Control of decoherence in the generation of photon pairs from atomic ensembles

We report an investigation to establish the physical mechanisms responsible for decoherence in the generation of photon pairs from atomic ensembles, via the protocol of Duan et al. for long-distance quantum communication [Nature (London) 414, 413 (2001)] and present the experimental techniques necessary to properly control the process. We develop a theory to model in detail the decoherence process in experiments with magneto-optical traps. The inhomogeneous broadening of the ground state by the trap magnetic field is identified as the principal mechanism for decoherence. The theory includes the Zeeman structure of the atomic hyperfine levels used in the experiment, and the polarization of both excitation fields and detected photons. In conjunction with our theoretical analysis, we report a series of measurements to characterize and control the coherence time in our experimental setup. We use copropagating stimulated Raman spectroscopy to access directly the ground-state energy distribution of the ensemble. These spectroscopic measurements allow us to switch off the trap magnetic field in a controlled way, optimizing the repetition rate for single-photon measurements. With the magnetic field off, we then measure nonclassical correlations for pairs of photons generated by the ensemble as a function of the storage time of the single collective atomic excitation. We report coherence times longer than $10\mu \mathrm{s}$, corresponding to an increase of two orders of magnitude compared to previous results in cold ensembles. The coherence time is now two orders of magnitude longer than the duration of the excitation pulses. The comparison between these experimental results and the theory shows good agreement. Finally, we employ our theory to devise ways to improve the experiment by optical pumping to specific initial states.

Figure 1

Relevant level structure of the atoms in the ensemble for (a) writing and (b) reading processes, with $\mid g⟩$ the initial ground state and $\mid s⟩$ the ground state for storing an excitation. $\mid a⟩$ and $\mid b⟩$ are excited states. The transition $\mid g⟩\to \mid a⟩$ is initially coupled by a classical laser pulse (write beam) detuned from resonance, and the forward-scattered Stokes light (field 1) comes from the transition $\mid a⟩\to \mid s⟩$, which has different polarization or frequency from the write light. A classical read pulse then couples the transition $\mid s⟩\to \mid b⟩$, leading to the emission of forward-scattered anti-Stokes light (field 2) from the transition $\mid b⟩\to \mid g⟩$.

Figure 2

(Color online) Energy-level scheme considered for the atomic ensembles.

Figure 3

(a) Experimental Raman-spectroscopy setup. The Raman beams and the probe beam are coupled into a polarization-maintaining (PM) fiber and sent through a beam spitter cube (BS). The reflected part is focused into the sample with an angle of 3° with respect to the quadrupole-field $z$ axis, while the transmitted part is used as a reference. (b) Relevant level structures and laser frequencies for Raman spectroscopy.

Figure 4

(a) Raman trace with the quadrupole MOT magnetic field on. The trace represents the absorbtion of the probe pulse following the Raman beams, as a function of the Raman detuning $\delta$. The linewidth (full width at half maximum FWHM) is around $5\mathrm{MHz}$. (b) Raman trace $4\mathrm{ms}$ after the quadrupole field has been switched off. The fitted linewidth is $20\mathrm{kHz}$, including $10\mathrm{kHz}$ of power broadening due to the Raman beams.

Figure 5

Evolution of (a) the ground-state linewidth and (b) the optical depth of the sample as a function of the delay from the time when the current is switched off in the MOT coils. The linewidth is measured with Raman spectroscopy. The dashed line represents the measured power broadening due to the Raman beams. The OD is determined by measuring the absorption of a probe pulse in the sample. In both graphs, the dashed area represents the window used for measuring correlations at the single-photon level.

Figure 6

Diffusion of atoms out of the excitation region. The solid line is an exponential fit with a time constant of $900\mu \mathrm{s}$. The Raman beam diameter is $150\mu \mathrm{m}$.

Figure 7

(Color online) Experimental setup. Write and read pulses propagate sequentially into a cloud of cold Cs atoms (MOT), generating pairs of correlated output photons 1 and 2. The write and read pulses have orthogonal polarizations, are combined at polarizing beam splitter PBS1, and then focused in the Cs MOT with a waist of approximately $30\mu \mathrm{m}$. The output fields are split by PBS2, which also serves as a first stage of filtering the (write, read) beams from the (1,2) fields. For example, field 2 is transmitted by PBS2 to be subsequently registered by detector D3 or D4 while the read pulse itself is reflected at PBS2. Further filtering is achieved by passing each of the outputs from PBS2 through separate frequency filters. SM stands for single mode.

Figure 8

Measurement of $g_{12}$ as a function of the storage time with the quadrupole field (a) on (taken from [17]) and (b) off. The observed decay in (b) is consistent with the residual magnetic field in the chamber, as measured by Raman spectroscopy.

Figure 9

(Color online) (a) Measurement of $g_{11}$ (open squares) and $g_{22}$ (open circles) as a function of the storage time. (b) Measurement of the coefficient $R$ as a function of the storage time. The big statistical errors are mainly due to statistical uncertainties in the measurement of $g_{11}$ and $g_{22}$. The points at $10\mu \mathrm{s}$ have been measured for a much longer time and exhibit smaller statistical error.

Figure 10

Theory and experiment for two-photon wave packets $P_{\tau }(t_1,t_2)$. (a) Measured two-photon wave packets for the case where write and read pulses are overlapped with a delay of $50\mathrm{ns}$, with the quadrupole magnetic field on. (b) Theoretical predictions for the same conditions as in (a). (c) Measured two-photon wave packets for the case of consecutive (nonoverlapping) write and read pulses with a delay of $200\mathrm{ns}$, with quadrupole field on. (d) Theoretical predictions for the same conditions as in (c). (e) Measured two-photon wave packets for nonoverlapping write and read pulses, with quadrupole field off. The delay between write and read pulses is $1\mu \mathrm{s}$. (f) Theoretical predictions for the same conditions as in (e). The vertical scales are given in arbitrary units proportional to the joint probability of detecting photons 1 and 2. See text for further details.

Figure 11

Variation of ${\tilde{p}}_{1,2}$ with the delay $\Delta t$ between write and read pulses for (solid curve) $K=1.1\mathrm{MHz}$ and an unpolarized sample, (dashed curve) $K=12\mathrm{kHz}$ and an unpolarized sample, and (dash-dotted curve) $K=12\mathrm{kHz}$ and an initially spin polarized sample with all atoms in $\mid F=4,m_F=0⟩$. The dotted curve corresponds to an initially spin-polarized sample classically excited by fields with polarizations such that only a magnetic-field-insensitive transition is allowed; see text for details. The same arbitrary scaling factor was used for all curves.