Theory of the Energy Variance in a Quantum Bit

Gilbert Reinisch

We show that the energy spectrum of a driven two-level quantum system-or driven quantum bit (qubit)-can be mathematically defined by its two "state energies" as the time derivatives of its large-amplitude, high-frequency internal as well as overall phases. While the mean values of these latter do fit of course Rabi's well-known harmonic (weak drive) as well as anharmonic (strong drive) dynamics, the standard deviations of these both state energies may become comparable to their mean values themselves. This remarkable property, which we check by use of the quantum expectation value of the variance operator, defines a large statistical time-dependent range of available values for the qubit energy in the presence of an external perturbation. We take the example of a recent experiment [Z. Minev et al. Nature, 570, 200 (2019)] and recover by use of the present theory the measured "time-of-flight" values.

Hans J. Kummer

Akoramurthy B, Surendiran B, Xiaochun Cheng

Charging a quantum battery through a collective non-adiabatic stroke stores energy in a shared bosonic mode, but part remains locked in correlations with the collective spin and is inaccessible to cyclic unitaries acting on the mode alone. A demon holding one bit can unlock this energy, quantified by the daemonic ergotropy. We investigate the value of one bit and its dependence on where the information is obtained. Two protocols are compared at matched stored energy and matched information, using balanced two-outcome measurements carrying exactly one bit. We find that one bit about the collective coordinate unlocks N ln(2) times as much work as one bit about a single ion. For three stored-energy settings and N = 4-24, the measured scaling exponent is 0.990 +/- 0.043, while double extrapolation gives a prefactor of 0.69298 +/- 0.00044, within 0.02% of ln(2). To leading order, the daemonic gain equals nu mu^2 times the between-outcome variance of Jx, verified numerically to 1.3%. A balanced single-ion measurement resolves 1/4 of this variance, whereas a balanced collective split resolves (ln(2)/4)N. The microscopic origin of the prefactor remains open; a Gaussian median-split estimate of 1/(2pi) is excluded by 9%. One bit recovers a constant fraction of the locked energy independent of N and yields roughly twice as much work per bit as a complete readout, indicating strong diminishing returns. We also show that unnormalized gain comparisons can reverse the conclusion and that the break-even ion number does not collapse onto the Dicke superradiant threshold when the mode frequency is varied.

GEORGE G. HALL

The theory of how a probability distribution may be estimated by sampling is applied to the problem of using approximate wavefunctions for quantum mechanical systems and estimating the errors involved. The most probable wavefunction of given mean energy is considered and some of its properties found. The energy variance, which gives a lower bound to the true energy, can be calculated. The theory is applied to the simple harmonic oscillator and the predicted relation between mean energy and variance is compared with the actual relations found using two different types of approximate wavefunction.

Yuriy Makhlin, Gerd Schön, Alexander Shnirman