A Method to Compute the Schrieffer–Wolff Generator for Analysis of Quantum Memory

Quantum illumination uses entangled light that consists of signal and idler modes to achieve higher detection rate of a low-reflective object in noisy environments. The best performance of quantum illumination can be achieved by measuring the returned signal mode together with the idler mode. Thus, it is necessary to prepare a quantum memory that can keep the idler mode ideal. To send a signal towards a long-distance target, entangled light in the microwave regime is used. There was a recent demonstration of a microwave quantum memory using microwave cavities coupled with a transmon qubit. We propose an ordering of bosonic operators to efficiently compute the Schrieffer–Wolff transformation generator to analyze the quantum memory. Our proposed method is applicable to a wide class of systems described by bosonic operators whose interaction part represents a definite number of transfer in quanta.


Introduction
Quantum memories are required to store and retrieve quantum states with high fidelity. To synchronize various events, quantum memories are essential for quantum information networks, including quantum computation [1], quantum communication [2], and quantum illumination [3]. Quantum illumination (QI), a target detection scheme using quantum entangled light with signal and idler modes, has as its objective enhancing the detection rate of a target with low-reflectivity in a highly noisy environment [3]. In QI, the signal mode is sent to the target while the idler mode is retained. Although the noisy environment destroys the entanglement between the signal and idler modes, we can take quantum advantage over the classical limit by jointly measuring the returned signal mode and the idler mode when the signal arrives [4][5][6][7]. During this process, it is highly appreciable to keep the idler mode in an ideal quantum memory. This was investigated using various systems, such as a microwave cavity [8], mechanical oscillators [9], or spin ensembles [10,11].
Here, we focus on quantum memories using microwave cavities that can have highquality factors and allow continuous-variable quantum information processes. By coupling a microwave cavity to a transmon qubit, it is able to write arbitrary states on the cavity and infer information about the cavity [8,12]. It is based on the cross Kerr effect, where the energy gap of neighboring levels of the cavity (transmon qubit) depends on the excitations of the transmon qubit (cavity). The anharmonicity of the transmon qubit gives rise to the cross Kerr effect through coupling of the qubit and cavity [13,14].
In dealing with such systems, it is crucial to understand how the coupling affects the energy structure. The Schrieffer-Wolff transformation computes this shift in energy structure by using a basis change unitary to remove the coupling [15,16]. For multiple bosonic modes containing nonlinear terms, it is complicated to find the exact generator of the unitary for the Schrieffer-Wolff transformation. Here, we propose a systematic approach to find the generator and compute the energy corrections induced by this transformation. An ordering of operators, which we call computational ordering, greatly simplifies the commutation structure of operators making it suitable in calculating the Schrieffer-Wolff transformation generator.

Equivalent Circuit Analysis of a Quantum Memory
The quantum memory demonstrated in Ref. [8] couples two microwave cavities through a transmon qubit. One cavity is used as a memory (storage) to store quantum states and the other cavity is used as a readout port whose response depends on the memory-cavity state through the transmon qubit. Such a system can be described by an equivalent circuit depicted in Figure 1b. Two LC oscillators represent the microwave cavities, while the middle oscillator represents the transmon qubit. The oscillators are labeled as s, t, and r for storage, transmon, and readout, respectively, as in Ref. [8]. The transmon qubit is coupled to both cavities by capacitors. The Hamiltonian describing this system isĤ whereâ s ,â t ,â r are bosonic annihilation operators corresponding to each oscillator mode. A detailed derivation of this Hamiltonian and expressions of ω s , ω t , . . . in terms of L i , C i , C c1 , C c2 (i = s, t, r) are given in Appendix A. We assume that the system is in the dispersive regime, where the couplings g 1 and g 2 are much smaller than the detunings |ω s − ω t | and |ω t − ω r |. The elimination of capacitive couplings in Equation (1) gives rise to cross Kerr effects among each cavity and the transmon. This is done by diagonalizing the Hamiltonian. There are two ways in achieving this; namely, second-order perturbation and the Schrieffer-Wolff transformation [16]. These methods were previously applied to systems of transmon qubits coupled with LC oscillators in evaluating the energy structure [13,14]. For the Schrieffer-Wolff transformation, one must find an operatorŜ, the Schrieffer-Wolff generator, which is an off-diagonal operator satisfying a given commutator equation. It is complicated to determine the operatorŜ and compute various commutators to obtain the second-order energy corrections. Thus, we introduce a method that simplifies the computation and apply it to Equation (1). After the computation, we truncate the transmon qubit to the lowest two levels to obtain a Jaynes-Cummings-like Hamiltonian [13]. Truncation of the transmon qubit should be done after diagonalization since virtual excitations of the transmon need to be considered.

Computational Ordering for Schrieffer-Wolff Transformation
We propose an ordering of bosonic operators, which gives direct computation of the Schrieffer-Wolff generator and second-order energy corrections. For a short recall of the Schrieffer-Wolff transformation, letĤ be the Hamiltonian in interest. We separate the Hamiltonian into diagonal and off-diagonal parts,Ĥ 0 ,V respectively, soĤ =Ĥ 0 +V. The Schrieffer-Wolff generatorŜ is defined as the off-diagonal operator which satisfies [Ŝ,Ĥ 0 ] = −V. Then, The energy separations ofĤ 0 must be larger thanV, so thatŜ becomes small and a perturbative approach is applicable [17]. This condition becomes evident when we write down the explicit form ofŜ in Equation (5). The second-order energy corrections toĤ 0 are given as the diagonal part of 1 2 [Ŝ,V]. We consider a system of N bosonic modes, whereâ i is the annihilation operator of the i-th mode and satisfies [â i ,â † j ] = δ ij . We propose an ordering of operatorŝ to efficiently compute the Schrieffer-Wolff transformation generator and second-order energy corrections toĤ 0 . In Equation (3), To ensure that f is unique, we require n, m to have disjoint support, i.e., n · m := (n 1 m 1 , . . . , n N m N ) = (0, . . . , 0). For example, the operatorâ † 1â † 1â † 1â 1â1 will be written aŝ The main motivation of this ordering is that diagonal operators in the Fock basis correspond to functions defined on N N 0 , with N 0 = {0, 1, 2, 3, . . . }, and functions are in general easier to manipulate than operators. The computational ordering is then equivalent to writing a given operator in terms of number operators as much as possible. Explicit expressions for writing normal-ordered or antinormal-ordered operators in this ordering are given in Appendix B. Note that operators that have n = m = 0 are exactly the diagonal operators in the Fock basis.
We write the HamiltonianĤ in this ordering: The sum, here and henceforth, is over all n, m satisfying n · m = (0, . . . , 0) and n, m are not both 0. This automatically splits the Hamiltonian into diagonal and off-diagonal parts. The hermitian condition onĤ forces f to be real valued and g * nm = g mn , z * being the complex conjugate of z. The main results arê where x n = x(x − 1) . . . (x − n + 1) is the falling factorial and the falling factorial of tuples is defined element-wise. The subscript (d) means to take the diagonal part, so 1 2 [Ŝ,V] (d) is the second-order correction to energy. Since f , g nm are essentially functions defined on N N 0 as noted before, the computation of Equations (5) and (6) is straightforward. In the end, the original Hamiltonian is transformed via the Schrieffer-Wolff transformation as where the omitted terms are off-diagonal terms of second-order inV or diagonal terms of third-order inV. The superscript (2) indicates that the term is second-order inV.
To obtain the main results Equations (5) and (6), we need a computational lemma.
Lemma 1. The commutators ofâ †n ,â m with f (â †â ) are as follows. [ Proof. It suffices to check on number states |k = |k 1 , . . . , k N . One can verify Here, is the rising factorial. The commutator withâ m follows from taking the adjoint. Now one can compute the commutator ofŜ withĤ 0 . WriteŜ aŝ with h * nm = −h mn so thatŜ is antihermitian. Then, one has The choice ofŜ as in Equation (5) The conditions on f , g nm ensure that h * nm = −h mn holds. This is well-defined as long as the diagonal part is nondegenerate, which is true when considering low excitations of transmons.
Using the generatorŜ defined as Equation (5), we can compute the correction to energies as the diagonal part of 1 2 [Ŝ,V]. The only term inV that gives a diagonal contribution with theâ †n h nm (â †â )â m term inŜ isâ †m g mn (â †â )â n . A pictorial representation of this statement is given in Figure 2. Hence, The calculation of the commutator can be done by using Lemma 1 and results in Appendix B.
Note that the summand in Equation (13) is symmetric under change of n, m, which leads to Equation (14). This result is equivalent to nondegenerate second-order perturbation energy correction, computed in our proposed ordering. . The Fock basis state |k must end up in |k to give a diagonal contribution. For example, in the productŜV, thê a †m g mn (â †â )â n term inV maps the state |k to |k + m − n . The only term inŜ which maps this back to |k isâ †n h nm (â †â )â m . This corresponds to the upper-half of the above diagram. The lower-half of the diagram represents theVŜ product.
In most cases, interaction terms are of formâ † i g ij (â †â )â j , which represent a single transfer of quantum excitations. If we restrict the interaction to only these terms, our main results Equations (5) and (6) are simplified tô where e i is the N tuple, which has 1 as its i-th component, and all other elements are 0 and i, j ∈ {1, 2, . . . , N}.
These functions give a full description of the Hamiltonian Equation (1). n, m, correspond tô a † sâs ,â † tâ t ,â † râr , respectively. The second-order energy corrections can be directly computed by our main result Equation (17).
with ∆ st := ω s − ω t , ∆ rt := ω r − ω t . To read off shifts in frequency, cross Kerr coefficients, and anharmonicities, we must put Equation (22) in normal order: Using the result from Equation (A16), the shifts are given aŝ where x n = x(x + 1) . . . (x + n − 1) is the rising factorial and factors ofh were omitted in the right-hand side for simplicity. Restoring these factors are done by replacing E C with E C /h. The shifts in physical quantities are found by simply reading off the coefficients of Equation (24): Hence, the total transformed Hamiltonian can be written as This extends the results using Bogoliubov approach to diagonalize the Hamiltonian of a coupled single LC oscillator and transmon [18] in the sense that the frequency shift of the transmon qubit is the sum of contributions from coupling to each LC oscillator. Such a system is described by a Hamiltonian H =hω 1â †â +hω 2b †b − E C 2b †b †bb +hg(â †b +âb † ).
Elimination of thehg(â †b +âb † ) term gives rise to cross Kerr coefficient betweenâ,b, where ∆ := ω 1 − ω 2 , which highly resembles the results in Equation (27). Note that there is a sign difference in the definition of ∆ compared with that of Ref. [18].
To obtain a form similar to that given in Refs. [13,14], we truncate the transmon Hilbert space to the first two levels. Such truncation is done by replacingâ † tâ t with (σ z + 1)/2 in Equation (22). The result iŝ up to an overall constant. Again, the contributions from each oscillator-transmon coupling stated in Ref. [13] are added independently. The frequency shifts of LC oscillators seem to be different compared with Equation (25), but this is due to a subtle difference of physical interpretation. The coefficients δ s , δ r in Equation (25) are the frequency shifts when the transmon is in the ground state, while the coefficients ofâ † sâs ,â † râr in Equation (31) are the average of the frequency shifts when the transmon is in the ground state and excited state. With theâ † sâs σ z ,â † râr σ z terms in consideration, both Equations (22) and (31) give the same energy spectrum when considering up to the first excitation of the transmon.

Discussion
We proposed an ordering of bosonic operators to efficiently compute the Schrieffer-Wolff transformation generator and energy corrections. This formalism was applied to a system with a transmon coupled to two different LC oscillators to model a quantum memory and readout device demonstrated in Ref. [8]. We solved the normal ordering problem for an operator that appears in the second-order energy correction, so that shifts in physical parameters such as frequency, anharmonicity, and cross Kerr coefficients can be directly read off from the normal-ordered form.
Our proposed method can be directly applied to systems consisting of LC circuits coupled with multiple transmons, and even to systems that have nonlinear couplings provided that the couplings represent a definite number of excitations or de-excitations in the Fock basis. It is possible to generalize this method to incorporate fermionic operators in this formalism, which can be used to reproduce the results of the original application of Schrieffer-Wolff transformation to the Anderson impurity model as in Appendix C. With such a general method, one can analyze a wide range of time-independent systems. Quantum illumination, an example of quantum information technology, uses entangled light to achieve higher detection rate of a target with low-reflectivity. The idler mode, a part of the entangled light, should be stored in a quantum memory for ideal operation.
Our method was used to analyze a demonstrated quantum memory and can be used to analyze other systems operating in various quantum technologies. For further research, it is required to find methods to store and release the propagating idler mode efficiently [19], leading to applications of quantum memories to quantum illumination.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Appendix A. Derivation of Hamiltonian from Circuit QED
In this section, we follow the quantization method of Ref. [20] and obtain various physical quantities, such as the normal mode frequency of each oscillator and the couplings of modes in terms of circuit parameters L i , C i , C c1 , C c2 (i = s, t, r). Let φ i be the flux variable at node i(i = s, t, r) as labeled in Figure 1b. The corresponding (linear) Lagrangian for this system is The conjugate variables are q i := ∂L/∂φ i and the Hamiltonian H is Elevating q i , φ i to canonical operatorsq i ,φ i with [φ i ,q j ] = ihδ ij and defining creation, annihilation operators as usual giveŝ Note that in the weak coupling limit (C c1 , C c2 C s , C t , C r ), the eigen frequencies and couplings simplify to The coupling betweenq s ,q r is off-diagonal of second-order, and hence, give energy corrections of third-and higher-order. Our analysis concerns up to second-order energy corrections, so we dropped this term. If one wants to consider higher-order corrections, then consideration of this coupling is necessary.
The nonlinearity of the transmon is introduced by replacingφ 2 where E C = e 2 C 2 t /2C 3 is the charging energy of the transmon, which is small compared withhω t in the transmon regime [18]. The correction to ω t is ignored, while it can be easily recovered in the final results by just replacing ω t with ω t − E C /h. We applied the rotating wave approximation to remove nonresonant terms such asâ † tâ t which represent the creation or destruction of two or more quanta.

Appendix B. Computational Ordering of Normal-Ordered and Antinormal-Ordered Operators
In this section, we give explicit formulas of writing normal-ordered and antinormalordered operators in our proposed computational ordering. They extensively use Stirling numbers of the first kind s(n, k), which are the matrix elements of the basis change of monomials x n and falling factorials x n [21] (p. 824).
x n = n ∑ k=0 s(n, k)x k . (A12) The relation of rising factorials and monomials is similar, only differing in sign.
Since the matrix elements of our computational ordered operators and normal-, antinormalordered operators in the Fock basis involve monomials, falling factorials, rising factorials, respectively, it is obvious that Stirling numbers will appear. The results are stated for a single bosonic operatorâ. Extension to several bosonic operators is trivial. To read off anharmonicity and cross Kerr coefficients as in Equation (23), one must know how to convert operators in our ordering into normal order. We end this section with showing that where c is not a nonpositive integer. This is equivalent to finding α k such that 1 n + c = n ∑ k=0 α k n k .
Since (n k ) nk is a lower diagonal matrix, its inverse exists. The inverse matrix is easily shown to be the lower diagonal matrix