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We propose a quantum speedup method for adiabatic generation of cat states in bosonic Josephson junctions via shortcuts to adiabaticity. We apply approximated counter-diabatic driving to a bosonic Josephson junction using the Holstein-Primakoff transformation. In order to avoid the problem of divergence in counter-diabatic driving, we take finite-size corrections into account. The resulting counter-diabatic driving is well-defined over whole processes. Schedules of the counter-diabatic driving consist of three steps; the counter-diabatic driving in the disordered phase, smoothly and slowly approaching the critical point, and the counter-diabatic driving in the ordered phase. Using the counter-diabatic driving, adiabatic generation of cat states is successfully accelerated. The enough large quantum Fisher information ensures that generated cat states are highly entangled. difficulty to find shortcuts depends on the sign of the nonlinear interaction. The ground state of the antiferromagnetic Lipkin-Meshkov-Glick model is the spin-squeezed Dicke state, which is unique and has been successfully produced using shortcuts to adiabaticity with high fidelity and within short time [77,79,81]. In contrast, the ground state of the ferromagnetic Lipkin-Meshkov-Glick model is the cat state. Shortcuts to adiabaticity in the ferromagnetic Lipkin-Meshkov-Glick model was first studied by Takahashi using the Holstein-Primakoff transformation in the thermodynamic limit [78]. Counter-diabatic driving was derived for both the disordered and the ordered phases. However, this counter-diabatic driving is ill-defined, i.e., diverges, at the critical point unless the fixed-point condition is satisfied. As discussed in literatures, especially in [80], this divergence is associated with the closing of the gap and the divergence of the correlation length. Campbell et al studied counter-diabatic driving around the critical point applying various approaches, especially in combination with optimal control [80]. By applying a small longitudinal field, which enables us to slightly avoid the critical point, mean-field prescription was applied both in the invariant-based inverse engineering approach [82] and the counter-diabatic driving approach [83].In this paper, we propose approximated counter-diabatic driving for bosonic Josephson junctions without energy imbalance, which is available across the critical point, using finite-size corrections in the Holstein-Primakoff transformation. Advantages of our method are that the counter-diabatic driving is well-defined over whole processes and that schedules of the counter-diabatic driving can be analytically obtained. Using our counter-diabatic driving, we can accelerate adiabatic generation of the cat state. Schedules of the counterdiabatic driving consist of three steps. The first one is the counter-diabatic driving in the disordered phase, where we aim to let the system be in the ground state. The second one is smoothly and slowly approaching the critical point, where we give up to be adiabatic but ai...

The strategy of shortcuts to adiabaticity enables us to realize adiabatic dynamics in finite time. In the counter-diabatic driving approach, an auxiliary Hamiltonian which is called the counterdiabatic Hamiltonian is appended to an original Hamiltonian to cancel out diabatic transitions. The counter-diabatic Hamiltonian is constructed by using the eigenstates of the original Hamiltonian. Therefore, it is in general difficult to construct the counter-diabatic Hamiltonian for quantum many-body systems. Even if the counter-diabatic Hamiltonian for quantum many-body systems is obtained, it is generally non-local and even diverges at critical points. We construct an approximated counter-diabatic Hamiltonian for the infinite-range Ising model by making use of the mean-field approximation. An advantage of this method is that the mean-field counter-diabatic Hamiltonian is constructed by only local operators. We numerically demonstrate the effectiveness of this method through quantum annealing processes going the vicinity of the critical point. It is also confirmed that the mean-field counter-diabatic Hamiltonian is still well-defined in the limit to the critical point. The present method can take higher order contributions into account and is consistent with the variational approach for local counter-diabatic driving.

We propose a simple construction of shortcuts to adiabaticity tracking instantaneous stationary states in classical spin systems without knowing tracked stationary states. In our construction, control fields of counter-diabatic driving are constituted by state-dependent magnetic fields, which can be easily determined with an aid of numerical calculations. Easiness of our construction is a remarkable feature since it is usually a hard task to determine explicit expression of required counter-diabatic terms in many-body systems. We also argue that our method can be applied to solve combinatorial optimization problems by considering classical spin dynamics under a timedependent Hamiltonian, which mimics the procedure of quantum annealing.

We investigate a superadiabatic scheme to produce a cat state in a bosonic Josephson junction in absence and presence of particle losses. The generation scheme is based on shortcuts to adiabaticity and strongly relies on the parity conservation. The parity conservation also ensures that the produced state is a superposition of cat states with various sizes, i.e., a "cats state". Parity is also the quantity to be measured in order to utilize the produced state in interferometry. The generation scheme still works even if a number of particle losses during generation are substantial.