquet states in the vicinity of (n is a positive integer). At the avoided crossings the lower and upper Floquet states are strongly mixed and resonance transitions between and occur, as shown in the plot of time-averaged transition probability, Fig. 2(e). Also indicates that these transitions are multiphoton resonance processes.

Figure 2
figure 2

Longitudinal coupling.

(a) Illustration of the ( is presented) and K2 graphs before the Cartesian production. models the discrete monochromatic laser field, and K2 represents the two-level system. (b,c) The schematic of the Cartesian product graph, and its corresponding adjacency matrix are given respectively. (d) Four branches of the quasienergies of the interacting system, and (e) Transition probability as a function of detuning energy, , for the two-level system driven by a single mode laser field with and , that is modeled by graph Cartesian product .

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The results presented in Fig. 3 show the appearance of three and four-photon resonance suppressions at longitudinal coupling between a superconducting qubit and a single mode resonator, when . The transmission blockade of more than two photons was reported before, by tuning the frequency of the driving field to be equal to the sum of the Kerr nonlinearity and the cavity resonance frequency16,38,39. As indicated by the solid and dashed arrows in Fig. 3(d), the three-photon and four-photon transitions are dramatically suppressed at tunnel splitting and , respectively. This nonlinear effect is emerged as the narrowed avoided crossings in the corresponding quasienergy plots, Fig. 2(a,b), and the slim transition probability distributions presented in Fig. 2(d,e). This effect is achieved by tuning the driving force to a suitably chosen frequency in the vicinity of an exact crossing of the corresponding two Floquet states. These multiphoton resonance suppressions can be explained by population trapping on the vertices connecting the final real states. For instance, in the case of three-photon resonance, these paths start from some initial state, eg. , passes some virtual intermediate states that do not conserve energy38,39, and arrive at the real final state . Figure 2(f,g) present the population flipping between the α and β rails on the ladder structure of the Cartesian graph product, Fig. 2(b). The red-dashed boxes indicate the α -vertices with the population trapping, which are responsible for the suppressed transition probability peaks in Fig. 2(d,e).

Figure 3
figure 3

Multiphoton resonance suppression.

The Appearance of multiphoton transition suppression at longitudinal coupling of a superconducting qubit with a single mode resonator. (a,b) The eight branches of the quasienergies of the system at the detuning energy of in (a), and in (b). The narrowed avoided-crossings at the third and fourth photon resonances in (a,b) are shown by solid and dashed green arrows, respectively. (c) The contour plot of the transition probability with respect to the tunnel splitting parameter and the detuning energy between the two states. The existence of the third and fourth-photon resonance suppressions are indicated by the arrows. (d,e) The transition probabilities corresponding to (a,b). The narrowed peak in each case, indicated by solid and dashed green arrows, demonstrate the suppression of that particular resonance. (f,g) Population flipping between the α and β chains in the graph for the cases presented in (d,e), respectively. Indicated by the red-dashed boxes are the α -vertices with the population trapping.

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Bidirectional coupling: Colored-weighted strong product

Superconducting qubits and the single mode field can have both transverse and longitudinal coupling40,41,42,43. The results presented so far have been focused on the use of linearly polarized (LP) laser fields. The use of elliptically polarized (EP) laser fields opens access to a number of strong-field processes that are either hindered or not present under the linear polarization44,45. The time-dependent Hamiltonian, which includes both the longitudinal and transverse coupling is given as,

Where is the ellipticity parameter. As expected, this system can be modeled by combining the above two cases where the transverse and longitudinal couplings were investigated separately. The graph-theoretical approach of modeling this interaction is given in Eq. (8) as the strong product with the resulting structure presented in Fig. 4(b). The corresponding adjacency matrix for this product graph is given in Fig. 4(c).

Figure 4
figure 4

Bidirectional coupling.

(a) Illustration of the ( is presented) and K2 graphs before the strong production. models the discrete monochromatic laser field, and K2 represents the two-level system. (b,c) The schematic of the strong product graph, and its corresponding adjacency matrix are given, respectively.

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In Fig. 5, we present the results on how the ellipticity of the field affect the multiphoton resonances between the two states of the qubit. Figure 5(d) is the contour plot of the transition probability with respect to the ellipticity parameter and the energy separation between the two states. The two ends of this plot, where the ellipticity is zero, correspond to the cases when we only have transverse coupling (left end) or only longitudinal coupling (right end). These two extreme cases were examined before separately. At the middle of this plot is the case when the both transverse and longitudinal couplings are fully engaged in the process. Indicated by the dashed red box in Fig. 5(f), as a result of such strong coupling, the three-to-five photon transition probabilities are maximized, compared to the other two case scenarios presented in Fig. 5(e,g). This can also be seen in the corresponding quasienergy plot, Fig. 5(b). The widely-expanded avoided crossing area between the three-to-five photon resonances in this figure indicates that the lower (blue) and upper (red) eigenstates are strongly connected and resonance transitions are well pronounced between the states of the two-level system. For comparison, in Fig. 5(a,c) the quasienergies for the cases with the contributions of and are presented, respectively.

Figure 5
figure 5

The effect of ellipticity on the multiphoton resonance.

(a–c) The four branches of the quasienergies of the system at the ellipticity values resulting the fields contributions of (a) , (b) , and (c) . (d) The contour plot of the transition probability with respect to the ellipticity parameter and the energy separation between the two states. The dashed green lines indicate the position of the presented three case scenarios. (e–g) The transition probabilities corresponding to the cases (a–c).

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Conclusion

In summary, we introduced a generalized graph theoretical method to investigate some of the characteristic multiphoton resonance processes and quantum interference phenomena for the superconducting qubit systems driven by intense ac fields. The various interacting designs at arbitrary coupling strengths are modeled by different graph products on colored-weighted graphs. The intuitive picture provided by this beyond rotating-wave approximation approach helps us to demonstrate some characteristic features of the superconducting qubit systems. The population analysis on the virtual dressed states of the product graphs is used to explain the nonlinear phenomenon of multiphoton suppression at longitudinal coupling case. One future step is to extend this method to higher dimensional systems and to study the topological features of their manifolds.

Additional Information

How to cite this article: Jooya, H. Z. et al. A graph-theoretical representation of multiphoton resonance processes in superconducting quantum circuits. Sci. Rep. 6, 37544; doi: 10.1038/srep37544 (2016).

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