Speeding up critical system dynamics through optimized evolution

The number of defects which are generated upon crossing a quantum phase transition can be minimized by choosing properly designed time-dependent pulses. In this work we determine what are the ultimate limits of this optimization. We discuss under which conditions the production of defects across the phase transition is vanishing small. Furthermore we show that the minimum time required to enter this regime is $T\sim \pi /\Delta$, where $\Delta$ is the minimum spectral gap, unveiling an intimate connection between an optimized unitary dynamics and the intrinsic measure of the Hilbert space for pure states. Surprisingly, the dynamics is nonadiabatic; this result can be understood by assuming a simple two-level dynamics for the many-body system. Finally we classify the possible dynamical regimes in terms of the action $s=T\Delta$.

Figure 1

(Color online) Instantaneous excitation energy in the LMG model for an optimized (green dashed line, total time $T\sim T_{\mathrm{QSL}}$), a non optimized (red dot-dashed line, $T\sim T_{\mathrm{QSL}}$) and a linear adiabatic process (orange continuous line, $T\gg T_{\mathrm{QSL}}$). Continuous (blue) lines represents the lowest energy levels as a function of the driving field $\Gamma =-t/T$.

Figure 2

(Color online) Infidelity $I$ as a function of the adimensional scaling variable $T/T^{*}$ for the LMG model (red squares), the Grover model (blue circles), and the LZ model (green triangles). Data correspond to half of the maximum size analyzed ($N=64$).

Figure 3

(Color online) Schematic representation of the ray $|{\psi }_{\theta }⟩$ evolving along the geodesic connecting the initial state $|{\psi }_i⟩$ and the target $|{\psi }_G⟩$.

Figure 4

(Color online) The action $s^{*}=T^{*}\Delta$ as a function of the size $N$ (for the LZ model we define an effective size $N={\Delta }^{-1}$), where $\Delta$ is the minimum spectral gap and $T^{*}$ the time required to reach an infidelity $I^{*}\sim {10}^{-3}$ for the linear (solid symbols) and optimized (open symbols) driving field for the LMG model (red squares), the Grover model (blue circles), and the LZ model (green triangles).

Figure 5

(Color online) The different dynamical regimes as a function of the action $s$.

Figure 6

(Color online) Instantaneous eigenvalues (blue dashed lines) and uncoupled energies of the spin-up, spin-down configurations (red continuous lines) of the LZ model for a linear quench, $\Gamma \sim t/T$. The time $\tilde{t}$ represents the boundary between Region I with high transformation rate ($\left|t\right|⩽\tilde{t}$) and Region II where the system is in the instantaneous ground state ($\left|t\right|⩾\tilde{t}$).