Amplification of the coupling strength in a hybrid quantum system

Realization of strong coupling between two different quantum systems is important for fast transferring quantum information between them, but its implementation is difficult in some hybrid quantum systems. Here we propose a scheme to enhance the coupling strength between a single nitrogen-vacancy center and a superconducting circuit via squeezing. The main recipe of our scheme is to construct a unitary squeezing transformation by directly tuning the specifically-designed superconducting circuit. Using the experimentally accessible parameters of the circuit, we find that the coupling strength can be largely amplified by applying the squeezing transformations to the system. This provides a new path to enhance the coupling strengths in hybrid quantum systems.


I. INTRODUCTION
Hybrid quantum systems, with the goal of harnessing the advantages of different subsystems to better explore new phenomena and potentially bring about novel quantum technologies (see Ref. [1,2] for a review), can have versatile applications in quantum information. Among various hybrid systems, the nitrogen-vacancy (NV) center in a diamond coupled to a superconducting circuit has attracted special attention (see, e.g., Refs. [3][4][5][6][7][8][9][10][11]), because it has distinct advantages, such as high tunability, long coherence time, and stable energy levels. In addition, superconducting circuits exhibit macroscopic quantum coherence, promise good scalability, and can be conveniently controlled and manipulated via external fields (see, e.g., Refs. [12,13]).
However, the coupling strength between a single NV center and a superconducting circuit is too small to coherently exchange mutual quantum information [6,7,9,14]. One solution to overcome this drawback is the use of an ensemble containing a large number (e.g., N ∼ 10 12 ) of NV centers, where two lowest collective excitation states of the ensemble encode a qubit (i.e., a pseudospin). Thus, the coupling strength between the NVcenter ensemble and the superconducting circuit can be effectively enhanced by a factor of √ N [15][16][17]. This makes it possible to reach the strong-coupling regime of the hybrid system. However, it is difficult for the ensemble to implement direct single-qubit manipulation and also the coherence time is greatly shortened due to the inhomogeneous broadening [18][19][20]. Therefore, significantly coupling a single NV center to a superconducting circuit has been longed for.
Here we propose an experimentally feasible method to * Corresponding author. jqyou@zju.edu.cn effectively amplify the coupling strength between a single NV center and a superconducting circuit. The main recipe of our scheme is to prepare the unitary one-mode squeezing transformations. After applying these squeezing transformations to the hybrid system, the effective coupling strength can be enhanced by two orders of the magnitude using the experimentally accessible parameters of the circuit. The methodology dates back to the amplification of Kerr effect [21], where a rather complicated circuit was exploited. Recently, a simpler squeezing-transformation circuit has been proposed for the cavity mode to amplify the coupling in an optomechanical system [22], but the generation of the squeezing terms in the system Hamiltonian requires an additional driven nonlinear medium. Here we specifically design a superconducting circuit that enables one to engineer the squeezing transformations by directly tuning the circuit.
The paper is organized as follows. Section II introduces the Hamiltonian of the proposed hybrid quantum system. In Sec. III, we design two basic gates by tuning the magnetic flux through the smaller loop of the circuit. In Sec. IV, we use these two basic gates to construct the squeezing operator and then apply the squeezing transformations to amplify the coupling strength between the single NV center and the superconducting circuit. Finally, we give a brief discussion and conclusion in Sec. V.

II. THE HYBRID QUANTUM SYSTEM
We propose a hybrid system which is composed of a superconducting loop embedding a superconducting quantum interference device (SQUID) and encircling a single NV center (see Fig. 1). Here we consider a symmetric SQUID with identical junction capacitances and Josephson coupling energies, i.e., C 1 = C 2 = C, and E J1 = E J2 = E J . In addition, we suppose that the main arXiv:1709.06433v3 [quant-ph] 16 Apr 2018 loop of the superconducting circuit is fabricated with a non-negligible inductance L, while the SQUID loop is small enough to have a negligible inductance. Also, two static magnetic fields in opposite directions are applied, respectively, to the small and main loops. The fluxoid quantization conditions for these two loops are where f s(m) = Φ s(m) /Φ 0 , with Φ 0 = h/2e being the flux quantum, ϕ i (i = 1, 2) is the phase drop across the ith Josephson junction in the SQUID, and I is the total circulating current in the main loop.
The kinetic energy of the superconducting circuit corresponds to the electrostatic energy stored in the capacitors [23]: T = 1 2 C(V 2 1 +V 2 2 ), where V i = (Φ 0 /2π)φ i is the voltage across the ith Josephson junction in the SQUID. Using the fluxoid quantization conditions in Eq. (1), this kinetic energy can be written as where ϕ ≡ (ϕ 1 +ϕ 2 )/2. We consider a static external flux for Φ s , soḟ s = 0. Then, the kinetic energy T is reduced to T = C(Φ 0 /2π) 2φ2 . Also, it follows from Eq. (1) that The inductive energy related to the inductance L is given by where E L = Φ 2 0 /(8π 2 L). When including this inductive energy, the total potential energy of the superconducting circuit is where E J (f s ) = 2E J cos(πf s ) is the flux-dependent effective Josephson energy. The Lagrangian of the superconducting circuit is L = T − U . Assigning ϕ as the canonical coordinate, we have the canonical momentum . Hence the Hamiltonian of the superconducting circuit is given by where E c = (2e) 2 /2C is the charging energy of a single Cooper pair and n = −i∂/∂ϕ is the number operator of Cooper pairs.
An NV center consists of a substitutional nitrogen atom next to a vacancy in the diamond lattice [24]. It has a spin triplet ground state and a zero-field splitting D ≈ 2.87 GHz [25] between the sublevels with the spin z components m s = 0 and m s = ±1. The strain-induced splitting is negligible in comparison with the Zeeman effect [26]. In our proposal, the crystalline axis of the NV center is set as the z direction. By applying a weak static magnetic field B ext z along the z direction, the two degenerate sublevels m s = ±1 are split due to the Zeeman effect. The sublevels m s = 0 and −1 can be well isolated from other levels by tuning B ext z and they act as a pseudo-spin. The pseudo-spin Hamiltonian is (we set = 1 hereafter) where is the energy difference between the lowest two sublevels with m s = 0 and −1, respectively. The corresponding Pauli operators are τ ≡ (τ x , τ y , τ z ). As shown in Fig. 1, a single NV center is located at the coordinate z NV , starting from the left edge of the main loop and along the z direction on the midline. The interaction Hamiltonian H int of the hybrid system is [27] where the magnetic field B SC x (z NV ) is associated with the persistent current in the main loop. According to the Biot-Savart law, B SC x (z NV ) can be written as where The total Hamiltonian H of the hybrid quantum system is given by H = H SC + H NV + H int .

III. TWO BASIC GATES
We tune the external magnetic field B ext z to have ω NV = 0, so as to achieve the two basic gates for constructing squeezing operations. Denote ω sc as the transition frequency between the lowest two energy levels of the superconducting circuit and g as the coupling strength between the single NV center and the superconducting circuit. Now the two subsystems become effectively decoupled due to |g/(ω sc − ω NV )| = |g/ω sc | 1. Also, we tune the two external magnetic fields in opposite directions to satisfy Φ m − Φ s /2 = 0. Because ω NV = 0 and |g/ω sc | 1, the total Hamiltonian can be approximately written as Note that if L → 0, E L → ∞, so it is required that ϕ → 0 in Eq. (12). However, L = 0 for a realistic circuit. Thus, in this nonzero L case, the phase drop ϕ is not constrained by the loop inductance but mainly by the effective Josephson energy of the SQUID. By tuning the magnetic flux in the SQUID loop (now denoted as Φ (12) is reduced to a harmonic oscillator In second quantization, where m = 1/(2E c ), and is the angular frequency of the harmonic oscillator. The creation (annihilation) operator a † (a) obeys the bosonic commutation relation a, a † = 1, and the Hamiltonian in Eq. (13) can be written as Evolving the hybrid system for a time t, a quantum gate is achieved.
Here we consider a circuit with |E J (f s )/E c | 1. For this circuit, we can define a quantity α to characterize its anharmonicity: where E 01 is the energy level difference between the ground state energy E 0 and the first excited state energy E 1 of the circuit and E 12 is the energy level difference between the first and second excited states energies (E 1 and E 2 ) of the circuit. We use α F to denote the relative anharmonicity of the full Hamiltonian H sc in Eq. (12). Note that the phase ϕ is constrained to be small for the circuit with |E J (f s )/E c | 1, so we can write cos ϕ ≈ 1 − ϕ 2 /2! + ϕ 4 /4! as a good approximation. Then, the Hamiltonian H sc in Eq. (12) is reduced to For this approximated Hamiltonian, we use α A to denote its relative anharmonicity. In Fig. 2(a), we show the lowest three energy levels of the circuit as a function of the normalized magnetic flux f s in the SQUID loop, where the solid and dotted curves are calculated using the Hamiltonians in Eq. (12) and Eq. (19), respectively. The parameters are chosen to be E c = 0.12 GHz, E J = 58 GHz, and E L = 58.6 GHz (corresponding to L = 1.4 nH [28]). In Fig. 2(b), we also show the dependence of the relative anharmonicity α F (α A ) on the normalized magnetic flux f s . From these results, we can see that the approximate Hamiltonian in Eq. (19) well matches the Hamiltonian in Eq. (12). Away from Φ (0) s = Φ 0 /2, where the gate U 0 (t) is achieved, we again tune the magnetic flux in the SQUID loop (now denoted as Φ (1) s ) to, e.g., Φ s /Φ 0 = 0.9) to obtain another quantum gate. As shown in Fig. 2, this flux is sufficiently away from Φ (1) s ≈ 0.9Φ 0 . Also, the Hamiltonian (19) has a larger relative anharmonicity at this flux. In second quantization, the quartic anharmonicity ϕ 4 in Eq. (19) corresponds to the Duffing terms [29] (a + a † ) 4 , where the main contributions arise from the doublephoton scattering processes, a † aaa and a † a † a † a. We neglect the high-order four-photon scattering processes (a † ) 4 and a 4 , and use a mean-field approximation [ under the thermal equilibrium. At a very low temperature T (e.g., ∼ 20 mK ), ω sc /k B T 1 and therefore N a ≈ 0. The Hamiltonian (19) can then be reduced to where with Obviously, the two parameters ω 1 and η 1 are both controllable by the magnetic flux Φ s . Owing to the presence of the inductance L, E J (f s ) can reach the regime of E J (f s ) < 0 for a harmonic oscillator, where we only ensure E L + 1 2 E J (f s ) ≥ 0. However, the oscillator becomes unstable when E L + 1 2 E J (f s ) < 0. With the Hamiltonian in Eq. (20), by evolving the hybrid system for a time t, another quantum gate is then obtained. Note that a series of quantum gates are used to achieve the coupling amplification between the single NV center and the superconducting circuit (see the next section). To have a high fidelity for each quantum gate, sudden switching between successive gates is needed. In the present case, one should be able to fast tune the magnetic flux in the SQUID loop. Currently, it is easy to implement such sudden switch as quickly as in just ∼ 1 ns using conventional techniques (see, e.g., [31]). With fast developing quantum technologies, much quicker sudden switch is expected to be implementable.

A. Squeezing operator
To enhance the coupling between the single NV center and the superconducting circuit, we need to construct a photon-squeezing operator using the two propagators in Eq. (17) and Eq. (23). With the annihilation operator a and the creation operator a † , we can define three operators with commutation relations Therefore, three new operators in Eq. (24) can be regarded as the three generators of SU(1,1) group that is non-compact and does not have any finite unitary representation.
Following the method used in Ref. [21], we can write these operators, in a simple two-dimensional non-Hermitian representation, as where τ = (τ x , τ y , τ z ) are Pauli matrices, and then where γ 1 , γ 2 and γ 3 are three parameters. Also, the propagators in Eq. (17) and Eq. (23) can be rewritten, respectively, as (28) Then, we can employ these two propagators to generate another propagator U s (t) = exp(i2η 1 Γ 1 t), which can be approximately constructed using where t = ω 0 t/ω 1 , and M is the operation times of the gate U 0 (U 1 ). Considering the specific parameters used in Fig. 2, we numerically compare the four matrix elements of U s with those of U s and find that U s can be well approximated to U s in the regime of t/M ≤ 0.15 (see Fig. 3). Once obtaining the propagator U s , we can combine it and the achieved propagator U 0 to produce the desired squeezing operator where ω 1 t = (4k + 1)π/4, with k = 0, 1, 2, . . ., and η 2 = −η 1 t. In the present proposal, we choose the parameters of the circuit to obey E J (f s ) < 0, but E L + 1 2 E J (f s ) > 0, so as to have η 1 < 0. To make η 2 large, we can use a longer evolution time t.

B. Coupling enhancement
Outside the time periods for achieving squeezing transformations S and S † , we tune ω NV to be nonzero to satisfy the near-resonance condition ω NV ∼ ω sc . In contrast to the sudden switching between successive quantum gates, the tuning of ω NV to satisfy the near-resonance condition ω NV ∼ ω sc should be adiabatic. For the NV center given in Eq. (7) and Eq. (8), this adiabatic process can still be achievable by fast tuning the magnetic field on the NV center, because the level difference has a simple, linear dependence on the applied magnetic field and no level anticrossing occurs there.
During this period of near resonance, the coupling between the single NV center and the superconducting circuit becomes important. Also, the magnetic flux in the SQUID loop remains at Φ tonian H tot of the hybrid system reads where To estimate the value of the coupling strength g, we choose the experimentally accessible parameters of the superconducting circuit as in Fig. 2, i.e., E c = 0.12 GHz, E J = 58 GHz, and L = 1.4 nH. Using z NV = 0.01 µm and f (0) s = 0.5, we have g ∼ 2π × 10 kHz. Applying the unitary squeezing transformations S and S † to the Hamiltonian H tot , we obtain an effective Hamiltonian for the hybrid system, where ω eff = ω 0 cosh (4η 2 ) is the transformed frequency of the circuit, χ = 1 2 ω 0 sinh(4η 2 ) is the strength for squeezing photons, and is the effective coupling strength between the single NV center and the superconducting circuit. Obviously, it is enhanced exponentially by a factor of 2η 2 .
In Fig. 4, we plot the effective coupling strength g eff versus the normalized magnetic flux f s for different ratios of E L /E J , where E c = 0.12 GHz and t = 1 ns. It shows that the coupling enhancement is very sensitive to the ratio of E L /E J . Comparing four curves in Fig. 4, we can see that the coupling strength between the single NV center and the circuit can be enhanced by two orders of the magnitude. Namely, the enhanced coupling strength can approach a few magehertz. Also, it can be further improved by prolonging time or using larger E c , but a long time requires more operation times.

V. DISCUSSION AND CONCLUSION
Without coupling amplification, the coupling strength g between a single NV center and a superconducting circuit can reach g ≈ 2π × 10 kHz in our proposed hybrid quantum system. This value of the coupling strength was also estimated in Ref. [6]. Experimentally, the coupling strength between a single NV center and the superconducting circuit was reported to be g ≈ 8.8 kHz in Ref. [7] and g ≈ 4.4 kHz in Ref. [9]. Note that either the theoretically estimated or experimentally achieved value of the coupling strengh is larger than the decoherence rate of a single NV center (which is about γ NV ∼ 1 kHz in Ref. [32]), but it is still too weak in comparison with the decoherence rate of the superconducting circuit (which is γ sc ∼ 1 MHz in Ref. [7]). This indicates that the coupling between the single NV center and the superconducting circuit is in the weak-coupling regime, and the decoherence time of this hybrid system is limited by the decoherence time of the superconducting circuit (i.e., T sc = 1/γ sc ∼ 1 µs for γ sc ∼ 1 MHz [7]).
For the superconducting circuit system governed by the Hamiltonian (20), the energy difference of the lowest two levels is about ∆E = 1.5 GHz at the point f  [32], the typical π/2-and π-pulse durations of manipulating an NV center are 15 ns and 30 ns, respectively, which are much longer than the characteristic time T c . Thus, manipulating the frequency of an NV center (ω NV ) in resonance with the frequency of the superconducting circuit (ω sc ) can be nearly adiabatic. Moreover, the typical pulse durations [32] are much shorter than the decoherence time of the superconducting circuit. Tuning ω NV to be resonant with ω sc can be implemented before the system decoheres. In our superconducting circuit, the Josephson coupling energy E J should be larger than the charging energy E c and a superconducting loop is introduced. These characteristics are analogous to those of a flux qubit. A recent experiment [33] shows that the decoherence time of the flux qubit can be increased to 85 µs when shunting a large capacitor to the smaller Josephson junction of the circuit to reduce the effect of the charge noise [34]. This idea can be applied to the superconducting circuit here to improve its quantum coherence.
It is shown in Fig. 4 that the effectively coupling strength g eff can be enhanced by two orders of the magnitude when using squeezing transformations. For example, given g ∼ 2π ×10 kHz, when the parameters in Fig. 4 are used, g eff is enhanced to ∼ 2π × 4 MHz at f s ∼ 1. It has reached a few megahertz.
In conclusion, we have proposed an experimentally feasible method to effectively enhance the coupling strength between a single NV center and a superconducting circuit. The main recipe of our scheme is to use the unitary squeezing transformations constructed by system evolution. This idea dates back to the amplification of Kerr effect and it can provide a new path to enhance the coupling strengths in hybrid quantum systems.