Tuned Transition from Quantum to Classical for Macroscopic Quantum States

The boundary between the classical and quantum worlds has been intensely studied. It remains fascinating to explore how far the quantum concept can reach with use of specially fabricated elements. Here we employ a tunable flux qubit with basis states having persistent currents of $1\mu \mathrm{A}$ carried by a million pairs of electrons. By tuning the tunnel barrier between these states we see a crossover from quantum to classical. Released from nonequilibrium, the system exhibits spontaneous coherent oscillations. For high barriers the lifetime of the states increases dramatically while the tunneling period approaches the phase coherence time and the oscillations fade away.

Figure 1

Experiment to test macroscopic quantum coherence: (a) Potential energy of the flux qubit. The barrier height $E_B$ is large compared to the zero-point energy $E_0$ resulting in the strong localization of states $|L⟩$ and $|R⟩$ which are connected to macroscopically distinguishable magnetic moments induced by the persistent currents in the qubit loop. The tunneling $|L⟩\leftrightarrow |R⟩$ can be observed by a DC-SQUID sensitive to a change of the magnetic flux. (b) Scanning electron micrograph of the flux qubit.

Figure 2

Tunable flux qubit: (a) Schematics. The qubit is formed by three Josephson junctions one of which is a tunable double junction. The arrowed lines show the persistent currents connected to the states $|L⟩$ and $|R⟩$. The qubit state can be controlled by the bias lines $I_{\epsilon }$, $I_{\epsilon ,\mathrm{dc}}$, $I_{\alpha }$ and measured by the DC-SQUID. (b) Potential energy (in units of the Josephson energy of the regular junctions) as a function of the two independent phase differences ${\gamma }_1$ and ${\gamma }_2$. (c) Sketch of the cross section of the potential energy along the white line connecting left and right wells through the saddle point [see (b)]. The energy eigenstates $|0⟩$ and $|1⟩$ are superpositions of the persistent current states $|L⟩$ and $|R⟩$. (d) Energy diagram of the qubit vs magnetic bias $ϵ$.

Figure 3

Qubit properties. (a) Gap vs magnetic frustration $f_{\alpha }$. The solid line is a guide to the eye with exponential dependence of $\Delta$ on $f_{\alpha }$. The dashed line indicates the expected thermal noise level of 50 mK; the dotted line shows the border between formally quantum tunneling (qubit energies are below the tunnel barrier) and quantum scattering (energies are above the barrier) regimes. (b) Spectrum of the qubit in the regular regime for $\Delta =5.2\mathrm{GHz}$. The white line is a fit with $I_p=544\mathrm{nA}$. (c) Spectrum of the qubit in the deep tunneling regime for $\Delta =375\mathrm{MHz}$. The white line shows a fit with $I_p=344\mathrm{nA}$. (d),(e) numerical simulations of the double-well potential and the two lowest states corresponding to the spectra (b) and (c). The color scale represents the SQUID switching probability minus 0.5.

Figure 4

Energy relaxation time $T_1$ vs qubit frequency ${\nu }_{\mathrm{qb}}$ at $\Delta =200\mathrm{MHz}$. The dots show $T_1$ obtained by fitting to the experimental traces measured at each qubit frequency. The error bars are the confidence intervals of the fits. The line indicates the expected relative dependence $T_1^{-1}\propto {cos}^2\theta {\nu }_{\mathrm{qb}}\coth (h{\nu }_{\mathrm{qb}}/2k_B{T}_e)$ with $\tan \theta \equiv \Delta /\epsilon$ with the absolute magnitude being a fit parameter. The noise was assumed to be Ohmic with $T_e=50\mathrm{mK}$. The inset shows the actual data trace for ${\nu }_{\mathrm{qb}}=4\mathrm{GHz}$ with $T_1=73\mu \mathrm{s}$ (the line is fit to the measurement data). $PSW$ is the switching probability of the SQUID minus 0.5.

Figure 5

Macroscopic quantum coherence of the persistent current states. (a) Measurement protocol: preparation of the qubit in $|L⟩$ by strongly tilting the double-well potential; a fast shift to the symmetry point (symmetric double well); free evolution for time $t$ and a fast shift back followed by the SQUID measurement pulse. (b) Time-resolved measurement of the tunneling of the persistent current states for $\Delta =200$, 150, and 90 MHz (circles) and fit to $e^{-t/T_2}\cos (\Delta t)$ (line) with $T_2=65$, 45, and 35 ns, respectively. (c)–(e) The colors indicate the switching probability of the SQUID. The horizontal scale represents the amplitude of the current pulse $I_{\epsilon }$ sweeping the qubit through the symmetry point where the macroscopic quantum coherence oscillations are observed.