Universal Control of Symmetric States Using Spin Squeezing

The manipulation of quantum many-body systems is a crucial goal in quantum science. Entangled quantum states that are symmetric under qubits permutation are of growing interest. Yet, the creation and control of symmetric states has remained a challenge. Here, we introduce a method to universally control symmetric states, proposing a scheme that relies solely on coherent rotations and spin squeezing. We present protocols for the creation of different symmetric states including Schrödinger’s cat and Gottesman-Kitaev-Preskill states. The obtained symmetric states can be transferred to traveling photonic states via spontaneous emission, providing a powerful approach for engineering desired quantum photonic states.

Figure 1

Universal control of quantum states using spin squeezing and rotations. (a),(b) Each step of the sequence consists of a coherent rotation and squeezing. The coherent rotation is described by the unity vector of the axis $\widehat{n}$ and the rotation angle $\theta$ around this axis: $R({\theta }_i,{\widehat{n}}_i)=\exp [i({\widehat{n}}_{ix}{S}_x+{\widehat{n}}_{iy}{S}_y+{\widehat{n}}_{iz}{S}_z){\theta }_i]$. The squeezing operation is characterized by squeezing in the $\widehat{x}$ and $\widehat{y}$ directions: $S({\alpha }_i,{\beta }_i)=\exp [i({\alpha }_i{S}_x^2+{\beta }_i{S}_y^2)]$. These two operations are sufficient for efficiently creating any arbitrary state in the Hilbert space of the permutationally symmetric states. (c) Example Wigner functions of the created states can be plotted on Bloch spheres [50], each representing the joint state of indistinguishable emitters. (d) The spontaneous emission by such emitters can be tailored to desired states of light, as illustrated by the corresponding Wigner functions of the emitted light pulses.

Figure 2

Protocol for creating target symmetric states. As an input, we set the target state $|{\psi }_{\text{target}}⟩$ that we want to get, the desired number of sequences $M$, and the operations used in each sequence (as shown in Fig. 1). The optimization protocol finds the parameters used in each sequence to maximize the fidelity between the final state and the target state. Although we use pure states in the figure, the code works with density matrices.

Figure 3

Example sequence that creates a square GKP state. Sequence for the creation of a GKP symmetric state with 10 dB squeezing using 11 steps displaying the symmetric state of the emitters after each step. Each $i$th step consists of coherent rotation and squeezing with $S_x^2$ and $S_y^2$ operators. The number of emitters in the simulation is 40, and the fidelity of the final state is 98.37% compared to a square GKP symmetric state with 10 dB squeezing. For other parameters, the movie of GKP formation can be found in the supplementary video [54].

Figure 4

Example states that can be created using sequences of squeezing and rotation operations. (a) Two-legged and four-legged cat states, as well as square and hexagonal GKP, states with 10 dB squeezing generated with our protocol (the third column corresponds to Fig. 3). (b) Corresponding Wigner functions of the emitted photonic states, assuming unity efficiency of coupling to a waveguide, using the model in [33]. (c) The fidelity between the target light states and the ones achieved by the optimization protocol, denoting the number of steps. (The full list of parameters found by the optimization protocol can be found in Supplemental Material [54]).