Nanomechanical cat states generated by a dc voltage-driven Cooper pair box qubit

We study a nanoelectromechanical system consisting of a Cooper pair box qubit performing nanomechanical vibrations between two bulk superconductors. We demonstrate that a bias voltage applied to the superconductors may generate states represented by entanglement between qubit states and quantum ’cat states’, i.e. a superposition of the coherent states of the nanomechanical oscillator. We characterize the formation and development of such states in terms of the corresponding Wigner function and entropy of entanglement. Also, we propose an experimentally feasible detection scheme for the effect, in which the average current that attains the specific features created by the entanglement is measured.


INTRODUCTION
Electro-mechanical and mechano-electrical transduction phenomena acquire whole new dimensions on the mesoscopic scale. A number of novel functionalities have resulted from new physics that gains relevance on nanometer length scales where quantum mechanics and Coulomb correlations play a crucial role. Phononassisted tunneling [1][2][3][4][5][6] , the rectification of ac current [7][8][9] , and the mechanical transportation of single electrons 10 are just a few examples of mechanically assisted electronics. Nanoelectromechanical (NEM) devices further allow for control over the mechanical subsystem by electron transportation. Ground state cooling 11,12 and shuttle instability 4,13 present good illustrations of such control.
Superconducting ordering brings new qualitative features to the functioning of NEM devices. It has been shown that the mechanical transportation of Cooper pairs can facilitate a superconducting current and generate Josephson coupling between remote superconductors [14][15][16] . A natural question, then, arises: how will the coherent dynamics of a superconducting circuit affect the nanomechanical performance of a NEM system? The coherent interplay between the states of superconducting qubits and mechanical excitations is a focus of modern frontline research in quantum communication [17][18][19][20][21][22] . Recently, it was demonstrated that individual phonons can be controlled and detected by a superconductor qubit, enabling the coherent generation and measurement of a non-classical superposition of the zero-and one-phonon Fock states 21,22 . This control allows one to accomplish phononmediated quantum state transfer and establish remote qubit entanglement. On the other hand, a mechanical resonator provides the possibility to store quantum information in complex multi-phonon states, so-called (Schrödinger) 'cat states'. Such states, rather resilient to external perturbation, allow quantum information to be encoded in such a way that mechanical losses can be detected and corrected, as opposed to single-phonon states where such losses irreversibly delete the quantum information 23 . The realization of cat-states is also vastly utilized in optics 24 . In this letter, we consider the possibility to generate quantum entanglement between a charge qubit and a nanomechanical resonator, and to encode the qubit states in the nanomechanical cat states.

Schematic diagram of experimental setup
A schematic of the nanomechanical charge qubit under consideration is presented in Fig. 1. The system consists of a superconducting island, positioned between two bulk superconductors, able to perform mechanical motion between them. We suppose that the island is attached to, for example, the top of a cantilever whose mechanical motion is described by a harmonic oscillator. The gate electrode induces an electrical potential on the island. In this work, we treat the superconducting island as a charge qubit, i.e., a Cooper pair box (CPB) whose basis states are charge states, for example, states that represent the presence or absence of one excess Cooper pair on the island; we refer to these states as charged and neutral states. Tunnel coupling between the CPB and the bulk superconducting electrodes induces transitions between the qubit states, with a transition amplitude that depends on the distance between the island and electrodes as well as on the value of the superconducting phase. Bias voltage V, applied between the bulk superconducting electrodes, generates a time evolution of the superconducting phase difference Φ according to the Josephson relation _ ΦðtÞ ¼ 2eVðtÞ=_.

Time evolution of entangled ground state
Hamiltonian b H, describing the joined dynamics of the charge qubit and the cantilever (for details see Supplemental material, Section II), can be presented as b The first term in b H 0 describes the position-independent part of the tunnel coupling between the CPB and bulk superconductors, where E J is the Josephson energy. Here,σ i , i = 1, 2, 3 are the Pauli matrices acting in charge qubit Hilbert space, particularly in the basis where vectors (1, 0) and (0, 1) represent charged and neutral states, respectively. The second term in b H 0 describes the mechanical subsystem with momentum b p and coordinate b x of the mechanical oscillator normalized to the amplitude of the zero mode vibrations where m is the mass of the oscillator and ω = 2π/T is its frequency. Hamiltonian b H 1 describes the coupling between the nanomechanical and electronic subsystems, induced by the position-dependant part of the tunnel coupling. Here, ε ≡ x 0 /x tun ≪ 1 is a small coupling parameter (x tun is the tunneling length). We consider the regime in which Φ = 0 for t < 0 and V(t) = VΘ(t), where Θ(t) is the Heaviside step function. For t < 0, the interaction between the subsystems is switched off and the system is in the pure state e in 0 j i. Here, 0 j i is the ground state of the mechanical oscillator, and e in ¼ P κ c κ i e i κ is the initial state of the CPB where e i κ are eigenvectors of Pauli matricesσ i with eigenvalues κ = ± 1. For t > 0, a constant bias voltage is switched on and the Josephson phase linearly increases in time, i.e., Φ(t) = Ω V t, where Ω V ≡ 2π/T V = 2|eV|/ℏ. To describe the time evolution of the system at t > 0, one can use the interaction representation for the time evolution operator in the form Here, b wherexðtÞ ¼xðt þ TÞ andσ 2 ðtÞ ¼σ 2 ðt þ T V Þ are operators in the interaction picture. If the frequencies ω and Ω V are incommensurable, then the product b xðtÞσ 2 ðtÞ oscillates quasi-periodically in time, and as a consequence, the mechanical subsystem remains in the vicinity of the ground state if εE J ≪ ℏω. However, in the 'resonant' case when ω = lΩ V and l is positive integer, the number of phonons increases in time and entanglement arises between the qubit states and the macroscopically excited states of the nanomechanical resonator. We perform analysis within the framework of rotating wave approximation, namely, we change the periodic functions of time, appearing in Eq.
where A l ¼ ðÀ1Þ l sgnðeVÞ J lÀ1 2EJ _ΩV , and J l (x) are the Bessel functions of the first kind. In this paper, we only consider the case l = 1, i.e. ω = Ω V .
Nanomechanical cat states Applying the operators from Eqs. (2) and (4) to the initial state, we obtain the state of the system at time t: In this equation, e 2 κ ðtÞ ¼ exp Ài EJ hω sinðωtÞσ 1 À Á e 2 κ , and α κ ðt; ξÞ j i denotes the coherent states such asâ α κ ðt; ξÞ j i¼ κνðt À ξÞ expðiωtÞ α κ ðt; ξÞ j i with ν = εA 1 E J / h, whereâ is the phonon annihilation operator. We should note that the rotating wave approximation is valid only if the energy of interaction between the qubit and the resonator, being of the order of εA 1 E J ffiffiffiffiffiffiffiffif x 2 q , is smaller than hω. This sets a restriction on time t, for which Eq. (5) is valid, to t ≤ (εν) −1 . Time t should also be shorter than any relaxation and dephasing time t 0 , which is discussed in the last section. There we give an estimation of the experimental feasibility of the proposed effect for typical values of parameters.
In the resonant case, numerical computation of the time evolution of the ground state shows entanglement of the CPB charged states with the mechanical coherent states whose amplitude linearly increases in time. On the other hand, the off-resonant case (ω ≠ lΩ V ) exhibits an evolution of the ground state that does not form coherent states, providing that the amplitude does not increase monotonically over time. In numerics, we prepare the Hilbert space in the basis of Fock states e 2 κ N n j i whereâ yâ n j i ¼ n n j i. The initial ground state is taken as e 1 À1 N 0 j i. To analyze the numerical results, we use the functional form for an arbitrary quantum state as ΨðtÞ j i¼ P nmax n¼0 ðC n e 2 þ1 N n j i À D n e 2 À1 N n j iÞ, and we find that D n = (−1) n C n due to the symmetry ½ b H; b P ¼ 0, where b P ¼ ðÀ1Þnσ 1 . To compare the resonant and off-resonant cases, we plot two numerical results obtained for ω = Ω V and ω = 1.01Ω V in Fig. 2. Numerical analysis of the distribution of |C n |, using D n = (−1) n C n , shows that the time-evolved state in the resonant case is an entangled quantum state between the coherent states of the mechanical resonator and the states of the CPB qubit.
Since ν = 0 when V = 0, one can control the entanglement described above by switching the bias voltage on and off. Let us consider the following time protocol: we switch off the voltage at time t s = N s T, where N s is integer, and we keep V = 0 during one period T, after which we switch on the voltage again (a general case for the arbitrary switch-on and switch-off times is presented in Supplemental material, Section III). During that single period, the time evolution is governed by operator UðtÞ ¼ exp½ i _ ðt À t s ÞðE J σ 1 þ _ωâ yâ Þ. Using Eqs. (2), (4), and (5), for t > t s + T, we find that the state of the system is given by the expression fρ α κ ðt À T; 0Þ j i þ iτ α Àκ ðt À T; 2ðt s À TÞÞ j i g ; One can see that for t > t s + T the state of the system is represented by the entanglement of the qubit states with two cat states of the mechanical resonator.

Entropy of entanglement
To characterize the entanglement between the qubit and the mechanical subsystem, we use the reduced density matrix b ϱ qðmÞ ðtÞ ¼ Tr mðqÞ b ϱðtÞ, where b ϱðtÞ ¼ ΨðtÞ j i ΨðtÞ h j. b ϱ qðmÞ are reduced density matrices with respect to tracing out mechanical or qubit degrees of freedom, respectively. In particular, using Eq. (6) we find that when e in ¼ e 1 À1 , the reduced density matrix b ϱ q ðtÞ takes the form b ϱ q ðtÞ ¼ 1 2 À 1 2 λðt; t s Þσ 1 ; λðt; t s Þ ¼ e À2ν 2 t 2 ; 0 < t t s λðt; t s Þ ¼ ρ 2 e À2ν 2 ðtÀTÞ 2 þ τ 2 e À2ν 2 tÀ2tsþT ð Þ 2 ; t > t s þ T: This expression allows us to calculate the entropy of entanglement, S en ðtÞ Tr b ρ q ðtÞ log b ρ q ðtÞ ¼ Tr b ρ m ðtÞ log b ρ m ðtÞ. A plot of S en (t) for νt s = 0.5 and different values of ρ is presented in Fig. 3. One can find that S en (t) monotonically increases in time before the bias voltage is switched off at t = t s . For t s < t < t s + T, S en = const due to the switched-off interaction between the CPB and mechanical subsystem. For t > t s + T, the entanglement entropy depends on ρ, as follows: for ρ>1= ffiffi ffi 2 p , S en (t) continues to grow monotonically, while for ρ<1= ffiffi ffi 2 p , S en (t) starts to decrease, reaches some minimal value (which equals zero for ρ = 0), and then grows again. However, regardless of the value of ρ, S en (t) saturates to a maximal value of log 2 for t → ∞.

The Wigner function of cat states
To characterize the state of the mechanical subsystem, it is convenient to use the Wigner function Wðx; p; tÞ ¼ ð_πÞ À1 R ϱðx þ y; x À y; tÞ expði2py=_Þdy, where ϱ(x 1 , x 2 , t) is the density matrix in x − representation. A plot of W(x, p, t) is presented in Fig. 4 for ρ ¼ 1= ffiffi ffi 2 p at two different times, t = t s in (a) and t = 3t s in (b). From Fig. 4, one can see that at t = t s , the Wigner function is positive and has two peaks, demonstrating entanglement between two-qubit states and two coherent states. This feature exists for all times t < t s . After switching off the bias voltage at t = t s and then switching it back on at t = t s + T, the Wigner function takes both positive and negative values, demonstrating entanglement between two-qubit states and two cat states of the mechanical subsystem.
Electric current through the junction Following the discussion above, the amplitude of the mechanical vibration increases in time, and the energy stored in the mechanical subsystem also increases. This energy supply comes from the electronic subsystem and may give rise to a rectification of the ac Josephson current induced by the bias voltage. To investigate this phenomenon, we calculate current I(t), produced by the bias voltage, and take its average over the period T, i.e.
NT IðtÞdt, N = 0, 1, 2, . . . The power, P = V ⋅ I, absorbed by the system is determined by the change of mechanical energy ΔE m and qubit energy ΔE q after the (N + 1)th period. Using the reduced density matrix for mechanical b ϱ m and qubit b ϱ q subsystems, we calculate IðNÞ h i T for e in ¼ e 1 À1 and different values of ρ with where ∇ N f(N) ≡ f(N) − f(N − 1) is the first difference, and the first and second contributions in Eq. (8) are ΔE m and ΔE q , respectively. Results for different values of ρ with νt s = 3 and E J /ℏω = 1.4 are presented in Fig. 5. From Eq. (7), one can see that for νt ≫ νt s ≫ 1, the eigenvalue λ decays exponentially, and therefore the main contribution to the current comes from the mechanical subsystem, and it linearly increases in time (see Fig. 5). The non-monotonic feature of t of the order of t s comes from the non-monotonic absorption of power by the qubit subsystem. It is pronounced around t = 0 (before t s ) when the evolution of the system from the ground state begins. During the period with switched-off voltage, the current is zero, i.e., IðN s Þ h i T ¼ 0. At the moment t = t s + T, the current exhibits a jump originating from the fact that it changes depending on amplitudes ρ and τ. For ρ = 1, the current continues to flow in the same direction, while for ρ = 0 it changes direction completely thus switching its sign. For , two contributions of the current flowing in opposite directions exactly cancel each other, giving zero current at t = t s + T.

Experimental feasibility
To give an estimation of the experimental feasibility of the proposed effect, we take typical values of the charge CPB qubit: decoherence time t 0 ≈ 1 μs (for different possible mechanisms leading to decoherence see Supplemental material, Section IV: A -phonon loss of the mechanical oscillator, B-charge flip of the jC n jÞ is linearly increasing as a function of n in the resonant case ω = Ω V , which is equivalently jC n j / jαj n = ffiffiffiffi n! p . The slope of logð ffiffiffiffi n! p jC n jÞ determines α. We use the following parameters: E J = 1, ω = 1/1.4, ϵ = 0.01, n max = 120; ℏ ≡ 1. CPB, C-both mechanisms, as well as refs. 25,26 ), Josephson energy E J % 0:1 Ä 1K, tunneling length x tun ≈ 1Å, and mass of the oscillator m % 10 À22 Ä 10 À21 kg 27 , providing the coupling strength εE J $ 0:01 Ä 0:1K. Then, using an asymptotic form of the Bessel function in A l coefficient in Eq. (4), we estimate a shift of the mechanical coherent state with respect to the origin for a given time t 0 , X % ffiffiffiffiffi EJ πm q x0 xtun t 0 $ 10 2 Ä 10 3 x 0 . Furthermore, using numerical simulations we show that coherent dynamics is preserved even above validity interval of the rotating wave approximation that we used to obtain the analytical results (see Fig. 2 and Supplemental material, Section IV, Figs. 2-7), i.e., during~10 3 periods of mechanical vibrations. For mechanical oscillator at 10 GHz (such frequency corresponds e.g. to the 3rd bending mode of the suspended CNT or some other setup [27][28][29][30] ) and a corresponding bias voltage of the order of 10 μV, which can be controlled fairly well to preserve the resonance to accuracy 0.1% (commercially available voltage sources, e.g., Keysight B2961A manufactured by Keysight Technologies) we obtain current, containing the predicted feature, of the order of nanoamp. Our preliminary analysis has revealed that effects qualitatively similar to those presented in this letter likewise take place in the transmon design of charge qubits (when the Josephson energy is larger than the charging energy) if the mechanical frequency is much smaller compared to the charging energy, and if the degeneracy of the qubit energy level at a certain phase difference is preserved.

DISCUSSION
We have analyzed the quantum dynamics of a NEM system comprising a movable CPB qubit that mechanically (harmonically) oscillates between two bulk superconductors coupled to the CPB via Cooper pair tunneling controlled by bias voltage. We demonstrate, analytically and numerically, that when the ac Josephson frequency of the superconductors is in resonance with the mechanical frequency of the CPB, the initial pure state (direct product of the CPB ground state and the oscillator state with zero phonons) evolves in time as the coherent states of the mechanical oscillator entangled with the qubit states. Furthermore, we establish a protocol for bias voltage manipulation that results in the formation of entangled states incorporating so-called cat states, i.e., a quantum superposition of the nanomechanical coherent states. The formation of such states is confirmed by analysis of the corresponding (negative) Wigner function, while their specific features provide for the possibility of experimental detection by measuring the average current. The discussed phenomena may serve as a foundation for the encoding of quantum information from charge qubits into (superpositions of) coherent mechanical states. As such, this work may constitute an important tool in the field of quantum communications due to the robustness of the multi-phonon states with respect to external perturbation, compared to the single-phonon Fock state. Discussion of the specific protocols for such encoding, though, are out of the scope of this work and will be presented elsewhere. In contrast to suggestions for mechanical cat states 31-34 generated either "electrically" (via strong polaronic effect), or "optically" (driving the mechanics by electromagnetic field), the phenomenon discussed in this paper allows entangling the NEM cat states with a superconducting charge qubit, providing the transfer of quantum information between the superconducting qubit to mechanical vibrations. Potentially, this may lead to a chance of exploring the possibility of achieving quantum communication utilizing the semiclassical mechanical (vibrational) cat states.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.