Time-tronics: from temporal printed circuit board to quantum computer

TL;DR

The paper addresses how time-domain crystalline structures can be harnessed for practical quantum devices by introducing temporal printed circuit boards (time-tronics). Using resonantly driven ultra-cold atoms in a 1D box, it shows how to program tunneling $J_{ij}$ and interactions $U_{ij}$ between arbitrary wave-packet sites through Bragg scattering and Raman transfers, yielding an effective Hamiltonian $H_{\rm eff}$ that remains fully reconfigurable over time. It then constructs a universal quantum computer on this platform by encoding qubits in pairs of wave-packets and implementing all required single-qubit gates and a CZ gate between any pair of qubits via controlled encounters, demonstrating high predicted fidelities (e.g., CZ fidelity ~0.993 for $^{39}$K with realistic parameters) and favorable quantum volume. The approach promises a scalable, dynamically reconfigurable architecture that bypasses spatial transport challenges inherent to conventional crystals, with concrete realizations and pathways for extending to other trapping potentials and fermions.

Abstract

Time crystalline structures can be created in periodically driven systems. They are temporal lattices which can reveal different condensed matter behaviours ranging from Anderson localization in time to temporal analogues of many-body localization or topological insulators. However, the potential practical applications of time crystalline structures have yet to be explored. Here, we pave the way for time-tronics where temporal lattices are like printed circuit boards for realization of a broad range of quantum devices. The elements of these devices can correspond to structures of dimensions higher than three and can be arbitrarily connected and reconfigured at any moment. Moreover, our approach allows for the construction of a quantum computer, enabling quantum gate operations for all possible pairs of qubits. Our findings indicate that the limitations faced in building devices using conventional spatial crystals can be overcome by adopting crystalline structures in time.

Figures (6)

• Figure 1: Controlling tunneling and interactions between resonantly driven quantum wave-packets.(a): Red ball illustrates a classical particle moving in a 1D box potential in the presence of an oscillating spatially periodic potential. If the period of particle motion is $s$ times longer than the period of the oscillating potential, $T=2\pi/\omega$, we have an $s:1$ resonance. In such a case, $s$ classical non-interacting particles can be positioned so that they follow one another along the resonant trajectory. In the quantum description, the $s:1$ resonance manifests itself as $s$ localized wave-packets, which evolve one after another along the resonant trajectory. Here, we assume that the wave-packets are sufficiently localized so that natural tunneling between neighboring wave-packets is negligible. (b): If during the encounter of the $i$-th and $j$-th wave-packets moving in opposite directions, one broad laser beam (labelled $\mathbf{k}_2$) and one narrow laser beam (labelled $\mathbf{k}_1$) focused on the meeting point of the wave-packets are briefly turned on and the Bragg scattering Muller2008 condition is satisfied, we can realize atom tunneling between the wave-packets with an amplitude $J_{ij}$, the modulus of which depends on the beam parameters and the interaction time with the atom, and a phase depending on the relative phase between the beams. With a controllable array of focused laser beams available (e.g., using a digital micromirror device Wang2020DMD as in the figure), which can be independently activated at different time points, one can control the tunneling amplitudes $J_{ij}$ between any pair of wave-packets. (c): Ultra-cold atoms can be prepared via a Feshbach resonance in states where they do not interact. In such a situation, we have the possibility of selectively controlling interactions between atoms occupying any pair of wave-packets. If just before the encounter of atoms occupying two wave-packets, we perform a Raman transfer Levine2022 (using broad and focused laser beams) from the initial internal states of the atoms to states where the atoms interact, then during the passage of the wave-packets, interaction between atoms will occur. After the wave-packets pass each other, we perform a Raman transfer again, but back to non-interacting internal states of the atoms. With a digital micromirror device available, interaction between atoms occupying any pair of wave-packets can be realized.
• Figure 2: Temporal printed circuit board. (a): The $s$ wave-packets evolving along the resonant trajectory [Fig. \ref{['fig1']}(a)] can be treated as $s$ states of a lattice with $s$ sites. If we arrange the sites of the lattice in a 1D chain and using Bragg scattering [Fig. \ref{['fig1']}(b)] we realize tunneling between nearest neighbors, we will have a 1D crystalline structure with nearest neighbor hoppings. If we arrange the sites in a 2D lattice and induce tunneling between nearest neighbors, we will have a 2D crystalline structure. Similarly, we can realize 3D and higher-dimensional crystalline structures. (b): Let us consider $s$ periodically evolving wave-packets as states of a 2D lattice. Using a broad laser beam and having only two focused laser beams at our disposal [Fig. \ref{['fig1']}(b)], all nearest neighbor tunnelings of the 2D lattice can be realized. For example, a 2D sublattice can be coupled to another 2D sublattice via a 1D chain as shown in the figure (realized tunnelings are marked with short blue arrows). Control over the phases of tunneling amplitudes $J_{ij}$ allows for the generation of an artificial magnetic field in the system. With an array of focused laser beams available, any sites in the lattice can be connected via tunneling, as illustrated by the long blue arrow. More exotic geometries can also be realized, such as the Klein bottle, where the left and right edges of the 2D lattice are connected normally, but the top edge is twisted before connection to the bottom edge. Using selective Raman transfer [Fig. \ref{['fig1']}(c)], interactions between atoms occupying any pair of sites can be realized and controlled (as indicated by red arched arrows in the plot). (c): Example building block of two connected $\operatorname{SU}(2)$ systems. The number of bosonic atoms decides the representation in each system $i$ --- if there is one atom, we deal with the spin-1/2 system. Blue vertical arrows correspond to tunneling (ladder operators), the (anti-) ferromagnetic interactions can be supplied by programmed interactions depicted by horizontal or diagonal lines. (d): Example of 1D Bose-Hubbard system of 5 sites (labelled 1 to 5 in the plot) connected to two engineered reservoir systems. The reservoirs represented by all-to-all connected systems allow for extremely fast mixing and their spectral density can be designed. During an experiment, the parameters can undergo arbitrary reconfigurations. Here, the purely dephasing interactions $U(t)$ with one reservoir and particle exchange with the other reservoir $J(t)$ can be, e.g., periodically switched during the experiment allowing for study of quantum thermodynamics in a fully closed system.
• Figure 3: Quantum computer. (a): Initially, $s/2$ bosonic atoms are prepared in a Mott insulator phase in a 1D box potential in the presence of a static optical lattice. We assume there are $s/2$ potential wells in the box, with one atom in each well. After preparing the Mott insulator phase, the interactions between atoms are turned off by means of a Feshbach resonance Pethick2002. The next step is to impart momentum to the atoms that satisfies the $s:1$ resonance condition with the frequency of the optical lattice oscillation, which we simultaneously switch on. Shortly after the previously described procedure, we observe $s/2$ wave-packets occupied by single atoms moving to the right in the figure and $s/2$ unoccupied wave-packets moving to the left. We assign one unoccupied wave-packet to each occupied wave-packet, forming the $|0\rangle$ and $|1\rangle$ states of qubits. In total, we have $s/2$ qubits. When two wave-packets corresponding to the $|0\rangle$ and $|1\rangle$ states of the same qubit meet during evolution along the resonant trajectory, a single-qubit gate can be performed using Bragg scattering. Control over the relative phase of the laser beams used in the Bragg scattering allows us to control whether the $\sigma_x$ or $\sigma_y$ operation is performed. High fidelity single-qubit operations are achieved by dividing them into several stages, meaning the entire single operation requires multiple encounters of wave-packets, and hence spanning several periods of the resonant trajectory, see Fig. \ref{['fig4']}(a). (b): When wave-packets corresponding to the states $|1\rangle_i$ and $|1\rangle_j$ of the $i$-th and $j$-th qubits meet during evolution, a controlled-Z gate can be realized. Atoms initially do not interact, but just before the wave-packets pass each other, we change the internal states of the atoms to states where they interact. We do this using Raman transfer with two beams. After the wave-packets pass each other, another Raman transfer restores the initial internal states of the atoms. If the interaction strength between atoms and the duration of interaction are appropriately chosen, after the wave-packets pass, the state $|1\rangle_i|1\rangle_j$ acquires a phase $e^{i\pi}$, and the controlled-Z gate is completed. Similarly to single-qubit operations, a high fidelity controlled-Z gate can be realized by dividing it into several encounters of the appropriate wave-packets, see Figs. \ref{['fig4']}(b)-\ref{['fig4']}(c).
• Figure 4: (a): Error of atom transfer between two wave-packets using Bragg pulses versus the assumed number of cycles of the resonant trajectory needed to achieve full transfer (two Bragg pulses per each cycle). Faster transfer requires stronger Bragg pulses, $\lambda_{\rm Bragg}$, increasing the coupling of the atom to other undesired states. Generally, apart from additional coherent oscillations, the longer the realization of the transfer, the smaller the error. (b) and (c): Fidelity of the controlled-Z gate (CZ), in which the interaction imparts a phase of $\pi$ to the state where one atom occupies one wave-packet and the second atom occupies another wave-packet, versus the number of required cycles (two interaction meetings of the wave-packets per cycle). The two curves represent the cases without (black) and with (red) the inclusion of Raman transfer errors Bluvstein2022. For a small number of cycles, stronger interactions are needed, and the coupling to other states limits the CZ fidelity. For a larger number of cycles, more Raman transfers are needed, and their imperfections limit the fidelity. (b) is for a variable interaction $g_0$ and fixed Raman transfer time ($T_{\rm Raman} \approx 0.12T$, before and after the center-point meeting time) and (c) is for a fixed interaction ($g_0=10$) and variable Raman transfer time. (d): Quantum volume Cross2018ValidatingQC calculated based on the B-gate decomposition of a generic SU(4) two-qubit operation Zhang2003MinimumCO. In this plot we assume no error in the single-qubit operations. We plot a square of fidelilty of decomposition of a single B-gate using optimal points selected from the CZ-fidelity plot (b). For higher Raman transfer fidelity one can use a longer multi-cycle realization of the CZ-gates leading to quick increase in the quantum volume. The results presented correspond to $^{39}$K atoms driven resonantly by an optical lattice potential, created by laser radiation with a wavelength $10.6\;\mu$m, which oscillates with a frequency 5.46 kHz and with Bragg pulses of wavelength 266 nm.
• Figure 5: Fidelity of the initial state preparation. Initially, the atom is prepared in the ground state of a static optical lattice potential. It is then accelerated to the resonant momentum $p_{\rm res} = \omega/2$ using counter-diabatic driving. The figure shows the fidelity of the desired final state as a function of the amplitude of the optical lattice oscillations. Blue dots correspond to the resonant momentum $p_{\rm res} = 30$, while orange squares correspond to $p_{\rm res} = 60$.