Soft quantum vibrations of a $\mathcal{PT}$-symmetric nonlinear ion chain

We theoretically study the quantum dynamics of transverse vibrations of a one-dimensional chain of trapped ions in harmonic potentials interacting via a Reggeon-type cubic nonlinearity that is nonunitary but preserves $\mathcal{P}\mathcal{T}$ symmetry. We propose the notion of quantum fragility for the dissipative structural phase transition that spontaneously breaks the $\mathcal{P}\mathcal{T}$ symmetry. In the quantum fragile regime, the nonlinearity dominates the response to mechanical perturbations and the chain supports neither the ordinary quantum phonons of a Luttinger liquid, nor the supersonic solitons that arise in classical fragile critical points in the absence of fluctuations. Quantum fluctuations, approximately captured within a one-loop renormalization group, give rise to mechanical excitations with a strongly momentum-dependent phonon velocity and dissipative spectral behavior. Observable signatures of the quantum fragile chain in trapped ion systems are discussed.

Figure 1

Zero-temperatures phase diagram of a Reggeon quantum chain. For large harmonic potential (to the right of the plot), the chain exhibits a massive zigzag mode and exponentially decaying correlations $\langle {\varphi }_i{\varphi }_j\rangle \sim e^{-|x_i-x_j|m}$. In the regime around the quantum critical point (blue dot), the chain is quantum fragile with long-ranged, power-law correlations among transverse vibrations $\langle {\varphi }_i{\varphi }_j\rangle \sim \frac{1}{|x_i-x_j{|}^{2-{\eta }_A^{*}}}$ with ${\eta }_A^{*}$ computed in Eq. (19).

Figure 2

Sketch of the model Eq. (4) for the transverse vibrations of a chain of trapped ions in a harmonic potential connected by nonlinear “springs.” Blue, straight springs denote the effectively one-dimensional harmonic restoring forces arising due to a combination of trapping potential and Coulomb interaction between the charged ions. Red, curved springs denote the (imaginary) cubic nonlinearity between neighboring ions. The main physical effect of the blue, harmonic force is to generate phonons as in the textbook harmonic chain. The red, nonlinear forces tend to make the chain fragile and eventually lead to the dissipative structural phase transition of the zigzag mode. We have chosen the spring representation to highlight cross connections to (floppy) polymer networks in soft matter physics.

Figure 3

Feynman diagrams for the flow equations to capture quantum fluctuations of the transverse vibrations in the chain. The straight line denotes the zigzag propagator and the crossing of three lines the cubic interaction vertex.