Driven one-particle quantum cyclotron

A quantum cyclotron is one trapped electron or positron that occupies only its lowest cyclotron and spin states. A master equation is solved for a driven quantum cyclotron with a QND (quantum nondemolition) coupling to a detection oscillator in thermal equilibrium, the latter making this an open quantum system. The predicted rate of a cyclotron and spin quantum jumps as a function of drive frequency, for a small coupling between the detection motion and its thermal reservoir, differs sharply from what has been predicted and used for past measurements. The calculation suggests a ten times more precise electron magnetic moment measurement is possible, as needed to investigate current differences between the most precise prediction of the standard model of particle physics, and the most accurate measurement of a property of an elementary particle.

Figure 1

Comparison of the measured electron magnetic moment [3] with the standard model predictions [2, 4, 11].

Figure 2

Quantum states of a particle in a Penning trap. Cyclotron transition and anomaly transition of $n_z=0$ state are also shown with dotted lines. Each cyclotron state has infinite number of axial substates.

Figure 3

Comparison of quantum calculation (solid) and classical calculation (dashed) with the different ${\gamma }_z^{\prime }s$ for weak drive (${\mathrm{\Omega }}_c=0.1{\gamma }_c$) in cyclotron transition. The two calculations agree when ${\overline{n}}_z{\gamma }_z>{\delta }_c$ as illustrated for damping rates ${\gamma }_z$ for (a)–(c) that are 1000, 100, 10 times the value in Table 2. The line shape for the ${\gamma }_z$ in the table is presented later in Fig. 5.

Figure 4

Time evolution in response to a weak and resonant cyclotron drive applied for 10 cyclotron damping times, indicated by vertical gray lines. The probability to be in the $\left|2,n_z\right.〉$ states (blue) reaches a steady state after transients die out on a time scale give by the cyclotron damping time, $1/{\gamma }_c$. The probability to be in the $\left|1,n_z\right.〉$ states is shown in black with unit probability subtracted out.

Figure 5

Quantum cyclotron line shape (solid) with clearly resolved axial quantum states (for a weak cyclotron drive with the larges peak normalized to 1) for the quantum calculation (solid), but not for the classical Brownian motion line shape (dashed). The quantum line shape is a huge improvement on the line shape used for the best measurement (dotted).

Figure 6

Cyclotron line shape for the resolved $n_z=0$ axial state and a weak drive (solid curve, ${\mathrm{\Omega }}_c=0.1{\gamma }_c$) has a full-width at half maximum of about $3{\gamma }_c$. The master equation integrated for 10 cyclotron damping times and the steady-state line shape (solid) coincide. A 10 times stronger drive (dashed, ${\mathrm{\Omega }}_c={\gamma }_c)$, only slightly increases the linewidth.

Figure 7

Time evolution in response to a weak (${\mathrm{\Omega }}_a=0.1{\gamma }_c$) and resonant anomaly drive (at a drive detuning ${\epsilon }_a=5{\delta }_a$) with a cyclotron damping times indicated by vertical grid lines.

Figure 8

(a) Anomaly line shape for spin flip transition induced by a weak anomaly drive. The integrated solution of the master equation for time $10/{\gamma }_c$ (solid) is compared to the quasi-steady-state solution (dashed) and the classical Brownian motion line shape (dotted). (b) The integrated and steady-state solutions coincide when normalized to their peak probability. These line shape is much narrower than the $\pm 300$ ppt uncertainty of the best measurement (represented by the ”error bar”).

Figure 9

Comparison of quantum calculation (solid) and classical calculation (dashed) with the different ${\gamma }_z^{\prime }s$ for weak drive (${\mathrm{\Omega }}_c=0.1{\gamma }_a$) and $10{\gamma }_c^{-1}$ drive time in anomaly transition. The damping rates ${\gamma }_z$ for (a)–(c) are 1000, 100, 10 times the value in Table 2, respectively.

Figure 10

Probability of a spin-flips caused by a driven anomaly transitions for cyclotron and axial reservoir temperatures of 100 (solid), 50 (dashed) and 25 mK (dotted). The black curves are the experimentally accessible parameters in Table 2. The blue curves are for a tenfold reduction in the cyclotron damping rate below the value in the table.

Figure 11

Probability of a spin-flips caused by a driven anomaly transitions for a 100 mK temperature bath (solid). If the axial motion is initially cooled to the $T=0$ limit so that only $n_z=0$ is initially populated, then the linewidth narrows from $\pm 190$ to $\pm 130$ ppt, becomes more symmetric and has a slightly smaller offset frequency.

Figure 12

Probability of a spin-flips caused by a driven anomaly transitions for the typical magnetic bottle in Table 2 (solid), 10 times larger bottle (dashed), and 100 times larger bottle(dotted).

Figure 13

The probabilities for spin up (red) and and spin down minus 1 (black) in (a), and for the derivatives of these probabilities (b) as a function of time for a spin flip drive tuned to the $n_z=0$ cyclotron resonance, with Rabi frequency ${\mathrm{\Omega }}_s/\left(2\pi \right)=0.01$ Hz and other parameters in Table 2. The horizontal grids are spaced by ${\left({\overline{n}}_z{\gamma }_z\right)}^{-1}$ The vertical grids show quasi-steady-state values.

Figure 14

(a) Quantum spin-flip line shape (solid) for a weak drive with the largest peak normalized to 1 and for the classical Brownian motion line shape (dashed). The line shape used for the best measurement (dotted) is also shown. (b) Normalized probability of a spin-flip transition (solid) compared to the cyclotron transition (dashed). The horizontal scale is ${\epsilon }_s/{\delta }_s$ for the spin transition and ${\epsilon }_c/{\delta }_c$ for the cyclotron transition.

Figure 15

Time evolution driven by weak and resonant cyclotron and anomaly drives applied for 10 cyclotron damping times, the latter indicated by vertical gray lines. The probability to be in the $\left|2,n_z\right.〉$ states (blue) reaches a steady state after transients die out on a time scale give by the cyclotron damping time, $1/{\gamma }_c$. The probability to be in the initial $\left|1,n_z\right.〉$ states, with unit probability subtracted out, is shown in black. The probability to be be driven to the final $\left|3,n_z\right.〉$ states is shown in red.

Figure 16

Rate $d{P}_3/dt$ for simultaneously applied weak cyclotron and anomaly drives (${\mathrm{\Omega }}_c={\mathrm{\Omega }}_a={\gamma }_c/10$) that are resonant. The quasi-steady state estimate (dashed) slightly overstates the transition rate.

Figure 17

The two-drive line shapes change dramatically as a function of the axial damping rate, ${\gamma }_z$. Contours of the quasi-steady state $d{P}_3({\epsilon }_c,{\epsilon }_a)/dt$ are shown as a function of the scaled detunings from ${\omega }_c$ and ${\omega }_a$ of the cyclotron and anomaly drives. Parameters other than ${\gamma }_z$ are from Table 2.

Figure 18

(a) The quasi-steady-state line shape $d{P}_3({\epsilon }_c,{\epsilon }_a)/dt$ for the resolved peak corresponding to $n_z=0$ in Fig. 17. The the contours shown are at 75%, 50%, and 25% amplitudes relative to the peak value. The anomaly and cyclotron drive frequencies are given in terms of the scaled detunings of these frequencies from ${\omega }_a$ and ${\omega }_c$. The interior black lines indicate the drive frequencies scanned in Figs. 19 and 20. (b) and (c) shows the contribution from the first and second terms in Eq. (88b), respectively. The colors indicate the amplitude relative to the peak value in (a). The dotted line, ${\epsilon }_c+{\epsilon }_a={\delta }_s/2$, indicates when the cyclotron and anomaly drive frequencies sum to the resonance ${\omega }_s+{\delta }_s/2$.

Figure 19

Cyclotron resonance line shape that is a horizontal slice through the maximum spin-flip probability of the contour plot in Fig. 18. The quasi-steady-state line shape (dashed) has the same shape as is numerically calculated (solid) but with a slightly different amplitude.

Figure 20

Anomaly line shape that is a vertical slice through the maximum spin-flip probability of the contour plot in Fig. 18. The quasi-steady-state line shape (dashed) has the same shape as is numerically calculated (solid) but with a slightly different amplitude.