# Phase gate of one qubit simultaneously controlling qubits in a cavity

## Abstract

We propose how to realize a three-step controlled-phase gate of one qubit simultaneously controlling qubits in a cavity or coupled to a resonator. The two-qubit controlled-phase gates, forming this multiqubit phase gate, can be performed simultaneously. The operation time of this phase gate is independent of the number of qubits. This phase gate controlling at once qubits is insensitive to the initial state of the cavity mode and can be used to produce an analogous CNOT gate simultaneously acting on qubits. We present two alternative approaches to implement this gate. One approach is based on tuning the qubit frequency while the other method tunes the resonator frequency. Using superconducting qubits coupled to a resonator as an example, we show how to implement the proposed gate with one superconducting qubit simultaneously controlling qubits selected from qubits coupled to a resonator (). We also give a discussion on realizing the proposed gate with atoms, by using one cavity initially in an arbitrary state.

###### pacs:

03.67.Lx, 42.50.Dv, 85.25.CpMarch 6, 2018

I. INTRODUCTION

Quantum information processing has attracted considerable interest during the past decade. The building blocks of quantum computing are single-qubit and two-qubit logic gates. So far, a large number of theoretical proposals for realizing two-qubit gates in many physical systems, such as trapped ions, atoms in cavity QED, nuclear magnetic resonance (NMR), and solid-state devices, have been proposed. Moreover, two-qubit controlled-not (CNOT), controlled-phase (CP), SWAP gates, or other two-qubit operations have been experimentally demonstrated in ion traps [1], NMR [2], quantum dots [3], and superconducting qubits [4-8].

In recent years, analogs of cavity quantum electrodynamics (QED) have been studied using superconducting Josephson devices. There, a superconducting resonator provides a quantized cavity field, superconducting qubits act as artificial atoms, and the strong coupling between the field and superconducting qubits has been demonstrated [9]. Theoretically, many approaches for performing two-qubit logic gates using superconducting qubits coupled to a superconducting resonator or cavity have been proposed [e.g., 10-15]. Moveover, experimental demonstrations of two-qubit gates or two-qubit operations using superconducting qubits coupled to cavities have recently been reported [7,8].

Attention is now shifting to the physical realization of multi-qubit gates (see, e.g., Ref. [16]) instead of just two-qubit gates. It is known that any multi-qubit gate can be decomposed into two-qubit gates and one-qubit gates. When using the conventional gate-decomposition protocols to construct a multi-qubit gate [17,18], the procedure usually becomes complicated as the number of qubits increases. Therefore, building a multi-qubit gate may become very difficult since each elementary gate requires turning on and off a given Hamiltonian for a certain period of time, and each additional basic gate adds experimental complications and the possibility of more errors. During the past few years, several methods for constructing multi-qubit phase gates with -control qubits acting on one target qubit based on ion traps [19], cavity QED [20,21], or circuit QED [22] have been proposed. As is well known, a multi-qubit gate with multiple control qubits controlling a single qubit plays a significant role in quantum information processing, such as quantum algorithms [e.g.,23,24] and quantum error-correction protocols [25].

In this work, we focus on another type of multi-qubit gates, i.e., a multi-qubit phase gate with one control qubit simultaneously controlling target qubits. This multi-qubit gate is useful in quantum information processing such as entanglement preparation [26], error correction [27], quantum algorithms (e.g., the Discrete Cosine Transform [28]), and quantum cloning [29]. In the following, we will propose a way for realizing this multi-qubit gate using () qubits in a cavity or coupled to a resonator. To implement this gate, we construct an effective Hamiltonian which contains interaction terms between the control qubit and each subordinate or target qubit. We will denote this -target-qubit control-phase gate as an NTCP gate [Fig. 1(a)]. We present two alternative approaches to implement this NTCP gate. One approach is based on tuning the qubit frequency, while the other method tunes the cavity (or resonator) frequency. We present these two alternative methods because some experimental implementations might find it easier to tune the qubit frequency, while others might prefer to tune the cavity frequency. For solid state qubits such as superconducting qubits and semiconductor quantum dots, the qubit transition frequency (or the qubit level spacings) can be readily adjusted by varying the external parameters [30-34] (e.g., the external magnetic flux for superconducting charge qubits, the flux bias or current bias in the case of superconducting phase qubits and flux qubits, see e.g. [30-33], and the external electric field for semiconductor quantum dots [34]). Also, the cavity mode frequency can be changed in various experiments (e.g., [35-39]).

As shown below, our proposal has the following advantages: (i) The two-qubit CP gates involved in the NTCP gate can be performed simultaneously; (ii) The operation time required for the gate implementation is independent of the number of qubits; (iii) This proposal is insensitive to the initial state of the cavity mode, and thus no preparation for the initial state of the cavity mode is needed; (iv) No measurement on the qubits or the cavity mode is needed and thus the operation is simplified; and (v) The proposal requires only three steps of operations.

Note that a CNOT gate of one qubit simultaneously controlling qubits, shown in Fig. 1(b), can also be achieved using the present proposal. This is because the -target-qubit CNOT gate is equivalent to an NTCP gate plus two Hadamard gates on the control qubit [Fig. 1(b)].

To the best of our knowledge, our proposal is the only one so far to demonstrate that a powerful phase gate, synchronously controlling qubits, can be achieved in a cavity or resonator, which can be initially in an arbitrary state. This proposal is quite general and can be applied to physical systems such as trapped atoms, quantum dots, and superconducting qubits. We believe that this work is of general interest and significance because it provides a protocol for performing a controlled-phase (or controlled-not) gate with multiple target qubits.

This paper is organized as follows. In Sec. II, we introduce the -target-qubit control phase gate studied in this work. In Sec. III, we discuss how to obtain the time-evolution operators for a qubit system interacting with a single cavity mode and driven by a classical pulse. In Sec. IV, using the time-evolution operators obtained, we present two alternative approaches for realizing the NTCP gate with qubits in a cavity or coupled to a resonator. In the appendix, we provide guidelines on how to protect multi-level qubits from leaking out of the computational subspace. In Sec. V, using superconducting qubits coupled to a resonator, we show how to apply our general proposal to implement the proposed NTCP gate, and then discuss its feasibility based on current experiments in superconducting quantum circuits. In Sec. VI, we discuss how to extend the present proposal to implement the NTCP gate with trapped atomic qubits, by using one cavity. A concluding summary is provided in Sec. VII.

II. ONE QUBIT SIMULTANEOUSLY CONTROLLING TARGET QUBITS

For two qubits, there are a total of four computational basis states, denoted by and Here, and are the two eigenstates of the Pauli operator . A two-qubit CP gate is defined as follows

(1) |

which implies that if and only if the control qubit (the first qubit) is in the state , a phase flip happens to the state of the target qubit (the second qubit), but nothing happens otherwise.

The NTCP gate considered here consists of two-qubit CP gates [Fig. 1(a)]. Each two-qubit CP gate involved in this NTCP gate has a shared control qubit (labelled by ) but a different target qubit (labelled by or ). According to the transformation (1) for a two-qubit CP gate, it can be seen that this NTCP gate with one control qubit (qubit ) and target qubits (qubits ) can be described by the following unitary operator

(2) |

where the subscript represents the control qubit , while represents the th target qubit, and is the identity operator for the qubit pair (), which is given by , with r, s . One can see that the operator (2) induces a phase flip (from the sign to the sign) to the logical state of each target qubit when the control qubit is initially in the state , and nothing happens otherwise.

It should be mentioned that the NTCP gate can be defined using the eigenstates and of the Pauli operator However, we note that to construct a -target controlled-NOT gate (based on the NTCP gate defined in the eigenstates of ), a Hadamard gate acting on each target qubit before and after the NTCP gate (i.e., a total of Hadamard gates) would be required. In contrast, the construction of the -target controlled-NOT gate, using the NTCP gate defined in the eigenstates of requires only Hadamard gates [see Fig. 1(b)]. Therefore, the NTCP here, defined in the eigenstates of , makes the procedure for constructing a -target controlled-NOT gate much simpler.

III. MODEL AND UNITARY EVOLUTION

Consider () qubits interacting with the cavity mode and driven by a classical pulse. In the rotating-wave approximation, the Hamiltonian for the whole system is

(3) |

with

(4) | |||||

(5) | |||||

(6) |

The Hamiltonian (3), together with the Hamiltonians (4-6), can be implemented when the qubits are atoms [40,41], quantum dots or superconducting devices (e.g., see section V below). Here, is the free Hamiltonian of the qubits and the cavity mode, is the interaction Hamiltonian between the qubits and the classical pulse, and is the interaction Hamiltonian between the qubits and the cavity mode. In addition, is the transition frequency between the two levels and of each qubit; () is the photon annihilation (creation) operator of the cavity mode with frequency is the coupling constant between the cavity mode and each qubit; and are the Rabi frequency, the frequency, and the initial phase of the pulse, respectively; and and are the collective operators for the qubits, which are given by

(7) |

where and with and () being the ground state and excited level of the th qubit. In the interaction picture with respect to , the above Hamiltonians and are rewritten respectively as (assuming )

(8) | |||||

(9) |

where

(10) |

is the detuning between the transition frequency of each qubit and the frequency of the cavity mode.

We now consider two special cases: and negative detuning , as well as and positive detuning The detailed discussion of these two cases will be given in the following subsections III.A and III.B. The results from the unitary evolution, obtained for these two special cases, will be employed by the two alternative approaches (presented in section IV) for the gate implementation.

A. Case for pulse phase and detuning

In this subsection, we consider the negative detuning case [Fig. 2(a)]. When the above Hamiltonian (8) reduces to

(11) |

where

(12) |

with Performing the unitary transformation we obtain

(13) |

with

(14) | |||||

Assuming that we can neglect the fast-oscillating terms [40-42]. Then the Hamiltonian (14) reduces to [40-42]

(15) |

The evolution operator for the Hamiltonian (15) can be written in the form [19,40,43,44]

(16) |

where

(17) | |||||

When the evolution time satisfies we have and . Then we obtain

(18) |

where The evolution operator of the system (in the interaction picture with respect to ) is thus given by

(19) |

In section IV, we propose two alternative approaches for the NTCP gate implementation, each one involving three steps. The evolution operator in Eq. (19) will be needed for the first step for either one of the two alternative approaches.

B. Case for pulse phase and detuning

In the following, we consider the positive detuning case . The detuning can be adjusted from to by changing either the qubit transition frequency or the cavity mode frequency , or both. In general, the qubit-cavity coupling constant varies when the detuning changes. To distinguish this case ( and ) from the previous case ( and ), we replace the previous notation and by and , respectively [see Fig. 2(b)]. For simplicity, we use the same symbols and for the pulse frequency, the qubit transition frequency, and the cavity mode frequency [Fig. 2(b)].

Suppose now that qubit is largely detuned (decoupled) from both the cavity mode and the pulse. This can be achieved by adjusting the level spacing of qubit (this tunability of energy levels can be achieved in different types of qubits, e.g., solid-state qubits [30-34]), or by moving qubit out of the cavity mode (e.g., trapped atoms [20,45,46]). Thus, after dropping the terms related to qubit (i.e., the terms corresponding to the index ) from the collective operators and in Hamiltonians (8) and (9), and replacing and (for the case of and ) with and (for the case of and ), we obtain from the Hamiltonians (8) and (9) (assuming )

(20) | |||||

(21) |

which are written in the interaction picture with respect to in Eq. (4). Here, and are the collective operators for the qubits (), which are given by

(22) |

In addition, the detuning is

(23) |

When choosing the Hamiltonian (20) reduces to

(24) |

where

(25) |

Note that the Hamiltonian (24) has a form similar to Eq. (11) and that the Hamiltonian (21) has a form similar to Eq. (9). Therefore, it is straightforward to show that under the condition when the evolution operator of the qubits (in the interaction picture with respect to ) is

(26) |

where The evolution operator in Eq. (26) will be needed for the second step of either one of the two alternative approaches (presented in section IV below) for implementing the NTCP gate.

One can see that the operator described by Eq. (19) or Eq. (26) does not include the photon operator or of the cavity mode. Therefore, the cavity mode can be initially in an arbitrary state (e.g., in a vacuum state, a Fock state, a coherent state, or even a thermal state).

For the gate realization, we will also need to have the qubits decoupled from the cavity mode and apply a resonant pulse to each qubit. Suppose that the Rabi frequency for the pulse applied to qubit is while the Rabi frequency for the pulses applied to qubits () is The initial phase for each pulse is . The free Hamiltonian of the qubits and the cavity mode is the given in Eq. (4). In the interaction picture with respect to , the interaction Hamiltonian for the qubit system and the pulses is given by

(27) |

For an evolution time , the time evolution operator for the Hamiltonian (27) is

(28) |

The evolution operator in Eq. (28) will be needed for the third step for either one of the two alternative approaches below.

IV. IMPLEMENTATION OF THE NTCP GATE

The goal of this section is to demonstrate how the NTCP gate can be realized based on the unitary operators (19), (26), and (28) obtained above. We will present two alternative methods for the gate implementation: one method based on the adjustment of the qubit transition frequency and another method which works mainly via the adjustment of the cavity mode frequency.

A. NTCP gate via adjusting the qubit transition frequency

Let us consider qubits placed in a single-mode cavity or coupled to a resonator. For this first method, the cavity mode frequency is kept fixed. The operations for the gate implementation, and the unitary evolutions after each step of operation, are listed below:

Step (i): Apply a resonant pulse (with ) to each qubit. The pulse Rabi frequency is The cavity mode is coupled to each qubit with a detuning [Fig. 3(a) and Fig. 3(a)]. This is the case discussed in subsection III.A. Thus, the in Eq. (19) is the evolution operator for the qubit system for an interaction time

Step (ii): Adjust the qubit transition frequency for qubits (), such that the cavity mode is coupled to qubits () with a detuning [Fig. 3(b)]. Apply a resonant pulse (with ) to each of the target qubits () [Fig. 3(b)]. The pulse Rabi frequency is now In addition, adjust the transition frequency of qubit such that qubit is decoupled from the cavity mode and the pulses applied to qubits () [Fig. 3(b)]. It can be seen that this is the case discussed in subsection III.B. Thus, the in Eq. (26) is the evolution operator for the qubit system for an interaction time

The combined time evolution operator, after the above two steps, is

In the last line of Eq. (29), we assumed (i.e., ) and (i.e., ), which can be achieved by adjusting the detunings and (i.e., changing the qubit transition frequency) as well as the Rabi frequencies and (i.e., changing the intensity/amplitude of the pulses). Note that and where is the identity operator for qubit Thus, Eq. (29) can be written as

(30) |

Step: (iii) Leave the transition frequency unchanged for qubit [Fig. 3(c)] but adjust the transition frequency of qubits () [Fig. 3(c)], such that the cavity mode is largely detuned (decoupled) from each qubit. In addition, apply a resonant pulse (with ) to each qubit. The Rabi frequency for the pulse applied to qubit is now [Fig. 3(c)] while the Rabi frequency of the pulse applied to qubits () is [Fig. 3(c)]. In this case, the interaction Hamiltonian for the qubit system and the pulses is given by Eq. (27) and thus the time evolution operator is the in Eq. (28) for an evolution time given above.

It can be seen that after the above three-step operation, the joint time-evolution operator of the qubit system is

Under the following condition

(32) |

(which can be obtained by adjusting the Rabi frequencies and ), Eq. (31) can be then written as

(33) |

with

(34) |

It can be easily shown that for the qubit pair (), we have

(35) |

where an overall phase factor is omitted. Here and below, and are the two eigenstates of the Pauli operator for qubit while and are the two eigenstates of the Pauli operator for qubit ( or ). By setting i.e.,

(36) |

( is an integer), we obtain from Eq. (35)

(37) |

which shows that a quantum phase gate described by

(38) |

is achieved for the qubit pair (). Here, is the identity operator for the qubit pair (), which is given by with Note that the condition (36) can be achieved by adjusting the detuning (i.e., via changing the qubit transition frequency ).

Combining Eqs. (33) and (38), we finally obtain

(39) |

which demonstrates that two-qubit CP gates are simultaneously performed on the qubit pairs (), (),…, and (), respectively. Note that each qubit pair contains the same control qubit (qubit ) and a different target qubit (either qubit or ). Hence, an NTCP gate with target qubits () and one control qubit (qubit 1) is obtained after the above three-step process.

From the description above, one can see that the method presented here has an advantage: it does not require the adjustment of the cavity-mode frequency.

B. NTCP gate via mainly adjusting the cavity-mode frequency

In this subsection, we present a different way for realizing the NTCP gate, which is mainly based on the adjustment of the cavity mode frequency .

Let us consider qubits placed in a single-mode cavity or coupled to a resonator. Note that now the transition frequency of the target qubits () is kept fixed during the following entire process. The operations for the gate implementation and the unitary evolutions after each step are listed below:

Step (i): Apply a resonant pulse (with ) to each qubit. The pulse Rabi frequency is The cavity mode is coupled to each qubit with a detuning [Fig. 4(a) and Fig. 4(a)]. It can be seen that this is the case discussed in subsection III.A. Thus, the in Eq. (19) is the evolution operator for the qubit system for an interaction time

Step (ii): Leave the transition frequency of qubits () unchanged while adjusting the cavity mode frequency . The cavity mode is coupled to qubits () with a detuning [Fig. 4(b)]. Apply a resonant pulse (with ) to each of qubits () [Fig. 4(b)]. The pulse Rabi frequency is now In addition, adjust the transition frequency of qubit such that qubit is largely detuned (decoupled) from the cavity mode as well as the pulses applied to qubits () [Fig. 4(b)]. One can see that this is the case discussed in subsection III.B. Hence, the in Eq. (26) is the evolution operator for the qubit system for an interaction time

Step (iii): Leave the transition frequency of qubits () unchanged while adjusting the transition frequency of the control qubit back to its original setting in step (i) [Fig. 4(c)]. Adjust the cavity mode frequency , such that the cavity mode is largely detuned (decoupled) from each qubit [Fig. 4(c) and Fig. 4(c)]. In addition, apply a resonant pulse (with ) to each qubit. The Rabi frequency for the pulse applied to qubit is now [Fig. 4(c)] while the Rabi frequency of the pulses applied to qubits () is now [Fig. 4(c)]. Therefore, the interaction Hamiltonian for the qubits and the pulses is in Eq. (27) and thus the time evolution operator is the in Eq. (28) for an evolution time given above.

Note that the time evolution operators and , obtained from each step discussed in this subsection, are the same as those obtained from each step of operations in the previous subsection. Hence, it is clear that the NTCP gate can be implemented after this three-step process.

From the description above, it can be seen that the transition frequency for each one of the target qubits () remains unchanged during the entire operation. Thus, adjusting the level spacings for the target qubits () is not required by the method presented in this subsection. What one needs to do is to adjust the cavity mode frequency and the level spacings of the control qubit .

C. Discussion

We have presented two alternative methods for implementing an NTCP gate. Note that when coupled to a cavity (or resonator) mode and driven by a classical pulse, physical qubit systems (such as atoms, quantum dots, and superconducting qubits) have the same type of interaction described by the Hamiltonian (3) or a Hamiltonian having a similar form to Eq. (3), from which the four Hamiltonians (8), (9), (20), and (21), i.e., the key elements for the proposed NTCP gate implementation, are available. Therefore, the two alternative methods above are quite general, which can be applied to implementing the NTCP gate with superconducting qubits, quantum dots, or trapped atomic qubits.

As shown above, the first method is based on the adjustment of the qubit level spacings while the second method works mainly via the adjustment of the cavity mode frequency. For solid-state qubits, the qubit level spacings can be rapidly adjusted (e.g., in 1 2 nanosecond timescale for superconducting qubits [47]). And, for superconducting resonators, the resonator frequency can be fast tuned (e.g., in less than a few nanoseconds for a superconducting transmission line resonator [36]).

In the appendix, we provide guidelines on protecting qubits from leaking out of the computational subspace in the presence of more qubit levels. As discussed there, as long as the large detuning of the cavity mode with the transition between the irrelevant energy levels can be met, the leakage out of the computational subspace induced due to the cavity mode can be suppressed. In addition, we should mention that since the applied pulses are resonant with the transition, the coupling of the qubit level or with other levels, induced by the pulses, is negligibly small. Therefore, the leak-out of the computation subspace, due to the application of the pulses, can be neglected.

As discussed above, during the gate operation for either of the two methods, decoupling of the qubits from the cavity was made by adjusting the qubit frequency or the cavity frequency . This particularly applies to solid state qubits because the locations of solid-state qubits (such as superconducting qubits and quantum dots) are fixed once they are built in a cavity or resonator. However, it is noted that for trapped atomic qubits, decoupling of qubits from the cavity can be made by just moving atoms out of the cavity and thus adjustment of the level spacings of atoms or the cavity frequency is not needed to have atoms decoupled (largely detuned) from the cavity mode (e.g., as shown in Section VI).

For both step (i) and step (ii) of the two methods above, it was assumed that the detuning () is much smaller than the qubit Rabi frequency () and that the Rabi frequencies are equal for all the qubits. In reality, the Rabi frequency for each qubit might not be identical, and the effect of the small Rabi frequency deviations on the gate performance may be magnified when the operation time is much longer than the inverse Rabi frequency. Hence, to improve the gate performance, the Rabi frequency deviations should be as small as possible, which could be achieved by adjusting the intensity of the pulses applied to the qubits.

We should mention that when the mean photon number of the cavity field satisfies , the multi-photon process can be neglected. In the case that the condition does not meet, one can adjust the qubit level spacings or choose the qubit level structures appropriately such that the multi-photon process is negligible in the process of quantum operations.

Before closing this section, it should be mentioned that the present method is based on an effective Hamiltonian

(40) |

which can be found from Eqs. (33) and (34). One can see that this Hamiltonian contains the interaction terms between the control qubit (qubit ) and each target qubit, but does not include the interaction terms between any two target qubits. Note that each term in Eq. (40) acts on a different target qubit, with the same control qubit, and that any two terms for different ’s commute with each other. Therefore, the two-qubit controlled-phase gates forming the NTCP gate can be simultaneously performed on the qubit pairs (), (),…, and (), respectively.

V. REALIZING THE NTCP GATE WITH SUPERCONDUCTING QUBITS COUPLED TO A RESONATOR

The methods presented above for implementing the NTCP gate are based on the four Hamiltonians (8), (9), (20) and (21). In this section, we show how these Hamiltonians can be obtained for the superconducting charge qubits coupled to a resonator. We will then show how to apply the first method above to implement the NTCP gate with charge qubits selected from charge qubits coupled to a resonator (). A discussion on the experimental feasibility will be given later.

A. Hamiltonians

The superconducting charge qubit considered here, as shown in Fig. 5(a), consists of a small superconducting box with excess Cooper-pair charges, connected to a symmetric superconducting quantum interference device (SQUID) with capacitance and Josephson coupling energy In the charge regime (here, and are the Boltzmann constant, superconducting energy gap, charging energy, and temperature, respectively), only two charge states, and , are important for the dynamics of the system, and thus this device [e.g., 30-32] behaves as a two-level system. Here, we denote the two charge states and as the two eigenstates and of the spin operator The reason for not using the eigenstates and of to represent the two charge states is that: in order to obtain the four Hamiltonians (8), (9), (20), and (21), in the following we will need to perform a basis transformation from the basis to the basis The Hamiltonian describing the qubit is given by [30,31]

(41) |

where , , and Here, is the gate capacitance, is the gate voltage, is the external magnetic flux applied to the SQUID loop, is the flux quantum, and is the effective Josephson coupling energy. We assume that where () is the dc (ac) part of the gate voltage and is the quantum part of the gate voltage, which is caused by the electric field of the resonator mode when the qubit is coupled to a resonator. Correspondingly, we have

(42) |

where and By inserting Eq. (42) into Eq. (41), we obtain the following Hamiltonian for the qubit-cavity system

(43) | |||||

where When and , the Hamiltonian (43) becomes [11,48]

(44) | |||||

where is the Rabi frequency of the ac gate voltage and is the coupling constant between the charge qubit and the resonator mode.

Let us now consider identical charge qubits coupled to a single-mode resonator [Fig. 5(b)]. One can select a subset of qubits for the gate, while the remaining qubits, which are not controlled by the gate, are decoupled from the resonator mode by setting their and to have their free Hamiltonian equal to zero. Without loss of generality, we assume that the set of qubits selected for the gate are the qubits labelled by 1, 2, …, and (here, ). From the discussion above, it can be seen that the Hamiltonian for the qubits and the resonator mode is

(45) | |||||

where and with and By setting (i.e., ) for each qubit and defining , the Hamiltonian (45) reduces to

(46) | |||||

Define now the new qubit basis and i.e., perform a basis transformation from the basis to the basis for the th qubit. Thus, in the new basis, the Hamiltonian (46) becomes

(47) |

where

(48) | |||||

(49) | |||||

(50) |

Here, the collective operators and are the same as those given in Eq. (7) and Eq. (12). In the interaction picture with respect to we obtain from Eqs. (49) and (50) (under the rotating-wave approximation and assuming )