Components of long-distance quantum Communication. Part 1
Where is a photon creation operator in the considered mode at the point of time t, is the vacuum state of field and . Squared amplitude determines the probability density function to detect the photon at the point of time t. The detailed discussion of such description of singe photon pulses can be found in the work . To form a quantum bit of single photon wave packet it is possible to proceed as follows : a single photon pulse is passed through misbalanced Mach-Zehnder interferometer so that at the output we have the superposition of two wave packets that are equal in form but different in time: , where α and β are complex numbers describing phase difference and amplitudes of two packets that meet the normalization condition . Let us define the quantum state corresponding to the first packet as , and define the second packet as . In this case the state of such a two-peaked pulse takes the form of , that corresponds to arbitrary state of time-bin qubit.
The most important characteristics of quantum memory are the efficiency and fidelity of reconstruction of quantum states. The efficiency h is defined as the probability of single photon state reconstruction that corresponds to ratio of pulse average energy at output to pulse average energy at input. Formal definition has the following form:
where Ain (t) and Aout (t) are the amplitudes of pulses at input and at output of the device, respectively. As to fidelity of quantum memory F, this characteristic describes the level of field state correspondence at device output to field state at input. In terms of single photon pulse it is referred to correlation of envelope and phase of wave packets that are time and coordinates functions, so it is possible to write the following formula:
where T is the time of information storage. The fidelity F equals to 1 in case of full compliance of the states and equals to 0 in case of maximum noncompliance (when the states are orthogonal).
Ideal quantum memory is the device designed to write and reconstruct single-photon wave packets with efficiency and fidelity equal to 100%. At that the time of information storage is supposed to exceed duration of single photon pulses significantly. Surely, the ideal device cannot be created in real life, therefore it is important to understand what requirements are imposed to optical quantum memory concerning protocols of long-distance quantum communication. The estimations, performed in the work  for the quantum repeater where the active multiplexing is used can be used as the example. If multiplexing degree (number of channels) is equal to 100, multi-channel storage and retrieval of single photon states for time about 50 msec with the efficiency of 90% can be required for implementation of quantum communication at a distance of 1 000 km. More realistic assessment can be found in the work . In any case, it is important to note that it is not necessary to keep quantum information for seconds or minutes for solution of practical tasks. If protocols with multiplexing are used, the storage period is proved to be about 100 msec that is significantly closer to abilities which are provided by available carriers of quantum information. However in doing so it is necessary to use the multimode quantum memory designed for storage and reconstruction of single photon states set at one cycle.
Nowadays the major attention is paid to development of optical quantum memory on the basis of polyatomic ensembles since their usage as information carriers allow storing and reconstructing various quantum states of electromagnetic field, in particular single photon wave packets, in a multimode regime. The most promising schemes are those based on the phenomena of electromagnetically-induced transparency, off-resonant Raman scattering and photon echo (see reviews [11–16]). Among the most significant experimental results are: demonstration of quantum memory protocols of high efficiency (more than 50%) on the basis of electromagnetically-induced transparency  and photon echo [18–23], storing and reconstructing of broadband photons (THz) in diamond crystal via off-resonant Raman scattering , storage of large pulse sequences (up to 1 000) [25, 26], and also demonstration of large time-bandwidth product (up to 5 000) [27, 28]. It is necessary to note that the possibilities of quantum memory protocols were shown using weak laser pulses in many cases and this fact certainly ensured the high signal-to-noise ratio. Nowadays the attention has refocused on carrying out experiments just with single photon states. From this point of view we can mention the solid-state experiments on storage and reconstruction of polarization qubits [29–31] and time-bin qubits , and also implementation of spin-based control of optical quantum memory during storage of single photon states , and experiments on multiplexing [34–35]. The significant attention is paid to development of integrated quantum memory with use of wave guides [36–38], photon chips [39, 40] and optical fibers [41–43]. In the context of spatial multiplexing usage, the great interest is aroused concerning experiments on storage and retrieval of quantum light states with orbital angular momentum [44–46].
The basic aspect is the search of the new materials which allow storage of recorded quantum information for a long period. Nowadays much attention is paid to research of rare-earth ion doped crystals  as at low temperatures (~2K) they provide large times of the phase relaxation both on optical, and on spin transitions of impurity ions, giving at the same time ample opportunities for control of transition frequency by means of external electric or magnetic fields, and also for integration in photonic circuits. Besides, the rare-earth-ion doped crystals are considered to be the promising systems for implementation of microwave quantum memory that is of great importance for creation of hybrid quantum schemes using superconducting qubits. As the information is stored in these crystals in the form of atomic superposition states, the quantum memory time is limited to phase relaxation time T2. Typical T2 values at low temperatures are equal to tens of microseconds on optical transitions and equal to hundreds of microseconds on spin transitions. Thus, advanced arrangements of optical quantum memory shall be based on use of long-lived spin states. The urgent research problem is the development of methods allowing increasing of phase relaxation time due to impact on spin states with the sequence of radio-frequency pulses (dynamic decoupling) (see the review ) or due to the usage of so-called clock resonance transitions of impurity ions with the frequency that in the first order does not depend on fluctuations of crystal magnetic field .
Integrating both approaches, it is possible to increase the phase relaxation time by several orders. Nowadays the greatest time of information storage (up to one minute) is shown in Y2SiO5:Pr 3+ crystal , and the greatest time of the phase relaxation on spin transitions (up to 6 hours) is shown in Y2SiO5:Eu 3+ crystal . In addition, isotopically pure crystals activated by rare-earth ions are of great interest (see the review ). Monoisotopic content both of matrix, and of impurity particles, allows achieving limit values of phase relaxation times, optical density and more complete control of material parameters. Besides, such crystals can have very small inhomogeneous broadening of optical transitions (~50 MHz) which allow implementing the quantum memory schemes based on off-resonant Raman scattering [53–58] in solid-state materials. These schemes appear to be particularly useful at usage of clock transitions with the longest coherence times.
In particular, a new method of storage and retrieval of weak light pulses based on continuous phase-matching control in extended multiatomic system due to control field angular modulation was proposed in works [56, 57]. Angular modulation allows implementing the quantum memory without controlling inhomogeneous broadening of resonant transitions by means of external electric or magnetic fields used for implementation of gradient photon echo and this fact is appeared to be the fundamental aspect when working with clock spin transition. One of possible implementation diagrams is shown in Fig. 1. The control and signal fields interact with information carrier (impurity crystal) which is placed in the resonator. In doing so, the signal field corresponds to one of resonator modes, and the control field propagates perpendicularly to the resonator axis. Let the control field wave vector be turned during off-resonant Raman interaction with atoms in the presence of weak (single photon) signal field (Fig. 2). Then the information on the temporal form of the single photon pulse is reversibly projected on spin coherence spatial grating generated on transition between two low levels. The amplitude of the input field in different time points is transferred to spin waves with different wave vectors. Such projection is possible in extended multiatomic system due to phase-matching condition allowing changing the collective atoms’ interaction with input field from one spin wave to another.
As a result, at the end of the storage process the coherence grating on spin transition which represents superposition of orthogonal spin waves in the interaction volume is created. Reading of the single photon state is reached by spontaneous Raman scattering of the control field on created grating. The resultant output pulse is the superposition of pulses generated from different spin waves which meet the phase-matching condition for different wave vectors of the control field at different time points.
When using the resonator, the off-resonant Raman scattering can be implemented in the impurity crystals activated by rare-earth ions. At that the most convenient crystals are those isotopically pure crystals where narrow inhomogeneous broadening lines of optical transitions can be achieved. The example of such material is YLiF4 crystal enriched with Li-7 isotopes and doped with trivalent ions of Erbium [59, 60] or Neodymium . At small concentration of impurity particles the inhomogeneous width of optical transitions is decreased up to 10 MHz in this crystal that is, perhaps, the smallest value considered in impurity crystals. The first experiments on implementation of quantum memory protocols in this crystal are performed in works [62, 63].
The work is done with financial support of the Russian Science Foundation (grant No. 14-12-00806).
In the following part of the review the urgent directions of research in the field of the single photon and correlated two-photon states of light creation intended for usage in systems of long-distance quantum communication will be considered.