Generation of a macroscopic entangled coherent state using quantum memories in circuit QED

Abstract

W-type entangled states can be used as quantum channels for, e.g., quantum teleportation, quantum dense coding, and quantum key distribution. In this work, we propose a way to generate a macroscopic W-type entangled coherent state using quantum memories in circuit QED. The memories considered here are nitrogen-vacancy center ensembles (NVEs), each located in a different cavity. This proposal does not require initially preparing each NVE in a coherent state instead of a ground state, which should significantly reduce its experimental difficulty. For most of the operation time, each cavity remains in a vacuum state, thus decoherence caused by the cavity decay and the unwanted inter-cavity crosstalk are greatly suppressed. Moreover, only one external-cavity coupler qubit is needed, which simplifies the circuit.

Unlike bipartite systems, it has been proven that there exist two inequivalent classes of multipartite entangled states, such as GHZ states¹ and W states², which cannot be converted to each other by local operations and classical communications. Relative to the tripartite entangled states, GHZ states are fragile: if any one qubit is traced out, the remaining bipartite states are separable states. However, W states are robust against qubit loss and qubit-flip noise because they maintain bipartite entanglement. W states are important for quantum communications. For example, W states can be used as quantum channels for quantum teleportation³, quantum dense coding⁴, and quantum key distribution⁵.

Over the past years, a number of theoretical ideas have been proposed for creating a discrete-variable W-class entangled state of qubits (i.e., two-state particles or two-level quantum systems)⁶,⁷,⁸,⁹,¹⁰,¹¹,¹²,¹³, where P_z is the symmetry permutation operator for the qubits (1, 2 ···n), and denotes the totally-symmetric state in which (n − 1) qubits out of a total of n qubits are in the state |0〉, while the remaining qubit is in the state |1〉. As an example, consider a three-qubit case (i.e., n = 3), for which the W state is . Experimentally, the discrete-variable W states |W_n−1,1〉_DV have been created with up to eight trapped ions¹⁴, four optical modes¹⁵, three superconducting phase qubits coupled capacitively¹⁶, atomic ensembles in four quantum memories¹⁷, and two superconducting phase qubits plus a resonant cavity¹⁸.

On the other hand, there is much interest in entangled coherent states (ECSs)¹⁹,²⁰,²¹,²²,²³,²⁴,²⁵,²⁶,²⁷,²⁸. In this work we focus on a macroscopic W-type ECS (i.e., continuous-variable W state), described by

where , with c_i ≠ 0 (i = 0, 1,..., n − 1), () is a coherent state, α is a complex number, and , when |α| is large enough. The W state (1) is of fundamental interest in quantum mechanics and plays an important role in quantum information processing (QIP) and quantum communications. For instance, the W state (1) can be used to test quantum nonlocality without inequality²⁹,³⁰ and the violation of the Bell inequalities because such state is greater than that for any states involving two spin-1/2 particles³⁰,³¹. In addition, ref. 32 has shown that there exists a quantum information protocol which is not suitable for GHZ-type ECSs but can only be accomplished with the W state (1). Moreover, the W state (1) is a necessary resource for remote symmetric entanglement³², which allows two distant parties to share a symmetric entangled state. For the past years, theoretical methods have been proposed for generating the W state (1) in some physical systems³³,³⁴,³⁵,³⁶,³⁷. Refs ^{32, 33, 34} have proposed how to generate the W state (1) of three/four modes with linear optical devices, and refs ³⁶ and ³⁷ have discussed how to create the W state (1) of three-cavity fields based on cavity QED. However, in these schemes, the W ECSs were prepared with photons or cavity fields, and thus decoherence may pose a problem due to photon loss or cavity-field decay.

Hybrid quantum systems, composed of superconducting qubits, nitrogen-vacancy centers (NVCs), nitrogen-vacancy center ensembles (NVEs), or/and superconducting microwave resonators/cavities, have attracted tremendous attention³⁸,³⁹,⁴⁰,⁴¹. Recently, much progress has been made in this field. For instance, coherent coupling between a superconducting flux/transmon qubit and an NVE⁴²,⁴³ or between an NVC/NVE and a superconducting resonator⁴⁴,⁴⁵ has been experimentally demonstrated. Moreover, based on the hybrid systems, various quantum operations, such as entanglement preparation, quantum logic gates, and information transfer, have been investigated in theory⁴⁰,⁴⁶,⁴⁷,⁴⁸,⁴⁹ and demonstrated in experiment⁴²,⁵⁰,⁵¹.

Inspired by previous works and the long decoherence time of NVEs, we here consider a hybrid system composed of one-dimensional transmission line resonators (TLRs) each hosting an NVE and a qubit and connected to a coupler qubit A [Figs 1(a) and 2]. We then propose a way to generate a continuous-variable W-type entangled coherent state, described by Eq. (1), by using NVEs each located in a different cavity. Because of the long decoherence time of NVEs, the prepared W state can be stored for a long time. Note that NVEs have been recently considered as good memory elements in quantum information processing³⁹,⁴⁰,⁴²,⁴⁵,⁴⁶,⁴⁷,⁴⁸,⁴⁹,⁵¹.

Figure 1.

(a) Setup of the hybrid system consisting of a coupler qubit A and three cavities each hosting a qubit (a dark dot) and a nitrogen-vacancy center ensemble (a green oval). C₁, C₂ and C₃ represent capacitors. An intracavity qubit can be an atom or a solid-state qubit. The coupler qubit A can be a quantum dot or a superconducting qubit. (b) Illustration of the decoupling among qubit A, cavity j, NVE j and qubit j (j = 1, 2, 3) before the W-state preparation. (c) The resonant interaction between qubit A and cavity j with coupling constant (used in step 1). (d) The resonant interaction between qubit j and cavity j with resonant coupling constant g_rj (used in step 2). (e) The resonant interaction between qubit j and the pulse with Rabi frequency (applied for step 3). (f) The dispersive interaction between cavity j and qubit j with coupling constant g_j and detuning , the dispersive interaction between cavity j and NVE j with coupling constant and detuning , as well as the resonant interaction between qubit j and the pulse with Rabi frequency Ω_j (applied for step 4). Here, , with the transition frequency of qubit j and the frequency of cavity j, and , with being the frequency of a bosonic mode describing NVE j. Since qubit A is not involved during the operation of step 4, qubit A is dropped off in (f) for simplicity. Note that in (b,e), the frequency of cavity j is highly detuned from those of qubit A, qubit j and NVE j, while in (f) the frequency of cavity j is adjusted such that cavity j is dispersively coupled to qubit j and NVE j. The bottom dark solid line in (b–f) also represents the ground state (i.e., the vacuum state) of cavity j.

Figure 2. Diagram of a coupler qubit A and n cavities each hosting a qubit (a dark dot) and a NVE (a green oval).

Qubit A is capacitively coupled to each cavity.

As shown below, this proposal has the following features and advantages: (i) Different from the previous works³³,³⁴,³⁵,³⁶,³⁷, the W state is prepared using NVEs (quantum memories) instead of cavity photons. Thus, the prepared W state can be stored for a long time due to the long decoherence time of the NVEs. (ii) Because cavity photons are virtually excited for most of the operation time, decoherence caused by the cavity decay and the unwanted inter-cavity cross talk is greatly suppressed. (iii) Each NVE is initially in the ground state. Thus, there is no need to initially prepare each NVE in a coherent state, which should greatly reduce its experimental difficulty. (iv) Moreover, only one external-cavity coupler qubit is needed, which simplifies the circuit. This method is quite general and can be applied to prepare the proposed W state with atomic ensembles or other spin ensembles based on cavity/circuit QED.

There are several additional motivations of this proposal:

1. Planar superconducting TLRs with internal quality factors above one million (Q > 10⁶) have been recently reported⁵², for which the lifetime of microwave photons can reach ~1 ms. Comparably, a lifetime of ~1 s for an NVE has been experimentally reported⁵³. Hence, a NVE is a good memory element for storing quantum states, superior to using cavity photons as memories.

2. By local operation, the prepared W state of the NVEs can be mapped onto the cavities (see the “Quantum state transfer” subsection).

3. The NVEs could be prepared in the ground state at a 40–50 mK or higher temperature⁴²,⁴⁴. The strong coupling of a superconducting qubit with a microwave resonator (e.g., g/2π ~ 360 MHz for a transmon qubit coupled to a TLR⁵⁴,⁵⁵) has been reported in experiments, and the strong coupling (~11 MHz) of an NVE to a TLR has recently been experimentally demonstrated⁴⁴. Moreover, superconducting qubits, capacitively or inductively coupled to TLRs¹³,⁵⁶,⁵⁷,⁵⁸,⁵⁹,⁶⁰,⁶¹,⁶²,⁶³,⁶⁴,⁶⁵,⁶⁶,⁶⁷,⁶⁸, were previously employed for QIP. Hence, the model considered in this work is reasonable and physical.

Note that based on circuit QED, a number of proposals have been presented for creating entangled states (e.g., Bell states, NOON states, and GHZ states) of microwave photons distributed in different TLRs/cavities⁵⁷,⁵⁸,⁶⁰,⁶³,⁶⁵,⁶⁷. Instead of preparing entangled states of cavity microwave photons, this work focuses on preparing the NVEs in a continuous-variable W-type entangled coherent state.

In this work we will also discuss possible experimental implementation of our proposal and numerically calculate the operational fidelity for generating a W-type entangled coherent state of three NVEs. Our numerical simulation shows that highly-fidelity implementation of W-type entangled coherent states with three NVEs is feasible with rapid development of circuit QED technology. The numerical calculations in this work were performed using the QuTiP software⁶⁹,⁷⁰.

Results

W-state preparation

Consider a hybrid system consisting of a coupler qubit A and three cavities, each hosting a qubit and an NVE [Fig. 1(a)]. Each cavity here is a one-dimensional transmission line resonator. The qubit and the NVE placed in cavity j are labelled as qubit j and NVE j (j = 1, 2, 3). The two levels of qubit A are denoted as |g〉_A and |e〉_A, while those of qubit j as |g〉_j and |e〉_j. The coupling and decoupling of each qubit from its cavity (cavities) can be achieved by prior adjustment of the qubit level spacings or the cavity frequency. For superconducting devices, their level spacings can be rapidly (within 1–3 ns⁶⁵,⁷¹,⁷², (Yu, Y. & Han, S. private communication.)) adjusted by varying external control parameters (e.g., via changing the external magnetic flux threading the superconducting loop of phase, transmon, Xmon or flux qubits; see, e.g. refs ⁷¹ and ^{73, 74, 75, 76, 77, 78, 79, 80}). In addition, as described in the Methods section, the coupling and decoupling of an NVE with a cavity can be made by rapidly adjusting the cavity frequency⁸¹,⁸².

Assume that the qubits, cavities, and NVEs are initially decoupled from one another [Fig. 1(b)]. The state-preparation procedure consists of four basic operations followed by a measurement on the state of each intra-cavity qubit, which is described below:

Step 1. Adjust the level spacings of the coupler qubit A so that it is resonantly coupled to each cavity [Fig. 1(c)]. Assume that the coupling constant of qubit A with cavity j is . In the interaction picture, the Hamiltonian reads

where and are the raising and lowering operators for qubit A, while a_j and are the annihilation and creation operators for the mode of cavity j (j = 1, 2, 3). We set , which can be met by a prior design of the sample with appropriate values of the coupling capacitance C₁, C₂, and C₃. Assume now that qubit A is initially in the state |e〉_A and each cavity is initially in the vacuum state. It is easy to show that the state of the system, under the Hamiltonian (2), evolves into

Here, the state |W_2,1〉_c of the three cavities (1, 2, 3) is given by

where |i〉| j〉|k〉 is the abbreviation of the state of cavities (1, 2, 3) with i, j, k ∈ {0, 1}; |0〉 and |1〉 represent the vacuum state and the single-photon state, respectively. From Eq. (3), it can be seen that when the interaction time equals to , we can create the state |W_2,1〉_c of the three cavities (1, 2, 3). Note that the coupler qubit A is in the ground state |g〉_A after the operation here and will remain in the ground state |g〉_A during the rest of the operations below.

Step 2. Adjust the level spacings of qubit A back to the original level structure such that it is decoupled from each cavity. In addition, adjust the level spacing of intra-cavity qubit j such that qubit j is resonantly coupled to cavity j [Fig. 1(d)]. The resonant coupling constant of qubit j with cavity j is denoted as g_rj. In the interaction picture, the Hamiltonian can be written as

where and are the raising and lowering operators for qubit j. For simplicity, we set g_r1 = g_r2 = g_r3 = g_r, which can be achieved by tuning the level spacings of qubit j or adjusting the position of qubit j in cavity j (j = 1, 2, 3). It is easy to show that under this Hamiltonian (5), the time evolution of the state of qubit j and cavity j is described by

where and are the photon-number states of cavity j. Assume now that qubit j is initially in the state |g〉_j. Choosing t = π/(2g_r), one obtains the transformation . As a result, the state |W_2,1〉_c of the three cavities turns into the following state of the three intracavity qubits (1, 2, 3)

where |i〉| j〉|k〉 is the abbreviation of the state |i〉₁| j〉₂|k〉₃ of intracavity qubits (1, 2, 3) with i, j, k ∈ {g, e}. It should be noted that each cavity returns to its original vacuum state after the operation here and will remain in the vacuum state during the following operations.

The condition g_r1 = g_r2 = g_r3 = g_r is unnecessary. For the case of g_r1 ≠ g_r2 ≠ g_r3, one can still obtain the state (7) from the state (4), by adjusting the level spacings of qubit j to bring qubit j on resonance with cavity j for a time t_j = π/(2g_rj) (j = 1, 2, 3).

Step 3. Adjust the level spacings of intracavity qubits back to the original level configuration, such that they are decoupled from their cavities. Then apply a classical pulse to qubit j. The pulse is resonant with the |g〉_j ↔ |e〉_j transition of qubit j [Fig. 1(e)]. The interaction Hamiltonian in the interaction picture is given by

where and ϕ are the Rabi frequency and the initial phase of the pulse, respectively. Set which can be readily met by adjusting the pulse intensities. It is easy to find that under the Hamiltonian (8), one can obtain the following rotations

We set t = π/(4Ω_eg) and ϕ = −π/2 to pump the state |e〉_j to |−〉_j and |g〉_j to |+〉_j. Here, are the rotated basis states of qubit j. Thus, the state (7) becomes

Step 4. Adjust the frequency of each cavity such that cavity j interacts with qubit j and NVE j [Fig. 1(f)]. Then apply a classical pulse (with frequency ω_j equal to ) to qubit j [Fig. 1(f)]. Here, is the |g〉 ↔ |e〉 transition frequency of qubit j. The system Hamiltonian in the interaction picture yields

where and are the frequency detunings ( being the frequency of cavity j while being the frequency of a bosonic mode describing NVE j), b_j is the bosonic operator for NVE j, g_j is the off-resonant coupling constant of qubit j with cavity j, is the coupling constant of NVE j with cavity j, and Ω_j is the Rabi frequency of the pulse applied to qubit j [Fig. 1(f)]. Note that the second term of Eq. (11) describes three NVEs interacting with their respective cavities (see the Methods section). In a rotated basis {|+〉_j, |−〉_j}, one has and , where , and . Hence, the Hamiltonian (11) can be expressed as

In a new interaction picture under the Hamiltonian , one obtains from Eq. (12)

In the strong-driving regime , one can apply a rotating-wave approximation and eliminate the terms that oscillate with high frequencies. Thus, the Hamiltonian (13) becomes

Consider now the large detuning conditions and It is straightforward to show that the Hamiltonian (14) changes to (for details, see ref. 84)

where and . As mentioned previously, each cavity is in the vacuum state after the first three steps of operation above. In this case, the Hamiltonian (15) reduces to

where the first term is the vacuum contribution Stark shift of NVEs, while the second term describes the coupling between qubit j and NVE j, mediated by the mode of cavity j. Because of using the large detuning technique, the effective coupling λ_j is smaller than g_j or by at least one order of magnitude. Accordingly, the operation time for this last step of the operation (essentially based on a model via virtual transitions) would become longer by one order of magnitude, when compared with each of the first three steps of operation via resonant interaction.

In a new interaction picture under the Hamiltonian , the effective Hamiltonian (16) can be rewritten as

where .

Let us now assume that the NVEs are initially in the state . Thus, under the Hamiltonian (17), the joint state of the three intracavity qubits and the three NVEs evolves into

with

Here, () is a coherent state and we have set α₁ = α₂ = α₃ = α for simplicity (which can be met for identical qubits, NVEs, and cavities). After returning to the original interaction picture by performing a unitary transformation , the state (18) becomes

where a common phase factor is discarded, () is a coherent state, and

for

The condition (21) is automatically satisfied for identical NVEs and cavities. The state (20) can be expressed as

where |W₁〉, |W₂〉, |W₃〉 and |W₄〉 are the macroscopic W-type entangled coherent states of three NVEs, given by

Now a measurement is separately performed on each intra-cavity qubit along a measurement basis {}. If qubits (1, 2, 3) are measured in the state (i) |e〉|e〉|e〉 or |g〉|g〉|g〉, (ii) |e〉|e〉|g〉 or |g〉|g〉|e〉, (iii) |e〉|g〉|g〉 or |g〉|e〉|e〉, and (iv) |e〉|g〉|g〉 or |g〉|e〉|e〉, one can see from Eq. (23) that the three NVEs are respectively prepared in the W states |W₁〉, |W₂〉, |W₃〉 and |W₄〉, respectively.

This method can be extended to a more general case. Consider a hybrid system composed of n cavities, each hosting a qubit j and an NVE j (j = 1, 2 ···n) and connected to a coulper qubit A, as shown in Fig. 2. Assume that the initial state of the system is . Employing the four-step procedure described above, it is straightforward to show that the n NVEs can be prepared in a W-type entangled coherent state. Let m_j = 0 represent qubit j being measured in the state |g〉, while m_j = 1 indicates qubit j being measured in the state |e〉. If the n intracavity qubits are measured in the state |m₁m₂ ···m_n〉, the n NVEs will be prepared in the macroscopic W-type entangled coherent state

Before ending this section, several points need to be addressed as follows:

1. From the description given above, one can see that only resonant interactions are used for the first three steps of operation, which can thus be completed within a very short time (e.g., by increasing the pulse Rabi frequencies and the qubit-cavity coupling constants). In contrast, the last step of operation employs a large detuning, leading to a relatively long operation time. However, cavity photons were virtually excited during this step of operation. Hence, in the present proposal each cavity remains in a vacuum state for most of the operation time.

2. Coupling/decoupling the NVE with the cavity in step 4 requires dynamic tuning of the cavity frequency⁸¹,⁸². It is known that tuning the cavity frequency (e.g., via the insertion of the flux-tunable inductor in the resonator) can significantly reduce the quality factor of the cavity, which would decrease the fidelity of the prepared W state. Alternatively, to have the cavities coupled with or decoupled from the NVEs, one can choose to adjust the level spacings of the NVEs (e.g., by varying the external magnetic fields applied to the NVEs⁴⁸,⁸⁵). However, it is an experimental challenge to change the NVE level spacings quickly. Typically, it takes more than 1 ms to adjust the NVE level spacings (Saito, S. private communication.), which would significantly prolong the entire operation time and thus the operation fidelity would be expected to be low due to decoherence. Therefore, we choose using the cavity of adjustable frequency in this paper.

3. As shown above, the intracavity-qubit W state of Eq. (7) can be produced within a very short time, because the first two steps of operation, for producing this intracavity-qubit W state (7), employ resonant interactions. Alternatively, this intracavity-qubit W state (7) can be prepared via a detuned interaction between the coupler qubit A and each cavity¹³,⁶⁴,⁶⁸. Thus, there are no cavity photons excited during the entire state preparation. However, the time required for preparing the W state (7) becomes much longer due to the use of a detuned interaction, and thus decoherence from the qubits may pose a significant problem.

4. Placing a qubit in each cavity [Fig. 1(a)] is necessary in view of energy conservation. During the last step, each cavity remains in a vacuum state and thus there is no energy transfer from each cavity onto the NVEs. Note that the intracavity qubits are the ones that absorb energy from the pulses applied to them and then transfer their energy to the NVEs through interaction with the NVEs. Thus, in spite of initially being in the ground state, the NVEs can be prepared in a W-type entangled coherent state.

5. As discussed previously, a measurement of the states of each intra-cavity qubit is needed during preparation of the W-class entangled coherent states. To the best of our knowledge, all existing proposals for creating entangled coherent states of two components and based on cavity QED or circuit QED require a measurement on the states of auxiliary qubits or qutrits⁶³,⁸⁵,⁸⁶,⁸⁷,⁸⁸,⁸⁹,⁹⁰,⁹¹,⁹²,⁹³,⁹⁴.

Possible experimental implementation

Superconducting qubits play important roles in quantum information processing⁷³,⁷⁵,⁷⁶,⁹⁵,⁹⁶,⁹⁷. In addition, circuit QED is a realization of the physics of cavity QED with superconducting qubits or other solid-state devices coupled to a microwave cavity on a chip and has been considered as one of the most promising candidates for quantum information processing⁷⁵,⁷⁶,⁹⁵,⁹⁶,⁹⁷,⁹⁸,⁹⁹,¹⁰⁰. Above, we considered a general type of qubit for both the intracavity qubits and the coupler qubit. As an example of experimental implementation, let us now consider each qubit as a superconducting transmon qubit.

The dynamics of the lossy system, with finite qubit relaxation and dephasing and photon lifetime included, is determined by the following master equation

where H_Ik is either H_I1, H_I2, H_I3, or H_I4; j represents qubit j (j = 1, 2, 3); and , with . In addition, κ_j is the decay rate of cavity j, is that of NVE j, γ_j (γ_A) is the energy relaxation rate of the level of qubit j (A), and γ_j,φ (γ_A,φ) is the dephasing rate of the level of qubit j (A).

The fidelity of the operation is given by¹⁰¹

where is the output state of an ideal system (i.e., without dissipation and dephasing), while ρ is the output-state density operator of the system when the operations are performed in a realistic physical system.

We now numerically calculate the fidelity of operation. Since the first three steps employ resonant interactions, we will look at the operational fidelity for each of these steps to see how short one should make the typical operation time for each step to combat decoherence while still being able to generate the entanglement with high fidelity. For simplicity, we will consider the ideal output state of the previous step of operation as the input state of the next step of operation when we analyze the operational fidelities for the first three steps. In addition, we will investigate the fidelity for the entire operation, which will be calculated by numerically solving the master equation with the initial state of the whole system as an input, but without making any approximation. Without loss of generality and for simplicity, we will consider identical transmon qubits, cavities, and NVEs. In this case, we have g_Aj = g_A, g_rj ≡ g_r, g_j ≡ g, and (j = 1, 2, 3). We set and Ω_j = Ω (j = 1, 2, 3). The decoherence times of transmon qubits and NVEs used in the numerical simulation are: μs, μs (which is a conservative estimate compared with those reported in experiments¹⁰²,¹⁰³,¹⁰⁴). In addition, we choose μs and ms in the numerical simulation (j = 1, 2, 3).

A. Fidelity for the first three steps

The operation fidelities are plotted in Fig. 3(a–c), which are for step 1, step 2, and step 3, respectively. Figure 3 shows that the fidelity for step 1, step 2, or step 3 increases drastically with g_A, g_r, or Ω_eg and reaches a high value for g_A/(2π), g_r/(2π), Ω_eg/(2π) ∈ [5 MHz, 50 MHz], which corresponds to the operation time ~3–30 ns. The analysis given here demonstrates that in order to combat decoherence while obtain the entanglement with a high fidelity ~1, one should make the typical operation time within a few nanoseconds for each of the first three steps, and a high fidelity ≥0.998 can be achieved even by increasing the operation time to ~30 ns.

Figure 3.

(a) Fidelity for step 1. (b) Fidelity for step 2. (c) Fidelity for step 3.

B. Fidelity for the entire operation

The fidelity for the entire operation is calculated based on Eq. (27), where the ideal output state is [with given by Eq. (20) or Eq. (23)] and ρ is obtained by numerically solving the master equation (26) for an initial input state We choose g_A/(2π) = 50 MHz, g_r/(2π) = g/(2π) = 5 MHz, and g_b/(2π) ~ 4 MHz⁴⁴. We here select g_r = g because the resonant coupling constant g_r and the off-resonant coupling constant g are both the same order of magnitude for superconducting qubits. Other parameters used in the numerical simulation are: Ω_eg/(2π) = 50 MHz, Ω/(2π) = 100 MHz (available in experiments¹⁰⁵,¹⁰⁶), and (obtained by numerically optimizing the system parameters). With the choice of these parameters, the fidelity versus is plotted in Fig. 4, which demonstrates that for D ~ 9, a high fidelity ~93.2% can be achieved for the state with |β| = 1.2. For D ~ 9, the entire operation time is estimated to be ~1.14 μs, much shorter than the decoherence times of transmon qubits and NVEs used in our numerical simulation but a little longer than the cavity decay time. Figure 4 also shows that the fidelity heavily depends on D (or the detuning ). The fidelity reaches its maximum as D increases to 9. However, it drops down when D becomes larger than 9. This means that further increasing the detuning will have an adverse effect on the fidelity. The interpretation for this is: As the detuning becomes larger than the optimum value (2π × 36 MHz) (i.e., the value where the large detuning is well satisfied), the NVE-cavity coupling becomes weaker, which increases the operation time and thus the effect of decoherence from transmon qubits and NVEs on the fidelity becomes more apparent.

Figure 4. Fidelity versus reduced detuning .

The red squares correspond to the case without considering the errors and decoherence for the first three-step operation, while the blue dots correspond to the case after the errors and decoherence for the first three-step operation are taken into account. The parameters used here are described in the text.

Note that although the entire operation time is longer than the cavity decay time used in our numerical simulations, the effect of the cavity decay on the fidelity is negligible. This is because: the first three steps are completed within a very short time due to using the resonant interaction, and (as illustrated in Fig. 5) the number of photons occupied in each cavity during the last step of operation is quite low due to using a large-detuning technique. Indeed, to reduce decoherence from the cavity decay, one can employ a longer cavity-decay time in the numerical simulation, which however would require cavities with a higher-Q quality factor and thus may pose a challenge in experiments.

Figure 5.

(a) The operational fidelity , the amplitude |β| (or |−β|), and the photon number of each cavity versus t − t₀ (i.e., the time required for the last step of operation), for the case of considering the identical NVE-cavity coupling strengths and NVE-cavity frequency detunings. The blue, black, and green curves are plotted for reduced detuning and parameters used in Fig. 4. The blue curve represents the operational fidelity, which is calculated for an ideal state () with |β| = 1.2. For t − t₀ = 1.08 μs (i.e., the time required for preparing the state with |β| = 1.2 during the last step of operation), the fidelity reaches the maximum ~93.2%. The black curve represents the value of |β|/2 or |−β|/2. The green curve indicates the photon number (enlarged 10 times) of each cavity. (b) The operation fidelity versus t − t₀ for inhomogeneous NVE-resonator couplings and unequal NVE-resonator frequency detunings. The brown, purple, and red curves represent the operational fidelities for NVE spin coherence times t = 10 μs, 100μs, and 1 ms, respectively. The maximum fidelities (corresponding to the peak values of the brown, purple, and red curves) are, respectively, 81.5%, 89.0%, and 90.3%.

Figure 5 is plotted by choosing the detuning D = 9 and using the same parameters for Fig. 4. For simplicity, Fig. 5 only shows the curves corresponding to the operation time t − t₀ required for the last step of operation. Here, t is the entire operation time while t₀ is the time required for the first three steps of operation. For the values of g_A, g_r, and Ω_eg chosen above, t₀ is ~36 ns. The blue curve represents the fidelity, which is calculated for an ideal state () with |β| = 1.2. The black curve represents the value of |β|/2 or |−β|/2. The green curve indicates the average photon number for each cavity. The blue curve indicates that the fidelity increases when t − t₀ approaches 1.08 μs (which is the time required for the last step of operation for preparing the desired state with |β| = 1.2). The maximum fidelity depicted by the blue curve of Fig. 5 is in good agreement with that shown in Fig. 4 for D = 9. In addition, the green curve shows that the average number of photons excited in each cavity is less than 0.02, implying that the cavity photons are almost not excited during the last step of operation.

In a realistic situation, it may be a challenge to obtain identical NVE-resonator frequency detunings and homogeneous NVE-resonator coupling strengths. Thus, we numerically calculate the fidelity by setting and , and . As shown in the brown, purple, and red curves of Fig. 5, one can see that a high fidelity 81.5%, 89.0%, 90.3% can be obtained for the NVE spin coherence times 10 μs, 100 μs, and 1 ms, respectively.

According to experimental reports⁸¹,⁸², the cavity frequency can be rapidly adjusted by Δω_c/(2π) = 500~740 MHz. As a conservative consideration, for Δω_c/(2π) = 500 MHz, the detuning changes to MHz, which can be further written as, e.g., for the identical NVE-cavity coupling strengths g_bj/(2π) ≡ g_b/(2π) = 4 MHz chosen above. This result shows that the decoupling of the cavities with the NVEs, which was required during the W-state preparation, can be well met by adjusting the cavity frequency. As discussed previously, the coupling or decoupling of the qubits with the cavities can be readily made by adjusting the level spacings of the qubits.

T₁ (energy relaxation time) and T₂ (dephasing time) can be made to be on the order of 20–80 μs for state-of-the-art superconducting transmon devices¹⁰²,¹⁰³,¹⁰⁴. The typical transition frequency of a transmon qubit is between 2 and 10 GHz⁷⁷,¹⁰⁷. As an example, consider each cavity of frequency ν_c ~ 5 GHz. Hence, for the used in the numerical calculation, the required quality factor of each cavity is Q_j ~ 3.1 × 10⁴, which is accessible in experiments because a quality factor Q ~ 5 × 10⁴ for CPW resonators with loaded NVEs has been experimentally demonstrated⁴⁴. The analysis given here shows that a high-fidelity implementation of the three-NVE W-type entangled coherent state , , , or described by Eq. (24) is feasible with rapid development of circuit QED techniques.

Quantum state transfer

Consider a cavity and an NVE inside the cavity. Based on Eq. (35) (see the Methods section), the NVE-cavity interaction Hamiltonian can be written as

where we set δ = ω_b − ω_c = 0. Assume now that the initial state of the cavity and the NVE is given by |0〉_c⊗|β〉_NVE, where |0〉_c is the vacuum state of the cavity while |β〉_NVE is the coherent state of the NVE, given by . In terms of , one can describe the system initial state as

Making use of the Hamiltonian (28), we can obtain the transformations . For g_bt = π/2, one has . Under the Hamiltonian (28) and after an evolution time t = π/(2g_b), the state of the system can be written as

where we have used and .

In the same manner, after an evolution time t = π/2g_b, the state of the cavity and the NVE is transformed to . Given the above results, one can transfer a macroscopic W-type entangled coherent state from the NVEs into the cavities. For instance, the above state of the three NVEs is transferred onto the three cavities, becoming

Discussion

We should mention that in 2015, Song et al. proposed a scheme to generate a GHZ-type macroscopic entangled coherent state of NVEs that are coupled to a superconducting flux qubit⁸⁵. In contrast, we here proposed a protocol for creating a macroscopic W-type ECS with NVEs coupled to different cavities. Hence, compared with⁸⁵, our proposal is for a different system and it differs in both the prepared states and the coupling structure.

A method has been presented to generate a continuous-variable W-type entangled coherent state of NVEs in circuit QED. As shown above, this proposal offers some distinguishing features and advantages. The prepared W state of NVEs can be mapped onto the cavities by local operations. Our numerical simulations show that the high-fidelity implementation of W-type entangled coherent states with three NVEs is feasible with rapid development of circuit QED technology.

Methods

NVE-cavity interaction Hamiltonian

As shown in Fig. 6(a), the energy levels of an NV center consist of a ground state ³A, an excited state ³E and a metastable state ¹A. Both ³A and ³E are spin triplet states while the metastable ¹A is a spin singlet state¹⁰⁸,¹⁰⁹. The NV center has an S = 1 ground state with zero-field splitting D_gs/(2π) = 2.88 GHz between the |m_s = 0〉 and |m_s = ±1〉 levels [Fig. 6(a)]. By applying an external magnetic field along the crystalline axis of the NV center⁴⁸,⁸⁴, an additional Zeeman splitting between |m_s = ±1〉 sublevels occurs [Fig. 6(b)].

Figure 6.

(a) Schematic diagram of electronic and spin energy levels of a nitrogen-vacancy center. (b) The ground electronic-spin levels of an NV center in the presence of an external magnetic field parallel to the crystalline axis. Here B and E represent the magnetic field and energy, respectively. (c) Illustration of the cavity decoupled from the NV center. Here, ω_c is the cavity frequency, while ω_0,−1 (ω_0,+1) is the energy gap between the |m_s = 0〉 and |m_s = −1〉 (|m_s = +1〉) levels of the NV center. The cavity frequency ω_c is sufficiently larger than ω_0,+1 and ω_0,−1, such that the cavity mode is highly detuned (decoupled) from both the |m_s = 0〉 ↔ |m_s = −1〉 transition and the |m_s = 0〉 ↔ |m_s = +1〉 transition. (d) Illustration of the cavity being coupled to the |m_s = 0〉 ↔ |m_s = +1〉 transition with a detuning δ = ω_c − ω_0,+1, but decoupled from the |m_s = 0〉 ↔ |m_s = −1〉 transition of the NV center.

If we need to eliminate the coupling of the cavity with the NV center, one can adjust the cavity frequency ω_c to have ω_c sufficiently larger than ω_0,+1 and ω_0,−1, such that the cavity mode is highly detuned (decoupled) from both the |m_s = 0〉 ↔ |m_s = −1〉 transition and the |m_s = 0〉 ↔ |m_s = +1〉 transition [Fig. 6(c)]. Here, ω_0,+1 (ω_0,−1) is the transition frequency between the two levels |m_s = 0〉 and |m_s = +1〉 (|m_s = −1〉). On the other hand, one can adjust the cavity frequency such that the cavity mode is coupled with the transition between the ground level |m_s = 0〉 and the excited level |m_s = +1〉, but still decoupled from the transition between the two levels |m_s = 0〉 and |m_s = −1〉 [Fig. 6(d)]. Note that for a superconducting transmission line resonator, the rapid tuning of cavity frequencies by a few hundred MHz in 1–2 nanoseconds has been demonstrated in experiments⁸¹,⁸²). During the W-state preparation described in the Results section, we assume that the level splitting of the NV center is fixed.

An NV center is usually treated as a spin while an ensemble of NV centers is treated as a spin ensemble (i.e., an NVE). Let an NVE be placed at an antinode of a single mode of the electromagnetic field. When the cavity is coupled to the |m_s = 0〉 ↔ |m_s = +1〉 transition, but decoupled from the |m_s = 0〉 ↔ |m_s = −1〉 transition [Fig. 6(d)], the system Hamiltonian in the interaction picture reads (in units of ħ = 1)

where δ = ω_c − ω_0,+1, ω_c is the eigenfrequency of the cavity mode, a (a^†) is the corresponding annihilation (creation) operator of the cavity mode, and are the raising and lowering operators for the kth spin, and g_k is the coupling strength between the cavity and the kth spin. We then define a collective operator

with , and is the root mean square of the individual couplings.

Under the condition of a large N and a very small number of excited spins (compared to the number N), b^† behaves as a bosonic operator and the spin ensemble behaves as a bosonic mode. Thus, we have [b, b^†] ≈ 1, and b^†b|n〉_b = n|n〉_b⁴⁸,¹¹⁰, where

with |0〉_b = |m_s = 0〉₁|m_s = 0〉₂ ···|m_s = 0〉_N. It is easy to verify that the frequency ω_b of the bosonic mode describing the NVE is equal to the transition frequency ω_0,+1 between the ground level |m_s = 0〉 and the excited level |m_s = +1〉 of each spin (i.e. ω_b = ω_0,+1). For simplicity we have defined |m_s = +1〉 = |+1〉 and |m_s = 0〉 = |0〉.

Therefore, the Hamiltonian (32) can be further rewritten as

with . Based on Eq. (35), one can find that for the case of three NVEs each placed in a cavity, the Hamiltonian for the three NVEs interacting with their respective cavities would be the second term of Eq. (11).

NVE-cavity coupling selection

During the last step of the W state preparation, we would require the coupling of each cavity with the |m_s = 0〉 ↔ |m_s = +1〉 transition while decoupling each cavity from the |m_s = 0〉 ↔ |m_s = −1〉 transition. The advantage of this is that the created W state has a mode frequency equal to ω_0,+1, which is adjustable by varying the magnetic field applied to the NVEs [Fig. 6(c,d)]. Instead of using the coupling of each cavity with the |m_s = 0〉 ↔ |m_s = +1〉 transition, one can employ the coupling of each cavity with the |m_s = 0〉 ↔ |m_s = ±1〉 transition (i.e., the transition between the ground state |m_s = 0〉 and the degenerate excited states |m_s = ±1〉). However, there is an inevitable shortcoming, i.e., the created W state has a fixed mode frequency, which is equal to ω_0,±1 = 2π × 2.88 GHz [Fig. 6(a)] and thus cannot be adjusted.

Additional Information

How to cite this article: Liu, T. et al. Generation of a macroscopic entangled coherent state using quantum memories in circuit QED. Sci. Rep. 6, 32004; doi: 10.1038/srep32004 (2016).