Coherent manipulation of cold Rydberg atoms near the surface of an atom chip

Coherent superpositions of the $49s_{1/2}$ and $48s_{1/2}$ Rydberg states of cold ${}^{87}$Rb atoms were studied near the surface of an atom chip. The superpositions were created and manipulated using microwaves resonant with the two-photon $49s_{1/2}-48s_{1/2}$ transition. Coherent behavior was observed using Rabi flopping, Ramsey sequences, spin echo, and spin locking. These results are discussed in the context of Rydberg atoms as electric field noise sensors. We consider the coherence of systems quadratically coupled to noise fields with $1/f^{\kappa }$ power spectral densities ($\kappa \approx 1$).

Figure 1

(a) Schematic illustrating the Rydberg excitation geometry. This illustration is upside down—the atoms are below the chip in the experimental apparatus. (b) Schematic of the microwave modulation system. QH: quadrature hybrid. The monitor output $V_{\mathrm{mon}}$ is useful for correcting modulation imperfections [14]. It is used to determine the in-phase quadrature (IQ) dc offset corrections required to minimize the IQ-modulator local oscillator leakage, and in the case of multiple pulses, for adjusting the I and Q inputs to obtain accurate relative phases for the different pulses.

Figure 2

Rabi flopping of initially excited $49s_{1/2}$ states as a function of microwave pulse duration (a) after release from the MOT, and (b),(c) after release from the microtrap with the $780\mathrm{nm}$ excitation beam (b) centered on the standing-wave maximum of the microwaves, and (c) displaced 1 mm from the maximum. The difference in the Rabi frequencies of (b) and (c) illustrates the spatial inhomogeneity of the rf field.

Figure 3

Ramsey sequences for atoms released from the MOT, initially excited to $49s_{1/2}$, as a function of free-evolution time between two $\pi /2$ pulses ($50\mathrm{ns}$ each), with the two-photon energy of the rf pulses detuned by $+1\mathrm{MHz}$; the rf detuning is $+0.5\mathrm{MHz}$. The different cases illustrate the role of the rf phase difference between the two $\pi /2$ pulses of a Ramsey sequence.

Figure 4

(a)–(c) Control sequences for (a) Ramsey, (b) spin-echo, and (c) spin-locking experiments. The initialization pulse rotates the Bloch vector by an angle of $\pi /2$ around the $x$ axis to create a coherent superposition of the $49s_{1/2}$ and $48s_{1/2}$ Rydberg states. The measurement pulse rotates the Bloch vector by an angle of $\pi /2$ around an axis lying in the equator; for the measurements shown in this figure the initialization and measurement pulses are along the same axis. In the spin-echo sequence, coherence is with a refocusing $\pi$ pulse as shown in (b); the spin-locking sequence requires continuous rotation around the $y$ axis as shown in (c). (d)–(f) Time evolution of coherence for atoms released from the MOT, $\approx$$3\mathrm{mm}$ from the surface. Control sequences are (d) Ramsey, (e) spin echo, and (f) spin locking. (g)–(i) Time evolution of coherence for atoms released from the microtrap, $\approx$$150\mu \mathrm{m}$ from the surface. Control sequences are (g) Ramsey, (h) spin echo, and (i) spin locking. In (d)–(i), red traces with triangles are experiments with identical initialization and measurement pulses, and blue traces with circles are experiments where the initialization and measurement pulses rotate along the same axis in opposite directions.

Figure 5

Rabi flopping (right-hand panels) as a function of the duration of a resonant pulse, observed at the end of (a) a spin-echo sequence of $20\mu \mathrm{s}$ free-evolution time and (b) a $20\mu \mathrm{s}$ duration spin-locking pulse sequence (both after release from a MOT). The left-hand panels indicate the populations observed at the end of the sequences with no Rabi flopping pulse applied. As expected, in (b) the correct rf phase leads to Rabi flopping of comparable initial amplitude to the measured coherence at the end of the spin-echo sequence, whereas with the rf phase shifted by ${45}^{\circ }$ (rotating about the $y$ axis) leads to no detectable oscillations.

Figure 6

Populations measured after a $\pi /2$ initialization pulse, $20\mu \mathrm{s}$ long spin-lock pulse, and various $\pi /2$ measurement pulses (inset), as a function of the Rabi frequency of the spin-lock pulse.

Figure 7

(a)–(c) Coherence decay for Ramsey sequences (solid lines) and Hahn spin-echo sequences (dashed lines) as a function of the scaled time ${\Gamma }_{f}t$. The power spectrum of the noise is of the form $S_{\lambda }\left(\omega \right)=S_0/{\omega }^{\kappa }$, with (a) $\kappa =1$, (b) $\kappa =3/2$, and (c) $\kappa =1/2$, with quadratic coupling to the energy-level separation. The lower cutoff frequency ${\omega }_{\text{ir}}=2\pi \times 1\mathrm{kHz}$, and the fractional uncertainty of the calculated coherences is $1/\sqrt{n=10000}=0.01$. (d) Time constant of the coherence decay, scaled to ${\Gamma }_f$, as a function of $\kappa$. The error bars show the variation of ${\Gamma }_f\tau$ at each $\kappa$ value as the noise amplitude $S_0$ is varied over about two orders of magnitude.