Precision Spectral Manipulation: A Demonstration Using a Coherent Optical Memory

The ability to coherently spectrally manipulate quantum information has the potential to improve qubit rates across quantum channels and find applications in optical quantum computing. In this paper, we present experiments that use a multielement solenoid combined with the three-level gradient echo memory scheme to perform precision spectral manipulation of optical pulses. These operations include separate bandwidth and frequency manipulation with precision down to tens of kHz, spectral filtering of up to three separate frequency components, as well as time-delayed interference between pulses with both the same, and different, frequencies. If applied in a quantum information network, these operations would enable frequency-based multiplexing of qubits.

Popular Summary

Quantum computers entered the spotlight in the 1990s when it was shown theoretically that they might be employed in attacking important cryptographic problems such as prime-number factoring. At the same time, quantum information processing promised new kinds of encryption and secure communications. In essence, the unusual rules of quantum mechanics permit a kind of parallel computation that enables substantial speed-ups in solving certain kinds of difficult tasks. As a result, the pace of discovery and development has accelerated, and the search is on for ways to go beyond proof of principle to create the components of a scalable and useful quantum computer. To fulfill these goals, quantum bits (qubits) must be stored and manipulated much the way conventional RAM in a desktop computer holds 1s and 0s for later processing. Several quantum computing schemes center on photons as a way to transport and process qubits, and in our paper, we experimentally demonstrate a way to condition and manipulate optical pulses in a quantum memory.

Our studies relied on a type of high-efficiency low-noise quantum storage called “gradient echo memory” (GEM). Photon echoes are a phenomenon in which an optical pulse is absorbed by an ensemble of two-level systems, such as a gas of atoms. Collectively, the atoms store the information of the light pulse. The individual atoms, however, evolve at different rates, so that very quickly they are all out of phase with each other. Echo memories employ a method to reverse this process, causing the atoms to rephase and thus re-emit the pulse at a later time as an “echo.” In GEM, this rephasing is achieved by placing a reversible frequency gradient along the path of the pulse. Due to this frequency gradient, different spectral components of the pulse are absorbed at different spatial locations in the material. The frequency gradient is established by a series of solenoids that apply a spatially varying magnetic field along the pulse path in a rubidium vapor cell, causing the local energy levels to shift and absorb different frequencies. Reversing the gradient causes the stored pulse to be recalled exactly, while adjusting the output gradient can effect a variety of spectral modifications.

Among the alterations to the optical pulse spectrum we have achieved with GEM are changing the pulse center frequency, increasing and decreasing the pulse bandwidth, filtering out selected spectral components, and controlled interaction of two stored pulses. These transformations could be used to increase qubit transmission rates across quantum information networks, allowing for faster communications and processing speeds for quantum computing, as well as the potential to act as operations in a quantum computer.

Figure 1

Basic two-level gradient echo memory operation. (a) A pulse with an input Gaussian envelope ${\mathcal{E}}_i(t)$, bandwidth ${\mathcal{B}}_p$, and center frequency ${\omega }_{ci}$ enters the storage medium, at time $t=0$, where the optical information is stored in the atomic excitation ${\sigma }_{12}$. The memory has a linear frequency gradient placed along it in the $z$ direction and a input frequency bandwidth ${\mathcal{B}}_i$. (b) At time $t=\tau$, the sign of the frequency gradient is reversed, with the memory output bandwidth ${\mathcal{B}}_o={\mathcal{B}}_i$. In this scheme, the echo is emitted at time $t=2\tau$ with pulse shape ${\mathcal{E}}_o(t)={\mathcal{E}}_i(-t)$ and center frequency ${\omega }_{co}={\omega }_{ci}$.

Figure 2

Experimental setup. Probe electric field envelope (${\mathcal{E}}_p$), nonpolarizing beam splitter (BS), and $I_1–I_8$ currents supplied to the individual solenoids. The inset shows the level scheme and the equivalence between the $\Lambda$ system used and a two-level atom: one-photon detuning ($\Delta$), two-photon detuning ($\delta$), coupling-field Rabi frequency (${\Omega }_c$), decay rate from the excited state ($\gamma$), decoherence rate between the ground states (${\gamma }_0$), coupling strength between ground and excited states ($g$), and the effective coupling strength for the equivalent two-level system ($g^{\prime }$).

Figure 3

Manipulating pulse frequency. (a) Two-photon detuning $\delta$ as a function of position $z$ along the memory (normalized to length $l$) due to (i) the input gradient and (ii) output gradients, with minimum and maximum gradient offset ${\delta }_{\mathrm{os}}$ (as noted on figure). The blue (dashed) line corresponds to the desired field, and points correspond to the measured magnetic field. (Error bars are due to the sensitivity of the Gauss meter.) (b) Heterodyne data showing (i) input pulse, (ii) echo for recall with ${\delta }_{\mathrm{os}}=0$, and (iii) echo for recall with ${\delta }_{\mathrm{os}}=1400\mathrm{kHz}$ offset. Orange points correspond to raw data, black lines correspond to modulated Gaussian fit to data, and ${\omega }_c$ values correspond to the center frequencies of pulses extracted from the fits. (c) The change in the center frequency of the output pulses relative to the input pulse $\Delta {\omega }_{\mathrm{c}}$ as a function of ${\delta }_{\mathrm{os}}$. Points represent the measured center frequency (error bars are from standard deviation of 100 traces), and the dashed line corresponds to the theoretical behavior.

Figure 4

Manipulating pulse bandwidth. (a) Two-photon detuning $\delta$ as a function of position $z$ along the memory (normalized to length $l$) due to (i) the input gradient and (ii) minimum and maximum output gradients (ratios noted on figure). The blue (dashed) line corresponds to the desired field, and points correspond to the measured magnetic field. (Error bars are due to the sensitivity of the Gauss meter.) (b) Amplitude plot, normalized to the size of the input pulse, showing (i) the input pulse (in red, scaled by a factor of $1/2$) and (ii) output pulses recalled with varying output gradients as noted. Points correspond to demodulated data, lines correspond to Gaussian fit to data, and ${\omega }_c$ values correspond to the center frequencies of pulses relative to the LO. The bracketed ratios indicate ${\eta }_o/{\eta }_i$. (c) The FWHM of the output pulses ${\mathcal{W}}_o$ normalized to the FWHM of the input pulse ${\mathcal{W}}_i$, as a function of input gradient over output gradient $|{\eta }_i/{\eta }_o|$. Points represent measured FWHM (error bars are from standard deviation of 100 traces), the red (dashed) line corresponds to Eq. (3), and the blue (solid) line corresponds to a numerical simulation, with free parameters: $g=0.066{\mathrm{s}}^{-1}$, ${\gamma }_0=0$, and $gN/c=1000$.

Figure 5

Spectral filtering. (a) Two-photon detuning $\delta$ as a function of position $z$ along the memory (normalized to length $l$) due to gradients (i)–(iii) corresponding to times (i)–(iii) in (b). For traces (a) and (c) blue (dashed) lines correspond to the desired field, and points correspond to the measured magnetic field. (Error bars are due to the sensitivity of the Gauss meter.) (b) Spectral filtering of (i) a Gaussian envelope containing two frequency components separated by 700 kHz (red, nondemodulated, scaled by $1/2$) and the demodulated retrieval (blue) of (ii) higher- and (iii) lower-frequency components averaged over 100 traces. For traces (b) and (d), points correspond to data, lines correspond to fit to data, and ${\omega }_c$ values correspond to center frequencies of pulses. (c) Two-photon detuning due to (i) input and (ii), (iii), and (iv) output gradients corresponding to times (i)–(iv) in (d), which shows the conversion from the time to the frequency domain of (i) a Gaussian pulse with two modulation sidebands at $\pm 700\mathrm{kHz}$ (red, nondemodulated, scaled by $1/2$), and the demodulated retrieval of (ii) the higher-frequency sideband, (iii) the carrier, and (iv) the lower-frequency sideband averaged over 100 traces (blue).

Figure 6

Interference with pulses of different frequencies. (a) Two-photon detuning $\delta$ as a function of position $z$ along the memory (normalized to length $l$) due to (i)–(ii) input gradients and (iii) the output gradient corresponding to times (i)–(iii) in (b). Blue (dashed) lines correspond to the desired field, and points correspond to the measured magnetic field. (Error bars are due to the sensitivity of the Gauss meter.) (b) Interference of two pulses that are initially time separated, (i) $P_1$ and (ii) $P_2$, shown in red, which are also separated in frequency by 700 kHz. Panel (iii) in (a) shows the superposition of the two pulses. The inset in (b) shows the output from the memory for storage of only a single pulse: $P_1$ recall ($E_1$, green) or $P_2$ recall ($E_2$, blue). Points correspond to demodulated data averaged over 100 traces, lines correspond to Gaussian fit to data, and ${\omega }_{\mathrm{c}}$ values correspond to center frequencies of pulses. (c) The change in the relative phase of the fitted interference pulse $\Delta {\theta }_{\mathrm{op}}$ as a function of the relative phase of the input pulses $\Delta {\theta }_{\mathrm{ip}}$. Points represent data extracted from the fit (error bars are from standard deviation of 100 traces), and the dashed line corresponds to the theoretical behavior.

Figure 7

Interference with pulses of the same frequency. (a) Two-photon detuning $\delta$ as a function of position $z$ along the memory (normalized to length $l$) due to (i)–(ii) input gradients and (iii)–(iv) output gradients corresponding to times (i)–(iv) in (b). Blue (dashed) lines correspond to the desired field, and points correspond to the measured magnetic field. (Error bars are due to the sensitivity of the Gauss meter.) (b) Interference of two pulses, initially time separated, (i) $P_1$ and (ii) $P_2$ (red dashed lines), which have the same center frequency. (iii) Initial $E_1$, and (iv) secondary $E_2$ superpositions of the two pulses. Blue points and solid line correspond to maximum constructive interference for $E_1$, while green crosses and dashed line correspond to maximum constructive interference for $E_2$. Points correspond to demodulated data averaged over 100 traces, lines correspond to the Gaussian fit to the data, and ${\omega }_{\mathrm{c}}$ values correspond to center frequencies of pulses. (c) The change in area (normalized to the maximum intensity of the individual echoes) as a function of the relative phase of the input pulses $\Delta {\theta }_{\mathrm{ip}}$ for (i) $E_1$, and (ii) $E_2$. Points represent data extracted from the fit (error bars are from standard deviation of 100 fits), and the dashed line corresponds to a fit to the data.